Obsidian Vault

Kafka

Fri, 26 Jun 2026 03:27:24 GMT

Apache Kafka is a distributed messaging system for logging or messaging with low latency.Kafka uses a distributed system of data where producers send data to the brokers, the consumers sends requests to the brokers to get the data. The data is synced via Apache Zookeeper.The data is set up as a Queue within the broker. The metadata is stored within the client and not the server which saves overhead. However, because of this, the server can't determine when all clients have consumed the message. The file is deleted after 7 days as a measure to flush old data.txtParquet: Colum based Data Avro: Row based Data (Different Params)RS: Hedwig NATS (Go) Red Panda (C++) Partition Management Spark (Data) Avro (File Format) Protocol Buffer

Method of Least Squares

Fri, 26 Jun 2026 02:06:55 GMT

The Method of Least Squares determines the line of best fit by minimizing the total mean squares.Consider a company that is capable of producing 400 units of product per year. Any demand above 400 is lost.Let the demand over the past 8 years be as followsThe trend, when plotted, appears to increase in linear fashion.The demand data has components of a trend and randomness

F-Statistic

Thu, 18 Jun 2026 00:04:02 GMT

The F Statistic is a measure of whether using a constant value or a linear trend is more effective for describing a trendFor a sample mean of the sum of the differences between all values of and is given by

Inventory Decision Analysis

Wed, 10 Jun 2026 18:59:01 GMT

For a business that handles product with a deterministic demand, the following considerations will need to be taken for stock The cost to buy enough inventory The cost to order inventory the cost to hold inventory The stockroom can buy enough inventory to last the entire year (but the cost to hold inventory might be costly) or can place smaller orders thorough the year (though the cost of ordering might be costly)Inventory Decision Analysis examines the optimal ordering size, Economical Order Quantity and frequency to extract the maximum valueEconomical Order Quantity: (EOQ) The order size that will minimize the total cost of ordering and holding for a given demand Lead Time: (LT) The amount of time between placing an order and the arrival of that orderA corporation buys its inventory of product (tires) at an average cost of per tire, the demand is fixed at 180 tires per day There is an ordering cost of per order There is an holding cost of per tire per yearFind the Economical Order QuantityFor this example, assume the cost of ordering and holding is independent of other productsThe total quantity of tires the company would need to buy annually isThe cost of the tire itself will not depend on the size of the order nor its frequency so we can get a baseline of the cost of the tires alone annuallyFor the deterministic inventory decision, let the single decision variable be the size of the order each time a order is placedThe objective now is to minimize the total annual cost which can be given byThere can also be maximum and minimum quantity limits on the order, for this example let's assume that tire manufacturer has a minimum and maximum limit on the size of the orderThe annual cost of ordering and holding can then be formulatedInventory on Hand In this example, the inventory on hand is simplified to be half of the order size , this assumes constant demand and is an average of the real cost of holding per ordering cycle After which, the total annual cost can be determinedThis function is then minimized to achieve an optimal order size of tiresSimilar to Sensitivity Analysis for a Make Buy Decision, the Sensitivity Analysis for Inventory Decision Analysis deals with the effects of changes in the parameters to its decisionConsider the example from above, if the demand is lower, the annual cost of ordering the tires will differIn this case the optimal order size will be tires
If the demand of the for a product becomes variable then the Inventory Decision becomes a Stochastic DecisionFor a deterministic demand , the reorder rate (also known as the Reorder Point ) can also be deterministicIn the case of stochastic demand, the demand is at a variable rateFor a inventory decision analysis with stochastic demand, two decisions are required What is the Working Stock that minimizes total annual cost? What is the amount of Safety Stock that should be carried? The stockroom can have two different types of stocks, though the product is identical: the Working Stock and the Safety StockWorking Stock: The general stock used for EOQ order size or POQ production sizeSafety Stock: Additional overhead outside of the working stock in cases of abnormally high demand

Sharpe Ratio

Sat, 06 Jun 2026 21:47:13 GMT

A Sharpe Ratio measures the volatility and associated return per risk for an investment

Sensitivity Analysis

Sat, 06 Jun 2026 02:11:04 GMT

The action taken in a Make Buy Decision depends on the fixed prices of costs, Sensitivity Analysis measures the effects of changes to these prices
Consider the Hospital Example from the Make Buy Decision and how the DOG machine costs monthlyExamine the sensitivity of the make buy decision if the monthly cost of the machine ranges from to $$1.2MRecall that the linear equation for the cost and revenue of the machine can be given as soBy varying the fixed cost of the machine, we can achieve a range of costsWith this, we can analytically find the new break even pointsThe effect of the fixed monthly cost of the machine to the required demand can be calculated as

Net Present Value Method

Sat, 06 Jun 2026 00:46:49 GMT

The Net Present Value Method accounts for the fact that returns in the future may not have a much value as returns now, the initial cost is then subtracted to get the total present value profitIf then the investment is a worthwhile investment financially, other considerations may need to be taken to accountIf then the investment is not worthwhile financially but may be implement for other reasonsA company is investing in a machine to help with manufacture a new series of product the machine costs and the prevailing annual interest rate is Determine the Net Present Value of the expected Revenue Flow for the next years (lifespan of the lathe) belowWe can find the Present Value of the revenue for each yearSo that we get a sum of Subtracting the initial cost of , we can get the final present value profitThe interest rate which makes is called the internal rate of return

Capital Recovery Factor

Fri, 05 Jun 2026 23:26:15 GMT

The Capital Recovery Factor is used to amortize a lump sum debt over equal payments and can be used to model mortgages or general loansA house developer plans to mortgage a home over years at yearly interest What is the annual payment for a potential buyer of that home?The Uniform Present Value in years of today at an annual interest is centsThe buyer must pay cents annually over years at interest to match dollar loaned todayTherefore the buyer mortgaging the home can expect to pay annually

Present Value

Fri, 05 Jun 2026 23:22:36 GMT

The Present Value (PV) of is the value today of a future revenue or a flow of future revenuesConsider an interest account with a yearly interest What is the present value of in yearsTo find the present value, the target amount must be divided by the compounded interest rateThe amount of now and in years is equal when considering present valueThe Uniform Present Value is the present value of a sequence of future revenuesIf the return on investment is fixed at yearly, determine the uniform present value of the following sequence of equal year end revenuesThe present value for year one isAnd such for year twoThe sums of all of the present values for all years is then

Time Value of Money

Fri, 05 Jun 2026 23:22:32 GMT

The concept of the Time Value of Money relies on the principle that Money today may have greater value than money in the future A smaller amount of money today may be equal to more great value than a larger amount in the future

Compound Interest

Thu, 04 Jun 2026 19:52:00 GMT

Compound Interest determines the amount of revenue that accrues if a sum is left in an interest bearing accountIf were to be invested at interest and compounded annually, what will be revenue accrued after years?At the end of the first year the balance on the account would beAt the end of the second yearThis can be generalized to yearsWhere after years the total amount of revenue can be computed

Make Buy Decision

Thu, 04 Jun 2026 19:27:29 GMT

A Make vs. Buy Decision is when there is a choice to producing a product in house verses buying it from a manufacturer. The cost of producing a product in house entails a recurring fixed cost and a variable cost per productfixed cost: Cost of overhead, that does not change with the amount of goods produced variable cost: Cost of each product that changes with the amount of goods producedThe Break Even Volume is the volume of units sold in a Make Buy Decision needs to exceed both the fixed cost and variable costA break even volume can be the volume of demand that a product needs to meet to be economically viable A hospital is considering purchasing a Digitalized Orthogonal Gastrointestinal (DOG) scanner. The fixed cost associated with purchasing the DOG, remodeling, hiring and training totals approximately per year over the next ten years— the expected life of the DOG. The variable cost to Mercy Hospital per DOG scan averages however, the hospital can charge per scan.Assume the expected demand for DOG scans is a Normal Random Variable with expected value and standard deviation , what is the probability the hospital will lose money in a given year?The equation for the expected cost of the operation of the DOG per year can be model as soAnd likewise for the expected revenue from the DOG per yearFrom this we can see that the break even volume is at DOG scansThe probability that the hospital can lose money can be modeled as the probability that the Normal Distribution will have a demand of less than scans

Stochastic Decision

Wed, 03 Jun 2026 22:31:14 GMT

A Stochastic Decision involves a probabilistic environment where the parameters of actions are not exactly knownA stochastic decision might be that a supermarket can only place orders of cereal in box orders, if the monthly demand is provided the Normal Distribution, , when should the next order of cereal take place so that the demand can be met with a confidence intervalA Simple Stochastic Decision has few parameters and/or few possible alternative actions. It is assumed that there is uncertainty in a small amount of parametersInventory Analysis with Stochastic Demand is an example of a simple stochastic decisionA complex Stochastic Decision have a large amount of parameters and actions. It can be assumed that there is uncertainty in numerous parameters in the analysisQueueing Analysis is an example of a complex stochastic decisionAnalytical Analysis For a Complex Stochastic Decision, there are only a few techniques in which the decision can be solve analytically, the most complex decision must be made using simulation

AC Steady State

Wed, 03 Jun 2026 22:24:31 GMT

In AC Steady State, the transient responses have died out and the circuit is driven fully by a sinusoidal sourceTo analyze AC steady state circuits Replace all components with its Impedance Analyze the circuit with impedance like a DC non time-dependent circuit
Instead of analyzing resistance in a like in a DC Steady State circuit, we analyze ImpedanceIn AC circuits, filters can cut out waves of certain frequenciesA lowpass filter, as the name suggests, only allows lower frequencies to pass
The voltage of a low pass filter can be calculated by using steady state analysisFind phase of the magnitudeThen thereforeThe frequency response for a low pass filter is shown belowFrequency Response The graph depicts a response in the frequency domain in decibels and not voltage Similar to a lowpass filter, a highpass filter only allows higher frequencies through
Similar to a low pass filter, the voltage of a high pass filter can be calculatedThereforeThe frequency response for a high pass filter in decibels is shown below

Deterministic Decision

Wed, 03 Jun 2026 21:22:35 GMT

A Deterministic Decision involves a choice of actions in which the parameters of actions are fixedA deterministic decision might be that a supermarket can only place orders of cereal in box orders, if the demand if exactly per month, how often should the store place an order.A Simple Deterministic Decision is one where there is few parameters per action or few alternative actions to considerA Complex Deterministic Decision is one where there can be a large number of parameters or possible actions but the parameters of the analysis are assumed to be fixedLinear Programming is an example of how a very large number of possibilities can be sorted to determine the best course of action

Expected Value of Information

Mon, 01 Jun 2026 00:52:12 GMT

In certain situations there exists an option to pay for more information, if the information gathered increases the likelihood that better decision is made, that information has an Expected Value

Utility

Mon, 01 Jun 2026 00:46:53 GMT

Utility is the payoff or a cost of an outcome that is either difficult or impossible to measure In this case, instead of using an absolute amount as a payoff or cost, a utility can be usedA utility is based on its relativistic worth to the decision maker and is by definition SubjectiveThe utility of an outcome can be represented by the notationConsider a set of outcomes as follows
While the Expected Value of each outcome can not be calculated the Utility of all outcomes can be compared as follows

Memory Layout

Mon, 01 Jun 2026 00:38:05 GMT

In a C executable, the memory is stored as below

Sequential Decision Trees

Mon, 01 Jun 2026 00:37:56 GMT

Sequential Decision Trees can be used in a situation where multiple, sequential decisions are to be made in a stochastic environment

Decision Trees

Mon, 01 Jun 2026 00:22:54 GMT

Decision Trees are an alternative way to visualizing a Decision Matrix Like decision matrices, decision trees include actions with its associated outcomes, probabilities, and payoff or costsDecision trees are however, a generalization of a decision matrix where each action may not exactly have the same outcomes and probabilities
Decision trees typical use the Expected Value Decision criterion and does not typically use Maximin or Minimax Decision criterionsA decision tree typically has a as an action node And typically has a as an outcome node
A supermarket chain is deciding whether to make or buy a type of cereal for their storesThe cost to make the cereal is a fixed monthly cost with a variable cost of per boxThe cost to buy from an outside source is per boxThe monthly demand for that type of cereal is found to be belowIn this scenario there are two actions that can be taken which will originate from a Each out decision will have three separate outcomes which will originate from a
The Expected Value (of the cost) can be given via the equationThe cost is then computed for each outcome in a decision tree
The decision can then be simplified at each decision pointFrom this the lowest cost decision can be made

Decision Trees Example_1

Mon, 01 Jun 2026 00:21:56 GMT

Decision Trees Example_0

Mon, 01 Jun 2026 00:19:42 GMT

Decision Trees Example

Sun, 31 May 2026 17:48:09 GMT

Decision Matrix

Sun, 31 May 2026 03:01:32 GMT

A Decision Matrix is a type of Matrix that lists possible actions along its outcomes, probabilities (if any), and payoff or costs

Expected Value Decision

Sun, 31 May 2026 03:00:18 GMT

The Expected Value Decision is used to chose the best of many Actions in a stochastic environment, the expected value decision choses the action determined by the maximum expected payoff and the lowest expected cost
Optimal Action While expected value decision chooses the best actions in any given action set, it may not necessarily be the most Optimal Action The most optimal action will only be chosen when it is in the set of all available actions Consider the a decision where someone is trying to go to work but there is a chance of raining The available actions are: Stay at Home Go to work without umbrella Go to work with an umbrella For this scenario, the umbrella is large and cumbersomeThe probabilities for rain are: Rain: No Rain:
And us the Decision Matrix can be given byGiven this there should never be a reason to stay at home since it will always result in a net loss
The Expected Value of the rest of the action/outcome pairs can be calculatedAccording to the expected payoff table, going without an umbrella is the best possible action in this scenario with an expected payoff of

Maximin Decision

Sun, 31 May 2026 03:00:08 GMT

A Maximin Decision is often used when where certain stochastic outcome probabilities are unknownMinimax Decision Some analysis requires the use of minimax decision instead of maximin Maximin decision factors payoff while minimax decision factors cost Consider the Decision Matrix of the scenario introduced in Expected Value DecisionIf the probabilities for rain is not given then we must consider the minimum payoff for each action Stay at Home Go Without Umbrella Go With Umbrella The maximin decision chooses the maximum of the set of minimum payoff of the available actions which would equate to going to work with an umbrellaBecause of this, the maximin decision usually leads to a conservative decision

Bayer's Theorem

Fri, 29 May 2026 22:21:35 GMT

Consider a situation where several mutually exclusive events have been determined along with their initial probabilities Now consider an events that changes the probabilities of these events, now the updated probabilities are Bayer's Theorem can be used to update the probabilities as soWhich can be simplified using the Law of Conditional ProbabilityFrom set theoryAnd by substituting the law of conditional probability So the full form of Bayer's Theorem for updating a single probability can be given byFor all and Note that Bayer's Theorem will only work when

Law of Conditional Probability

Fri, 29 May 2026 21:58:47 GMT

The Law of Conditional Probability can be used to find the probability that an event will occur given event has occurred which can be written as The law as followsThe law only works when Two coins are flipped, what is the probability that both coins lands on heads given that one coin has landed on headsThe sample space is as followsThe original probability of both coins landing heads is one out of four outcomes or If one coin lands head, the sample space then becomesThe probability then becomes one out of two outcomes or This is given by the law of conditional probability as soWhereThe because we're in the original sample space , the event of one heads and either heads of tails and only heads only share one outcome out of four Therefore

Exponential Distribution

Fri, 29 May 2026 21:38:14 GMT

The Exponential Distribution for a Stochastic process is a continuous, non negative distribution specified by one parameter: its Expected Value The probability distribution for a random variable is given byThe expected value is given by The variance is given by The standard deviation is also given by
Unlike both the Normal Distribution and Poisson Distribution, the cumulative distribution can be given as a closed form expressionWhere
Consider a server that processes information with an exponential distribution with an Expected Value of What is the probability that a request gets processed in less than From taking the equation of an exponential distributionAnd substitution our parameters

Poisson Distribution

Fri, 29 May 2026 21:25:15 GMT

The Poisson Distribution, which means fish in French, is a distribution for a Stochastic process represented by only one parameter: its Expected Value For a random variable with the poisson distribution
Unlike a Normal Distribution, the poisson distribution is not symmetric along its expected value, instead it has a tail towards

Electric Field of Continuous Charge

Fri, 29 May 2026 21:23:27 GMT

Where is the distance from to is the vector form to Consider a ring along the axis

Poisson Distribution Example

Fri, 29 May 2026 21:21:34 GMT

Normal Distribution

Fri, 29 May 2026 21:13:07 GMT

A normal or gaussian distribution is a distribution of a real-world random variable specified by two parameters, the Expected Value and the Standard Deviation The normal distribution can be represented via The position of the curve is determined by the The height of the curve is determined by the
Probability in a Normal Curve The area of a normal curve is equal the probability of that domain occurring The total area of a normal curve is always equal to

Normal Distribution Example

Fri, 29 May 2026 21:06:06 GMT

Expected Value

Fri, 29 May 2026 20:59:13 GMT

Expected Value is the averaged expected value among all numbered outcomes of Sample SpaceA an expected value of a random variable is often denoted as or The expected value of a common die of six sides can be given as belowHowever, is not rollable number on the die, it just represents the average value of the dice roll as the amount of rolls approach infinity

Event Operations

Fri, 29 May 2026 04:10:18 GMT

Operations can be applied to Events similar to Set OperationsThe union between events and describes the set of outcomes of the events or or bothA union is generally notated as The intersection between events and describes the set of outcomes shared by both and An intersection is generally notated as If the intersection between and is empty, then and are mutually exclusive
The complement of event describes the set of outcomes in the Sample Space that are not the outcomes of A complete is often represented as a hat The union of an event and its complements can be given as so SinceComplements also follow the equality
Where is the entire Sample SpaceConsider three coins being flipped , and let event be at least one heads being flipped Find the followingThe outcomes of is any case where there are at least one heads and at most two tails And the outcomes of is all outcomes that doesn't contain Since contains out of the outcomesThe probability of can be calculated two ways

Event

Thu, 28 May 2026 21:28:21 GMT

An event is a subset of all possible outcomes in a Sample SpaceIt is often denoted as a capital letter after a probability functionThe event contains the sum of the probabilities of all outcomes Consider three coins being tossed with being the probability for heads and being the probability for tailsThe probabilities of these eventsCan be given as such

Basic Probability

Thu, 28 May 2026 21:11:44 GMT

A probability is the change that an outcome will occur from a space of all possible outcomesConsider a coin toss, the probability that heads will land is and that probability that tails will land is also

Faraday's Law Example

Wed, 13 May 2026 18:44:51 GMT

Faraday's Law

Wed, 13 May 2026 17:32:47 GMT

The change in magnetic flux over a closed loop causes an induced voltage as an Electromotive ForceWhere
Negative EMF The EMF generated is negative of the change in magnetic flux because of Lenz's Law Consider two parallel wires with a resistor connecting it and a moving conductive rod
Consider a wire with a current and a moving closed loop of wire

Current

Wed, 13 May 2026 17:26:16 GMT

CurrentCurrent is flow of positive charge, it's denoted by ( for fixed currents) Current is measured in Amperes and is a standard SI Unit
Flow of positive charge Current is the flow of positive charge while electrons contain negative charge, in reality electrons flow the opposite direction from the current Since charge carriers do not travel in a straight line, drift velocity measures the general velocity of charge carriers
The current density is the measure of the flow of current per unit area and is dependent on drift velocity

Ampere's Law

Wed, 13 May 2026 17:23:48 GMT

The magnetic field along a closed path is proportional to the enclosed currentWhere is the Current Density
Amperean Loop Similar to a Gaussian surface, the loop in which the magnetic field is integrated over is arbitrary Consider a solenoid where the region within the solenoid is a straight magnetic field
The inner magnetic field can be found using Ampere's law

Poynting Vector

Wed, 13 May 2026 16:22:22 GMT

The poynting vector conveys the direction and intensity of an Electromagnetic WaveThe direction of the poynting vector is always in the direction of the electromagnetic wave no matter the phase

The time average of the Poynting vector is the intensity of a WaveWhere is the intensity And where is the power And where is the area

Poynting Vector Example

Wed, 13 May 2026 16:20:37 GMT

Capacitance

Wed, 13 May 2026 16:05:15 GMT

Capacitance is a measure of potential charge given in units of faradsThe capacitance of a system only depends on the geometry of the system.To find the potential of conductorTherefore the equations of the potential across a conductorFor capacitors in seriesFor capacitors in parallelConsider two parallel plates of area and charge that are distance away
The Electric Potential of the two plates can be found via Consider bringing in charges from infinity, the work needed to bring the first charge is zero The work needed to bring the second charge is
Generally it can be given asConverting to energyAnd since
Where is the energy stored The energy within a capacitor is stored within the Electric Field
Consider two spherical shells
The time it takes to charge a capacitor is proportional to the total resistance and capacitance in the circuitUsing , we can find the charge of a capacitor as function of time

Electric Potential

Tue, 12 May 2026 22:28:55 GMT

The electric potential is the energy held by a charge due to surrounding Electric FieldsFor a point chargeElectric potential has units of or or voltsIn mechanics, work is dependent on displacementWorkWork is defined as the amount amount of force applied over a distanceTherefore the total work done over a distance is equal to the change in kinetic energy between the two distances Potential Energy Kinetic EnergyIn a system where energy is conserved
For Conservative Forces, the path of the object does not affect the total work done by the force Since electric fields are conservative, the work is equal to the total displacement done by electric fieldsThe change in potential energy of given work isThen the change electric potential given the potential isPotential vs Electric Potential Potential energy and electric potential energy is not the same, while potential energy behaves similarly to mechanics, electric potential energy depends on the charge of the object Find the electric potential between two charged spherical shells
The electric potential and the electric field has the relation ofMeaning the electric potential can be found from the electric field and vise versa
Find the electric potential of a source then use it to solve for the Electric Field
Find the electric potential of a uniformly charged rod

Magnetic Dipole

Tue, 12 May 2026 21:52:06 GMT

Magnets usually have a north and a south pole which is called the dipoleThe magnetic field over an enclosed area must be zero Therefore, magnetic monopoles can not existThe the magnetic dipole moment is the direction and strength of a the magnetic field from a source that induces a magnetic field The magnetic dipole moment is defined as

Coulomb's Law

Tue, 12 May 2026 21:12:53 GMT

Coulomb's Law represents the Electric Field created by two charges

Node Analysis Example_0

Mon, 11 May 2026 05:35:34 GMT

Serotonin

Mon, 11 May 2026 04:20:19 GMT

NH₂NHHO

Oxidation Numbers

Mon, 11 May 2026 04:16:52 GMT

Formal Charge and oxidation numbers are two extremes of election assignmentWhen a bond splits homolytically, the electrons are hypothetically split apart evenly so each atom receives one electron. This gives a formal charge of for both atoms.When a bond splits heterolytically, one atom, usually the more electronegative one receives both electrons. This gives a oxidation charge of and The actual charge is intermediate between the two extremes, both of these models are hypothetical representations.homolytically: splits evenly heterolytically: splits unevenly, favoring one side Fluorine always has an oxidation number of Oxygen usually has an oxidation number of Hydrogen has an oxidation number of when bounded to nonmetals but when bounded to metals The sum of the oxidation numbers is the charge on a molecule.Find the charge of is assigned The molecule has a charge of

Equations

Mon, 11 May 2026 01:25:31 GMT

Current Law: Voltage Law: Series: Parallel: Series: Parallel: Voltage Divider: Current Divider: RC circuit: RL circuit: Natural Response (1st order): Resonant Frequency: Ideal assumptions: (virtual short), (no input current)Diode (Shockley): where at room tempBJT: MOSFET (Saturation): where MOSFET (Triode/Linear): De Morgan's: Key Identities: Euler's: → |z| = \sqrt{a^2+b^2}\theta = \tan^{-1}!\left(\frac{b}{a}\right)$Multiply: magnitudes multiply, angles add. Divide: magnitudes divide, angles subtract.Transfer Function:

Diode

Sat, 09 May 2026 23:24:38 GMT

A diode is an active component that only allows current through one way of the component

AC Steady State Example_0

Sat, 09 May 2026 22:37:55 GMT

AC Steady State Example

Sat, 09 May 2026 22:15:24 GMT

Impedance

Sat, 09 May 2026 21:04:10 GMT

Impedance is a generalization of resistance in AC circuits, it is generally represented byWhere is a sinusoidal voltage And is a sinusoidal currentSimilar to resistance, impedance is measured in
A Resistor will always have an impedance of its resistanceFor dynamic elements such as capacitors and inductors, their resistance can be calculated via using Let's consider the voltage of both componentsWe assume that the current is always a complex sinusoidal function Simplifying Per the form

DC Steady State

Sat, 09 May 2026 20:36:28 GMT

In time dependent circuit, a steady state is achieved after a significant amount of time when there is no longer a change in voltage or currentTo analyze DC steady state circuits Replace all Inductors with open circuits
Replace all Capacitors with short circuits Analyze the resulting circuit like a non time-dependent circuit In a steady state circuit the Consider this time dependent circuit
After converting all time-dependent components into their steady state equivalents, we get this circuit
Now solving for the remaining systemAnd since the remaining system is in steady state (no voltage or current changes)The capacitor has no current (open circuit) and the voltage of And the inductor has no current (short circuit) and the current of Because this system has both a capacitor and an inductor we need to take into account two time constantsSince the circuit reaches steady state in approximately , we take the longer value

DC Steady State Example_0

Sat, 09 May 2026 18:08:40 GMT

DC Steady State Example

Sat, 09 May 2026 18:08:31 GMT

Capacitor

Fri, 08 May 2026 04:33:49 GMT

Capacitors store charge between its two plates and stores energy between the plates via an Electric FieldThe capacitor is given by this symbol
The capacitance is read from a three-digit code XYZ where: X and Y are significant figures Z is the multiplier (power of 10 to multiply by) Examples 103K = 10 × 10³ pF = 10,000 pF = 10 nF, ±10% tolerance 105 = 10 × 10⁵ pF = 1,000,000 pF = 1 µF (tolerance typically ±20% if not specified) 221 = 22 × 10¹ pF = 220 pF The charge stored in a capacitor can be determined via
Where is the Capacitance of the capacitorAnd as Capacitors store energy equal to one half of the conductivity of the inductor times voltage squared
This energy is usually stored within a dielectric as a Electric Field
Unlike, Resistors, capacitors in series divide all voltage across all capacitorsVoltage and Current While resistors in parallel share a voltage split current, capacitors in series will usually share a common current and split voltage Capacitors in parallel combine their capacitance
When paired with a resistor in a RC Circuit, a capacitor charges and discharges according to the time constantA capacitor reaches approximately 63% of its final voltage after one and is considered fully charged after

Capacitor Example

Fri, 08 May 2026 04:23:43 GMT

Resistor

Fri, 08 May 2026 04:12:32 GMT

A resistor adds resistance to a line of current, any medium that can resist a current (like a wire) can be treated like a resistor, resistors always absorb energy Resistors are often the most ideal componentsideal: Ideal components do not draw current nor supply current The resistor is given by this symbol
Europe Standards In Europe, resistors are defined by a different symbol The resistance of a resistor can be determined by using a color code system.Ohm's Law relates voltage and current through a resistor Ohm's Law Ohm's Law is only meant to be used for resistors, therefore it can not be generalized to non resisting components The root mean square gives the mean of a set of, of continuos, numbers This is useful for measuring sine/alternating waves in AC voltagesResistors in series combine their resistanceResistors in parallel divide the current between all resistorsJustification for resistors in parallel
If then If then For resistors in series, the total resistance is always greater than the largest value For resistors in parallel the total resistance is always lesser than the smallest valueIn certain cases, the resistance of an entire circuit (in series) can be combined into a single resistor
The resistance of a resistors can be variable by shifting the contact between two points, changing the resistance
An electric guitar uses variable resistors to control volume and tone
The temperature coefficient of a resistor is the variation in resistance given changes in temperature

An ohmeter measures the resistance by measuring the voltage of a predictable current source

Operational Amplifier

Fri, 08 May 2026 04:11:37 GMT

An operation amplifier (Op Amp) takes in a positive and negative voltage then outputs a multiple the difference of the two inputsThe op amp is drawn as a triangle, the voltage source is often emitted The op amp does not draw current, instead, it uses a outside voltage sourceThe gain is often large but never exceeds the external power supplyThe gain of the Op Amp is heavily variable, it subject to changes in temperature, humidity, and background conditionsConsider this op-amp circuit
If is slightly higher than then will be Vice versa, if is slightly lower than then will be Consider the system
Component multiples input into while component takes a fraction of and feeds it into the negative of the input sumThe system can be modeled asSo to double the input , must be In electronics, an op amp can be used to double voltage as follows
For Op Amps with feedback, if the voltage then there is no gain or
This is called a virtual shortZero Voltage If the voltage is zero, then the voltage output is not zero, it would be solved differently than Consider a sensor and a speaker output with a
Because the sensor will often not have enough voltage to power the speaker An op amp can be used to amplify current in a circuit known as the Unity Gain Circuit
The voltage can be model as followsBecause And therefore the input of the sensor is unchanged while the current is amplifiedIf a system is linear, an input signal will produce equivalent output signal However, for non-linear systems, the output will not be equivalent
The op amp can filter distortion by using a feedback loop
An op amp can be tested for negative feedback using the following process Remove op amp from circuit Set all voltage and current sources to zero Place a voltage source at op amp's former output node If any part of the that voltage gets to the input's negative terminal, then there is negative feedback Consider this circuit
Since the voltage of the test source reaches the negative terminal The circuit has a feedback loopNow consider this circuit
While the current reaches the negative terminal, the negative terminal is connected to ground so therefore no voltage reaches the negative terminal The circuit does not have a negative feedback loopNow consider this cat
Isn't he silly.Finally, consider this circuit
Since neither the current nor the voltage reaches the negative input terminal The circuit does not have a negative feedback loopThere are two types of amplifiers using Op Amps, inverting and noninvertingNoninverting amplifiers amplify the signal voltage in phase with the input
Where the amplification factor is a function of the resistorsVoltage Follower A buffer can be created when is close to 0 and is close to infinity (or removed) Inverting amplifiers amplify the signal voltage opposite of the phase of the input
Where the amplification factor is a function of the resistors

Operational Amplifier Example_13

Fri, 08 May 2026 04:08:12 GMT

Operational Amplifier Example_12

Fri, 08 May 2026 03:59:07 GMT

Operational Amplifier Example_11

Fri, 08 May 2026 03:40:01 GMT

Thevenin and Norton Theorems

Thu, 07 May 2026 21:14:37 GMT

Both Thevenin and Norton theorems both rely on the principle that any linear circuit can be treated as a simplified circuit with two terminals.linear circuit: A circuit with output that's linearly proportional to the inputThevenin equivalence assumes the circuit has an open circuit voltage and a Parasitic Resistor in series (assuming all voltage sources are turned off)

Norton equivalence assumes the circuit has a short circuit current and a Parasitic Resistor in parallel
Consider this circuit to find the voltage difference between A and B
The voltage difference between A and Ground is
The Node C acts as a port to a voltage divider between A and Ground
Since no current flows through the Resistor, the Thevenin voltage equivalence is

Thevenin and Norton Theorems Example_1

Thu, 07 May 2026 21:10:20 GMT

Thevenin and Norton Theorems Example

Thu, 07 May 2026 20:58:32 GMT

Thevenin and Norton Theorems Example_0

Thu, 07 May 2026 20:58:19 GMT

Node Analysis

Thu, 07 May 2026 19:49:58 GMT

Node Analysis

Thu, 07 May 2026 19:49:55 GMT

Node Analysis is a set of procedures to solve simple circuitIt's steps are as follows: Identify all Nodes Chose one node as a reference Asking variable names to the rest of the nodes
Write KCL at each unknown node Solve resulting linear system for the the voltages Use the voltages to solve for any needs currents Consider a system of resistors
Using these equalities, solve for voltages via linear algebraThen use voltages along with KCL to solve for currents

Node

Thu, 07 May 2026 19:08:15 GMT

A node is a junction with several paths, nodes are subject to Kirchhoff's Laws
Nodes are area of wire that connect two or more components, uses KCL
Super Nodes are multiple nodes linked together (often linked with an ideal voltage source), uses KCL
Loops can be thought of as an independent system of voltages, all voltage additions must be accompanied by voltage drops, uses KVL

Sockets

Sun, 03 May 2026 06:51:06 GMT

Sockets are channels of communication between two endpointsSome types of sockets includes Internet Sockets, UNIX Sockets, and sockets over Bluetooth, Radio, or other mediums of informationUDP is a single socket method of communication that's analogous to postal mail. While the size of information transmitted might be high, there is no guarantee of delivery, and the route of each packet might be differentTCP is a single socket method of communication that's analogous to a telephone call. The size of each individual packet is small but the stream is continuous Sockets can be identified by a combination of their host IP Address and Port NumberWhile multiple services can exist within a host, only one process can occupy a port number at a timeThe connection of a socket follows a series of protocols starting with the socket function The socket function initializes the socket connection between the hostsint socket(int domain, int type, int protocol)Where domain is the protocol family AF_INET for IPv4 AF_INET6 for IPv6Where type is the semantic of the communication SOCK_STREAM for TCP SOCK_DGRAM for UDPWhere protocol is the specific protocol and is usually 0After the socket() is established the connection can be bound via bindint bind(int sockfd, const struct sockaddr *addr, socklen_t addrlen)Where sockfd is the socket identifier, the return value of socket() Where sockaddr is the protocol specific struct Where addrlen is the size of the address in bytesByte Ordering With little endian, the least significant byte is stored at the lowest address. With big endian, the least significant byte is stored at the highest address. While LINUX uses little endian, TCP/IP uses big endian After the bind(), the listen function can be called to mark the server as a passive server, a passive server, or a server that can accept incoming connections via acceptint listen(int sockfd, int backlog)The accept function waits for a request to come in, this blocks the process until a request arrivesint accept(int sockfd, struct sockaddr *addr, socketlen_t *addrlen)

Process

Sun, 03 May 2026 03:50:49 GMT

A process is a executable unit that handles a single programA zombie process is a process where termination of the process has not been handledA zombie process can be created by failing to wait before the child process is finishedAn orphan process is a process where the parent process no longer existsThis can happen is the parent process exits before the child process is finished or if the parent process does not wait for the child processOrphans are created when their parents die (either through killing them directly or by natural causes). Because their parents do not exist, orphans are useless entities that can take up space and resources within the system at large while contributing nothing.It is then important to kill orphans as soon as possible to save on resources. Orphans can be killed in any number of ways.The most popular way to kill orphan is via killkill -9 [Name of Orphan] Below is a orphan killing algorithm#!/bin/bash # Kill orphan processes: user-space processes re-parented to init (PPID=1) GRACE=5 MIN_AGE=60 while IFS= read -r pid comm; do [[ ! -s "/proc/$pid/cmdline" ]] && continue uid=$(awk '/^Uid:/{print $2}' "/proc/$pid/status" 2>/dev/null) [[ "${uid:-0}" -le 999 ]] && continue age=$(( $(date +%s) - $(stat -c %Y "/proc/$pid" 2>/dev/null || echo "$(date +%s)") )) [[ "$age" -lt "$MIN_AGE" ]] && continue echo "Killing orphan: PID=$pid CMD=$comm UID=$uid AGE=${age}s" kill -TERM "$pid" 2>/dev/null sleep "$GRACE" kill -0 "$pid" 2>/dev/null && kill -KILL "$pid" 2>/dev/null done < <(ps -eo pid=,ppid=,comm= | awk '$2 == 1 && $1 != 1 {print $1, $3}')

Protocol

Sun, 03 May 2026 01:37:29 GMT

A protocol is an agreement between two parties on a method of communication

Signal

Sun, 03 May 2026 01:36:33 GMT

A signal is a type of interrupt that change the flow of a program, they can communicate hardware level problems (such as division by 0), OS level exceptions (such as alarm) and process level requests (such as kill)Signals can be generated via functions such as the alarm and kill functionsunsigned int alarm(unsigned int seconds) sets and alarm and sends the SIGALRM signalint kill(pid_t pid, int sig) sends the specified signal to the process with the pidSignals can be handled with both signal and signactionWhen a signal is called, the operating system stops the program then passes the signal to the signal handler, the handler then processes the signal and returns control back to the programSignals used to be handled via the signal handler#include #include #include void sighandler(int signo) { if (signo == SIGINT) { printf("Received SIGINT\n"); } } int main() { if (signal(SIGINT, sighandler) == SIG_ERR) { printf("Can't catch SIGINT\n"); } // A long sleep so we can issue a signal. while (1) { sleep(1); } return 0; } However, this way of handling lacks the ability to block other signals from being called during the handle process, signal also have different behavior on different systemsModern programs use sigaction which changes the way the process operates The structure of the function call is as belowsint sigaction(int signum, const struct sigaction *act, struct sigaction *oldact) Where signum can be any signal except SIGKILL or SIGSTOPstruct sigaction { void (*sa_handler)(int); void (*sa_sigaction)(int, siginfo_t *, void *); sigset_t sa_mask; int sa_flags; void (*sa_restorer)(void); } The mask sa_mask is the set of signals to block while the signal is being handled, the signal that triggered the handler is blocked by default#include #include #include void handle(int sig, siginfo_t *siginfo, void *context) { printf("Sending PID: %ld, UID: %ld\n", (long)siginfo->si_pid, (long)siginfo->si_uid); } int main() { struct sigaction act = {0}; // Use the sa_sigaction field because the handle has two additional // parameters. act.sa_sigaction = handle; // The SA_SIGINFO flag tells sigaction() to use the sa_sigaction field, not // sa_handler. act.sa_flags = SA_SIGINFO; if (sigaction(SIGTERM, &act, NULL) < 0) { perror("sigaction"); return 1; } while (1) { sleep(1); } return 0; } A reentrant function is function that can safely be interrupted and restarted without causing issues To be a reentrant function, the function must satisfy the following It may not use global/shared resource It should operate only with the data provided by the caller It must call only other reentrant functions

Lorentz Transformation Example_1

Tue, 28 Apr 2026 14:41:33 GMT

Lorentz Transformation

Tue, 28 Apr 2026 14:34:35 GMT

Lorentz Transformation Example_0

Tue, 28 Apr 2026 14:33:54 GMT

Lorentz Transformation Example

Tue, 28 Apr 2026 14:26:56 GMT

Electromagnetic Wave

Tue, 21 Apr 2026 15:08:23 GMT

As a wave travels, the Electric Field and the Magnetic Field are orthogonal to the wave and each other
The speed of the wave is equal to The wavelength of the wave is equal to Consider Faraday's Law

Electromagnetic Wave Example

Tue, 21 Apr 2026 14:40:04 GMT

Self Inductance

Tue, 14 Apr 2026 14:47:11 GMT

Coils with a current source induces Magnetic Fields that then induces a current when that current source is off

Self Inductance Example

Tue, 14 Apr 2026 14:37:50 GMT

Mutual Inductance

Tue, 14 Apr 2026 14:32:59 GMT

A Magnetic Field induced from a Current in one coil induces a magnetic flux within another, coupled coil
Consider two concentric circles with current flowing through the other coil

Mutual Inductance Example_0

Tue, 14 Apr 2026 14:29:21 GMT

Mutual Inductance Example

Tue, 14 Apr 2026 14:24:53 GMT

Linear Independence

Tue, 07 Apr 2026 18:20:25 GMT

A set vectors can be can be determined independent by using its corresponding Linear FunctionFor a set of vectorsIf only has a trivial solution then all vectors are linearly independentIf has non trivial solutions, then the vectors are linearly dependentAlternatively, the vectors in form must have a trivial solutionConsider the vectorsTo determine linear independency, must be a trivial solution
To show this, we can turn into a upper triangular matrix in Echelon formBecause has pivots for a matrix then is a trivial solutionTherefore, vectors ,, and are all linearly independent

Matrix Factorization

Tue, 07 Apr 2026 18:05:11 GMT

A matrix factorization expresses a Matrix as a product of two or more matricesLU Factorization is a kind of matrix factorization that uses two matrices and Where is A lower triangular matrix With 's on its diagonal Where is A upper triangular matrix
LU Theorem Assume is an matrix that can be reduced to Echelon form without Row Interchange Then, can be uniquely written in the form Consider a matrix
Using Elementary Matrix Operations, reduce into Echelon form (without interchange)
Pivot Row Like in Gaussian Elimination, the pivot row should always be used first when performing replacement operations Failing to do so may not generate a upper triangular matrix This upper triangular matrix is now
can then be represented as a a product of Elementary Matrices
By taking the Inverse all of the matrices, we can rewrite Then the inverses becomes Then finally, we can write as soWhen factorizing a matrix into an matrix, the interchange and scaling operations are not allowedIf interchange is necessary, then a helper matrix can be used to transform into a matrix that does not need scalingThen A can be represented as soScaling When scaling, the matrix, by definition, can not have the same identity No matrix should necessitate scaling during factorization

Geometrical Model for Gauss Jordan

Tue, 07 Apr 2026 18:05:11 GMT

The Gauss-Jordan Elimination can be solved geometrically via using vectorsConsider a two equation systemAfter graphing the system, we can determine there is only one solution for the system
A system is made out of equations and unknown variables Theses systems can be consistent or inconsistent based on their solutionsConsider the system and it's matrix equivalentConsider the pivot column It's variable is called the basic variable while the other variables , are called free variables
pivot: Leading entry in Echelon form The solution to the system can have multiple formsHowever, generally, we would have to solve for the basic variable in terms of free variablesConsider the systemWhen constructing equations, ensure basic variables are solvedSo that for the equations in matrix form, the basic variables are Consider the systemBecause the solution is a pivot, there are no solutions There is not enough equations
In a reduced Echelon form of If a row matching Then the system has no solutions and is inconsistent

Inverse Elementary Matrix Operations

Tue, 07 Apr 2026 17:32:53 GMT

All Elementary Matrix Operations can be inversed
For a Identity Matrix An inverse replacement operation subtracts values from one row to anotherAn inverse interchange operation behaves exactly the sameAn inverse scaling operation scales the middle value of the matrix inversely

Elementary Matrix Operations

Tue, 07 Apr 2026 17:32:48 GMT

Elementary matrix operations are simple operations that can be applied to matrices All elementary operations are Inversible
For a Identity Matrix An replacement operation adds values from one row to anotherAn interchange operation swaps one row for anotherA scaling operation scales the middle value of the matrix

Memory Errors

Mon, 06 Apr 2026 15:35:39 GMT

Memory errors are errors relating to management of Memory within a C ProcessOccurs when a memory is read from a location that does not existSimilar to invalid read but occurs when memory is written to a location that does not existThe message "conditional jump or move depends on uninitialized value(s)" occurs when the operation depends on a variable that is uninitializedconditional jump: a statement that controls flow such as if, for, or while statements move: any other kind of read from memoryA segmentation happens when the operation system intervenes before the program crashes, among other things, a common cause of it is dereferencing a null pointer

Biot-Savart Law

Mon, 06 Apr 2026 02:47:29 GMT

Given a wire of charge and a position , the Magnetic Field at point can be given as
Consider a Current carrying wire and a position P

Since the magnetic field is a Cross Product of the change in the wire length and the vector to , the magnetic field is pointing into the pageConsider a straight finite wire and a point that is on the bisector of that wire
Now consider a situation where is going to infinitySo the magnetic field of an infinite current carrying wire isConsider a ring of wire and a point

Faraday's Law Example_0

Fri, 03 Apr 2026 00:22:03 GMT

Echelon

Thu, 02 Apr 2026 18:51:54 GMT

An echelon or an echelon form is a Matrix All rows with zeros are below nonzero rows All nonzero leading entries must be below and to the right of all others All entries below a leading entry must be zeros Below are examples of echelon form matricesThe identity matrix is in echelon formEverything right and above the leading entries can be any number

Pasted image 20260402142806

Thu, 02 Apr 2026 18:28:06 GMT

No title

Gaussian Elimination Example_2

Thu, 02 Apr 2026 17:40:47 GMT

Gaussian Elimination Example_1

Thu, 02 Apr 2026 17:31:01 GMT

Gaussian Elimination Example_0

Thu, 02 Apr 2026 17:29:40 GMT

Gaussian Elimination Example

Thu, 02 Apr 2026 17:16:34 GMT

Lenz's Law

Thu, 02 Apr 2026 14:44:55 GMT

Lenz's Law is made out of two parts that deal with forces generated by changes Induced EMF will always be in the direction that opposes the direction of the change in magnetic flux that caused it Induced currents, torque, and force will always oppose the direction of the change in magnetic flux that caused it

Lenz's Law Example

Thu, 02 Apr 2026 14:34:48 GMT

Matrix Functions

Tue, 31 Mar 2026 18:03:34 GMT

A matrix function is a Matrix composed of component functions is differentiable at if all of its components are differentiable at The derivative of a matrix function can be taken by taking the derivative of its components Sum Rule Product Rule

Linear System

Tue, 31 Mar 2026 18:03:08 GMT

A linear system of linear algebraic equations is a set of linear equations without differentialsWhere the system can be given as a MatrixComposed of unknown vector and constant vectorsSo that the system can be given asWhere is the coefficient matrix and are unknown vectors

Electromotive Force Example

Tue, 31 Mar 2026 14:57:10 GMT

Electromotive Force

Tue, 31 Mar 2026 14:42:24 GMT

The electromotive force (EMF) is the generalized force per unit charge along a closed pathThe force is measured in voltsConsider a closed loop of wire partially in a Magnetic Field

We can use Faraday's Law to calculate the change in magnetic flux, which is equal to the EMF

DC and RC Circuits

Tue, 31 Mar 2026 14:22:04 GMT

DC: Direct Current AC: Alternating Current RC: Resistor CapacitorThe Electromotive Force (EMF) is the representation of the push of current from a battery It is often represented as and represents the work per unit charge across a closed pathAn ideal battery maintains the same EMF across its terminals
Kirchhoff's Current Law states that the current going into a node or component must be the same as the current going out of the node or componentConsider these two scenarios
In both cases the total current into the system equals the total current out of the system Kirchhoff's Voltage Law states that all voltage rises must be accompanied by voltage dropsAnd therefore the sum of all voltage across a circuit must sum up to zero
Consider this Loop containing three components
Since the total rises must equal the total drops , therefore Consider a circuit with resistors, find the current of all nets
Using Kirchhoff's Laws, we can find three independent equations to solve for current

Interprocess Communication

Mon, 30 Mar 2026 21:58:37 GMT

Inter-process communication, IPC, is the way that Processes can communicate with each otherUnix pipes are one of the oldest forms of IPC Pipes are half duplex meaning that data can only flow in one direction Pipes can only communicate between processes with a common ancestorConsider this commandcat file.txt | sort | uniq > sorted_file.txt The output of cat (stdout) is inputted into sort (stdin), then the output of sort is inputted into uniq which is then written to a filePipes can be created via the function call pipe(int[2])#include int pipe(int fd[2]) In this example, fd[0] is open for reading, and fd[1] is open for writing

Digital Logic

Mon, 30 Mar 2026 21:28:03 GMT

Digital Logic is a system of design where data is represented by a on or off stateA NOT Gate reverses the state of the inputAn AND Gate checks if both inputs are onMulti-input Gates AND and OR gates traditionally have two input but can have variants of multiple inputs. Below is a example of a 26 input AND gate An OR Gate checks if any of the inputs are onAn XOR or an Exclusive Or Gate checks if only one of the inputs are onDigital Logic can have common rules Consider a truth table as belowThe logic for this circuit can be given via Boolean Logic as soConsider that And also The circuit is given below

Gate X

Mon, 30 Mar 2026 20:22:50 GMT

The X gate, similar to a NOT Gate, denoted as X is a gate that flips the spin of the qubit

Switch

Mon, 30 Mar 2026 20:22:01 GMT

A switch is a type of component that can start or stop the flow of currentA switch has different symbols depending on its state

Switch Example

Mon, 30 Mar 2026 20:21:39 GMT

String

Sat, 28 Mar 2026 19:22:35 GMT

Strings in C are represented by arrays of characters that ends with a \0 or a null terminatorA string requires bytes where is the amount of charactersCommon string functions includeint strlen(const char *s) which returns the length of the string char *strcpy(char *t, const char *s) which copies string into an already allocated array char *strdup(const char *s) which returns a heap copy of the string

Heap

Sat, 28 Mar 2026 19:21:52 GMT

The heap is a section of memory managed directly by the programAs the name suggests the memory is stored as a HeapThe malloc() function reserves a section of the heap by the amount of bytes provided and returns a pointer to the beginning of the block The free() function frees a section of memory by taking the pointer of the first block as input

Ampere's Law Example_0

Sat, 28 Mar 2026 19:19:48 GMT

Command Line Argument

Sat, 28 Mar 2026 19:17:18 GMT

The C main program has two command line argument variablesint argc number of command line arguments char **argv command line arguments in a String Array

Keywords

Sat, 28 Mar 2026 19:11:39 GMT

Keywords are words that have functional properties and can not be used as variablesThere are as follows

Maxwell's Equations

Thu, 26 Mar 2026 15:19:27 GMT

Maxwell's Equations are a set of equations that describes the relationship between the Electric Field and the Magnetic Field
Gauss's Law
Magnetic Flux Law
Ampere's Law
Faraday's Law Gauss's Law Magnetic Flux Law Ampere's Law Faraday's Law

Ampere's Law Example

Thu, 26 Mar 2026 14:53:55 GMT

Fork Pathway

Wed, 25 Mar 2026 22:26:40 GMT

A fork pathway is a way to visualize forks within a programConsider the functionvoid function1() { pid_t pid; if ((pid = fork()) == 0) { sleep(1); printf("C1 "); fflush(stdout); } if ((pid = fork()) == 0) { sleep(2); printf("C2 "); fflush(stdout); } if ((pid = fork()) == 0) { sleep(3); printf("C3 "); fflush(stdout); } // Small hack to wait for all descendent processes while (wait(NULL) != -1) { } } The Fork Pathway for this function would look like the following:

Kernel Call Exec

Wed, 25 Mar 2026 22:20:49 GMT

The exec kernel call changes the current process into a different programThe execl family takes in a list of elements, the first argument being being the file path and subsequent arguments being the argument list execl("cat", "cat", "/src/a.txt"); Program name repetition The program name is repeated twice in execl due to different program paths and alias'

Fork Pathway Example

Wed, 25 Mar 2026 22:15:40 GMT

Classes

Wed, 25 Mar 2026 22:03:26 GMT

C has multiple base classes to store different structures of dataA struct is a collection of variables under the same classstruct my_struct { int x; int y; char c; } Consider the structstruct a_struct { char a; int b; long c; } While 13 bytes are being stored, the actual size of the struct is not 13Padding needs to be used added to make accessing easier and follows these rules: The offset of each member must be a multiple of that member's size The total size of the struct must be a multiple of it's largest member's size The order of the members in memory stays the same as the code Therefore the struct a_struct will takes up 16 bytesArrays behave similarly, consider the structstruct b_struct { int[3] a; char[5] b; long c; } Will have the memory format ofUnions are used to store different types of data in the same memory locationunion my_union { int i; char c; } Variables in unions can be overwritten by one anotherSimilar to java Arrays, C Arrays are used to store many of the same type of dataint[3] my_array = { 0, 0, 0 }; my_array[1] = 1000;

Matrix Multiplication Rules

Tue, 24 Mar 2026 18:56:59 GMT

Like regular numbers, Matrices, follow associative and distributive laws during Matrix Multiplication Associative Distributive However, they do not follow the communicative law during multiplicationIn general: Consider the matrix The matrix is not the same as

Matrix Multiplication

Tue, 24 Mar 2026 18:51:27 GMT

Two matrices can be multiplied together

Matrix Transposition

Tue, 24 Mar 2026 18:40:30 GMT

Matrices can be transposed, where the Row Vector become the Column Vector and vice versaA transposition is reversable, meaning two transposition operations result in the same matrix

Matrix

Tue, 24 Mar 2026 18:39:28 GMT

A matrix is a multidimensional set of numbersWhereSpecial matrices A column matrixA row matrixColumn vector and row vectors are not equivalent A vector by default is a column vectorA matrix is square if its columns match it's rowsA diagonal of matrix

Matrix Exponents

Tue, 24 Mar 2026 18:39:24 GMT

Matrices can be have exponentsWhereMatrices can have an inverse form whereWhere is the Identity MatrixLet , be integers

Scaling Matrix

Tue, 24 Mar 2026 18:16:28 GMT

A fixed matrix can be "scaled" via using a scaling matrixConsider these examples of scaling matricesGet the same matrix Pick up the first row of B and put on the first row of C Pick up the first row of B put on the second row of C Pick up the second row of B and put on all rows of C Put the sum of each column on the second row of C Keep the first row of B, multiply the second row of B by 2, and divide the third row of B by 2

Inductor Example_0

Mon, 23 Mar 2026 17:50:56 GMT

Inductor

Mon, 23 Mar 2026 17:49:02 GMT

Inductors are a circuit component that uses Magnetic Fields to buffer current
During the energizing phase, the inductor creates and sustain a magnetic field from the flow of CurrentDuring the energizing phase, the created magnetic field pushes currentThe inductor is given by this symbol
The inductance of a inductor is measured in Henrys, the inductance of a inductor can be determined by a color code systemThe voltage of an inductor is equal to its inductance and the rate of change of the currentThe density of an inductor is equal to its inductance and the current
All Electrical Fields induce a Magnetic Field Orthogonally
Inductors, similar to Resistors, combine inductance when in series but split inductance in parallelBut this is only the face if they're magnetically isolated from each other.

Like Capacitors in a RC circuit, inductors have a phase in which the buffer builds upHowever, unlike capacitors, inductors deal with current, not Voltage Current through an energizing inductor Voltage across the inductor Voltage across the resistor Notice the L/R in the Voltage across the resistor Current through a deenergizing inductor Voltage across the inductor Voltage across the resistor Now notice the R/L in the Voltage across the resistor

Inductor Example

Mon, 23 Mar 2026 17:46:29 GMT

Gate Z

Wed, 18 Mar 2026 21:15:34 GMT

The Z Gate, denoted as Z, flips the sign of the value and leaves the value unchanged

Gate Hadamard

Wed, 18 Mar 2026 16:59:15 GMT

The Hadamard gate, denoted as H, creates a superposition where all probabilities are equally likely

Wave Function

Wed, 18 Mar 2026 16:49:51 GMT

The wave function of a single Qubit system can be defined asIn the function, and are complex numbers Where is the probability that the qubit is a and is the probability that the qubit is a Consider the wave functionThe function has exactly a chance of being a and chance of being a The form of a wave functionCan be expanded to the formWhere , , and are all real numbersThe coefficient has no observable effect on the spin of the qubit and thus can be ignored Giving the complex equation

Qubit

Wed, 18 Mar 2026 16:44:40 GMT

A qubit is the information stored in the spin of an atomIn classical computation, a bit can be either 0, or a 1 However, in quantum computation, the bits are measured as down spin or up spin Dirac Notation Instead of a discrete bits, quantum uses Dirac notation to measure the information of the bit

Ohm's Law

Thu, 12 Mar 2026 20:37:20 GMT

Gfg

Gauss's Law

Thu, 12 Mar 2026 15:19:27 GMT

The flux through a surface is equal to the enclosed charge over Gauss's Law can be applied to uneven surfaces since it only relies of volume enclosedConsider a spherically symmetrical charge configuration

Current Loop in External Magnetic Field

Thu, 12 Mar 2026 15:03:06 GMT

Consider a loop of Current in an external Magnetic Field
The force of the magnetic field on the wire can be given byThe force can be broken up into four legs whereSo that

Current Loop in External Magnetic Field Example

Thu, 12 Mar 2026 14:54:37 GMT

Magnetic Dipole Example

Thu, 12 Mar 2026 14:38:08 GMT

Biot-Savart Law Example_1

Wed, 11 Mar 2026 18:10:40 GMT

PLACEHOLDER

Biot-Savart Law Example_0

Tue, 10 Mar 2026 14:39:58 GMT

Biot-Savart Law Example

Tue, 10 Mar 2026 14:24:41 GMT

Magnetic Field Example_0

Thu, 05 Mar 2026 16:12:40 GMT

Magnetic Field

Thu, 05 Mar 2026 16:09:39 GMT

The magnetic field is a Vector Field that describes the magnetic force experienced by a moving chargeThe force from a magnetic field can be given
Where the force depends on the charge, the charge's speed, and its Cross Product
Similar to Electric Fields, magnetic field lines point from North to SouthGiven a charge in a uniform magnetic field, the charge will bend towards or away from the magnetic field
If the charge bends towards the magnetic field Centripetal motion is achieved
The radius of the motion is given as followsThe change in force of a wire carrying charge in a magnetic field can be given as followsWhereSoIn the case the is uniformIn the case that the wire is straightGiven a moving charge, the induced magnetic field can be seen as followsWhere in

Magnetic Field Example

Thu, 05 Mar 2026 15:46:12 GMT

Cross Product

Thu, 05 Mar 2026 15:34:47 GMT

The cross product describes the vector of a parellogram plane from two vectors and is denoted by

Electric Field

Thu, 05 Mar 2026 15:23:39 GMT

The electric field is a Vector Field that dictates the force a single, positive, test charge experience at a pointThe electric field can be found by using the formulaBy substituting into , another form of the equation can be usedThe term depends on the source of the chargeThe force from an electric field can be given byConsider a ring of charge of charge , find the electric field induced by the ring at point

The electric field can be solved from the Electric Potential

Gradient

Mon, 02 Mar 2026 21:17:13 GMT

Chain rule of can be defined as a dot product of a Gradient and a derivative
Let be completely contained in the Level Set Then is always
Since the gradient is orthogonal to every tangent vector in the Level SetFind an equation for the tangent plane to the surface at the point The magnitude of the gradient is the maximum rate of change in the direction of Only maximized when or when

Nonhomogeneous Equations

Sat, 28 Feb 2026 15:14:49 GMT

ConsiderTheorem: Let and be two independent solutions of the second equation Let be any particular solution of the first equation Then, the general solution of the first equation isConsider the equationFind the homogenous equationWhere the solutions areTake the original equation and divide by Find general solution

Higher Order Linear

Thu, 26 Feb 2026 19:21:44 GMT

Consider an equation of higher order derivativesIf and are continuous on an open internal , , then there exists a unique solution to both equations and the solution is defined throughout .

Ordinary Differential Equation

Thu, 26 Feb 2026 18:50:34 GMT

Differential equations are equations that involve derivatives Ordinary differential equations only involves only ordinary derivatives Ordinary: Not involving PartialsWhen solving for ODEs the whole solution set must be found Solutions for ODEs must include an arbitrary constant For example, is a valid solution to the ODEHowever the solution is only valid for Consider gravity and a drag force affecting a ball
If the net force is , the system will be in equilibrium, therefore the equation becomes This implies that velocity is constant and is independent of Initial value problems (IVP) have a given differential and an initial conditionIVPs solves for the arbitrary constant
Can be solved by find the Integral Curve that passes the initial condition
Ordinary Differential Equation
Single Equation First Order
First Order Linear
Homogenous Equation
Nonhomogeneous Equations First Order Nonlinear Second Order
Second Order Linear Second Order Nonlinear Higher Order
System of Equations

Classification of Differential Equations

Thu, 26 Feb 2026 18:12:34 GMT

Differential Equations can be classified in many different waysSolutions to Ordinary Differential Equation only have one dependent variable while Partial Differential Equations have two or moreDifferential equations can be given in a single equation or a set of equationsThe order of different equations refer to the highest derivative involvedFor example,Would be a 2nd order partial derivative Differential equations can be classified as linear if all of derivatives of the unknown term are linearAny non-linear component makes the equation non-linear

Power

Thu, 26 Feb 2026 16:27:16 GMT

Power is given as the change in energy per timeFor circuits, power is given as the product of the current and voltageSince Batteries always supply powerResistors always dissipate power

Resistor

Thu, 26 Feb 2026 16:09:18 GMT

Resistors resists charge within a circuit by making it harder for positive charge to flow throughResistors in series combine their resistanceResistors in parallel divide the current between all resistorsJustification for resistors in parallelIf then If then For resistors in series, the total resistance is always greater than the largest value For resistors in parallel the total resistance is always lesser than the smallest value

Electrical Meter

Thu, 26 Feb 2026 16:07:19 GMT

Meters are instruments that measures a value in real timeAn ammeter measures current in units of mA, it has an in side and an out side and measures the current from the in side to the out sideA voltmeter measures voltage in units of V, it measures voltage form its positive side to the negative side Reverse Current If a voltmeter is supplied with current that travels from the negative side to the positive side, the voltage is reversed An Ohmeter measures resistance by measuring the voltage of a standard voltage source

Circuit Analysis

Thu, 26 Feb 2026 16:07:19 GMT

The current and voltage of components and Nodes within a circuit can be identified using Kirchhoff's Laws and Ohm's LawBecause of KCL, the sum current in into a system of resistors is the same a the sum current out of the system Current of a resistor in parallel $$ i_{a}=i_{\text{total}} \frac{R_{\text{parasitic}}}{R_{a}} $$

DC and RC Circuits Example_0

Thu, 26 Feb 2026 16:00:09 GMT

Kirchhoff's Laws

Thu, 26 Feb 2026 15:41:20 GMT

Kirchhoff's Laws dictate how electricity moves within a circuitKirchhoff's Current Law states that the current going into a node or component must be the same as the current going out of the node or componentConsider these two scenariosIn both cases the total current into the system equals the total current out of the system Kirchhoff's Voltage Law states that all voltage rises must be accompanied by voltage dropsAnd therefore the sum of all voltage across a circuit must sum up to zero
Consider this Loop containing three components
Since the total rises must equal the total drops , therefore

DC and RC Circuits Example

Thu, 26 Feb 2026 15:37:26 GMT

Ohm's Law Example

Thu, 26 Feb 2026 15:33:36 GMT

Current

Thu, 19 Feb 2026 16:20:50 GMT

Current Example_0

Thu, 19 Feb 2026 16:15:43 GMT

Current Example

Thu, 19 Feb 2026 16:08:33 GMT

Capacitance Example_2

Thu, 19 Feb 2026 16:01:22 GMT

Capacitance Example_1

Thu, 19 Feb 2026 15:43:57 GMT

Capacitance Example_0

Thu, 19 Feb 2026 15:34:08 GMT

Capacitance Example

Thu, 19 Feb 2026 15:26:22 GMT

Conductor and Insulators

Thu, 19 Feb 2026 15:12:17 GMT

Conductors and Insulators control the flow of chargeIn a conductor, charges are free to move around Materials such as metals are conductorsThe surface of conductors are equal-potential surfacesInsulators restricts the move move charges Electrons are strongly bond Materials such as plastic, wood, and rubber are insulatorsFor a conductor to be in equilibrium, any external Electric Field must be canceled by an electric field of the induced charge within the conductor, making the net internal electric field zero

In addition is to the surface of the conductor, the conductor itself a Equipotential Surface The electric field on the surface of the conductor in equilibrium is perpendicular to the surface therefore it is equipotentialConsider a a plate with surface charge density and a person with a grounded plate between
In these equation the surface charge density is divided by because the surface is a plane with no edge effectsSince And since
The grounded plate effectively shields the person from the external Electric FieldConsider a similar system except the plate is not grounded

Amplification with Op Amps

Wed, 18 Feb 2026 22:49:11 GMT

Amplification of Signals can be achieved with Operational Amplifiers
Inverting vs Noninverting This circuit is noninverting, meaning that the output is not a negative of the input Information for inverting amplification is below
In this circuit, is given by by Kirchhoff's Voltage LawSinceWe reach the equationThe resistors can then be changed to change the gain of the circuitAn inverting amplifier can be created by using a negative feedback loop
The output will be an amplified negative of the inputIn a typical current circuit, the current through the load is often not the same as the current source due to current drops from the source
An op amp fixes this by "adding" back lost current
A solar cell, modeled here as and outputs a voltage but is unreliable with currentThe op amp ensures that a consistent current passes through
A strain gauge measures strain by using resistance, the length changes based on strain and that influences resistance A system of 4 gauges can be used for surfaces
The strain is measured by the difference in voltage of two strain gauges, the circuit to measure the difference is shown below

Amplification with Op Amps Example_3

Wed, 18 Feb 2026 22:36:49 GMT

Amplification with Op Amps Example_5

Wed, 18 Feb 2026 22:27:08 GMT

Amplification with Op Amps Example_4

Wed, 18 Feb 2026 22:24:38 GMT

Amplification with Op Amps Example_1

Wed, 18 Feb 2026 22:10:04 GMT

Amplification with Op Amps Example_2

Wed, 18 Feb 2026 22:08:14 GMT

Amplification with Op Amps Example_0

Wed, 18 Feb 2026 21:58:00 GMT

Amplification with Op Amps Example

Wed, 18 Feb 2026 21:43:50 GMT

Operational Amplifier Example_9

Wed, 18 Feb 2026 21:27:09 GMT

Operational Amplifier Example_10

Wed, 18 Feb 2026 21:27:04 GMT

Operational Amplifier Example_8

Wed, 18 Feb 2026 21:25:01 GMT

Minimum Spanning Tree

Tue, 17 Feb 2026 21:04:27 GMT

A set of edges that produces no cycle and connects all vertices in the Graph The set is at minimal cost (or weight)Prim's algorithm starts with a vertex and considers the least weighted connected vertex, repeating until all vertexes are accounted for
Also known as Dijkstra's single source shortest path algorithmGiven a starting location in space, find the shortest path to another locations
Sorts edges first by edge weight then by lexicographical ordering, checks if the edge does not create a cycle, adds each edge to the minimum spanning tree
Finding cycles within the tree can be achieved through union-findmakeset(x): creates a one-element set find(x): traces the path from up to the root of the path, returning the rootunion(x,y): attaches the root of 's subtree to the root of 's subtreeThere is a cycle when find(x) == find(y)
Given the graph shown in Kruskal's algorithm, separate all nodes using makeset(_) into singletonsSince find(b) != find(c) Create a tree union(b,c)Since find(e) != find(f) Create another tree union(e,f)Continue this process for all edges

Homogenous Equations with Constants Example_5

Tue, 17 Feb 2026 18:59:12 GMT

Homogenous Equations with Constants

Tue, 17 Feb 2026 18:51:58 GMT

In a homogenous equation, coefficient are constantSecond Order Homogeneous equations can be formatted into several First Order Linear equations using reduction of orderConsider the equation
Assume the solutions are exponentials where is a constant Given that the Wronskian The solutions to the equation can be found by finding the solutions to Consider the same equation from above
Consider the equation and initial condition
If the detriment or then the solution will include complex roots and the Wronskian will be zero To solve complex roots the initial set up remains the sameIf thenApply Euler's formula
However these solutions are still complex, to find real solutions use the Superposition Principle
If two solutions are equivalentSo therefore
Apply Variation of ParametersSince
For a non characteristic equation, assume that the solution is ThereforeConsider the equation

The equation can not be solved via Reduction of Order

Homogenous Equations with Constants Example_0

Tue, 17 Feb 2026 18:49:19 GMT

Homogenous Equations with Constants Example_4

Tue, 17 Feb 2026 18:20:32 GMT

Conductor and Insulators Example_1

Tue, 17 Feb 2026 15:44:08 GMT

Priority Queue

Mon, 16 Feb 2026 23:01:29 GMT

A priority queue is a type of Queue where instead of the last element, the element with the most priority is removed first, it supports the following methodsinsert(x):
(Linked Lists)
(Red-Black Tree) deleteMin():
(Linked Lists)
(Red-Black Tree)
However, a better time cost can be achieved by using a COMS Data Structures/Heap getMin(x): (Deletion is ) insert(x): sort():
(Linked Lists)

Operational Amplifier Example_2

Mon, 16 Feb 2026 22:20:51 GMT

Operational Amplifier Example_7

Mon, 16 Feb 2026 22:20:34 GMT

Operational Amplifier Example_6

Mon, 16 Feb 2026 22:11:44 GMT

Operational Amplifier Example_5

Mon, 16 Feb 2026 22:08:05 GMT

Operational Amplifier Example_4

Mon, 16 Feb 2026 21:58:50 GMT

Operational Amplifier Example_3

Mon, 16 Feb 2026 21:56:28 GMT

Operational Amplifier Example

Mon, 16 Feb 2026 21:48:08 GMT

Operational Amplifier Example_0

Mon, 16 Feb 2026 21:44:15 GMT

Operational Amplifier Example_1

Mon, 16 Feb 2026 21:43:44 GMT

Node Analysis Example

Sun, 15 Feb 2026 20:18:00 GMT

Second Order Linear

Sat, 14 Feb 2026 20:52:19 GMT

Second Order Linear Ordinary Differential Equations follow the general form
Let , , and be continuous and on the open interval then there exists a unique solution for the Initial Value Problem, and Also called the superposition principle for homogenous equations If and are two solutions of a homogenous equationThen is also a solutionIf and are two solutions of a homogenous equation and Then is the general solutionLet and be two solutions of a homogenous equation, the Wronskian of the equation can be given asIf then and are linearly dependent If then and are linearly independentTheorem II vs Theorem III While both theorems give a similar form, theorem III gives the general solution while theorem II only gives a single solution
Let and be two solutions of a homogenous equationIf the Wronskian of the solutions at is not zeroFor the initial condition: and There exists a unique choice of and such thatProof for the condition that

Electric Field Example_0

Fri, 13 Feb 2026 01:25:44 GMT

Electric Potential Example_0

Fri, 13 Feb 2026 00:56:31 GMT

Homogenous Equations with Constants Example_3

Thu, 12 Feb 2026 19:13:04 GMT

Homogenous Equations with Constants Example_2

Thu, 12 Feb 2026 18:35:00 GMT

Conductor and Insulators Example_0

Thu, 12 Feb 2026 16:24:15 GMT

Conductor and Insulators Example

Thu, 12 Feb 2026 16:00:31 GMT

Equipotential Contours

Thu, 12 Feb 2026 15:53:49 GMT

Equipotential Contours is a map of contour lines that acts similarly to a height map They follow the following rules: Each point on a specific contour line has the same Electric Potential
The closer the controller lines are to each other the bigger the charge in Electric Potential for a radius
The direction of the Electric Field points form high to low potential is perpendicular to the contour lines

Equipotential Contours Example_1

Thu, 12 Feb 2026 15:51:04 GMT

Equipotential Contours Example_0

Thu, 12 Feb 2026 15:49:52 GMT

Equipotential Contours Example

Thu, 12 Feb 2026 15:49:35 GMT

Lewis Structure

Thu, 12 Feb 2026 15:34:59 GMT

A Lewis Structure minimizes the Formal Charge for all atoms The order of drawing a structure is as follows: Know or guess the molecular structure Produce a trial structure by drawing bonds by using their common valance requirements (the amount of hydrogens in their hydride) Draw bounds for all other atoms (using double bounds if necessary)
Aim to reach an Octet Configuration for all atoms but recognize that not all atoms will have an octet configuration Structures that is an average of several structures, molecules like ozone have several asymmetric formsO⁻O⁺OOO⁺O⁻The resonance structure takes the average of the two formsOOOIn certain situations the octet rule can be violated
Atoms with odd electrons like nitrogen can not expect to follow the Octet Rule all the time, for example, nitrogen oxide has an odd electron on the nitrogen
Molecules like Xeon Difluoride will have more than 8 electrons on the center, this is because the extra electrons are taking space in the hidden d orbitals of the atom Lewis Structure provide no indication of shape of the molecule
While VSEPR can be used to predict shapes, these predictions are not always accurate, for example Oxygen is predicted as diamagnetic while it is paramagnetic
In addition, many compounds can not be represented by Lewis Structures without the use of Resonance StructuresDiamagnetic: Contains no unpaired electrons Paramagnetic: Contains unpaired electrons and is attracted to magnets

Curl of an Electric Field

Thu, 12 Feb 2026 15:29:32 GMT

If the curl of an Electric Field is
Then is ConservativeThis means that the electric field can not loop onto itself

Homogenous Equations with Constants Example_1

Tue, 10 Feb 2026 19:10:00 GMT

Homogenous Equations with Constants Example

Tue, 10 Feb 2026 18:49:15 GMT

Second Order Linear Example_1

Tue, 10 Feb 2026 18:28:10 GMT

Geometry of Electric Fields

Tue, 10 Feb 2026 16:18:29 GMT

Consider Gauss's Law
By using Divergence Theorem, we can rewrite the law into another formThis is considered the Differential Form of Gauss's LawConsider the equation
Recall that is a Conservative Force

Divergence Theorem

Tue, 10 Feb 2026 15:53:42 GMT

Electric Potential Example_1

Tue, 10 Feb 2026 15:40:20 GMT

Type Casting

Mon, 09 Feb 2026 23:04:05 GMT

A cast is implicit when the complier auto casts a value while an explicit cast is called manually within the codeA cast is considered a promotion when the new type more specified than the old, an int is a very primitive type while a float is more specific

Coding Style

Mon, 09 Feb 2026 22:51:29 GMT

Different stylings for C can be used based on different best practices and standardsFor system's programming, most people follow the Linux Kernel Style

Alternating Current Example

Mon, 09 Feb 2026 21:33:03 GMT

Alternating Current

Mon, 09 Feb 2026 21:32:26 GMT

Circuit Analysis Example_0

Mon, 09 Feb 2026 21:24:55 GMT

Circuit Analysis Example

Mon, 09 Feb 2026 21:21:53 GMT

Second Order Linear Example_0

Thu, 05 Feb 2026 19:25:21 GMT

Second Order Linear Example

Thu, 05 Feb 2026 19:00:26 GMT

PLACEHOLDER

Initial Value Problem

Thu, 05 Feb 2026 18:45:27 GMT

Separable Equation

Thu, 05 Feb 2026 18:45:15 GMT

An Ordinary Differential Equation can be called separable if it can be written in this form Separable equations can be separated into two parts and can be integrated directly
Exactness All separable equations are Exact This is a separable equation since can be multiplied on both sidesThis is a separable since can be divided (given certain conditions) on both sides This is not separable however since can not be properly removed to fir the separable form

Integral Curve

Thu, 05 Feb 2026 18:45:15 GMT

Curves that map the solutions to an Ordinary Differential Equation

First Order Linear

Thu, 05 Feb 2026 18:45:15 GMT

First Order Linear Ordinary Differential Equation can be solved using either the Method of Integrating Factor or Method of VariationSome equations can have special forms and can be solved generally
These equations are ExactType one equations usually follow the format and can be easily integratedType two equations follow the format and can be reduced to a type oneSingle Separatable Type II

Exact Equation

Thu, 05 Feb 2026 18:36:24 GMT

Given an equation, if there exists functions such that Then the equation is an exact equationThe total derivative of a function can be givenConsider the equationAssume can be given as
By taking the Partial Derivatives, we can see that is equal to the original equationBy equating the partial to we can see that is constantRather than finding and to determine exactness, the following process can be usedLetThen take the complementary derivative of both equationAssume is continuous and differentiable If this condition is satisfied then the equation is exactTo determine exactness

Exact Equation Example_1

Thu, 05 Feb 2026 18:36:15 GMT

Electric Field Example

Thu, 05 Feb 2026 16:03:48 GMT

Electric Potential Example

Thu, 05 Feb 2026 15:55:30 GMT

Work

Thu, 05 Feb 2026 15:17:15 GMT

Work is defined as the amount amount of force applied over a distanceTherefore the total work done over a distance is equal to the change in kinetic energy between the two distances Potential Energy Kinetic EnergyIn a system where energy is conserved

Resistor Example_6

Wed, 04 Feb 2026 22:17:13 GMT

Nuclear Magnetic Resonance Spectroscopy

Wed, 04 Feb 2026 21:55:39 GMT

Current is flow of positive charge, it's denoted by ( for fixed currents) Current is measured in Amperes and is a standard SI Unit Flow of positive charge Current is the flow of positive charge while electrons contain negative charge, in reality electrons flow the opposite direction from the current Some nuclei can have spins, for example and nuclei have spins of these spins are considered degenerate in absence of a magnetic field but degeneracy is removed in a magnetic fieldIn a magnetic field the different spins have different energies, the difference in those energies is proportional to the strength of the magnetic fieldIf the energy difference between the two states only depended on the magnetic field then there will be no difference in the resonance at the same frequency However, electrons surrounding the nuclei establish an internal magnetic field that shields the applied magnetic field, giving variation to protons with differing chemical environmentsDifferent methyl groups have different strengths of shielding

Resistor Example_4

Wed, 04 Feb 2026 21:48:07 GMT

Resistor Example_5

Wed, 04 Feb 2026 21:38:50 GMT

Gauss's Law Example

Wed, 04 Feb 2026 20:09:57 GMT

Kirchhoff's Laws Example

Wed, 04 Feb 2026 19:51:50 GMT

Kirchhoff's Laws Example_0

Wed, 04 Feb 2026 19:47:29 GMT

Basics of Bash Scripting

Mon, 02 Feb 2026 23:35:52 GMT

Bash commands can be used in a script to perform complex functionsENDOFTEXT The keyword ENDOFTEXT can be used to denote a multiline string Standard Out and Standard In acts as buffers to for input and output If statements can be implemented via the syntax if ( [condition] ); then # function fi Integer comparisons are represented by -ne(!=), -lt(<), -gt(>), -le(<=), and -ge(>=)Mathematical expressions can be represented by (( [expression] ))Standard integer operations such as ++, // can be usedConstants can be denoted via the readonly modifierFunctions can be implemented via the syntax afunction() { # function } Arguments can be given by $1, $2, and so on.For advanced parsing of command line arguments, getopts can be used while getopts ":s" option; do case "$option" in s) size_flag=1 ;; ?) printf "Error: Unknown option '-%s'.\n" "$OPTARG" >&2 exit ;; esac done This will check for the command line argument -s, the : in front of the s suppresses all built-in errors, and since there is no : after the s, there is should be no value afterUpon fail this will output to Standard Out and output with an error, the "$OPTARG" is the variable of the flag and 2 is the standard error flagShifting Arguments can be shifted via shift [number] shift "$((OPTIND-1))" Shift up until the first available argument (removes all flags)

Complier

Mon, 02 Feb 2026 23:24:11 GMT

A complier is a program that turns code into binary code, complied programs are often faster and more portable than codeIn windows cmake can be used to compile C files In linux, gcc can be used Source file#include int main() { printf("%s\n", "Hello, world!"); return 0; } Expands #include and #define header Replaces all static variables with their values Headers The complier only adds the implementation of functions within a header when they are utilized within the code When using standard libraries like stdio and stdlib, only the used functions within the library are included to reduce compiled size Preprocessed Fileextern int ftrylockfile (FILE *__stream) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void funlockfile (FILE *__stream) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); # 949 "/usr/include/stdio.h" 3 4 extern int __uflow (FILE *); extern int __overflow (FILE *, int); # 973 "/usr/include/stdio.h" 3 4 # 5 "helloworld.c" 2 # 6 "helloworld.c" int main() { printf("%s\n", "Hello, world!"); return 0; } Translates preprocessed into readable assembly code Complied File.file "helloworld.c" .text .section .rodata .LC0: .string "Hello, world!" .text .globl main .type main, @function main: .LFB0: .cfi_startproc endbr64 pushq %rbp .cfi_def_cfa_offset 16 ... Translate the code into machine code Assembled File�UH��H�H��]�Hello, world!GCC: (Ubuntu 14.2.0-4ubuntu2) 14.2.0GNU �zRx E �U ��helloworld.cmainputs �� .symtab.strtab.shstrtab.rela.text.data.bss.rodata.comment.note.GNU-stack.not e.gnu.property.rela.eh_frame @ �0 ▒&^^1^90l&B �R�j�e@�▒ ▒�� ▒ �▒� Links object files and outputs the final file Overloading In C, the linker associates each symbol as a single definition, thus functions can not be overloaded Linked File@ @@@�▒▒▒��uu ��-�=�=X`�-�=�=�8880hhh��S�td8880P�td 44Q�tdR�td�- �=�=HH/lib64/ld-linux-x86-64.so.2 GNU��GNU��d��'�+��U��`~▒��FDO{"type":"deb","os":"ubuntu","name":"glibc ","version":"2.40-1ubuntu3","architecture":"amd64"}GNU��e�mH d s ▒"puts__libc_start_main__cxa_finalizelibc.so.6GLIBC_2.2.5GLIBC_2.34_ITM_deregisterTMCl oneTable__gmon_start___ITM_registerTMCloneTable'u▒i 1��=�@�?�?�?�?�?�? ��H�H��/H��t��H��5�/�%�/@��h��f��% �/fD��%v/fD��1�I��^H��H��PTE1�1�H�=��S/�f.�H�=y/H�r/H9�tH�6/H��t ��H�=I/H�5B/H)�H��H��?H��H�H��tH�/H��fD��=/u+UH�=�.H��t H�=�. ��d��.]��w��UH��H��H��]��H�H��Hello, world!0 0: f3 0f 1e fa endbr64 4: 55 push %rbp 5: 48 89 e5 mov %rsp,%rbp 8: 48 8d 05 00 00 00 00lea 0x0(%rip),%rax # f f: 48 89 c7 mov %rax,%rdi 12: e8 00 00 00 00 call 17 17: b8 00 00 00 00 mov $0x0,%eax 1c: 5d pop %rbp 1d: c3 ret In a more complex program, multiple file objects can be linked into one filea.c/a.h -> compile -> a.o b.c -> compile -> b.o a.o b.o -> link -> a.out

Resistor Example_3

Mon, 02 Feb 2026 21:53:21 GMT

Resistor Example_2

Mon, 02 Feb 2026 21:40:19 GMT

Resistor Example_1

Mon, 02 Feb 2026 21:36:02 GMT

Resistor Example_0

Mon, 02 Feb 2026 21:29:28 GMT

Resistor Example

Mon, 02 Feb 2026 21:28:45 GMT

Exact Equation Example_0

Mon, 02 Feb 2026 00:36:21 GMT

Separable Equation Example

Mon, 02 Feb 2026 00:35:25 GMT

Exact Equation Example

Tue, 27 Jan 2026 19:06:51 GMT

Electric Field of Surface Charge Example

Tue, 27 Jan 2026 18:06:28 GMT

Electric Field of Surface Charge

Tue, 27 Jan 2026 17:10:08 GMT

Electric Field of Line of Charge

Tue, 27 Jan 2026 15:56:43 GMT

Consider a uniformly distributed line charge

Line of Charge Example

Tue, 27 Jan 2026 15:55:36 GMT

Electric Field of Continuous Charge Example

Tue, 27 Jan 2026 15:33:00 GMT

Electric Field of Continuous Charge Example_0

Tue, 27 Jan 2026 15:32:53 GMT

Method of Integrating Factor Example

Tue, 27 Jan 2026 15:13:46 GMT

Method of Integrating Factor General Method

Fri, 23 Jan 2026 01:23:59 GMT

Method of Integrating Factor

Fri, 23 Jan 2026 01:02:02 GMT

For non integrating First Order Linear equations an integrating factor can be used to convert the equation into a Type I equation, the factor must also be solved then substituted before integrating to reach the final answer

Given an initial condition

Method of Integrating Factor General Method_0

Thu, 22 Jan 2026 18:58:52 GMT

Method of Integrating Factor Example_0

Thu, 22 Jan 2026 18:47:47 GMT

First Order Linear Example

Thu, 22 Jan 2026 18:23:24 GMT

SI Unit

Wed, 21 Jan 2026 22:22:16 GMT

The standard units for physics is the International Standard of UnitsThe system is built on 7 base units defined using the fundamental constants c, h, and the vibration of a Caesium-133 atom

Bond Length

Wed, 21 Jan 2026 22:14:09 GMT

Bond length is the average distance between the nuclei of two bonded atoms in a moleculeBond lengths are usually measured in picometers (pm) or angstroms (Å) using experiment techniques such as x-ray diffraction or Molecular Spectroscopy
Bond length usually decrease when Bond Order increase or when Bond Energy increase

Ordinary Differential Equation Example

Tue, 20 Jan 2026 18:39:58 GMT

Scalar Field

Tue, 16 Dec 2025 20:43:17 GMT

A scalar field is a function of several variables of the typeWhich takes in a multi-dimensional vector and outputs a scalar With the domainAnd the imageThe level set for a scalar field is the set of all input values that output The level set of is a circle Check continuity (plug it in) Check a few paths Any two paths differ then the limit DNE The limit can not be proved to exist via exhaustion Try Polar Coordinates Solving limits via polar cordinate

Valance Bond Theory

Tue, 16 Dec 2025 20:01:35 GMT

The valance bond theory is a quantum mechanical method of describing Covalent Bonding, where a bond is created through the atomic overlap to two atomsA constructive overlap of two orbitalsA single bond is a sigma bond A double bond is a sigma bond and a pi bond A triple bond is a sigma bond and two pi bondsValance bond theory so far, can not account for polyatomic molecules such as To account for this, carbon goes through hybridization,One core electron is promotedThen carbon goes through hybridization, creating 4 equivalent hybrids The superscripted indicates that p orbitals have combined with one s orbital
Though valance bond theory is in some ways better than using Lewis Structures, there are still some problems with the theory No indication of what hybrid to use Valance bond theory still require resonance
Molecular Orbital Theory was made to correct these

Line Integral Example

Tue, 16 Dec 2025 17:41:50 GMT

Flux

Tue, 16 Dec 2025 17:39:20 GMT

A flux integral of a vector field through a surface with Orientation can be shown asThis integrated the flow of a vector field through a surface at an orientationTo parameterize the integral, note that Since can be parameterized to

Chain Rule

Tue, 16 Dec 2025 17:37:56 GMT

The chain rule represents the behavior of derivatives in chained functions Chain Rule in 1D The chain rule can be generalized to dimensionsAssume has continuous derivatives of all order. Expressin terms of the Partial Derivative of Relationships that are not dependent on a single variableEvery implicit relation among variables and can be expressed as a level set

Application of Integrals

Tue, 16 Dec 2025 17:37:35 GMT

Density can be determined as a ratio of mass over volume, this ratio can vary continuously as a function of locationIn situations where density of an object is not uniform, integrals can be used to determine center of massWhere is the position of the point In non-uniform objects, the moment of inertia measures how spread apart the mass is from the center of mass, this can determine how hard it is to spin the objectGiven a solid ball, a hollowed out ball, and a hoop, all with the same mass, find the smallest moment of inertia Moment of Inertia of a Solid Ball Substitute

Angular Momentum

Sun, 14 Dec 2025 20:04:00 GMT

The change in angular momentum is defined as cross product between the radius and momentum The change in angular momentum is also the Torque Consider a rotating figure stature with their arms stretched, if they pull in their arms the angular momentum must be conserved
Since the moment of inertia decreases as the arms pull in, the angular momentum must increase, therefore the skater spins faster
However, consider the Kinetic EnergySince is constant and decreases, the kinetic energy must have increased, which leaves the question: what's doing the work?
When the figure skater does positive work to pull their arms against the centrifugal force, this internal force is capable of changing the system's kinetic energy

Precession Example

Sun, 14 Dec 2025 19:54:39 GMT

Precession

Sun, 14 Dec 2025 19:50:02 GMT

Precession is the change in orientation of the rotational axis as a response to external torque The angular speed of precession can be givenConsider a rapidly spinning bicycle wheel with on end of its axle supported on a pivot
Because the wheel is rapidly spinning, the Angular Momentum is in the direction of the axle
Gravity will pull down the wheel at the center of mass therefore the Torque must be perpendicular to the direction of the angular momentumThe torque will apply a small change in angular momentum but because of the already existing momentum, only the direction of the momentum will be changed slightly cause the wheel to precess around the pivot

Torque

Sun, 14 Dec 2025 19:25:44 GMT

Torque is the twisting force that causes angular acceleration While torque is a vector, the direction of the torque changes constantly so it is often not expressed as vectorIn 2D the change in work is defined asHowever, for rotations, this is defined asTorque in a direction depends on two perpendicular components, torque in the direction will depend on force in the and directionsMore simply, torque can also be defined as the change in Angular Momentum in the same directionThe torque can also be defined as the projection of onto the perpendicular to Lever Arm The perpendicular vector to is called the lever arm. : Lever arm

Angular Momentum Figure Skater

Sun, 14 Dec 2025 19:14:10 GMT

Rotational Energy

Sun, 14 Dec 2025 19:05:44 GMT

Minimum Spanning Tree Dijkstra's Example

Sun, 14 Dec 2025 01:49:22 GMT

Hash Table

Sat, 13 Dec 2025 19:59:50 GMT

A table that stores objects in hashed table indexesDefine a table (Array) of length Define a Hash Function that maps keys into an integer indexpublic static int hash( String key, int tableSize ) { int hashVal = 0; for( int i = key.length()-1; i >= 0; i-- ) hashVal = 37 * hashVal + key.charAt( i ); hashVal %= tableSize; if( hashVal < 0 ) hashVal += tableSize; return hashVal; Several parameters can be hashed by multiplying the hash function of each parameter by a unique prime numberInputs of Hash Functions While any object can be hashed, the object but be immutable in order to ensure the same object can retrieve the same index
If several objects return the same hashed index, the object is then prepended to a Linked List at that indexIdeally the the length of the linked list at every index will be 1, so the search function will be
In the worst case scenario, the hashing function will return the same index for all objects, the search function will be The average length of the linked lists is given by
In open hashing, each table cell holds a container (Linked List) and all items that hash to that cell are stored in that containerOpen Hashing provides better insertion and search speed at the cost of removal speed and memory cost `
In closed hashing, all items are stored directly in the table, and a Probing function is used to determine the exact cellClosed Hashing provides better deletion speed and lower memory cost

Car on a Circular Track

Sat, 13 Dec 2025 01:28:34 GMT

Consider a car on a flat circular trackThe three forces acting on the car are, gravity , the normal force , and the static friction of the track We can separate this into two equations, one for the vertical direction and one for the radial direction And since for the car to turn without slipping the static friction must be not exceed the maximum static friction forceNow consider the same car on a banked slope, the banking allows the car to turn at much higher speeds
Similarly, the three forces acting on the car are, gravity , the normal force , and the static friction of the track but the normal and frictional forces are slopedThe equations are motion can be given below

Car on a Circular Track Banked Sloped Setup

Sat, 13 Dec 2025 01:21:21 GMT

Car on a Circular Track Setup

Sat, 13 Dec 2025 01:13:18 GMT

Green's Theorem

Fri, 12 Dec 2025 01:27:42 GMT

Green's theorem is a way to find the line integral of a continuous path over and a conservative Vector FieldWhere is a (simple) region on the plane And is a counterclockwise-oriented, simple, closed curve, forming the boundary of D. And which is a continuously differentiable vector field

Ligand

Fri, 12 Dec 2025 00:50:57 GMT

Ligands are simple molecules or ions that combine with metals to make metal ionsIn this coordinated ion, both and are ligandsNH₂CoClClNH₂H₂N

Crystal Field Theory

Fri, 12 Dec 2025 00:39:02 GMT

Crystal field theory is based on how how atomic orbitals interact with the surrounding ligands

Transition Metals

Fri, 12 Dec 2025 00:28:07 GMT

Transitions metals are elements from Group 3 – 11 and are called d-block metals, it's important to note that and are not transition metalsTransition metals usually: Exhibit variable oxidation (valence) states. Form compounds that have incomplete d shells.
Form compounds that are paramagnetic (attracted to magnets) Form colored compounds. Form compounds that consist of complex ions Exhibit catalytic properties as the metal or in compounds

Rotation of Molecules

Thu, 11 Dec 2025 23:40:41 GMT

Molecules are considered to be Rigid Bodies where Rotation occurs about the center of Gravity
In order to excite a rotational transition, the molecule must have a permanent Dipole Moment, this is because light interacts with the electric dipole of the molecule, if there is no permanent dipole, light can not excite the moleculeIn addition, absorption of light is only possible for transitions that obey the selection rule Consider a microwave and a cup of waterBecause the water is polar, microwave radiation excites molecular rotations in the water, the rotation of the water hits other water molecules, causing them to translate and transfer the microwave radiation into heatThe cup however, is most likely not polar and therefore the molecule rotation has no effect on the cup

Rigid Body Dyanmics

Thu, 11 Dec 2025 23:14:31 GMT

We can define force as the rate of doing WorkForce can also be defined as the rate of change of momentumUsing the work definition, when separating variables for Using the momentum definition

Heisenberg Uncertainty Principle

Thu, 11 Dec 2025 23:13:09 GMT

For particles that exhibit Wave behavior, the more we know about the position, the less we know about the trajectory, this is true vice versa

Vibration of Molecules

Thu, 11 Dec 2025 23:12:52 GMT

While atoms can only exhibit translation, molecules can exhibit translation, rotation, and vibrationThe ability of a molecule to vibrate depends on the energy required to increase or decrease internuclear separation from the equilibrium valueThis behavior can be model by using springs and Hooke's LawIn classical mechanics, the oscillation frequency is given byWhere is the force constant And is the reduced mass ()
In Quantum Mechanics however, each level of energy is quantizedBecause the spacing between energy level is Independent of the electron, the change in energy level is restricted to
Some vibrations do not change the Dipole Moment of the molecule, for example, a stretching movement does not typically induce a change in the dipole moment, these vibrations are called IR inactiveFor a molecule to be IR active, the molecule must exhibit a change in the dipole momentFor polyatomic molecules, the number of vibration modes can be given belowConsider water (), using the non-linear vibrational equationWe end up with vibrational modes

Vibration of Molecules Water Vibration

Thu, 11 Dec 2025 22:58:17 GMT

Electronic Spectroscopy

Thu, 11 Dec 2025 22:37:48 GMT

Electronic Spectroscopy works by transferring an electron from a Molecular Orbital to another orbital

Pasted image 20251211172943

Thu, 11 Dec 2025 22:29:43 GMT

Molecular Spectroscopy

Thu, 11 Dec 2025 22:09:27 GMT

Because molecules are three dimensional, the energy of molecules also depend on rational and vibrational propertiesThe energy of a molecule can be represented by“Pasted image 20251211170925.png” could not be found.Absorption spectra for molecules are typically recorded by spectrophotometersIn an experiment the intensity of the light transmitted through the sample cell is compared to a reference cellSpectra can be presented in either transmittance or absorbance modes, 0% transmittance means no light is observedTransmittance can be defined by Absorbance can be defined by Absorbance can also be given generally byWhere is the absorption coefficient And is concentration And is the molar extinction coefficient And is the cell length

Molecular Orbital Theory

Thu, 11 Dec 2025 21:59:30 GMT

Molecular orbital theory is created to fix certain problems with Valance Bond Theory and VSEPR Theory Molecular orbitals are derived from a superposition of the atomic orbitals, the theory relies on the fact that as two orbitals overlap, they would interfere constructively or destructivelyElectrons would bond in pairs to either the bonding orbital or the antibonding orbital depending on the lowest energyThe number of molecular orbitals equal the sum of atomic orbitals The number of electrons in molecular orbitals equals the sum of the number of electrons in the atomic orbitalsA bonding orbital is the sum of two orbitals, and usually has lower energy than the orbital of any one atom
Consider two hydrogen atoms, as the atoms approach each other the orbitals will overlap and interfere constructively or destructively based on the phase
Not to be confused with antibonding orbitals, nonbonding orbitals are orbitals with inappropriate symmetry are non-bonding because the constructive and destructive interference cancel each other out
Unlike essentially nonbonding orbitals, rigorously nonbonding orbitals are unable to interact at all Identify orbitals from valance shell Combine pairs of atomic orbitals Atomic Orbitals combine most effectively with other atomic orbitals of the same or similar energy The effectiveness of their combination is their overlap Adjust for charge Accommodate electrons in order of increasing energy When molecular orbits have the same energy, electrons enter orbitals with the same spin before pairing the spins The orbitals from combining two atoms will produce 8 molecular orbitals The bond order of a molecular orbital can be given by
An increase in bond order generally increase Bond Energy and decrease Bond LengthTo predict the molecular structure using Molecular Orbital Theory, calculate all possible structures for a molecule then chose the structure with the lowest energyConsider the Molecular Orbital Diagram for fluorine ()
The 8 atomic orbital from each atom gives 8 molecular orbitals of which only 7 are filled, fluorine has 7 valance electrons which means only 14 electrons are filled in the molecular orbitalsThe final electron configuration of is Since the bond order can be given by the number of bonding orbitals minus the nonbonding orbitalsIn molecular orbital theory, a molecule is paramagnetic is there are unpaired electrons in any of the orbitals
For example, because fluorine doesn't contain any unpaired electrons it is diamagnetic
However, oxygen contains two electrons in the orbit, making it paramagnetic

Atomic Orbital

Thu, 11 Dec 2025 21:52:39 GMT

Atomic orbitals are complex mathematical functions that can be separated into two components: a radial wavefunction and an angular wavefunction While the energy is described by a single Quantum Number , the wavefunction is described by three quantum numbers , ,

Bond Energy

Thu, 11 Dec 2025 21:20:38 GMT

The average bond energies within a reaction can be used to predict if a reaction will be favorable or notEnergy is released in the formation of a bond meanwhile it is absorbed in the breakage of a bondEnergy change in the formation of one mole of a compound from its components Lattice energy only exists for ionic compounds and can be given byThe energy required to isolate a molecule is determined by it's energyAtoms are more stable in molecules since there are less valance electronsConsiderBreaking apart and requires energy to be spent respective to their bond energy Creating releases its bond energyThis equation will release overallFor similar systems, bond energy increase with increase in Bond Order while bond length decreases with increase in bond order

Molecular Orbital Theory Diatonic Example

Thu, 11 Dec 2025 19:57:35 GMT

Molecular Orbital Theory Nonbonding Example

Thu, 11 Dec 2025 19:43:51 GMT

Molecular Orbital Theory Example

Thu, 11 Dec 2025 19:33:00 GMT

Variation of Orbital Energies

Tue, 09 Dec 2025 23:54:17 GMT

In Valance Bond Theory, as the atomic number increases, 3s, 3p drops more rapidly because they penetrate the core more effectivelyHowever the 3d orbital remains unchanged up until around (Z = 19)This causes orbital energy to be more sensitive to the atomic number as the order decreases
Unusual configurations - - - - - There is a drop in the first ionization energy from to , this goes against the trend of ionization energy increasing across a period
The reason why lies within the Electronic Configurations of the two elements: has a configuration of , which is creates a half-filled shell making N more stable than 's configuration of which undergoes more electron propulsion

Electron Configuration

Tue, 09 Dec 2025 23:44:38 GMT

Electron configuration is the arrangement of electrons within the orbitalEach electron can be characterized by a set of four Quantum Numbers which defines the the electron Orbital's shape, orientation, and spin
Electron configurations are prone to follow principles such as the Aufbau's Principle, Hund's Rule, and the Pauli Exclusion PrincipleWhile most transition metals have the configuration , and have configurations of and respectively; this can be explain by the fact the half-filled shells are more stable because electrons inhibit parallel spin

Hund's Rule

Tue, 09 Dec 2025 23:32:17 GMT

If there are more than one Orbital in a subshell the electrons are added to different orbitals with parallel spins rather than pairing two electrons in the same orbital

Aufbau's Building Up Principle

Tue, 09 Dec 2025 23:32:17 GMT

The Electron Configuration of atoms is made by filling orbitals with electrons in order of increasing energy.The sequence of orbitals can be remembered using the *n + l rule The energy increases with the n + l value. For two subshells with the same value of n + l the lower value n is completed first The order is as follows 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d

Electron Spin

Tue, 09 Dec 2025 23:30:06 GMT

The spin of an electron can be considered the fourth quantum numberMultiple electrons in a system are hard to evaluate since each body affects the other electron Because all electrons affect each other in an orbit, the effective charge of an electron can be given minus the shielding constant Core penetration in Effective Nuclear Charge While 3s and 3p have higher average distances than 3d, they have local maximums within the core area, because of this, their effective nuclear charge is higher If a p orbital has a node () at the nucleus, how do electrons get from one side to the other?While in classical mechanics, the particles must move to via continuous translation, in Quantum Mechanics this is not the case, the expression is only true when the electron is forced to behave as a particle

Photoelectric Effect

Tue, 09 Dec 2025 23:19:28 GMT

The photoelectric effects happens as electrons are ejected from a metal surfaceClassical theory determined that the energy of light depends only on its intensity and not its frequency, however experimentation showed that electrons are emitted, even at a low intensity, given a high enough frequencyEinstein relates the energy of photons to its frequency rather than its intensity Where is the Planch constant and is the frequency of the photonAtoms emit or absorb light based on energy released or absorbed, this relation can be given byO

Euler Path

Sun, 07 Dec 2025 18:45:52 GMT

An Euler path is a path that visits every edge exactly onceAn Euler circuit is a Euler path that starts and ends on the same node

Spanning Trees

Fri, 05 Dec 2025 02:18:20 GMT

The depth first spanning tree is a search tree used by Depth First Search Add a tree edge for every graph edge taken by DFS Add a back edge for every skipped edgeIf the root of the DFS spanning tree has two outgoing edges then the root is an articulation point

Graph

Fri, 05 Dec 2025 02:16:55 GMT

Graphs An undirected graph is connected if there is a path for every vertex to every other vertexA graph that inhibits biconnectivity can not have any edge removed without losing connectivity of the graphAn adjacency matrix maps the relation between any two points on a graph
While an adjacency matrix plots all relations, the adjacency list is more space efficient by removing

Minimum Spanning Tree Example Union-Find

Tue, 02 Dec 2025 19:26:34 GMT

Minimum Spanning Tree Kruskal

Tue, 02 Dec 2025 19:04:11 GMT

Sorting

Tue, 02 Dec 2025 03:55:30 GMT

Sorting algorithms aim to take a disorganized array and arranges it into a sorted arrayBubble sort iterates over the Array, swapping value pairs that are not sorted, is then set to and the process repeats until is 0. is the length of the unsorted array is the highest index where a swap occursBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Basic Implementationpublic static void bubbleSort(int[] array) { int len = array.length; while (len > 0) { int n = 0; for (int i = 1; i < len; ++i) { if (array[i - 1] > array[i]) { swap(array, i - 1, i); n = i; } } len = n; } }
Selection sort iterates over the Array and repeatedly finds the smallest element in the unsorted array, then swaps it to it's resting index
Reverse Selection Sort Selection Sort can also move the maximum value to its resting index and still have the same Time Complexity Best: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Practical Time Even though bubble sort have better time complexity on paper, selection sort swaps less than bubble sort, equating to better practical speed Basic Implementationpublic static void selectionSort(int[] array) { for (int i = 0, i_bound = array.length - 1; i < i_bound; ++i) { int min_index = i, min = array[i]; for (int j = i + 1; j < array.length; ++j) { if (array[j] < min) { min_index = j; min = array[j]; } } if (min_index != i) { swap(array, i, min_index); } } } Insertion sort creates a subarray with the first elements in order in respect to other elementsBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Basic Implementationpublic static void insertionSort(int[] array) { for (int i = 1; i < array.length; ++i) { int k, current = array[i]; for (k = i - 1; k >= 0 && array[k] > current; --k) { array[k + 1] = array[k]; } array[k + 1] = current; } }
Instead of swapping every pair, like in Bubble Sort, Odd-Even Transposition considers all odd pairs using Parallelization, swapping when necessary, considers all evens When no swaps are made after both an odd and even passthrough, the algorithm endsParallelization Since Java can not support parallelization on the CPU, the Odd-Even Transposition sort can not be implemented natively in Java To use heap sort convert the array to a max heap then perform deleteMax() operations to get the array in reverse orderBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: (Distinct elements )private static void swap(int[] array, int i, int j) { int temp = array[i]; array[i] = array[j]; array[j] = temp; } private static void percolateDown(int[] array, int i, int n) { int child = 2 * i + 1, temp; for (temp = array[i]; child < n; i = child, child = 2 * i + 1) { if (child != n - 1 && array[child] < array[child + 1]) { child++; } if (temp < array[child]) { array[i] = array[child]; } else { break; } } array[i] = temp; } public static void heapsort(int[] array) { for (int i = array.length / 2 - 1; i >= 0; i--) { percolateDown(array, i, array.length); } for (int i = array.length - 1; i > 0; i--) { swap(array, 0, i); percolateDown(array, 0, i); } } Merge sort relies on two sorted arrays being able to combine into a single sorted array, merge sort divides the array into two arrays then recursively sorts the arraysBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: private static void mergesortHelper(int[] array, int[] scratch, int low, int high) { if (low < high) { int mid = low + (high - low) / 2; mergesortHelper(array, scratch, low, mid); mergesortHelper(array, scratch, mid + 1, high); int i = low, j = mid + 1; for (int k = low; k <= high; k++) { if (i <= mid && (j > high || array[i] <= array[j])) { scratch[k] = array[i++]; } else { scratch[k] = array[j++]; } } for (int k = low; k <= high; k++) { array[k] = scratch[k]; } } } public static void mergesort(int[] array) { int[] scratch = new int[array.length]; mergesortHelper(array, scratch, 0, array.length - 1); } Quick sort partitions Counting sort works by counting all appearances of a number within an array then rebuilds an array in orderBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Space: Topological Sort works by computing the indegree of all vertices and putting all indegrees of 0 in a min-heap, then it repeatedly extracts the lowest number indegree 0 then updates adjacent vertices indegree: amount of adjacent vertices that point towards the given vertex
O

Uniform Circular Motion

Sun, 30 Nov 2025 17:46:20 GMT

Uniform Circular Motion occurs when an object moves in a circular path at a constant speed, even though the speed remains the same, the direction of motion changesAn object moving in a circle of radius with angular velocity has a position vectorThe linear velocity is always tangential to the circle:The magnitude of velocity can be expressed asAlthough the speed remains constant, the direction of changes, producing an inward (centripetal) acceleration:or equivalently,This acceleration is responsible for changing the direction of motionUsing Newton’s second law,The net force required to maintain circular motion is thus inward (centripetal):This is known as the centripetal force, which is not a new type of force but the result of other forces The time taken to complete one revolution (the period) is given by:The frequency (number of revolutions per second) is:

General Roation in 3D

Sun, 30 Nov 2025 17:41:59 GMT

Force can be defined as a sum of radial and tangential forcesSince Because rotation induces an outward radial force on its subject, there must be a counteractive force to ensure the subject stays at the same radial positionAnd since Simplifying in polarBecause is constant, substitute the variable to constants13476554433231

Conservative Force

Sun, 30 Nov 2025 17:41:33 GMT

We can define work as the change in energy over a distanceBy equating What..Take the curl of

Newton's Law

Sun, 30 Nov 2025 17:41:15 GMT

Newton's Laws of motion describe the relationship between an object's motion and forces that act upon itThe first law states that an object in motion stays within that motion unless acted upon by an external forceThe second law states that the Force of an object is equal the object's mass times its acceleration.Newton also defines the force as the change in momentum at a given timeOne of the fundamental forces is the gravitational force Where is the direction from one body to the otherOn earthNewton's third law states that if two objects exert a force on each other, those forces will have the same magnitude but opposite directions

Friction on Inclined Planes

Sun, 30 Nov 2025 17:35:24 GMT

The normal force of an incline plane is given by The force in the direction down the slope is given by Static friction matches up until which is given by The critical point of where static friction fails to stop the object can be given by

Accelerating Frames

Sun, 30 Nov 2025 17:33:43 GMT

In an accelerating frame, is not The rotation change in is equal to And since we can simplifySo that for any vector its inertial Time derivative is equal toThe position in the inertial frame can be given by The Foucault Pendulum
It has a velocity of But because the earth is spinning on its axis

Accelerating Frames Example

Tue, 25 Nov 2025 16:10:01 GMT

Time derivative

Tue, 25 Nov 2025 15:54:01 GMT

The derivative of a vector in respect to time will always be perpendicular to the vectorProduct rule for vectors Given Then To rotate a vector to its time derivative multiply by a vector orthogonal to both vectors Where is orthogonal to both and

Orientation

Thu, 20 Nov 2025 23:36:27 GMT

A surface is oriented if there is a consistent choice of normal vectors This can also be achieved by parameterization of of the surfaceTo flip an orientation, negate a component, the surface will be flipped in respected to the flipped component

Show Flipped

Minimum Spanning Tree Example

Thu, 20 Nov 2025 18:30:59 GMT

Sorting Example

Thu, 20 Nov 2025 17:45:58 GMT

General Roation in 3D Example

Thu, 20 Nov 2025 15:44:43 GMT

Pauli Exclusion Principle

Wed, 19 Nov 2025 03:52:34 GMT

No two numbers can have the same set of four Quantum Numbers
This is the basis for opposite Electron Spin, an orbital can only hold up to 2 electrons which can not be the same spin

Quantum Number

Tue, 18 Nov 2025 23:36:14 GMT

Elements can be defined using a set of quantum numbers that are solutions to Schrödinger's Equation.
These are used in Valance Bond Theory to predict shape.
Quantum numbers can be used to find the Shape of Atomic Orbitals Given by the symbol Determines the overall size and energy scale Number of orbitals in shell Maximum electrons in shell Given by the symbol
Determines the orbital angular momentum and the shape of the Orbital Number of orbitals in a subshell = Maximum electrons in a subshell = S orbitals Even though S orbitals have 0 angular momentum, the electron still has motion, the electron can be imagined as having a closed linear path Given by the symbol Specifies the orientation of the orbital along an axis Values are integer steps; there are possible values for a given For example can have values of
Main Page: Electron Spin Given by the symbol Intrinsic angular momentum of the electron Two possible spin states per orbital Shown experimentally by the Stern–Gerlach experiment Splitting beams in an inhomogeneous magnetic field

Schrödinger's Equation

Tue, 18 Nov 2025 22:40:38 GMT

Because electrons are waves, it's position and movement in space can be represented by a wave equationThe Schrödinger's equation is underivable and is regarded as a postulate.Its general form can be given byThe solutions to this equation are the wavefunction and energy of a particleThe Probability Density of a particle can be given by The solution for a hydrogen and other one election ions gives the wavefunction and corresponding energiesThe wave functions are called atomic orbitals to differentiate them from Bohr orbitsWhere is the atomic number And is the Rydberg energy ()
Although the energies are the same as levels predicted by Bohr's theory, quantization is a natural consequence of the Schrödinger equation

Covalent Bonding

Tue, 18 Nov 2025 22:15:28 GMT

Atoms in the molecule are held together by sharing electronsCovalent bonds are often represented by a line (–) between two atomsCH₃OHOOH

Atomic Orbital Solutions

Tue, 18 Nov 2025 22:07:13 GMT

Shapes of Orbitals

Tue, 18 Nov 2025 21:13:13 GMT

Orbitals have differing shapes based on interactions between electrons in the Valance Bond Theory
Every electron has a wave function that determines the probability that the electron is at a certain radiusThe probability density can be determined by squaring the wave function Nodes are spaces in the wave function with a zero probability, meaning no electrons will occupy that spaceBecause the wavefunction goes to infinity, the size of an atom can not be fully defined A convention is to use to boundary where there is a 90% probability of the electron can be locatedWhile represents the probability at a point, represents the probability at a certain radius
Where is the radial wavefunctionIn a radial distribution function, spaces with zero probabilities are radial nodes, which appear as spheres without any electron occupation

Atomic Radii

Tue, 18 Nov 2025 21:11:26 GMT

In Bohr's Theory, the radius of an Element can be given byWhere is the Bohr radiusAtomic radius generally increases down a group and decreases across a period
Quantum Representation of Radii The wavefunction of an atom can be used to more accurately describe the radius of atoms

Wave

Tue, 18 Nov 2025 20:42:57 GMT

Forces like electromagnetic radiation and light behave similar to wavesWave-Particle Duality Electromagnetic radiation and light in reality exhibit Wave-Particle Duality and act like both waves and particles Waves that combine are in phase with each other interfere constructively, increasing the amplitude of the resulting wave, waves that interfere destructively result in a lower amplitude wayTraveling waves are waves that change travel along a distance, standing waves change in place A wave traveling along length of rope will exhibit standing wave behavior while a guitar string will exhibit standing wave behavior

Wave-Particle Duality

Tue, 18 Nov 2025 20:42:53 GMT

Small particles, at times can display Wave-like properties.
Electrons in a Bohr orbit may be associated with a circular standing wave wave, interfere is the reason why only certain wavelengths appear in each orbit
Einstein's equation can be linked to Planck's equation to form a relation between the wavelength and momentum of a photonAnd since is the speed of a photonAnd since
Heisenberg Uncertainty Principle Because of the Heisenberg Uncertainty Principle, the more known about a particle's position, the less known about its momentum and vice versa

Dipole Moment

Tue, 18 Nov 2025 20:37:24 GMT

If a shape of a molecule is known, we can determine the dipole moment of a molecule by combining the dipole moment of its individual bondsIn , a flat shape, the dipole moments of the bonds cancel outIn , a bent shape, the dipole moments of the bonds add together to be polar

Quantum Mechanics and Atomic Theory

Tue, 18 Nov 2025 20:23:34 GMT

Rutherford proposed a planetary model for the atom where most of the mass is concentrated in the nucleus while electrons "orbit" around the nucleusDeath Spiral of an electron A common critique of Rutherford's model is that an electron can not keep orbit forever, as the the electron orbits the electron loses energy, eventually spinning into the nucleus A central idea of quantum theory is that energy isn't continuous but comes in discrete packetsOriginally light was thought of as a particle, however Maxwell suggested that light consists can behave like a Wave.Amplitude Wavelength (): Frequency (): Speed ()Speed of a waveFrequency of a waveLight is a form of electromagnetic radiation, the electric component influences charged particles, the magnetic field influences moving charged particlesHumans can be seen in the dark by detecting heat, in a similar way, all objects emit radiation (infrared light) that can be detectedObjects that emit blackbody radiation emit energy dependent on the temperature of the object Incandescence: Light due to temperature Chemiluminescence: Light due to a chemical reaction Fluorescence: Light emitted by an object that absorbed light

Rotation

Tue, 18 Nov 2025 15:31:56 GMT

Given a system of particles with no external forcesWhere is the center of massRotation does not induce any external forcesIn a rigid body, the distance between the fixed center of mass and a point will be constant.Conversion from polar to cartesianThe change in position for rotationThe position in rotation for a given can be given as follows

Isomerization

Mon, 17 Nov 2025 15:34:43 GMT

General Time derivative

Fri, 14 Nov 2025 03:38:55 GMT

Given The general formula for a time derivative of a position vector can be defined asWhere is the time derivative of the angle Construct a unit vector to by using cross product

Green's Theorem Example_0

Fri, 14 Nov 2025 03:37:22 GMT

Line Integral

Thu, 13 Nov 2025 23:40:41 GMT

Also called a path integral, is an integral with the domain of , the image of a curve There are several version of line integrals with differing usesLine integrals in respect to arc length measures the area between the curve and the Scalar Field the integrated shape is a extruded path
In order to integrate the line integral, convert in terms of using Reparameterization
Line integrals in respect to a coordinate measures the area between the linear change in the curve and the Scalar FieldGiven , find the line integral in respect to , , and arc length

Line integrals in respect to the Vector Field measures the alignment between the curve by integrating the dot productIf and is a continuous path from to thenSolution to

On a closed curve, a line integral conservative vector field will always equal

Green's Theorem Example

Thu, 13 Nov 2025 23:07:36 GMT

Green's Theorem Example Not Simple Not Closed

Thu, 13 Nov 2025 22:53:29 GMT

Green's Theorem Example Simple Not Closed

Thu, 13 Nov 2025 22:53:12 GMT

Green's Theorem Example Not Simple Closed

Thu, 13 Nov 2025 22:52:42 GMT

Green's Theorem Example Simple Closed

Thu, 13 Nov 2025 22:52:25 GMT

Benzene

Thu, 13 Nov 2025 16:29:47 GMT

Types of Forces

Thu, 13 Nov 2025 15:57:02 GMT

Forces can be classified into different types depending on Gravitational Force $$ \vec{F}=-\frac{GmM}{r^2}\hat{r} $$ Electric Force $$ \vec{F}=\frac{1}{4\pi\epsilon} \frac{qQ}{r^2}\hat{r} $$Phenomenological motion is constraining motion that results from complex interactions between objects the force is observed and found experimentallyStatic Friction: Negates motion parallel to a surface Kinetic Friction: Opposes motion parallel to a surface Normal: Opposes motion normal to a surface Tension: Constraint of a string, force in direction in the direction of a string at its length Spring: Constraints of a spring, force in direction opposite to the resting position

Power

Thu, 13 Nov 2025 15:56:46 GMT

Power is the change in work over time or how fast the work is being doneSincePower can be defined asPower can be generalized in 2D by using vectors

Rotation Example

Thu, 13 Nov 2025 15:33:06 GMT

Tree Searching

Wed, 12 Nov 2025 23:26:00 GMT

Searches along a length firstSearches along the row first

Friction on Inclined Planes Pulley

Wed, 12 Nov 2025 23:24:45 GMT

Friction on Inclined Planes Walking

Wed, 12 Nov 2025 23:24:38 GMT

Friction on Inclined Planes Example

Wed, 12 Nov 2025 23:24:29 GMT

Vector Field

Wed, 12 Nov 2025 23:22:10 GMT

A vector field is a function where every vector input will output a vectorA vector field is conservative if there is a Scalar Field (potential) that satisfies
Conservative vector fields can be called gradient vector fields because a conservative vector field is a Gradient of a scalar fieldA vector field can be proven to be not conservative by using proof by contradiction
Non Conservative Vectors There are exceptions to this proof since there are non conservative vector fields that satisfies A potential can be found by defining as and using the equality

Graph Example_0

Wed, 12 Nov 2025 23:16:12 GMT

Tower of Hanoi

Wed, 12 Nov 2025 23:03:34 GMT

The tower of Hanoi is a proposed game where rings are placed in descending size onto a peg and the goal of the game is to move those rings onto a different peg, all rings must be in descending order at all times

Binary Tree

Wed, 12 Nov 2025 23:03:28 GMT

In Binary Trees, the number of children is at most twoThe amount of nodes within a Binary Tree is exponential and can be given by where is the total levels in the tree Binary trees are considered full if every node is a leaf or has two children Binary trees are considered complete if all levels (except possibly the last) is completely filled and every node is as far left as possibleIn Binary Search Tree the root is the key, all left children are less than the key while all right children are greater than the key

Graph Example

Wed, 12 Nov 2025 23:02:57 GMT

Time Complexity

Wed, 12 Nov 2025 22:58:38 GMT

The time complexity of a function can be solved by using a 5 step processAims to reduce the problem to a single constant If base case, return directly. Reduce input to a smaller instance. Recursively solve the smaller instance. Extend/adapt that solution to solve the original. Find the time complexity of given
]

Shapes of Molecules Using VSEPR

Wed, 12 Nov 2025 15:31:10 GMT

Lewis Structure molecules does not provide information about the shape of the molecule VSPR theory can be used to determine the shapeVSEPR, which stand for Valance Shell Electron Pair Repulsion, can be used to estimate a shape of a molecule. VSEPR is not in itself a theory but can be used as an aide Molecules will minimize electron pair repulsion which can determine its shapeSteric numbers can be found by adding the number of coordinated electrons with the number of lone pairs By using the steric number the shape of a molecule can be found2: Linear 3: Trigonal Planar 4: Tetrahedral 5: Trigonal bipyramidal 6: Octahedral 7: Pentagonal bipyramidalThe preferred rotation in any of these shapes minimizes interactions of lone pairs first then the interactions of line pairsShapes can be drawn in any rotation as long as the lone pairs minimizes interactionsBecause electrons bonded between two atoms take up less space, they are distributed more evenly, for example the Hydrogens in are evenly spaced Electrons in lone pairs take up more space and decreases the angle of other bonds, the Hydrogens in are closer to each other than

Line Integral Fundamental Theorem

Thu, 06 Nov 2025 23:51:06 GMT

Vector Field Conservative Vector Proof

Thu, 06 Nov 2025 23:29:15 GMT

Polar Vector

Thu, 06 Nov 2025 23:26:03 GMT

The variables and can be used to define coordinates in the polar plane The general equation for a polar vector value function is as followsThe variables and and can be used to define coordinates in the 3D polar planePolar vectors in 3D share similar dot and cross product rules to cartesian vectors in 3DOmega can be defined as the change in angle in the z directionSo thatThe equation for circular motion can be given by

Polar Coordinates

Thu, 06 Nov 2025 23:25:46 GMT

The magnitude of a vector is given by And its corresponding angle given by Such that And its time derivative Rotate a Polar Vector to its time derivative is similar its Cartesian equivalentThe period and frequency of an polar vector are as followsA Polar vector also be defined asAngular Direction The angular direction is by definition the perpendicular direction of

Cylindrical Coordinates

Thu, 06 Nov 2025 23:25:00 GMT

Volumes can be expressed in terms of Polar Coordinates with the terms The conversion for cylindrical coordinates are the same as polar coordinates in 2D

Spherical Coordinates Volume

Thu, 06 Nov 2025 22:19:43 GMT

Parallelization

Wed, 05 Nov 2025 23:44:12 GMT

Parallelization is the process of running multiple processes at the same time

Directional Derivative

Wed, 05 Nov 2025 23:39:10 GMT

Let be a unit vector, the directional derivative of can be given byFind the direction derivative of the function at the point in the direction of
Take the unit vector of
Take the Gradient of
Dot the Gradient with

Extrema

Wed, 05 Nov 2025 23:38:27 GMT

A local minimum of a Scalar Field is a value such thatSimilarly, a local maximum of a scalar field is a value such thatWhere the domain is an open set containing
The local extrema of a Scalar Field can be determined byFor all positions A critical point of a function is a position such that is not differentiableIf is a local extremum, then is a critical point

Set

Wed, 05 Nov 2025 23:36:50 GMT

A collection of objects that does not allow duplicates unlike Arrays that do allow duplicateinsert(element): remove(element): contains(index): allAll(set): union(set): removeAll(set): intersection(set): A set with defined orders, all pairs are greater than or lesser than other pairsContains all set operations with findMin(): findMax():

Lagrange Multiplier

Wed, 05 Nov 2025 23:35:13 GMT

In most applications the control variables are constrained This can be expressed with a level set, which is known as the constraint curveA local minimum of subject to the constraint occurs at only if a is a critical point for or if the two functions are parallel for some scalar Extreme Value Theorem A continuous function on a closed and bounded set must have an absolute minimum and maximum (Extrema) On a closed and bounded set Find all critical points Find all Lagrange points The maximum is the greatest value on the combined set The minimum is the least value on the combined set

Mathematics Review

Wed, 05 Nov 2025 23:33:05 GMT

In a given Set of n. The number of subsets that can be made from elements of that set equal to n2Because the sum of i can be represented as S = 1, 2, ... , n , reversing the expansion and adding the it back onto itself can give us 2S = n+1, n+1, ... , n+1 This can be shortened as:

Tree

Wed, 05 Nov 2025 23:32:19 GMT

Trees are data structures that are ordered in a hierarchical or nested structureRoot: A node at the top of the tree Subtree: A smaller tree under a root tree Leaves: Nodes at the end of branchesParent: Nodes that contain one or many sub nodes Child: Nodes that is contained in a parent node Sibling: Nodes that share a parents Ancestors: All nodes that contain the current node including itselfPath: The sequence of nodes where the two nodes are connected via the edges Length: The number of edges on the pathDepth: The length of the path from the root to the node Height: The length of the longest path from the root to a leafDiameter: The number of nodes on the longest path between any two nodes Maximum Width: The number of nodes on the level in the tree that contains the most nodes Data can be represented in trees to speed up algorithms, because it's organized only parts of a tree need to be considered to solve problems Tree can often speed up an algorithm from to An infix expression can be represented with a tree, for example the equation can be represented byA tree can be represented by a Linked List by referencing a sibling and the first childpublic class LinkedSiblingTree { private TreeNode root; private static class TreeNode { E element; TreeNode firstChild; TreeNode nextSibling; } } Any tree can be traversed in different waysIn level order, nodes are traversed from left to right, top to bottom

Searching

Wed, 05 Nov 2025 23:18:06 GMT

Probe each value in a list sequentially until target is foundBest Case: Worst Case: Basic Implementationpublic static int SequentialSearch(int target, int[] array) { for (int i = 0; i < array.length; i++) { if (array[i].compareTo(target)) { return i; } } return -1; } Sorted Only (Sorting) Probe in the middle, split into two groups, compare if the target is higher or lower, repeat search in the respective groupBest Case: Worst Base: Basic Implementationpublic static int BinarySearch(int target, int[] array) { int low = 0; int high = array.length - 1; while (low <= high) { int mid = low + (high - low) / 2; int result = target.compareTo(mid) if (result == 0) { return mid } else if (result < 0) { high = mid - 1; } else { low = mid + 1; } } return -low - 1; // Return possible index of target }

Binary Search Tree

Wed, 05 Nov 2025 23:17:42 GMT

The difference between binary search trees and Binary Tree are that all nodes to the left are less than the root while all nodes to the right are greater than the rootIt is possible to check whether a tree is a binary search tree by checking if every left node is less than and every right node is greater than
Binary Search Trees can be searched in time, similar to Binary Searchboolean checkBSTHelp(Node root, int min, int max) { if (root == null) return true; if (root.data > max || root.data < min) return false; return checkBSTHelp(root.left, min, root.data - 1) && checkBSTHelp(root.right, root.data + 1, max); } boolean checkBST(Node root) { return checkBSTHelp(root, 0, Integer.MAX_VALUE); } A binary search tree can have the potential to be very unbalanced, this can cause search times to approach rather than
This is often solved by height balanced trees, or Red-Black Tree

Lists

Wed, 05 Nov 2025 23:14:45 GMT

Lists are ordered data, it is implemented as List Interface in Java, The List Interface contains an Array Lists and Linked ListsA class that supports dynamic arrays, the size of the array grows as elements are inserted, it can store any Java objectList Instantiation Define as a List Reference instead of a ArrayList List list = new ArrayList<>(); So that the type of List can be changed easily in testing The Array List grows by every time the size of the array reaches its allotted size Array Lists do not resize when elements are removedIterator: A built-in object to loop elements in order Amortized: The expected run time over a Pre vs Post incrementer Different incrementers can be used based on necessity x++ is a post incrementer that returns the current value before modifying it ++x is a pre incrementer that modifies the value before returning it The Time Complexity of common Array List methods are as followsadd(element): : When array does not need to be expanded : When the array needs to be expanded : Amortized Time add(index, element): get(index): remove(index): indexOf(element): Linked Lists are dynamic lists made out of nodesAll objects in a Linked List must be encapsulated by a node, which contains the value and a pointer to the next node in the list They can be added and removed as needed, in any order from the ListAll Linked Lists must have a head, otherwise the linked list will be headless A headless list will not be referred and will be collected by the garbage collectorThe head refers to the starting node in the ListLinked Lists can have a tail which refers to the end of the ListJava Method Arguments When a method is called, all passed-in arguments are copies of a pointer of that value, meaning the argument is completely local int a = 8; boolean isNextEven(int number) { number++; // number will be 9; a will still be 8 return (number%2==0); }
The Time Complexity of common Array List methods are as followsadd(element): : With tail pointer : Without tail pointer add(index, element): get(index): remove(index): indexOf(element):

Heap

Wed, 05 Nov 2025 22:45:05 GMT

A heap is a complete Complete Binary Tree that is stored in Level OrderIn this way, given a node at address the address of its children and parent can be determined as such left child: right child: parent: (since integer division rounds, both children return the same parent) In Min Heaps, the root has the lowest value and all node below the root has values that are greater than or equal to the rootIn Max Heaps, the root is the highest node and all nodes are less than or equalIn order to store incomplete heaps, allocate the maximum amount of space the heap uses, leave any blank nodes as empty cellsArrays with Identical Elements Because array with identical elements does not need to be percolated, the run time for this type of array is

Hash Function

Sun, 02 Nov 2025 20:09:03 GMT

Hash functions returns a ideally unique value for every inputHash functions must follow these rules Ideally hash keys in a uniform distribution Potentially output all index values of the table Every Java object has a default implementation of hashCode The default hashCode returns a distinct integer for distinct objects but does not consider the contents of the object In order to consider the contents, the hashCode must be overridden

Probing

Sun, 02 Nov 2025 20:08:08 GMT

If the hash index of an object is already occupied, and the objects are not equal, probing can be performed to guaranteed a unique index for the functionThe next index of a cell can be given byWhere is the probing functionIn linear probing, each cell is checked linearly However, because linear probing tends to cluster around occupied cells, the function can be inefficient in near-filled tablesIn quadratic probing, the next checked cell is farther apart as increase This distributes the occupied cells more evenly and makes probing more efficientQuadratic Probing Theorem Given that is prime, then the first cells visited by quadratic probing are distinct. Therefore we can always find an empty cell if the table is at most half full. Another hashing function can be used to determine the next checked cell Where is the secondary hashing functionSimply setting a cell to null can cause problems with probing, if during search a cell is found empty, the algorithm will assume that following cells do not contain the elementInstead of setting the value to null, a cell can be lazily deleted (not removed but marked as deleted)

Line Integral Arc Length Example

Thu, 30 Oct 2025 22:28:14 GMT

Binary Tree Example

Thu, 30 Oct 2025 03:40:41 GMT

Kinematics in 1D

Thu, 30 Oct 2025 00:12:54 GMT

Given constant acceleration, the velocity at a time can be given by the following Given oscillating acceleration, the velocity at time must be integratedObjects moving thro

Time Complexity Example B

Wed, 29 Oct 2025 22:03:50 GMT

Time Complexity Example C

Wed, 29 Oct 2025 22:03:49 GMT

Orbital

Wed, 29 Oct 2025 19:45:48 GMT

Orbitals per subshell: 1 () Electron capacity: 2 electrons Shape: spherical shaped Nodes: Angular nodes = Radial nodes = Since the s orbitals penetrates the core better, s orbitals have higher effective energy than both p and d orbitals Orbitals per subshell: 3 () Electron capacity: 6 electrons Shape: dumbbell shaped Nodes: Angular nodes = Radial nodes = Similar to s orbitals, p orbitals also penetrate the core better having higher effective energy than d orbitals Orbitals per subshell: 5 () Electron capacity: 10 electrons Shapes: four cloverleaf-type or donut around a lobe Nodes: Angular nodes = Radial nodes = Orbitals per subshell: 7 () Electron capacity: 14 electrons Shapes: complex multi-lobed forms Nodes: Angular nodes = Radial nodes = f orbitals tend to be more shielded by inner electrons and less involved in simple covalent bonding compared with valence s and p orbitals. A node is a region where the wave function changes sign and becomes zero A node can either be angular or radialThe amount of angular nodes can be given by The amount of radial nodes can be given by

Formal Charge

Wed, 29 Oct 2025 19:08:01 GMT

A charge assigned to an atom within a molecule in relation to other atom which is not a real charge Its value can be determined byWhere is the total valance electrons, is the total non bonded electrons, and is the total bonded electrons

Periodic Table

Wed, 29 Oct 2025 19:04:26 GMT

The Periodic Table orders all Elements and is divided into groups and periods. Groups share the same amount of valence electrons Called Alkali because they combine with water to make alkaline solutions Soft and mailable, could be cut with a knife Highly reactive metals, especially towards water Harder than Alkali metals Magnesium reacts slowly with water Transition from semi-metallic () to metallic () Some elements such as gallium is almost a liquid at room temperature Transition from nonmetallic () to semi-metallic () to metallic ()
Carbon exhibits Allotropy Petroleum is a compound that contains carbon and hydrogen and is often used a source of fuel Transition from nonmetallic () semi-metallic (), to metallic () From Greek pnigein: to choke Nitric AcidNO⁻OHO Transition from nonmetallic () to semi-metallic () From calcogen: ore former Iron rusts, by being exposed to oxygen at room temperature, being turned into hydrated iron oxides From Greek halos: salt genes: former Relatively inert, non-reactive Often exists in monoatomic form

Avogadro's Number

Wed, 29 Oct 2025 18:49:05 GMT

Avogadro's number is the number of atoms in exactly 12 grams of Mole: A mole is one of Avogadro's number of any particleWhat is the mass of 6 atoms of americium (Am, 243amu)?1 mole of Am will have a mass of 243 gramsMass of one atom of Am = 243g / 243 / (60.22 * 10^23)

Octet Rule

Wed, 29 Oct 2025 18:32:46 GMT

The Octet rule states that during chemical reaction, molecules are more likely to combine to achieve full shells of electronsHydrogen, however, only needs to achieve a shell of two electronsAlso called coordinate covalent or donor-acceptor, dative bonds are ones where a single atom provides both electronsNHHBThe Formal Charge of a covalent bond can be equated byWhere is the number of valence electrons, are the lone electrons and are the bonded electrons

Ionic Bonding

Wed, 29 Oct 2025 18:28:27 GMT

Atoms are bonded when one electron is transferred from one atom to another, this creates ionsNa⁺Cl⁻

Element

Wed, 29 Oct 2025 18:28:07 GMT

Elements can still be broken down to simpler elements, i.e. Ozone to OxygenElements are defined as a substance that is comprised of atoms with the same atomic numberDifferent structural forms of an element that are in the same phaseCarbon : Diamond GraphiteA substance that contains two or more elementsTwo or more substances, which can be elements or compounds, in a physical combinationHomogenous: Substances are dissolved and can not be separated easily Heterogenous: Substances can be separated easily Elements consists of atoms Fusion and ionization can divide atoms All atoms of an given element are identical and mass and other properties Isotopes have different masses and properties Different elements have different masses and properties Compounds are formed when unlike elements are combined in a simple integer ratio (Law of Definite proportions) In a pure compound, the constituent elements are always present in a definite proportionThe ratio of the masses in two different compound are usually simple integersEqual volumes of different gases at the same temperature and pressure contain equal number of particlesCharge of a proton = - Charge of an electronMass of a proton ~= Mass of a neutron Mass of a proton = 1836 * Mass of an electron

Electronegativity

Wed, 29 Oct 2025 18:27:56 GMT

In a molecule of the two electrons are shared equally However, in a mole with a difference in electronegativity the share of electrons will be differentThese bond are called "polar"

Empirical and Molecular Formulas

Wed, 29 Oct 2025 18:22:10 GMT

The molecular formula shows the number of atoms of each element in one molecule For example, the molecular formula for glucose is And its corresponding empirical formula Not all reactants will be exhausted when a reaction occurs. The reactant that is used up first is called the limiting reactant

Ionization Energy

Wed, 29 Oct 2025 14:56:49 GMT

Red-Black Tree

Tue, 28 Oct 2025 22:28:08 GMT

A red-black tree is a binary search tree that satisfies these properties Every node is either red or black The root is black Every leaf (NIL) is black If a node is red then both its children are black For each node, all simple paths form the node to decedents leaves contain the same number of black nodes A red-black tree is an extension of the binary search treeBy constraining the node colors on any simple path from the root to a lead, no path will be more than twice as long as any otherIn fact the height of a tree with nodes will at most be The height of a red-black tree with internal nodes will at most be Base case Where is the black height of a sub tree, which is the maximum height that only counts black nodes not including the rootDuring insertions, these properties can be violated 2. The root is black 4. If a node is red, then both of its children are blackBecause of this knowing the "uncle/aunt" node is more useful than knowing the parentAll insertions follow the insertion rules for binary search trees and start off red Its color changes based on any violated rules(50R is inserted) 50R is red which violates 2.(50R 50B)50B Nothing is violated(99R is inserted) 50B NIL 99R Nothing is violated(70R is inserted) 50B NIL 99R 70R (y)NIL 70R is a child of 99R which violates 4 (2b)(Right Rotate 70R 99R) 50B (y)NIL 70R 99R 99R is a child of 99R which violates 4 (3b)(70R 70B) (50B 50R) (Left Rotate 70B 50B Left) 70B 50R 99R Nothing is violated

Recursion

Tue, 28 Oct 2025 22:27:43 GMT

A method of solving a problem where the overall solution depends on solutions to smaller instances of the problemBase Case: When the answer to the problem is directly known Recursive Call: When the function calls itself to get closer to the base case Pending Operation: Remaining operations that need to be finished before the function ends Tail Recursion: When there is no remaining pending operations when the base case is reachedEach time a function call is made, a stack frame is pushed onto the stackThis repeats until the base case is reachedTheta Notation O(n) measures the worst case scenario and has a chance to break out early (n) measures a constrained scenario with no chance to break early public static int pow(int x, int y) { if (y==0) { return 1; } return x * pow(x, y - 1); } private static int powHelp(int x, int y, int result) { if (y==0) { return result; } return powHelp(x, y - 1, x * result); } public static int pow(int x, int y) { return powHelp(x, y, 1); } A String builder is an array in Java that is made out of characters. It can add or remove characters in constant time There are three different symbols to notate time complexity: Worst Case Scenario : Constant Case Scenario : Best Case ScenarioNumbers that, when arranged, can make up a triangleEx. 1, 3, 6, 10 Equation for Triangular Numbers $$ \frac{i(i+1)}{2} > n $$

Linear Structures

Tue, 28 Oct 2025 22:27:19 GMT

Stacks are linear data structures where objects are only added/removed at the topQueues are linear data structures where objects are only added in the end but only removed at the start Queues are interfaces in Java while Stacks are a classInterfaces vs Classes Interfaces define a methods and properties but can not be instantiated Classes both define state and behavior and can be instantiated Queues do not shift all elements once a object has been removed Circular queues loop around the allocated memory while simple queues do not

Induction

Tue, 28 Oct 2025 22:06:42 GMT

Assumes that any induction is true for all integersAny integer has factorization into prime numbers is a prime number can be factored into itself is not a prime number can be written as a since and can be factored into primes, can be factored into primesTo prove that proof is true, assume is falseThere are infinite prime numbersAssume there are finite prime numbers. Prime numbers are Define integer One of the primes must but a factor of N (all integers can be factored into primes) There must exist an integer where So that divides into Since divides and product = 1 There can not be finite prime numbers

Time Complexity Example A

Tue, 28 Oct 2025 05:51:40 GMT

Time Complexity Example D

Tue, 28 Oct 2025 05:51:37 GMT

Rehashing

Mon, 27 Oct 2025 22:51:28 GMT

Map

Mon, 27 Oct 2025 21:54:20 GMT

A collection of key value pairs Every key is unique while values does not have to beget(key): put(key, value):

Tower of Hanoi Example

Mon, 27 Oct 2025 21:47:55 GMT

Derivative Rules for Vectors

Mon, 27 Oct 2025 21:26:22 GMT

Product rules for vectors are isometric to the scaler product ruleSimilarly the chain rule for vector is as follows

Optimization

Mon, 27 Oct 2025 21:23:40 GMT

To optimize on an open set, look for critical points For closed sets, use the Lagrange multiplier on boundary pointsFor optimization problems, identify The target function or desideratum The control variable () Description where appropriate Domain of admissible values Given Find the closest point to the Origin

Spherical Coordinates

Mon, 27 Oct 2025 21:23:29 GMT

Volumes can also be given using the spherical coordinates Difference between and Different system sometimes swap and , mathematically, the should match for and for The volume can be determined using spherical coordinates via the formula
Similar to in Cylindrical Coordinates, is the constant of integration for spherical coordinates

Second Derivative Test

Mon, 27 Oct 2025 21:23:25 GMT

Given that all 2nd order partial of are continuous, in the area of a critical point . the concavity of can be determined by if and < 0, then is a local maximum if and > 0, then is a local minimum if is a saddle point
Boundaries are open if they do not contain their boundary points Closed sets do contain their boundary point Boundaries can be neither open nor closed

Lagrange Multiplier Example

Mon, 27 Oct 2025 21:23:03 GMT

Second Derivative Test Example A

Mon, 27 Oct 2025 21:21:23 GMT

Second Derivative Test Example B

Mon, 27 Oct 2025 21:21:10 GMT

Spherical Coordinates Definition

Mon, 27 Oct 2025 21:20:15 GMT

Directional Derivatives Formula

Mon, 27 Oct 2025 21:14:26 GMT

Directional Derivatives Example

Mon, 27 Oct 2025 21:14:26 GMT

Reparameterization

Mon, 27 Oct 2025 21:12:33 GMT

A new function that maps another function into the same space with different domainsA smooth reparameterization of is a scaler function such that Where and traces the same pathDerivative of inverses Given The derivative of the inverse is The arc length of a parameterized curve can be defined asMagnitude in Integration The integral of a magnitude does not equate to the magnitude of the integral

Partial Derivative

Mon, 27 Oct 2025 21:12:28 GMT

A partial derivative is deriving a function with respect to a variable, this derives only one variable while the other remains "constant" Partial Derivative of f(a, b) in respect to x $$ f_{x}(a,b) = \lim_{ h \to 0 } \frac{f(a+h,b)-f(a,b)}{h} $$ Partial Derivative of f(a, b) in respect to y $$ f_{y}(a,b) = \lim_{ h \to 0 } \frac{f(a,b+h)-f(a,b)}{h} $$Notations of Partial Derivative Partial Derivatives are can be notated in a number of ways The partial derivative can be taken multiple timesDifferentiability If all partial derivatives of a function are differentiable, then the function itself is differentiable Partial differential equations are differential equations with partial derivativesThe heat equation, for example, gives temperature as a function of position and time Mixed Partials are partial derivatives where derivatives in respect to more than one variable is taken Mixed Partial of f in respect to x and y $$ f_{xy} = \frac{\partial^2f}{\partial x \partial y} $$Order of Mixed Partials The partial is equal to Similarly, is equal to and If the relevant mixed partials exists and are continuous around the particular position then

Order of Integration

Mon, 27 Oct 2025 21:12:27 GMT

In a triple integral, bounds can be written in different ways while producing the same resultGiven , integrate over volume Since has bounds between the first integral can be set toThe second variable can be set to and can be set to

Linear Function

Mon, 27 Oct 2025 21:12:24 GMT

A linear function on has the formIn a one variable linear function , the graph is a line In a two variable linear function, the graph is a planeFor one variable functions, is differentiable at if exists, meaning that if at looks like a straight line close up, it is differentiableFor two variable functions, is differentiable at if it resembles a plane close upA scalar field is differentiable at if there is a linear function such thatWhere as The linearization of function at point is given byLinearization of functions can be used to estimate values of functionsGiven Estimate Instead of estimating values, changes can be estimated using differentials

Integrals of Vectors

Mon, 27 Oct 2025 21:12:21 GMT

Because integrals are the limits of sums, and vectors can be added together, there is a direct translation to the integral of vectors Fundamental Theorem for vectors $$ \vec{r}(t)-\vec{r}(t_{0})=\int_{t_{0}}^t\vec{r'}(u)du $$ Show that a particle that changes direction has nonzero acceleration Suppose The velocity can be defined as And since The velocity can not change direction if A particle initially at rest at the origin is subject to an acceleration For Since Then is equal to For

Gradient Decent

Mon, 27 Oct 2025 21:12:17 GMT

The minimization (decent) or maximization (accent) of a cost/loss function of several variables

Curvature

Mon, 27 Oct 2025 21:12:08 GMT

The curvature of a function is how much the direction of a tangent line changes throughout the functionA curve is called a smooth curve when Smooth curves contains a well-defined unit tangent vectorCurvature can be defined using as followsThe curvature can also be given using

Changing Coordinates

Mon, 27 Oct 2025 21:12:03 GMT

Areas can also be integrated via Consider a polar rectangleArea of the rectangle can be determined the following equation Given where , can be expressed asThe area using Can be given as

Optimization

Mon, 27 Oct 2025 20:21:54 GMT

Index

Thu, 16 Jul 2026 04:42:53 GMT

Intro to Applied Math
Multivariable Calculus
Probability I
Gen Chemistry I
Advanced Programming
Data Structures
Intro to Electrical
Physics E&M
Physics Mech
Decision Analysis
Quantum Information Theory

AOE Ted Talk

Mon, 11 May 2026 16:01:12 GMT

The term side-questing has become more prevalent in today's culture as a descriptor for actions that are extraordinary. If you, for example, gone to midtown to see a play but accidently came back with a a bag full of Tibetan crystals and a belly full of samosas, that's side questing. If you have math and physics homework due in two days and you, with full intention of finishing them by the end of your day, find yourself in the middle of Chinatown at midnight? That's side questing. But despite how unhelpful these "side quests" may seem. I might argue that we don't "side quest" enough. It's understandable to get lost in the amount of work, clubs, frats--- Columbia is amazing in the sense that you can never get bored while within its gates. But these gates are doubled sided; We tend to focus on what's happening inside the gates that we sometimes fail to notice what's happening outside. This laser-focused mentality simply not the best way to grow as a person.All of us have unique experiences and perspectives that pertains only to us. And that's wonderful. But often we can end up getting too bunched up in these experiences, we become sheltered to how others live and only see the world a certain way. Seeing as much of the conditions of others will do wonders in helping us understand the conditions of the world we live in.Side questing does not have to be purely for personal growth, it can be applied academically as well. When we think of a scholar what do we think of? something something classroom engagement An ideal scholar should be one that's proficient in many fields. But due to how we naturally gravitate towards a specific field, we as scholars tend to over-specialize in a specific field. In that sense we tend to look at situations or problems a certain way, we fail to see it through the venues of others fields. Via side questing in other fields, (through clubs, jobs, or projects), we can better understand the approach that others might take to solve a certain problem. So all in all, go off on a tangent. Take all the opportunities, be spontaneous with your plans, and see the world by side questing, and you might find yourself a more complete person by the time you're done.

Wars Should Not be Fought in the Dark — The Myanmar Civil War

Wed, 10 Dec 2025 08:33:55 GMT

Lynn OngMy father woke up to perhaps one of the most disheartening birthday gifts on the morning of Feb 1st, 2021. Our family watched as the Myanmar military brought a caravan of armored personnel carriers and trucks into the state capital to instate a coup d'état, the fourth time coup det tat in our country's history (Irrawaddy).
Just months ago, Myanmar celebrated its second successful democratic elections after decades of fighting for democracy on Nov 5th of 2020. In the election, the National League for Democracy (from here to be referred to as NLD) won a unprecedented 501 seats in the parliament with the military-backed Union Solidarity and Development Party winning only 38 seats (MIMU). This of course, angered the military junta, who at first refused the election results as fraudulent and called for a re-election before calming down and claiming that they had accepted the election results (Irrawaddy). This acceptance lasted only three months as the military started its take over the following February. Since this, the civil war has gone on for almost six years, leaving millions of people under the looming threat of displacement, starvation, or death.Yet, much to my concern, I found that most people have no knowledge of this war. While I found myself regressing backing to the topic of the war whenever I speak Burmese, I rarely talk about it when I'm speaking English or other languages. People tend to recall a TikTok of a woman doing yoga while armored vehicles roll in the background. This TikTok though, fails to capture the gravity of the situation, the human aristocracies, and ongoing suffering happening within my country.
In their article on the media coverage of suffering on the Syrian War, Xu Zhang and Catherine A. Luther complied a definition for suffering to be a "severe distress that has a detrimental impact on personhood" (Zhang & Luther 3). It's is the lack of basic human needs or dignity that causes mass suffering in countries in Myanmar that are undergoing war. The United Nations estimate that 19.9 million persons are in need of some basic necessity within Myanmar with the most being food or shelter (UNOCHA).
Yet, this problem of unawareness seems to not effects countries such as Ukraine, for example. Since the Russian invasion in 2022, Ukraine has deservedly garnered much international media attention along with tangible military aid by western powers (USIP). Ukraine have already received armory "including tanks and advanced weapons systems" from western powers (Chang & Coppel 85) yet counties like Myanmar have only received encouragements. Chang and Coppel, in their book, "Myanmar's Digital Coup" claims that the "quick defence of a fellow Western state [sic] suggests a degree of racism although the stronger Ukrainian diaspora in the US and Europe was also a contributing factor" (87).While it may be easy to dismiss this as racism on the part of Western nations, this fails to capture the full extent of the problem. The discrepancy in aid between Ukraine and Myanmar can be attributed to the discrepancy in media attention between the two war zones. Ukraine enjoys continuous coverage in both US news sources and international news sources when compared to other on-going wars like in Syria, Sudan, or Myanmar. This is especially worse for the Rohingya peoples of Myanmar, who used to be internationally recognized but has completely been overshadowed by the civil war despite their continued struggle in being murdered and displaced by the military. While this lack of attention may not appear as important to resolving any conflict the difference in attention may be affecting the amount of humanitarian aid received and how foreign governments react to conflicts.
The United Nations Office for the Coordinate of Humanitarian Affairs (UNOCHA) estimates that in 2024, private persons and companies donated 23.5 thousand USD to Myanmar and 1.5 million USD to Ukraine (UNOCHA).
The difference is undoubtedly due to the fact that updates in the Ukraine war is more prevalent in international news while updates for Myanmar is often left unreported. Chang and Coppel finds that a possible explanation for this difference could be Ukraine's narrative of a "nuclear-armed power’s invasion of a sovereign neighbour is regarded as a greater security threat than an internal coup" (86). But invasion or coup, in both cases, real people are suffering. Which is why it is the responsibility of international media to report on these critical situations. The lack of knowledge by many citizens curbs any political pressure for governments to act on these atrocities. Without media coverage, Myanmar continues suffer in the darkness.Please excuse the different style of formatting; I'm using a different word processor for the Op-Ed This Op-Ed is not a revision of any previous essay so all of the content is completely new. Instead of talking about the revision, I hope you will find me talking about the process of writing the article along with its history of revisions satisfactory.I first researched the situation in Myanmar in the lens of media and geopolitics. I've been exposed to many stories on the ground through family and friends who are living there. But I did not know much about other countries reactions to the situation in Myanmar nor did I know why the media does not report on these stories.I was able to find several sources that were able to give me some answers along with many sources that gave context on the war in Myanmar. Using those sources I started drafting an article on how the lack of media reporting on Myanmar can cause harm to people suffering there. I wanted this piece to be more personal since I do have a personal stake so I was more liberal in using first person within the article. Given the word limit, I unfortunately had to cut some of the personal stories to give room to more pressing contents.The peer review gave me a better idea of what my essay needs works on. After the review, I rewrote most the introduction along with transitions within the article. Through research I picked Al-Jazeera for the publication because of reputation for journalistic freedom along with its mission statement to "give voices to the voiceless". I researched more on the publication guidelines and made small edits to align myself better. I still want to be open to other publications in the future so I tried to not overtly copy Al-Jazeera's styling.

EGNL1010 4c

Wed, 10 Dec 2025 05:19:53 GMT

(a) Context Possible Study Introduction (b) Representation Case Study? (c) Analysis Myanmar Digital Coup Routledge Handbook History of Successful Coup d’états (134) Terrance Lee Independence in 1948 First Coup in 1958 Second Coup in 1962 Third Coup in 1974 Myanmar Informational Management Unit My father woke up to perhaps one of the most disheartening birthday gifts on the morning of Feb 1st, 2021. Our family watched as the Myanmar military brought a caravan of armored personnel carriers and trucks into the state capital to instate a coup d'état, the fourth time coup det tat in our country's history (Irrawaddy).
Just months ago, Myanmar celebrated its second successful democratic elections after decades of fighting for democracy on Nov 5th of 2020. In the election, the National League for Democracy (from here to be referred to as NLD) won a unprecedented 501 seats in the parliament with the military-backed Union Solidarity and Development Party winning only 38 seats (MIMU). This of course, angered the military junta, who at first refused the election results as fraudulent and called for a re-election before calming down and claiming that they had accepted the election results (Irrawaddy). This acceptance lasted only three months as the military started its take over the following February. Since this, the civil war has gone on for almost six years, leaving millions of people under the looming threat of displacement, starvation, or death.Yet, much to my concern, I found that, most people have no knowledge of this war. While I found myself regressing backing to the topic of the war whenever I speak Burmese, I rarely talk about it when I speak English or other languages. People do recall a TikTok of a woman doing yoga while armored vehicles roll in the background but never the gravity of the situation, and never the human aristocracies and ongoing suffering happening within the country.
In their article on the media coverage of suffering on the Syrian War, Xu Zhang and Catherine A. Luther complied a definition for suffering to be a "severe distress that has a detrimental impact on personhood" (Zhang & Luther 3). It's is the lack of basic human needs or dignity that causes mass suffering in countries in Myanmar that are undergoing war. The United Nations estimate that 19.9 million persons are in need of some basic necessity within Myanmar with the most being food or shelter (UNOCHA).
Yet, this problem of unawareness seems to not effects countries such as Ukraine, for example. Since the Russian invasion in 2022, Ukraine has deservedly garnered much international media attention along with tangible military aid by western powers (USIP). Ukraine have already received armory "including tanks and advanced weapons systems" from western powers (Chang & Coppel 85) yet counties like Myanmar have only received encouragements. Chang and Coppel, in their book, "Myanmar's Digital Coup" claims that the "quick defence of a fellow Western state [sic] suggests a degree of racism although the stronger Ukrainian diaspora in the US and Europe was also a contributing factor" (87).While it may be easy to dismiss this as racism on the part of Western nations, this fails to capture the full extent of the problem. The discrepancy in aid between Ukraine and Myanmar can be attributed to the discrepancy in media attention between the two war zones. Ukraine enjoys continuous coverage in both US news sources and international news sources when compared to other on-going wars like in Syria, Sudan, or Myanmar. This is especially worse for the Rohingya peoples of Myanmar, who used to be internationally recognized but has completely been overshadowed by the civil war despite their continued struggle in being murdered and displaced by the military. While this lack of attention may not appear as important to resolving any conflict the difference in attention may be affecting the amount of humanitarian aid received and how foreign governments react to conflicts.
The United Nations Office for the Coordinate of Humanitarian Affairs (UNOCHA) estimates that in 2024, private persons and companies donated 23.5 thousand USD to Myanmar and 1.5 million USD to Ukraine (UNOCHA). The difference is undoubtedly due to the fact that updates in the Ukraine war is more prevalent in international news while updates for Myanmar is often left unreported. Chang and Coppel finds that a possible explanation for this difference could be Ukraine's narrative of a "nuclear-armed power’s invasion of a sovereign neighbour is regarded as a greater security threat than an internal coup" (86). But invasion or coup, in both cases, real people are suffering. Which is why it is the responsibility of international media to report on these critical situations. The lack of knowledge by many citizens curbs any political pressure for governments to act on these atrocities. Without media coverage, Myanmar continues suffer in the darkness.

Outline E3

Wed, 26 Nov 2025 19:25:26 GMT

(a) Intro Strate v. A-1 Contractors Federal Encroachment Indigenous Resistance (b) History 1980 (p) Power Imbalance (j.1) Montana (g) Recognition (g.1) Failure of recognition (g.2) Montana Rule 1990 (h) Establishment of Courts (j.2) Branched Government (c) Significance of Tribal courts (c.1) Peace making court (c.2) Relationships (j.3) Wilson Marchington Defined Montana Establishment of Criteria 2000 (j.4) Nev. Hicks (j.5) MacArthur San Juan (m) Modern Day

Sources

Wed, 26 Nov 2025 19:15:44 GMT

Tribal Courts Reduces crime rates in tribal lands Tribal Courts are underfunded Causes crimes to go unpunished for deferred Less authority in tribal lands Public Safety Need Reform Low sentencing limits Poor fundings Tribal Court Law and Order Act Passed in 2010 More Oversight for Tribal Courts Public Defenders Increases limit to 3 years Federal Gov not consulting tribes enough Discrepancy between federal communication and tribal communication 2012 Obama strengthened communication with tribal governments, requiring permitting for federal projects Meaningful contribution No One definition Must include face to face meetings Bear Ears Coalition New proposal Manager will consult a commission of tribal experts Violence Against Women Reauthorization Act Cherokee Nation v. Georgia Indian Tribes are not foreign nations but domesticated nations ISDEAA Federal funding for tribal programs MacArthur v San Juan 2005 (9th D) (Civil) Hospital open up in Navajo Nation, Utah Contracted by County Nation argued that affirmative action is not followed Billing practices also decreased total visitors Did not follow exception in Montana v United States Confirmed lower court dismissal Nation's argument was dismissed Wilson v. Marchington 1997 (9th D) (Civil) Plaintiff, a Blackfeet Indian member filled suit in tribal court over alleged damages Defendant, a nonmember argued that tribal courts do not have subject matter jurisdiction Full Faith and Credit Clause Argued for Sovereignty of tribal nations Nev v. Hicks 2001 (SCOTUS) (Civil) On matter of state sovereignty over tribal borders Nevada wardens suspected nation members of hunting protected game Followed into tribal land and unsuccessfully searched homes of suspects Members sued under tribal court Tribal court held claim over jurisdiction on case Warden claimed declaratory judgment that tribal court lacked jurisdiction Provided writ to district court Montana v. United States 1980 (SCOTUS) (Civil) Crow Tribe limited hunting and fishing in their reservation Non native land owners sued saying it infringed on right Court determined native land was held in trust, indigenous tribes do not "own" land Created Montana Rule All court cases under tribal jurisdiction must involve both parties to be indigenous Ex Parte Crow Dog

Questions for Conference E3

Thu, 13 Nov 2025 18:28:16 GMT

How to walk the line between generalization and not using voice I don't use my voice enough (contributions to the conversations) Tend to generalize, how to argue a point without citing everything Direction (Based on think but prove) Put yourself in Be more specific on claim Talk about why testimonies We don't know their problems Stakes Ask why? Put in essay Different ways that Navajo Nation govern themselves (different views of justice) NYPD Prove importance

Direction for Essay Three

Thu, 13 Nov 2025 18:16:47 GMT

Scholarly Problem: Tribal Courts field their power from the federal government, while recognizing sovereignty federal courts do not allow tribal courts to practice sovereignty Project: Primary Sources Court Cases Testimony News Paper Interviews Secondary Sources Articles Object of Analysis Strate v. A-1 (Outside) Montana v. United States (Outside) How the Judicial System Changed for Tribes within the Navajo Nation In Modern Times Pathways Courts Government Structure Information Send email Interview EBSCO

Coherence Excercises

Mon, 10 Nov 2025 17:50:07 GMT

Vegetation covers the earth [sic]. Plants grow most richly in fertilized plans and river valleys, but also grow at the edge of perpetual snow in high mountains. They grow at the edge of the ocean, in and around lakes, and swamps. They grow on the sidewalks of bustling streets as well as around barren cliffs. Vegetation has inhabited the earth long before humans existed and will inhabit the earth long after we are gone.In this essay, I will navigate the development and changes in tribal courts in the Navajo nation throughout modern times. More specifically, how the jurisdiction of these tribal courts grows and diminishes in regard to the relationship between the US government and tribal nations. To accomplish this, I will compare firsthand accounts from newspapers, testimonies, and journals, along with examinations into relevant ruling on court cases that deal with this issue. I will view these US court cases as external events, because rulings from these courts cases often fail to argue effective on the side of tribal courts. Instead, I will use reactions and testimony to these court cases to analyze the change within tribal courts.To understand the modern changes in tribal courts, first we must understand its foundation and its history under the context of indigenous autonomy within the United States. Before Tribal courts, different indigenous populations made use of different systems to handle disputes among their people. Within the Navajo tribes, there exists hozhoji' naat'aah, or roughly, peace leaders (Need to specify). These leaders are often chosen by the community for their wisdom and spirituality and will facilitate sessions where disputing parties communicate with each other and work out a compromise in what is known as meditative counseling (Yazzie). Courts that resemble western courts were introduced in around the 1880’s and primarily acted to “[further] the values and customs of the BIA, not tribes” (Wild 1609). These courts were implemented during the encampment of the Navajo people and targeted Navajo for. However, this system of oppression was later turned into a way to practice autonomy. Tribal courts in the modern sense were established in 1991 after a reorganization of the old tribal government into a tree branched government. This reorganization was primarily made in the image of the US government. (Navajo Nation). These courts enforce tribal code and other disputes between indigenous and non-indigenous persons on native land.

Indigenous Interruptions

Mon, 10 Nov 2025 04:39:54 GMT

The Road Ends Here Private Enterprise, Tribal Sovereignty, and the Making of the Modern

Mon, 10 Nov 2025 04:39:54 GMT

TRIBAL COURTS AND THE ADMINISTRATION

Mon, 10 Nov 2025 04:39:54 GMT

Tribal Consultation Toward

Mon, 10 Nov 2025 04:39:54 GMT

Notes for Essay Three

Mon, 10 Nov 2025 02:58:29 GMT

Can pick aspects of colonialism Language Space Government Law Can pick different different cultures Time, Space, PlaceAdjustments based on searches Meet with research librarian Seed Text, Formula Ask prof Run claim back US South West Effects of Law on Native Law Might be too big Lexus Nexus Case Law Newspaper, Tribal Publications Scholarly problem will need more context (Find Disconnect) Event with Legal State v Federation Send Email to Navajo Claim will result from scholarly problem Have a couple different idea Start Searches Meet with Librarian Find Context Go to office hoursFind conflict within Tribal Nation Unpack Struggle Claim Normal to write on primary and secondary sources Can not be equivalent Continue as normal Purpose Object Tribal governments in the American southwest, Especially the governmental structure and culture around governance Method Seeing how the structure changes through court cases and first hand account (news papers) Stakes

Oct 27

Mon, 27 Oct 2025 16:50:28 GMT

John Torfanelli jt628@columbia.eduWorkshops are open to all studentsDirectory of all librarians can be found in on the webpageArticles can be found by looking up a term in google scholar Articles that cite a particular source can also be found by clicking on the "Cited by" linkA Cultural History of Association Football in Scotland, 1865-1902 : Understanding Sports As a Way of Understanding Society research.ebsco.com/linkprocessor/plink?id=0b98007e-474e-3fbe-bcb8-4689e826d869.Global games: culture, political economy and sport in the globalised world of the 21st century
https://doi.org/10.1080/014365904200281302Cultivating kinship and refusal in Indigenous diaspora.
https://doi.org/10.1080/15595692.2024.2337940“Little Kingdoms”: Administrative Authority, Security, and Sovereignty Co-Optation in Indian Country
https://ezproxy.cul.columbia.edu/login?url=https://www.proquest.com/dissertations-theses/little-kingdoms-administrative-authority-security/docview/3232674911/se-2?accountid=10226Neither Settler nor Native
https://doi.org/10.4159/9780674249998

Oct 21

Tue, 21 Oct 2025 20:33:21 GMT

Male Gaze, affect See women as sex object or more complex How to explore Vermeer's What could they represent What are the ideas in the art Deeper than the subject How do they appear How contribute Points of Convergence Explain the actions and details Secondary to Primary Can be inverse Affect theory Revisit Claim Create SP Use Details and Secondaries to build up Given all art is political What is the painter arguing Use sources Help see (lens) Religion (projection) Subclaim Context of the movement Astro Agency Don't simplify too much How sources would respond to analysis Is empowerment adhering to the definition of beauty set by society? Can beauty be separated from its political context? (a) Vermeer's use of beauty Restrictive Hint at the claim (b) Beauty is political (b.1) Desirability (b.2) Objectification of marginalization Srinivasan (b.3?) Attunement Theory Ahmed "Social Club" (c) Therefore Empowerment is political? (c.1) Srinivasan's Empowerment Empowerment adheres to society's definition Vermeer's Smiling Lady Ahmed? (d) Can "Pure" beauty exists (d.1) Example of nature (d.2) Vermeer's Milkmaid Pure beauty Srinivasan? (e) Vermeer's beauty can be separated? (e.1) Intention vs Realization (e.2) Abstraction of women Vermeer's Glass of Wine Agency (church) (f) Is empowerment political (f.1) Srinivasan 'I became' (f.2) Vermeer's Sleeping Lady Srinivasan's trust in empowerment Is Agency restrictive?
Could represent the churchFully casted onto the woman But half casted on the man

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Sep 23

Tue, 23 Sep 2025 21:52:26 GMT

Structures are okay (1) Define "completely different scene" "Set up" the passage Introduce the passage Get into the passage later Introduce Howard in Intro Difference between experience at Howard and Street In this paper ___ placeholder