Definition - What does Drilling Formulas mean?
Drilling Formulas are sets of special formulas used by drilling engineers for a wide number of oil well related operations. These formulas form the base of drilling calculations. Also known as drilling formula sheets, these formulas are essential mathematical and engineering factors that help in drilling, managing and controlling an oil well. These formulas are taught in Petroleum engineering and in oil well control schools. The drilling industry makes use of almost 5000 formulas on a regular basis for well control, drilling wells and other operations.
Petropedia explains Drilling Formulas
Petroleum engineering makes use of special formulas to analyze and calculate the various aspects related to the drilling of an oil well and its later operations. Drilling Formulas and drilling calculations play an important part as they help graduate students to learn how to implement these dynamics on real sites. These formulas are regulated by the American Petroleum Institute and are available in several handbooks meant for well control. Nowadays, these formulas are also available in Smartphone applications which help offshore and onshore drilling experts to use them easily. Since all the phases of oil production require the maintenance of the oil well, Drilling Formulas remain pivotal to do so.


A Day in the Life of a YPP Math Literacy Worker
“We help [younger] kids to understand math in a way that we didn’t get to when we were their age. All MLWs have to be leaders. A great MLW is a great team member, someone who comes to the workshop and actually gives energy. Kids look up to teenagers, what they see from us, that’s what they’re going to do.”
The Young People’s Project (YPP) was envisioned and continues to conduct its work in the spirit of Ella Baker. Founded in 1996 by two generations of young people, YPP is an outgrowth of the Algebra Project which Bob Moses founded as part of the legacy of SNCC’s student-led grassroots organizing and voter registration efforts in 1960’s Mississippi. It has since grown into a nationally recognized, math literacy and youth empowerment nonprofit that serves over 600 students and teachers across 14 sites every year.
Through 25 years of partnering with schools and community-based organizations, YPP has demonstrated that it is possible to develop and implement a ground-up education program that engages young people who are most often underrepresented in STEM fields, in learning and teaching math through near-peer mentoring, in a way that increases their interests in math, their persistence, confidence and willingness to more broadly serve their communities. The program shows a path toward improving education for those who are not benefitting from the country’s public educational system. This year (2022) marks YPP’s 25th Anniversary as a nonprofit organization.
YPP is a solution that young people build with each other, under the leadership, support, and guidance of invested adults. MLWs learn pieces of math well enough to teach it and learn to facilitate math activities with middle school students. Together they learn and teach each other math in ways that are meaningful and engaging. As a result, they develop as a community, and become their own support. In doing so they invite and challenge the adults and systems around them to see and treat them differently.
As one middle school principal put it, “YPP may not be the answer for every child, but it is the answer for the community, that we begin to see our children differently, that we begin to believe in them.”
YPP seeks to recruit MLWs from high school students representing a diverse group of academic performers and peer leaders, from the same community as the middle school students. The training of high school students as paid Math Literacy Workers (MLWs) develops the abilities of the middle school students and the high school students themselves, to succeed in school and in life, and in doing so involves them in efforts to eliminate institutional obstacles to their success. MLWs care that the younger students learn and have fun.
YPP is Exploring STEM Literacy
In this video you will see YPP’s emerging Computer Science and STEM literacy
work in action at Excel High School in Boston, MA.
In this video you will see YPP’s National Youth “Flagway” Math Tournaments.
|Gates Foundation Targets Culturally Responsive Math Teaching With New Grants
|Young People's Project Empowers Youth With Math Skills


A wavelet is a kind of mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.
In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set of Frame of a vector space (also known as a Riesz basis), for the Hilbert space of square integrable functions.
Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). Note that both DWT and CWT are of continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid. The word wavelet is due to Morlet and Grossmann in the early 1980s. They used the French word ondelette, meaning "small wave". Soon it was transferred to English by translating "onde" into "wave", giving "wavelet".
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use discrete-time filterbanks. These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response (FIR) or infinite impulse response (IIR) filters. The wavelets forming a CWT are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency resp. scale to that event. The product of the uncertainties of time and frequency resp. scale has a lower bound. Thus, in the scaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. This is related to Heisenberg's uncertainty principle of quantum physics and has a similar derivation. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.
Wavelet transforms are broadly divided into three classes: continuous, discretised and multiresolution-based.
Continuous wavelet transforms (Continuous Shift & Scale Parameters)In continuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the Lp function space L^2(\R)). For instance the signal may be represented on every frequency band of the form [f,2f] for all positive frequencies f>0. Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components.
The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This subspace in turn is in most situations generated by the shifts of one generating function \psi \in L^2(\R), the mother wavelet. For the example of the scale one frequency band [1,2] this function is
The subspace of scale a or frequency band [1/a,\,2/a] is generated by the functions (sometimes called child wavelets)
- \psi_ (t) = \frac1\psi \left( \frac \right),
The projection of a function x onto the subspace of scale a then has the form
- x_a(t)=\int_\R WT_\psi\(a,b)\cdot\psi_(t)\,db
- WT_\psi\(a,b)=\langle x,\psi_\rangle=\int_\R x(t)\overline\,dt.
See a list of some Continuous wavelets.
For the analysis of the signal x, one can assemble the wavelet coefficients into a scaleogram of the signal.
Discrete wavelet transforms (Discrete Shift & Scale parameters)It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the affine system for some real parameters a>1, b>0. The corresponding discrete subset of the halfplane consists of all the points (a^m, n\,a^m b) with integers m,n\in\Z. The corresponding baby wavelets are now given as
A sufficient condition for the reconstruction of any signal x of finite energy by the formula
- x(t)=\sum_\sum_\langle x,\,\psi_\rangle\cdot\psi_(t)
Multiresolution-based discrete wavelet transforms
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. To avoid this numerical complexity, one needs one auxiliary function, the father wavelet \phi\in L^2(\R). Further, one has to restrict a to be an integer. A typical choice is a=2 and b=1. The most famous pair of father and mother wavelets is the Daubechies 4 tap wavelet.
From the mother and father wavelets one constructs the subspaces
- V_m=\operatorname(\phi_:n\in\Z), where \phi_(t)=2^\phi(2^t-n)
- W_m=\operatorname(\psi_:n\in\Z), where \psi_(t)=2^\psi(2^t-n).
- \\subset\dots\subset V_1\subset V_0\subset V_\subset\dots\subset L^2(\R)
From those inclusions and orthogonality relations follows the existence of sequences h=\_ and g=\_ that satisfy the identities
- h_n=\langle\phi_,\,\phi_\rangle and \phi(t)=\sqrt2 \sum_ h_n\phi(2t-n)
- g_n=\langle\psi_,\,\phi_\rangle and \psi(t)=\sqrt2 \sum_ g_n\phi(2t-n).
Mother waveletFor practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons, one chooses the wavelet functions from a subspace of the space L^1(\R)\cap L^2(\R). This is the space of measurable functions that are absolutely and square integrable:
- \int_^ |\psi (t)|\, dt and \int_^ |\psi (t)|^2 \, dt .
Being in this space ensures that one can formulate the conditions of zero mean and square norm one:
- \int_^ \psi (t)\, dt = 0 is the condition for zero mean, and
- \int_^ |\psi (t)|^2\, dt = 1 is the condition for square norm one.
For \psi to be a wavelet for the continuous wavelet transform (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform.
For the discrete wavelet transform, one needs at least the condition that the wavelet series is a representation of the identity in the space L^2(\R). Most constructions of discrete WT make use of the multiresolution analysis, which defines the wavelet by a scaling function. This scaling function itself is solution to a functional equation.
In most situations it is useful to restrict \psi to be a continuous function with a higher number M of vanishing moments, i.e. for all integer ''m\int_^ t^m\,\psi (t)\, dt = 0
Some example mother wavelets are:
The mother wavelet is scaled (or dilated) by a factor of a and translated (or shifted) by a factor of b to give (under Morlet's original formulation):
- \psi _ (t) = \psi \left( \right).
For the continuous WT, the pair (a,b) varies over the full half-plane \R_+\times\R; for the discrete WT this pair varies over a discrete subset of it, which is also called affine group.
These functions are often incorrectly referred to as the basis functions of the (continuous) transform. In fact, as in the continuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation uses a subtly different formulation (after Delprat).
Comparisons with Fourier Transform (Continuous-Time)The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The Short-time Fourier transform (STFT) is also time and frequency localized but there are issues with the frequency time resolution and wavelets often give a better signal representation using Multiresolution analysis.
The discrete wavelet transform is also less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform. This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT.
Definition of a waveletThere are a number of ways of defining a wavelet (or a wavelet family).
Scaling filterThe wavelet is entirely defined by the scaling filter - a low-pass finite impulse response (FIR) filter of length 2N and sum 1. In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.
For analysis the high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction filters the time reverse of the decomposition.
Daubechies and Symlet wavelets can be defined by the scaling filter.
Scaling functionWavelets are defined by the wavelet function \psi (t) (i.e. the mother wavelet) and scaling function \phi (t) (also called father wavelet) in the time domain.
The wavelet function is in effect a band-pass filter and scaling it for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html#note7 for a detailed explanation.
For a wavelet with compact support, \phi (t) can be considered finite in length and is equivalent to the scaling filter g.
Meyer wavelets can be defined by scaling functions
Wavelet functionThe wavelet only has a time domain representation as the wavelet function \psi (t).
For instance, Mexican hat wavelets can be defined by a wavelet function. See a list of a few Continuous wavelets.
Applications of Discrete Wavelet TransformGenerally, an approximation to DWT is used for data compression if signal is already sampled, and the CWT for signal analysis. Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.
Wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional Fourier Transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, ab initio calculations, astrophysics, density-matrix localisation, seismic geophysics, optics, turbulence and quantum mechanics. This change has also occurred in image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multifractal analysis. In computer vision and image processing, the notion of scale-space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.
One use of wavelet approximation is in data compression. Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets. This means that although the frame is overcomplete, it is a tight frame (see types of Frame of a vector space), and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details see wavelet compression.
A related use is that of smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or denoising operations can be performed.
HistoryThe development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Notable contributions to wavelet theory can be attributed to Zweig’s discovery of the continuous wavelet transform in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound), Pierre Goupillaud, Grossmann and Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's Harmonic wavelet transform (1993) and many others since.
Wavelet TransformsThere are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms but the common ones are listed below:
Generalized TransformsThere are a number of generalized transforms of which the wavelet transform is a special case. For example, Joseph Segman introduced scale into the Heisenberg group, giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.
Another example of a generalized transform is the chirplet transform in which the CWT is also a two dimensional slice through the chirplet transform.
An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example, darkfield electron optical transforms intermediate between direct and reciprocal space have been widely used in the harmonic analysis of atom clustering, i.e. in the study of crystals and crystal defects. Now that transmission electron microscopes are capable of providing digital images with picometer-scale information on atomic periodicity in nanostructure of all sorts, the range of pattern recognition and strain/metrology applications for intermediate transforms with high frequency resolution (like brushlets and ridgelets) is growing rapidly.
List of wavelets
- Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0-7503-0692-0
- Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-274-2
- A. N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets, Academic Press, 1992, ISBN 0-12-047140-X
- P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0-13-605718-7
- Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1-56881-041-5
- Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7
- Haar A., Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331-371, 1910.
- Ramazan Gençay, Faruk Selçuk and Brandon Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, 2001, ISBN 0-12-279670-5
- Donald B. Percival and Andrew T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, ISBN 0-5216-8508-7
- Tony F. Chan and Jackie (Jianhong) Shen, Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, Society of Applied Mathematics, ISBN 089871589X (2005)
- Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-466606-x
- Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the Making", AK Peters Ltd, 1998, ISBN 1568810725, ISBN-13 978-1568810720
- Wavelet Digest
- NASA Signal Processor featuring Wavelet methods Description of NASA Signal & Image Processing Software and Link to Download
- 1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)
- Binomial-QMF Daubechies Wavelets
- Wavelets made Simple
- Course on Wavelets given at UC Santa Barbara, 2004
- Wavelet Posting Board
- The Wavelet Tutorial by Polikar (Easy to understand when you have some background with fourier transforms!)
- OpenSource Wavelet C++ Code
- An Introduction to Wavelets
- Wavelets for Kids (PDF file) (Introductory (for very smart kids!))
- Link collection about wavelets
- Wavelet forums (French) Wavelet forum (English)
- Gerald Kaiser's acoustic and electromagnetic wavelets
- A really friendly guide to wavelets
- Wavelet-based image annotation and retrieval
- Very basic explanation of Wavelets and how FFT relates to it
wavelet in German: Wavelet
wavelet in Spanish: Wavelet
wavelet in Finnish: Wavelet-muunnos
wavelet in French: Ondelette
wavelet in Indonesian: Wavelet
wavelet in Italian: Wavelet
wavelet in Polish: Falki
wavelet in Portuguese: Wavelet
wavelet in Russian: Вейвлет
wavelet in Swedish: Vågelement
wavelet in Ukrainian: Вейвлет
wavelet in Chinese: 小波分析
wavelet in Japanese: ウェーブレット


By Matthew Piechalak | email@example.com
Two Parker Upper School students earned the highest possible score on their ACT test.
Alexandra Coyle McDonald and Alec Sheres, both Class of 2022, each earned a composite score of 36 on the standardized test, which is used for college admissions in the United States. Fewer than one percent of all test takers earn the top score, according to ACT. Roughly 1.67 million students nationwide took the test this year.
The ACT consists of tests in English, Mathematics, Reading, and Science, each scored on a scale of 1-36. A tester’s composite score is the average of the four test scores. According to ACT, a student who earns a composite score of 36 has likely mastered all of the skills and knowledge they will need to succeed in first-year college courses in the core subject areas.
“Earning a top score on the ACT is a remarkable achievement,” says ACT CEO Janet Godwin. “A student’s exceptional score of 36 will provide any college or university with ample evidence of their readiness for the academic rigors that lie ahead.”


To understand what 1 trillion to the 10th power means, it’s essential to comprehend the concept of exponentiation. When we raise a number to a certain power, we are essentially multiplying that number by itself multiple times. In this case, you want to find out what 1 trillion (1,000,000,000,000) raised to the 10th power is.
Mathematically, it can be represented as follows:
Calculating this value can be quite overwhelming, as it involves multiplying 1 trillion by itself ten times. However, modern computers and calculators can handle such large calculations with ease.
1,000,000,000,000^10 is an incredibly massive number, and it’s challenging to grasp its magnitude without the help of scientific notation or comparison.
To put this number into perspective, consider that:
1,000 is 10 to the 3rd power (10^3).
1,000,000 is 10 to the 6th power (10^6).
1,000,000,000 is 10 to the 9th power (10^9).
So, 1 trillion (1,000,000,000,000) is 10 to the 12th power (10^12).
Raising a number to the 10th power means you’re multiplying it by itself 10 times. In this case, 1 trillion to the 10th power is:
This results in an astronomical number, often expressed in scientific notation. Scientific notation allows us to represent very large or very small numbers in a more manageable form.
1 trillion to the 10th power in scientific notation is:
1 x 10^12 to the 10th power, which simplifies to:
1 x 10^120
So, 1 trillion to the 10th power is equal to 1 followed by 120 zeros:
This number is so incredibly large that it’s challenging to comprehend. It exceeds the number of atoms in the observable universe by many orders of magnitude. It’s a number that doesn’t often appear in practical calculations and is more of a mathematical concept to help us understand the power of exponentiation and the vastness of numbers.
Certainly, let’s explore further the magnitude and some context for a number as colossal as 1 trillion to the 10th power, or 10^120.
Cosmic Scale: To put this number in perspective, it’s helpful to look at it in the context of the universe. The observable universe contains an estimated 2 trillion galaxies, and each galaxy can have billions to trillions of stars. Even with such an astronomical number of stars and galaxies, 1 trillion to the 10th power (10^120) is vastly greater.
Particle Physics: At the other extreme of scale, consider the tiniest particles in the universe. A grain of sand contains roughly 1 sextillion (10^21) atoms. Even if you were to create a stack of grains of sand, each containing as many atoms as there are in our real universe, you would still fall woefully short of 10^120.
Combinatorics: The number 10^120 is also much larger than the total number of unique chess games that can ever be played. The estimated number of possible unique chess games is on the order of 10^120 itself, which demonstrates the staggering complexity of the game of chess.
Computational Limits: From a practical standpoint, modern computers, no matter how advanced, would struggle to process or store a number as enormous as 10^120. The sheer magnitude of this number is far beyond what current technology can handle.
Mathematical Utility: Numbers like 10^120 are often encountered in abstract mathematics and theoretical physics when dealing with cosmological or quantum-scale phenomena. They are used to represent the extreme boundaries of what can be conceived mathematically.
In essence, 1 trillion to the 10th power is a number that stretches the limits of human comprehension and practical utility. It serves as a mathematical concept that helps us appreciate the vastness and complexity of the universe and the mathematical structures used to describe it. While we might not encounter such numbers in our day-to-day lives, they play a crucial role in fields like cosmology, theoretical physics, and advanced mathematics, helping us explore the furthest reaches of our understanding of the universe and its underlying principles.


Does your child struggle with math? Would you welcome help to keep them inspired? Elizabeth Cousin, LEA’s math expert is here to take the stress out of math.
Math Class is a weekly class teaching concepts from Saxon’s 5/4 and 6/5 courses. Concepts will be chosen based on the student’s needs. Students will also meet individually with the mentor to set their own weekly math goals. This also includes a subscription to the award-winning online, self-directed program, Moby Max.
Math Lab is provides your student with a math mentor using their own math course. Whenever they need extra help, they contact the math mentor who sets up a private tutoring session with them. The mentor may also have the student email their weekly math goals to help motivate them to be consistent in their math studies.
Math Lab with ALEKS includes everything in Math Lab plus a 12-month subscription to ALEKS math which has math courses from elementary through calculus. Options for shorter subscriptions are available. Contact LEA for more info.
Pre-Algebra is a course for students who are preparing for algebra next year. Since we can’t cover every possible topic in only one hour per week, we will focus on a few subjects that are the most essential topics to be ready for algebra. We will cover: number properties, integers, exponents, factors and multiples, expressions and equations, and the coordinate plane. Each unit will be covered over 3 to 4 weeks so students will have a solid understanding of each topic.


Only made obsolete by the electronic variety, it was simple and cheap for the time. It also handles non-decimal measurements, like feet and inches, or pre-decimalization pounds, shillings, and pence. Addition and subtraction require different 'screens', handled by turning the instrument over, or flipping a front panel, or, later, by extended sliders and an extra lower panel. Although not always advertised (e.g. the Magic Brain Calculator mentions "add, subtract, multiply" on its front plate), procedures exist for multiplication (by repeated addition or by individual digit multiplications) and division (eg by repeated subtraction, or use of additions combined with complementary numbers).
More expensive versions have a built-in slide rule on the back.
This type of calculator was introduced by the Frenchman Troncet in 1889. The Addiator was one of the most popular calculators of this sort, and the name is often used to refer to the type generally.


Apple's Secret Calculator
The iMac’s Spotlight search is an amazing tool that lets you swiftly search your entire iMac for any media, program, or document. You can click the magnifying glass in the upper right hand corner, or just hit Command + Spacebar and the Spotlight search bar will drop down, ready for you to type any text in to begin your search. But did you know that the bar actually does basic math too? Instead of having to actually open your Calculator, just type in some basic math right into the spotlight search bar and… voila! You will see the Icon for Calculator appear, and your math problem will be solved right in the spotlight window.
One more secret - if your Mac doesn't have a number pad, you might make a few simple mistakes based on how the calculator keys appear visually. If you want to use multiplication, remember to use Shift-8 to get the star icon (*) and not the letter X. For keying in division problems, the forward slash (/) is the way to go.
Keep reading for more handy iMac tips and tricks!


- Colors of Infinity Alternative
Retraces the steps that led to the discovery of the Mandelbrot set in 1980. Dr Arthur C Clarke explains the mathematics and the basic computer operations that generate the Mandelbrot set and the associated imagery. He also demonstrates how the discovery of the set, which gave birth to the science of fractal geometry, is significant to all the sciences. The study of fractals is leading to a better understanding of chaotic systems such as weather and galaxy clusters. The fractal method of digital image compression developed by Michael Barnsley is also discussed. Includes animated images of the Mandelbrot set. Contributors include Dr Benoit Mandelbrot, Prof Stephen Hawking and Prof Ian Stewart.


Much of our time in 8th grade is spend analyzing the characteristics of linear functions. Linear functions are very useful in the “real world,” and understanding how they work is an important key to understanding any relationship between two changing quantities.
A linear equation has four parts: an independent variable, a dependent variable, slope and a y-intercept. Students are first introduced to these parts through the use of slope-intercept form.
In 8th grade, the equation y = mx + b is most often used to represent linear functions. The variable x represents an input value, and the variable y represents an output value. The variable m, represents the rate of change between the inputs and outputs, or the slope of the relationship.
We first investigate slope of a graph, by measuring the differences horizontally and vertically between two points.
We then move to finding those differences between two points by using subtraction.
Finally, we use the slope formula to calculate the slope without having to use a graph.
Don’t think we forgot about the last part of a linear function! The y-intercept is simply the coordinates of the point where the graphed line crosses the y-axis. It is represented by the b in slope-intercept form.


#3 Question: Does every single dollar of tax deductions means a $1 reduction in taxes due?
If you thought that every single dollar of your tax deductions actually equals a dollar reduction in taxes, you were wrong. The dollars you deduct only minimize your tax bill based on the amount of your actual marginal tax rate. The marginal tax bracket for you refers to the tax rate you actually pay on your last dollar of income.
For instance, if your marginal tax rate is 22 percent, and you have, let’s say, a $1,000 deduction, you have to multiply the amount of your deduction by your marginal rate in order to know exactly the amount of money you actually save (in this case the amount of money saved is $220).


Tuition for all school subjects
Howland Tutors (Corporate Members of the Tutors’ Association) was founded in 1997 and is owned by Andy and Barbara Howland.
For many years, they taught Maths (Andy) and French (Barbara) at Sir William Borlase’s Grammar School in Marlow. Earlier in their careers, Andy taught at Gillott’s (Henley) whilst Barbara taught at Chiltern Edge (Sonning Common) – both schools being Comprehensives. Borlase’s has always had a very high reputation in all school subjects at GCSE, AS and A2 Levels including in English, Modern Languages, Mathematics and Science.
Both Barbara and Andy have over thirty years’ teaching and tutoring experience.
The agency provides access to over ninety tutors in Maths, English, History, Geography, French, German, Spanish, Physics, Biology, Latin, Chemistry and General Science. Tutors are also available in Business Studies, Sociology, Psychology, Economics and Primary level subjects.
For more details check the website – http://www.howlandtutors.co.uk or call 01628 477164


ILP Statistical Analysis
Quantitative experimental data will generally need to be analysed using some form of statistics. There are a large number of different statistical tests that can be used depending upon your data, how it was collected and the types of comparisons you intend to make.
The information shown below is extracted from the "information for authors" for Nature Cell Biology submissions.
Nature Cell Biology - Guide to Authors and Referees
The description of all reported data that includes statistical testing must state the name of the statistical test used to generate error bars and P values, the number ( n ) of independent experiments underlying each data point (not replicate measures of one sample), and the actual P value for each test (not merely 'significant' or 'P < .05').
Descriptive statistics should include:
- clearly labeled measure of center (such as the mean or the median)
- clearly labeled measure of variability (such as standard deviation or range)
- Ranges are more appropriate than standard deviations or standard errors for small data sets. Standard error or confidence interval is appropriate to compare data to a control.
- Graphs should include clearly labeled error bars.
- Authors must state whether a number that follows the ± sign is a standard error (s.e.m.) or a standard deviation (s.d.).
Since for complex biological experiments the number of independent repeats of a measurement often has to be limited for practical reasons, statistical measures with a very small n are commonplace.
- Statistical measures applied to too small a sample size are not significant and they can suggest a false level of significance.
- Error bars should not be provided for n < 3. Instead, the actual individual data from each experiment should be plotted
- If n < 5 individual data should be plotted alongside an error bar.
- In cases where n is small, a justification for the use of the statistical test employed has to be provided.
It is admissible to present a single 'typical result' of n experiments.
- If n is not based on independent experiments (that is n merely represents replicates of a measurement), it may still be meaningful to present statistics, but a detailed description of the repeated measurement is required.
A basic description of n, P and the test applied should be provided in the figure legends, and a further discussion of statistical methodology should be provided in the methods section.
Authors must justify the use of a particular test and explain whether their data conform to the assumptions of the tests.
When making multiple statistical comparisons on a single data set, authors should explain how they adjusted the alpha level to avoid an inflated Type I error rate, or they should select statistical tests appropriate for multiple groups (such as ANOVA rather than a series of t-tests).
Many statistical tests require that the data be approximately normally distributed; when using these tests, authors should explain how they tested their data for normality. If the data do not meet the assumptions of the test, then a non-parametric alternative should be used instead.
Small Sample Size
When the sample size is small (less than about 10), authors should use tests appropriate to small samples or justify their use of large-sample tests.
Text extract source: Nature Cell Biology - Guide to Authors and Referees


Title: Adjacency Labelling of Planar Graphs
Adjacency labelling schemes, which have been studied since the 1980s, ask for short labels for n-vertex graphs G such that the labels of two vertices u and v are sufficient to determine (quickly) if uv is an edge of G. One of the long-standing problems in the area was the optimal length of labels for planar graphs. The problem is closely related to the size of the smallest universal graph for all n-vertex planar graphs. In this talk I will show how we resolved this problem (up to lower order terms) with the help of a new graph theoretic tool: a product-structure theorem for planar graphs. This new tool and our result are applicable not only to planar graphs but also to bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k-planar graphs.
Title: Algorithms for Explainable Clustering
An important topic in current machine learning research is to explain and/or interpret how models actually make their decisions. Motivated by this, Moshkovitz, Dasgupta, Rashtchian, and Frost recently formalized the problem of explainable clustering. A k-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the k leaves corresponds to a cluster.
In this talk, we see an algorithm that outputs an explainable clustering that loses at most a factor of O(log2 k) compared to an optimal (not necessarily explainable) clustering for the k-medians objective, and a factor of O(k log2 k) for the k-means objective. This improves over the previous best upper bounds of O(k) and O(k2), respectively, and nearly matches the previous Ω(log k) lower bound for k-medians and our new Ω(k) lower bound for k-means. Moreover, the algorithm is remarkably simple and, given an initial not necessarily explainable clustering, it is oblivious to the data points and runs in time O(dk log2 k), independent of the number of data points n.
This is joint work with Buddhima Gamlath, Xinrui Jia, and Adam Polak.


I spent a good chunk of Saturday and Sunday ignoring the NCAAs and spent my time on something only slightly less frivolous: crafting an algorithm to beat the awesomely addicting game 2048. Spoilers below
— byuhas (@byuhas) March 23, 2014
Wednesday update: I’ve updated this post to reflect recent improvements in the algorithm.
I learned of 2048 from this xkcd cartoon. I played a few times, and then on Thursday it struck me that the game is similar to the Towers of Hanoi problem. That night I spent way too many hours crafting the outlines of a winning strategy, proving to myself that I could beat the game. The basic idea is to align the tiles in a snakelike pattern from highest to lowest (from the upper-left, down, then over one, then up, etc) as shown below:
It’s especially crucial to keep your highest-numbered tile in a corner (e.g., the upper-left).
The computer scientist in me wasn’t satisfied with solving the puzzle by hand. Rather, I felt the need to develop an algorithm, thus allowing a computer to win the game for me. Overall, I did an adequate job — the computer wins the game, but I notice that it still makes small mistakes. You can check it out here. (On my personal machine, each move takes about a second, which provides a pleasant viewing experience — hopefully the same is true for you.)
Some notes on the algorithm:
- It looks four moves ahead and does not prune the game tree. (Speed options: Blazing = 3 moves ahead; Plodding = 5 moves ahead.)
- Grids are scored on:
- Better score if tiles are in exactly the right place for the “perfect” grid (accounting for gaps in the numbers available).
- Even if the tiles on the “snake path” aren’t perfect, it’s better if they are monotonic
- Points for having at most two of the same tile number.
- Points for having more blank spaces.
- Points for having a higher maximum tile number.
- End game: once there are enough points to win on the board: the snake path and monotonicity become less important, adjacent pairs are worth works, and blank spaces are worth more.
- I’m not sure what percent of the time it wins, especially since there are three speeds. It does make mistakes — likely a combination of deterministic new-number pop up, poor coding of a grid-score function, and not looking that far down the tree.
- It take about 1000 moves to win.
- Aside: finding end-game bugs is annoying. I had a build methods to test arbitrary boards. Cost of doing business I guess.
Comments welcome. I respond fairly quickly to tweets.


Steven H Strogatz
Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. He holds a joint appointment in the College of Arts and Sciences (Department of Mathematics) and the College of Engineering (Sibley School of Mechanical and Aerospace Engineering).
After receiving his bachelor's degree in mathematics from Princeton in 1980, Strogatz spent two years as a Marshall Scholar at Trinity College, Cambridge. He did his doctoral work in applied mathematics at Harvard, followed by a National Science Foundation postdoctoral fellowship at Harvard and Boston University. From 1989 to 1994, Strogatz taught in the Department of Mathematics at MIT, and then joined the Cornell faculty in 1994.
Strogatz works in the areas of nonlinear dynamics and complex systems, often on topics inspired by the curiosities of everyday life. He is perhaps best known for his 1998 Nature paper on "small-world" networks, co-authored with his former student Duncan Watts. As one measure of its impact it was the most highly cited paper about networks between 1998 and 2008, across all scientific disciplines, as well as the sixth most highly cited paper--on any topic--in all of physics.
He has received numerous awards for his research, teaching, and public service, including: a Presidential Young Investigator Award from the National Science Foundation (1990); MIT's highest teaching prize, the E. M. Baker Award for Excellence in Undergraduate Teaching (1991); the J.P. and Mary Barger '50 Teaching Award (1997), the Robert '55 and Vanne '57 Cowie Teaching Award (2001), the Tau Beta Pi Teaching Award (2006), and the Swanson Teaching Award (2009), all from Cornell's College of Engineering; and the Communications Award from the Joint Policy Board for Mathematics (2007), a lifetime achievement award for the communication of mathematics to the general public. In 2009 he was elected a Fellow of the Society for Industrial and Applied Mathematics for his "investigations of small-world networks and coupled oscillators and for outstanding science communication." In 2012 he was elected to the American Academy of Arts and Sciences.
Professor Strogatz has been lauded for his exceptional ability as a communicator. He received the Communications Award--a lifetime achievement award for the communication of mathematics to the general public, awarded by the four major American math societies--in 2007. He has also filmed a series of 24 lectures on Chaos for the Teaching Company's Great Courses series. Professor Strogatz has been a frequent guest on National Public Radio's RadioLab. He wrote a weekly column on mathematics for The New York Times in spring 2010 and fall 2012.
His books include Nonlinear Dynamics and Chaos (Perseus, 1994), the most widely used textbook in the field; the bestselling science book Sync (Hyperion, 2003), which was chosen as a Best Book of 2003 by Discover magazine and has been translated into six languages; and The Calculus of Friendship (Princeton University Press, 2009), which chronicles the story of his remarkable 30-year correspondence with his high school calculus teacher, and which reviewers have called "an intimate view of mentorship" (Nature), "wonderful" (American Scientist), "marvelous" (MAA Review), and "a genuine tearjerker" (Bookslut). His latest book, The Joy of x, appeared in October 2012.
Communication of Mathematics
- 2010. "Redrawing the map of Great Britain from a network of human interactions." PLoSONE 5 (12): e14248-1 -- e14248-6. .
- 2014. "Phase diagram for the Kuramoto model with van Hemmen interactions." Physical Review E 89: 012904. .
- 2012. "Education of a Model Student." PNAS 109: 1-6. .
- 2012. "Encouraging moderation: Clues from a simple model of ideological conflict." Physical Review Letters 109: 118702. .
- 2011. "Continuous-time model of structural balance." PNAS 108 (5): 1771-1776. .
Selected Awards and Honors
- Rouse Ball lecturer (University of Cambridge) 2009
- 2013 AAAS Public Engagement with Science Award 2013
- Elected member (American Academy of Arts and Sciences) 2012
- Department of Mathematics Teaching Award (Cornell) 2012
- Gerald and Judith Porter Public Lecturer (American Mathematical Society) 2010
- BA (Mathematics), Princeton University, 1980
- BA (Mathematics), Cambridge University, 1982
- Ph D (Applied Mathematics), Harvard University, 1986
- MA (Mathematics), Cambridge University, 1986


Numbers play a vital role in our day-to-day lives.
From the day we make our appearance known, right down to the minute, we enter into the visible and invisible realm of numbers.
Before we can count, it is being done for us. The date and time of our birth is recorded. We are whisked off to check vital statistics. Our length is measured, our weight is recorded and we are given an APGAR score.
Two eyes, check! Ten toes, yes! Ten fingers, great!
Later on in life, we learn to not only count on our fingers, but we learn to number our friends. We are very blessed if we have a handful of best friends we can count on.
Numbers Are Everywhere
Numbers permeate every area of life. There isn’t anything that does not involve them; everything from the microcosm to the macrocosm, and everything in between. They even encompasses The Secret of Life, which I will discuss below.
Even the hairs on our head are numbered. (Matthew 10:30)
We use numbers to:
- add, subtract, multiply and divide.
- chart progress and productivity.
- detect patterns.
- entice customers. Today only… two for one.
- find out the truth, to obtain proof. Deductive reasoning tells when the numbers don’t add up.
- keep track of time, to schedule appointments. Is the train late?
- measure area, distance, growth, intelligence, sizes of population, time, volume, and weight.
- provide excellence in service.
- record important events.
- report a crime. How many were involved? How long ago did this happen? How tall? What weight? Did you get a license number?
- solve problems. How much do they cost? How many are needed?
We could do none of these things without numbers.
Curious, Fascinating Numbers and The Secret Of Life
Some cultures treat numbers with a certain reverence; that they are lucky and control destiny.
Such is the popularity of such things as Fung Shui, horse races, and lottery tickets.
Some people treat numbers with superstition. A good example of this is the number thirteen.
Some think it is good luck, and others a curse.
The number sixteen is interesting. In the Old Testament, there were sixteen prophets.
In the New Testament, you will find sixteen apostles and evangelists.
The number 153 is also a curiosity. If you cube each number, it returns to itself.
1 + 125 + 27 = 153
The Beauty, Goodness & Truth of Numbers
There is a purity to numbers. They are the voice of truth. They never lie, even if they get fudged.
To date, no one has been able to figure out the determinant pattern of prime numbers.
Prime numbers are positive, non-zero numbers that have just two factors; the number one and the number itself.
They certainly have held my imagination captive for many years now. I have written high-level encryption programs, and others involving prime numbers. One such program treats prime numbers as linguistic elements.
Here is an example:
The phrase, “The Secret of Life” consists of eleven elements, each of which is a prime number.
When the elements are joined together, they produce six hundred eleven thousand nine hundred fifty one.
This is also a prime number. to see for yourself, check here.
But wait! Aren’t these eleven elements more than two factors?
No. The elements are added, whereas factors are multiplied.
In providing one more example, The Secret Of Life shows up in a different way.
The gathering of the eleven elements is a picture of unity in diversity.
Collectively, they are equally important.
For example, is it possible that three can be one? …. or any number of things be one?
When you add them together, one plus one plus one equals three.
But, when you multiply them, one times one times any number of ones will always equal one.
Perhaps, one day, the poet-philosophers will write about the day that humanity became one.
Teach Us To Number Our Days
The Bible says, “Teach us to number our days, that we may gain a heart of wisdom.”
When we are small, naïve, and our life experience matches our size, the number of our days
seems infinitely large. We have a small number of memories, and a lifetime to unearth beauty.
When we are older, wiser, and perhaps becoming small again, the number of our days seems to be
in short supply. The brevity of life looms larger. We have many memories, some which are buried and long forgotten.
If we are wise, we also learn to make the most of our days.
You Have A Voice… Let’s Hear It
Do you have a favorite number?
Do you enjoy the elegance and beauty of numbers?
Please contribute you thoughts here.
I appreciate each and every one. Thank you kindly! 🙂


In order to calculate the Customer Retention Rater, there are three pieces of information we need to use.
- E is the number of customer at the end of a period;
- N is the number of new customers acquired during that period;
- S is the number of customers at the start of that period.
We are interested in the number of customers remaining at the end of the period without counting the number of new customers acquired. Remaining customers can be calculated by subtracting N from E. To calculate the percentage, we divide that number by the total number of customers at the start and multiply by 100.
So, in formula, we have: ((E-N)/S)*100


Please use this identifier to cite or link to this item:
|Title:||Algorithm to determine the intersection curves between bezier surfaces by the solution of multivariable polynomial system and the differential marching method|
multivariable polynomial systems
|Publisher:||The Brazilian Society of Mechanical Sciences|
|Description:||The determination of the intersection curve between Bézier Surfaces may be seen as the composition of two separated problems: determining initial points and tracing the intersection curve from these points. The Bézier Surface is represented by a parametric function (polynomial with two variables) that maps a point in the tridimensional space from the bidimensional parametric space. In this article, it is proposed an algorithm to determine the initial points of the intersection curve of Bézier Surfaces, based on the solution of polynomial systems with the Projected Polyhedral Method, followed by a method for tracing the intersection curves (Marching Method with differential equations). In order to allow the use of the Projected Polyhedral Method, the equations of the system must be represented in terms of the Bernstein basis, and towards this goal it is proposed a robust and reliable algorithm to exactly transform a multivariable polynomial in terms of power basis to a polynomial written in terms of Bernstein basis .|
|Appears in Collections:||Technology and Engineering|
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Tim has been a tutor and mentor to our middle school son for nearly a year now. Our son’s attitude, work ethic and grades have improved beyond our expectations. Tim has a way of connecting and relating that facilitates a great learning experience. I thank our lucky stars we found him and highly recommend him.
Mr. Freeburg was my sophomore geometry teacher. Before this my toughest classes have always been math- I just couldn’t grasp certain subjects that are essential to moving forwards into Algebra 2 and Pre-Calc. When the SAT rolled around, I did terribly on the math section and I decided to give Mr. Freeburg a call. He helped me to finally grasp those concepts that had given me trouble and I found my confidence in math growing alongside my understanding. My grade in Physics even shot up and I finished the year off with A’s in both subjects. Not to mention my SAT score improved ? . I greatly appreciate everything that Mr. Freeburg has done for me and I highly recommend him to anyone who needs help!
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Vector graphics can be drawn directly in LaTeX using specialised packages. This improves the quality of simple designs and allows you to add mathematics to an image. The link below
provides examples of drawing sketches, graphs and charts directly using LaTeX code. It has been written by T. Morales de Luna of Universidad de Córdoba.
Useful Vector Graphic Tools for LaTeX Users
Feynman diagrams can be drawn with the TikZ package or the feynMF package. TikZ is a more general package that can draw any basic shape. FeynMF is specifically designed for Feyman diagrams. A very minimal introduction to TikZ written by Jacques Crémer of Université de Toulouse gives you the basics of TikZ but this fantastic tex document by Flip Tanedo (currently at University of California) gives you multiple examples of using TikZ to create Feynman diagrams. If you wish to use feynMF then Drawing Feynman Diagrams with LATEX and METAFONT written by Thorsten Ohl, Universität Würzburg will provide you with many practical examples.
There are many mathematical symbols and it is impossible to remember all of them. Therefore a cheatsheet is a useful reference sheet that allows you to look up the LaTeX code you need to create each symbol. There are also Apps available for smart devices that allow you to draw the symbol that you want and the App will find the LaTeX code that you need. One example of these from the Google Play Store is Dextify.


• Decision Dilemmas: Each course section is introduced with a real-world business vignette that presents a dilemma and related managerial or statistical questions. Solutions to these questions require the use of techniques presented in the section. A Decision Dilemma Solved feature concludes each section, giving students the opportunity to answer and discuss each question presented at the beginning of the section.
• Thinking Critically About Statistics in Business Today Exercises: Each course section features one or several of these exercises that give real-life examples of how the statistics presented in the section apply in the business world today.
• EXPANDED Databases: Twenty databases representing several industries including banking, consumer spending, energy, environmental, finance, manufacturing, healthcare, market research, retailing, stocks and more provide additional opportunities for students to apply the statistics presented in each chapter.
• NEW Big Data Case: Using data from the American Hospital Association, each chapter contains an activity asking students to perform several tasks using variables, samples, and data.
• NEW Visualizing Time-Series Data Section: helps students use historical data with measures taken over time to predict what might happen in the future.
• Ethical Considerations: This feature in each course section integrates the topic of ethics with applications of business statistics.
• Tree Taxonomy Diagrams: These diagrams illustrate the connection between topics and techniques and the ability to see the big picture of inferential statistics.
• Section Reorganization Options: This course was designed to allow for both one- and two-semester coverage.
• 900+ Practice Problems: A treasury of practice problems are available in this course.
1 Introduction to Statistics and Business Analytics
2 Visualizing Data with Charts and Graphs
3 Descriptive Statistics
5 Discrete Distributions
6 Continuous Distributions
7 Sampling and Sampling Distributions
8 Statistical Inference: Estimation for Single Populations
9 Statistical Inference: Hypothesis Testing for Single Populations
10 Statistical Inferences About Two Populations
11 Analysis of Variance and Design of Experiments
12 Simple Regression Analysis and Correlation
13 Multiple Regression Analysis
14 Building Multiple Regression Models
15 Time-Series Forecasting and Index Numbers
16 Analysis of Categorical Data
17 Nonparametric Statistics
18 Statistical Quality Control
19 Decision Analysis
<한빛아카데미> 도서는 한빛 홈페이지에서 더 이상 판매를 하지 않습니다. 도서 구입은 인터넷 서점을 이용하시기 바랍니다. 양해바랍니다.


Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers.
The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful.
Who is this class for: This course is primarily aimed at the Master of Science students of Mathematics department. Students are expected to have knowledge of one-semester courses of advanced calculus and linear algebra.
ENROLL IN COURSE


Dr. Laurel Travis received a both her MBA and her Ph.D. in Business-Quantitative Analysis from the University of Wisconsin-Madison. She obtained a bachelor’s degree from the University of Michigan in Mathematics. Since that time, she has taught at 7 universities in the United States, Canada, and most recently, China. She has taught at both the undergraduate and graduate levels, in subjects including management science, management information systems, engineering, mathematics, and computer science. She has published a textbook entitled Simulation for Decision Making, with Arne Thesen.
Laurel's research and teaching interests focus on international programs, gerrymandering, discrete event simulation, and applied optimization. Laurel’s recent volunteer activities include caring for the Hokie horses, and teaching STEM on a schooner in the Great Lakes.


We are kicking 3rd math off to a great start this year! Below are several math websites that your child can access for extra practice at home. These websites include fact fluency, place value skills, word problems, and more! The list will be updated as we cover more topics. We look forward to seeing each and every child grow to love math and become confident and successful in their math skills.
Topics we have already covered
Please allow your son/daughter to practice addition, subtraction, counting money, and now multiplication every day. We are going to have a contest at the end of the year where students can earn an ice-cream cup depending on their multiplication skills. Every week we are going to practice one fact. We’ve already covered the facts for 2s, 10s and for 5s. These are the foundations for learning to deduce the rest of the facts. We do not want them to memorize the facts but to understand how multiplication work and derive the products from what they already know. Memorization will eventually occur as a consequence of repeated use. Thank you for your support at home.
Perimeter class by Math Antics
This video explains what is perimeter and how to determine it.Area class by Math Antics
What is area and how to determine it. Make sure you stop at 5 minutes. We are interested in determining area of squares and rectangles only.Fun to learn numbers
Fun to Learn Numbers is for kids to practice and reinforce their concepts of number counting, sequencing and picture addition.Place Value class by Math Antics
Basic Math Videos made by a very clever and fun teacherPlace Value Games for 3rd Graders
This website provides several activities for your child to revisit and extend place value understanding.Spell the Number
A fun way to practice spelling numbers from standard form to word form and vice-versa using a check metaphor.Writing numbers in word form
Using this website, students practice reading numbers in word form and finding the same number in standard form and vice-versa.Expanded and word form
Here students will practice converting between expanded and word formsFruit Splat
This website will allow students to practice in a fun way how to convert numbers from expanded formABCYa
ABCYa provides a huge variety of short games involving math skills for your child to practice while having fun!Prodigy
Practice your Math skills at home with this awesome website!
We will be creating a Prodigy class for the 2019-2020 school year soon.
Johnnie's Math Page
Over 1000 math activities, tools and games are available to your child!
Johnnie's Math Page is the site to find fun math for kids. Over one-thousand math learning and teaching resources have been categorized and set out for you.Math Playground
This website provides games with advanced 3rd grade math concepts. Recommended for advanced students now or for all students after we’ve covered addition, subtraction, multiplication, and division.


Introduction This page or section is a stub. You can help the wiki by accurately contributing to it. Floating point numbers are a way to represent real numbers inside computer memory which usually operates with binary digits. As opposed to fixed point numbers which have a fixed number of digits before and after the decimal markfloating point numbers can be considered to have a certain amount of significant digits or 'accurate leading numbers' that we consider to carry an accurate approximation to some value.
The DX—Core2 microprocessors contain their own built-in arithmetic coprocessors. Be aware that some of the cloned microprocessors from IBM and Cyrix did not contain arithmetic coprocessors.
The instruction sets and programming for all devices are almost identical; the main difference is that each coprocessor is designed to function with a different Intel microprocessor. This chapter provides detail on the entire family of arithmetic coprocessors.
Because the coprocessor is a part of the DX—Core2, and because these microprocessors are commonplace, many programs now require or at least benefit from a coprocessor. The family of coprocessors, which is labeled the 80X87, is able to multiply, divide, add, subtract, find the square root, and calculate the partial tangent, partial arctangent, and logarithms.
Data types include, and bit signed integers; l8-digit BCD data; and, and bit floating-point numbers. With the improved Pentium coprocessor, operations execute about five times faster than those performed by the microprocessor with an equal clock frequency.
Note that the Pentium can often execute a coprocessor instruction and two integer instructions simultaneously. The Pentium Pro through Core2 coprocessors are similar in performance to the Pentium coprocessor, except that a few new instructions have been added: The multimedia extensions MMX to the Pentium—Core2 are instructions that share the arithmetic coprocessor register set.
The MMX extension is a special internal processor designed to execute integer instructions at high-speed for external multimedia devices. For this reason, the MMX instruction set and specifications have been placed in this chapter.
See Table 14—1 for a listing of all Intel microprocessors and their companion coprocessors. These data types include signed integer, BCD, and floating-point. Each has a specific use in a system, and many systems require all three data types. In order to accomplish any such modification, the instruction set and some basic programming concepts are required, which are presented in this chapter.
Signed Integers The signed integers used with the coprocessor are the same as those described in Chapter 1. When used with the arithmetic coprocessor, signed integers are worddoubleword integeror bits quadword integer wide. The long integer is new to the coprocessor and is not described in Chapter 1, but the principles are the same.
Conversion between decimal and signed integer format is handled in exactly the same manner as for the signed integers described in Chapter 1. Integer data types are found in some applications that use the arithmetic coprocessor.
See Figure 14—1, which shows these three forms of signed integer data. Data are stored in memory using the same assembler directives described and used in earlier chapters. Example 14—1 shows how several different sizes of signed integers are defined for use by the assembler and arithmetic coprocessor.
Each number is stored as an digit packed integer in nine bytes of memory as two digits per byte. The tenth byte contains only a sign-bit for the digit signed BCD number. Figure 14—2 shows the format of the BCD number used with the arithmetic coprocessor.
This form is rarely used because it is unique to the Intel coprocessor. Floating-Point Floating-point numbers are often called real numbers because they hold signed integers, fractions, and mixed numbers.
A floating-point number has three parts: Floating-point numbers are written in scientific binary notation. The Intel family of arithmetic coprocessors supports three types of floating-point numbers: See Figure 14—3 for the three forms of the floating-point number.
Please note that the single form is also called a single-precision number and the double form is called a double-precision number. Sometimes the bit temporary form is called an extended-precision number. The floating-point numbers and the operations performed by the arithmetic coprocessor conform to the IEEE standard, as adopted by all major personal computer software producers.
This includes Microsoft, which in stopped supporting the Microsoft floating-point format and also the ANSI floating-point standard that is popular in some mainframe computer systems.
The float is a bit version, double is the bit version, and decimal is a special version developed for Visual studio that develops a very accurate floating-point number for use in banking transactions or anything else that requires a high degree of precision.
The decimal variable form is new to Visual Studio and Converting to Floating-Point Form.Nov 08, · Lets an analysis of the floating point coprocessors in the design of a microprocessor in computer science start right off with a controversial claim: Forth is the hackers programming language Coding in Forth is a little bit like writing assembly.
The term floating point is derived from the fact that there is no fixed number of digits before and after the decimal point; that is, the decimal point can float. There are also representations in which the number of digits before and after the decimal point is set, called fixed-point representations.
Floating point number, normalization, exceptions, latency, etc. 1. INTRODUCTION An arithmetic circuit which performs digital arithmetic operations has many applications in digital coprocessors, application specific circuits, etc. Because of the advancements in the . For example, a math coprocessor performs mathematical computations, particularly floating-point operations.
Math coprocessors are also called numeric and floating-point coprocessors. Most computers come with a floating-point coprocessors built in. Note, however, that the program itself must be written to take advantage of the coprocessor.
smart-memory coprocessors to a microprocessor. This Introduction The Data-IntensiVe Architecture (DIVA) project  is building a workstation-class system using (Integer to floating-point) instruction, the fraction is first converted to sign-magnitude format by conditionally inverting if the sign is negative.
Then the result is shifted. Introduction to Many Integrated Core (MIC) Coprocessors on Stampede. Intel Xeon Phi™ SE10P coprocessors to change the power/performance curves of supercomputing • Over 70% provided by Xeon Phi moderate floating point throughput –2x E on Stampede: Tflop/s, W.


Master’s in Data Science
Master’s in Data Science Overview
From cybersecurity to health and healthcare, from marketing analytics to risk analysis, computing and data science are pervasive. They are transforming discovery and innovation in our economy, our society, our culture, our habits—and how we best prepare students for the future.
Data science is an interdisciplinary field in which a host of mathematical and computational techniques are used to extract information from data, and subsequently used to make informed decisions. Career opportunities are vast and growing everyday in this exciting, applied field of study.
Lehigh's Master's of Science in Data Science allows students from a wide range of backgrounds to gain the qualifications necessary to tap into the wealth of data science jobs, many of which require an advanced degree. The U.S. Bureau of Labor Statistics projects jobs in the field to grow more than 30 percent between 2020 and 2030. Lehigh's Data Science program provides access to the skills and tools of data science to people who want to develop that skill set, even if they don't have a background in computer science or statistics. Prospective students need a strong foundation in math and some basic training in computer science, but no prior data science coursework is required. Classes will be taught in a hybrid (both in-person and remote) format.
Master’s in Data Science Contact
Blake Plimpton | Graduate Recruiting Manager | email@example.com | (610) 758-5483


14 October 2020 10:00am–11:30am
We have secured funding through the Inclusive Learning Project in the South East for a four week online course hosted by the National Numeracy Project. This is a project which aims to advance maths in the workplace; there is absolutely no maths tutoring/ training or learning involved. The purpose of the sessions are to understand anxiety around and towards maths, looking at ways you can help members to overcome it (if they ask).
Within the UK 49% of adults are operating with a maths level of that expected of primary school children and we can help tackle this. The National Numeracy Project understand that as soon as you start talking maths people start to switch off, they propose a much more subtle approach to advancing maths learning in the workplace and as part of that you will learn how to run a successful project.
This course is open to Union Learning Reps and Lifelong Learning Co-ordinators.
How to attend
Simply email email@example.com with your membership number and the title of the event. Spaces are limited, so apply today!


|Posted by niatm on September 15, 2016 at 8:45 AM|
Monday, October 23, 2017
"Uniting Geometrical Constructions and Proofs" (Middle/HS)
Presenter: Mr. Matthew Foster, Waukegan Public Schools
As an architect, it gives me goose bumps to see geometric constructions that are so darned elegant. As a math teacher, I get them when students develop elegant proofs. Geeky, right? Can my 2 loves be combined from Day 1? Yes! Bring a compass (or use mine) & straightedge and let’s make connections!
Tuesday, December 5, 2017
"It's Not Magic - It's Algebra!" (Grades 6-10)
Presenter: Mr. David Spangler., Math Consultant, McGraw-Hill Education
Come away with a collection of motivational activities, card tricks and puzzlers. Your students will want to know WHY they work - and you will oblige by exploring the powerful role variables play in their justification. Students will thus discover how unknowns take the unknowns out of algebra!
Wednesday, February 21, 2018
"Theater and Improv for the Math Classroom" (K-12)
Presenters: Mr Steve Starr, Math Consultant, formerly Lake View HS, CPS
Math teacher Steve Starr explores theater games and improv exercises to enliven your classroom and help students connect with each other as active learners.
Thursday, April 19, 2018
"Create Your Own Khan Academy Style Video Library for Your Classes" (Grades 8-16)
Presenter: Dr. Angela Thompson, Governors State University and Ms. Michelle Hale, Aqsa School, Chicago
As a teacher, the ability to make your own videos on mathematical concepts is a powerful tool for student learning. In this session, we will demonstrate how easy it is to make Khan Academy style videos using a tablet and explain why your students need it.


Infinity puzzles are jigsaw puzzles with no beginning and no end. They have no fixed shape, no starting point, no edges, and can be assembled in thousands of different ways.
The “Infinity Puzzle” is based on the Klein bottle, a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.
The puzzle, created by Nervous System, is configured using science, math, and lasers. “Our infinity puzzles build on that tradition with a new mathematical twist that would be almost impossible with hand cutting, a puzzle that tiles in every direction,” the company wrote on their blog.
“The intricate branching shapes of our puzzle pieces emerge from a simulation of crystal growth and are lasercut from plywood.”
If you’re interested in purchasing this puzzle you can do so here. Be sure to give this post a thumbs up and a share with your friends on Facebook before you go.


Dusk years and earth years are very much not the same. Earth has 365(ish) days per year, and 24 hours per day. Dusk has 27 hours per day, and 243 days per year (both decimal, it's 30 and 300 days respectively in nonary). Further complicating matters, Dusk's most common number structure is nonary, so when they say "the year is 1120AV", they really mean the year is 828, and those 828 years were only 6,561 hours long compared to our 8,760 (so in reality, it's been ~620 Earth years since 0AV). The following is a calculator to convert ages between the two timekeeping systems.
For more information, see Time and Timekeeping.
This calculator assumes the current year is 1120AV on Dusk.


Mathematica Link for Microsoft Excel
There are dozens of useful features in the completely reworked Mathematica Link for
Microsoft Excel. Most of the limitations of the earlier version have been effaced, and
the user interface is much more developed. Among the items at the top of user's
wish list, Mathematica graphics can now be generated inside Excel. Arrays of any
type can be sent and received. Calculations can be interrupted, messages can be received,
and a terminal window for quick communication with the kernel is provided. An easy-to-use Mathematica
Wizard guides users through the process of entering Mathematica functions into a
spreadsheet. For Mathematica programmers, this simply means that their code has the
potential of being used by more than just the Mathematica literate. This session
will provide a brief tour of the new product and illustrate examples of functions written
in Mathematica that are conveniently used inside Excel.
More information on Mathematica Link for
Microsoft Excel is available in the Wolfram Research Store.


Recently I had to implement a median filter algorithm, I found Redis very powerful to accomplish this! A very simple solution and scalable.
As described in Wikipedia, Median filter works in this way: given a signal, output sample is the median of last N input samples, where N can be any positive integer number. Higher is N, more aggressive will be your filter.
We need a store for last samples, in a FIFO way, pushing a new one will drop an older one. Redis provides lists, which are perfect for this scope. Also we need to sort these samples in numerical order, to get every time we want the median value. Redis provides a SORT command, which does this job.
I implemented these two primitives:
add_sample(value)- called every time to add a new sample
get_median()- to get the median value
For the former we need to call two Redis commands:
MULTI LPUSH <yourkey> <value> LTRIM 0 <N>-1 EXEC
LPUSH will simply add a new sample to the list,
LTRIM ensures that at least
N elements will be stored, no more.
To get median value we need to call
SORT and then calculate median. SORT returns
all values on that list, sorted by numerical order. I have preferred
to use a script, so Redis avoids returning non useful data to clients:
local list_key = KEYS -- Key of samples list -- Sort values local sorted_values = redis.call("SORT", list_key) local size = #sorted_values local median = 0.0 -- Calculate median if size % 2 == 0 then median = (sorted_values[size/2]+sorted_values[size/2+1]) / 2 else median = sorted_values[(size+1)/2] end -- Use tostring because median value may be floating point return tostring(median)
Calling this script it’s easy:
EVAL <script> 1 <yourkey>
Copyright © 2014-2021 Luca Marturana. License.


Differential Equations Notes & Study Guide
This semester I am taking differential Equations and this is a study guide/cheatsheet I’m writing to help me better understand my notes and to help anyone else understand the topic who maybe in similar boat.
A First Course In DIFFERENTIAL EQUATIONS with Modeling Applications (11th Edition)
Florida Polytechnic University
Note: I will try to not use textbook language wherever possible but I will include Book definition in some form to allow me to elaborate on what the book definition means.
Calc 2 skills such as
- Logarithms and their properties
- Complete the square
- Partial Fraction decomposition
Table of contents:
ii. Exam 1
iii. Exam 1 Summary
iv. Exam 2
v. Exam 2 Summary
vi. Exam 3
vii. Exam 3 Summary
viii. Final Exam
What is a Differential Equation?
“An equation containing the derivatives of one or more unknown functions ( or dependent variables), with respect to one or more independent variables, is said to be a differential equation (DE).”
The video above goes into detail about what this definition means but basically:
Differential Equations are equations written to express real life problems where things are changing and with ‘solutions’ to these equations being equations themselves. We are using math to represent real life scenarios, usually meaning a rate of change.
ie. beats per minute (bpm) , meters per second (m/s), or anything that isn’t exactly a finite value but rather a changing one.
Now to the goal in this class is to find the original function that has a derivative of the one in the equation you are given.
That may sound complex but this class is usually taken after Calculus 2 and that means Im assuming you have a simple understanding of Derivatives and integrals.
d/dx (x²) = 2x
and your job is to find:
integral of (2x)
which is x²+c
Where your answer is still a function but that’s very basic calculus, I will go back and write a Calculus 2 and 3 Article later but for now thats all you need to know.
Types of Differential Equations
Now the equations you’re given in differential equations problems will involve 1 or more derivatives of many unknown functions as defined below as ODE’s but eventually you will be doing Differential Equations with partial derivatives (PDE’s) from Calc 3 (More on this later)
- If a differential equation contains only ordinary derivatives of one or more unknown functions with respect to a single independent variable, it is said to be an ordinary differential equation (ODE)
2. An equation involving partial derivatives of one or more unknown functions of two or more independent variables is called a partial differential equation (PDE)
Initial Value Problems:
“ An initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.”
To Solve the above:
y’ is equal to 𝑑𝑦/𝑑𝑥 in this case so:
Then we move the y+1 and 𝑑𝑥 around
Then Integrate both sides to get
Now to remove the ln we need to make each side the base of e^x
hint: e^[ln(y+1)] is just y+1
(𝑒^c) is still just C and we move the +1 to get:
𝑦= 𝑐𝑒^x -1
to arrive at the general solution of:
y = 𝑐𝑒^𝑥 -1
but the could be:
1𝑒^𝑥-1, 2𝑒^𝑥-1, 3𝑒^𝑥-1 etc.
The C represents a constant that we don’t know yet,
But we were given some initial condition in the problem y(0) = 5, then we can now use this to solve for our C value by plugging it in to our
5 = C𝑒⁰-1
5+1 = C𝑒⁰
5+1 = C(1)
and Then we get
which means C = 1
And this means we now have a single function answer of:
y = 6𝑒^𝑥 -1
as our solution, rather than a family of functions as a solution (The general solution)
We didn’t know that without knowing the initial condition but thats how we would solve for the C value given an initial condition but otherwise we just look for the general solution.
And thats only with there being one derivative which means the equation is of the first order.
First order differential equations:
The number of derivatives you have relates to the Order of the equation (equations with second derivates are said to be of the second order, equations with third derivative are said to be of the third order, etc)
for every extra derivative in your problem, you will get an extra C value which can be denoted
x being the order of derivative taken for that value
These Values are necessary for accuracy but eventually can be combined later so just make sure you never forget your +C
We will go into detailed examples but for now this is the basic theory behind the equations to help as it maybe a little tricky or obscure at first but we’ll get there.
Exam 1 :
Sections 1.1 to 4.2
Our professor is testing us on Chapter 1 through chapter 3 section 1 (3.1)
Here’s the notes we’ve covered of the chapters covered (not skipped)
This first exam will be on ODE’s of any kind using the different techniques covered so I will go through each of the techniques and when they would be used to solve a ODE
To solve Any Differential equation, you must first start by deciding on which technique is most suitable for the equation in question.
The most effective way solve ODE’s is through this list of techniques, checking to see if the equation can be solved by each method in this order.
Note: Most problems tend to be solvable through multiple techniques but the optimal technique will make your solution much easier which makes it so important to pick the correct and best method.
The later techniques above often will turn your original equation into an equation solvable by the earlier methods (ex. A Bernoulli can sometimes create a separable equation which are much easier to solve)
A differential equation is said to be separable if the variables can be separated. A separable equation is one that can be written in the form dy/dx. Once this is done, all that is needed to solve the equation is to integrate both sides. Our example in the introduction is considered a separable equation as with the one below:
- If you can split of the X and Y terms to be on opposite side of the equal sign, then it is a separable equation
- Once separated, take integral of both sides
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written f(x,y)dy=g(x,y)dx, where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form dx/x=h(u)du, which is easy to solve by integration of the two members.
- Every term must be of the same degree
A substitution problem is essentially a problem where you do a u-substitution for portion of the problem to make the problem easier to manipulate or solve.
Usually look for a term that when replaced with a u, makes the problem simpler
dy/dx =( some term)² could be u = some term
and hopefully you’d get
u = some term
dy/dx = du/dx + d/dx( some term)
You then substitute in your u term and new dy/dx term
and solve accordingly
- Can be made separable
4. Linear (y or x)
To start a linear equation, you must first write it in standard form
y’ P(x) + y = Q(x)
Once this is done, the integrating factor is
I(x) = e^integral of P(x)dx
And then the general solution is now
y = 1/I(x)[integral of I(x)Q(x) dx + C]
5. Bernoulli (y or x)
Basically, if in the form M(x,y) dx + N(x,y)dy = 0,
then you can separate the
M(x,y) and N(x,y) terms
Take the partial derivative of each of x and y and if they are equal,
your equation is exact.
7. Integrating Factor
Format your equation as such:
y’ + yP(x) = Q(x)
Integrating factor → I(x) = e^integral(
Use accordingly (Look at exact)
y = 1/I(x) * int[ I(x)*Q(x) + C ]
Growth & Decay
rate of change of x with respect to time
Modeling with First order Differential Equations
Now that you’ve practiced these different techniques for solving differential equations, Our next objective is to apply these differential equations to rate of change problems that we can now solve now that we’ve created am equation of the differential type.
Usually our differential equations are either separable or non-separable.
If separable, split up the variables and integrate
If not separabale, we must
- Label the variables in the equation
- List any given constants or given initial conditions
- Write out equation in terms of words/variables
- Write equation in standard form
- Solve this differential equation
- Plug in the initial condition in to get C
Exam 1 Summary
The best way to go about first order differential equations
- Does the problem fall under SHEILDS — Separable, homogeneous, Exact, Linear, Direct Integration, or Substitution. (I know the order is different but shields is a good acrynonm
- Separate the variables into the dy and dx terms on the respective sides of the equal sign and then integrate and solve in terms of y
- Replace y = ux or x = vy
- Find dy/dx and then substitute in y or x and dy/dx
- Integrate the two sides
- Look at the equation and see if you can identify a M(x,y) and N(x,y) term which is set equal to 0
- Find the partial derivative of M(x,y) and N(x,y) and see if they are equal
- Put the equation in standard form:
y’+yP(x) = Q(x)
Your integrating factor is now
I(x) = e^integral(P(x))
You then multiply the equation by your integrating factor
I(x)* y =integral ( I(x) * Q(x) dx
For Direct Integration
- Just integrate the equation, shouldn’t be too complicated
- If the equation can be made simpler, substitute in a u = some term and then find du and apply that to the equation
When given a modeling word problem, you need to memorize a couple of different problem types and their associated differential equation type so that you can solve for their general solution and for C if given the initial conditions
Bacterial Growth, Half-Life of Plutonium, Age of a Fossil
dx/dt = kx or dA/dt = kA
Cooling of a Cake
dT/dt = k(T-70)
Mixture of Two Salt Solutions
dA/dt = (input rate of salt) -(output rate of salt) =>R in — R out
Exam 2 :
Sections 4.2 to 7.1
Reduction of order
Homogeneous Linear Equations with Constant Coefficients
Undetermined Coefficients — Superposition Approach*
Undetermined Coefficients — Annihilator Approach
y’’ + y’ + … = 0
y c: complementary function
y p: particular solution
y p = u1(x)cosx + u2(x)sinx
u1(x) = negative integral of y2 f(x) / w(x) dx
u1(x) = integral of y1 f(x) / w(x) dx
w(x) is the Wronskian
To find wronskian we do a crossproduct of |y1 and y2|
y = yc + yp
Exam 2 Summary
At the present moment in time, classes were now moved completely online meaning we did receive a lecture video review which I will be breaking down into sections, steps and explanations as best I understand.
The video that was made to cover our second exam material is from sections 4.1 to 4.8 and it is mainly solving homogenous and non-homogenous linear differential equations.
I took hand written notes that can be found here
I will write out verbal explanations with light equations due to the inability to type the mathematics properly with sub notation but I will do my best to describe steps in full
Topics and Examples
Prove that given solutions are indeed solutions to the homogenous, linear, second order differential equation
Show that [e^x , e^-3x] is a fundamental set of solutions of the equation
y’’ + y’ — 6y=0
First we must check if the given solutions are linearly independent or not
so we use each of the provided solutions as inputs for y and take their respective first and second derivatives to see if they satisfy the equation.
In this problem the y’s and their derivatives satisfy the differential equation so now we must check if they are linearly linearly independent by finding the general equation and since the equation is a second order differential equation so there are only two solutions.
so e^(2x) & e^(3x) are our f(x)1 and f(x)2
To find the general solution we need to find the wronskian which is
the cross product of the f(x)’s and their derivatives.
If the final solution is never 0 then the solutions are linearly independent
In this case e^(2x)•-3e^(-3x) + -2^(2x)•e^(-3x) = -3e^(-x) — 2e^(-x)
- This equals -5e^-x so they are linearly independent
- Therefore both of the provided solutions are the fundamental set of solutions
Reduction of order formula
Given the equation: x²y’’ — 3xy’+4y=0 and the solution: y1 = x² ,
Find the other solution
This is a second order linear differential equation that has a non-constant coefficient so we take the original equation and divide by the term in front of the y’’ to get it by itself
In this case we divide by x² to get
y’’ — 3y’/x + 4/x²y = 0
The coefficient of y’ is P(x)
The coefficient of y is Q(x)
Now at this point we need to use the reduction of order formula
Then using this formula we can plug in and solve for the the second solution the to the differential equation.
Find the general solution of the homogenous linear equation
Solving non-homogenous differential equations using the method of undetermined coefficients
Solving non-homogenous linear differential equations with constant coefficients using the method of variation of parameters
1.1: Definitions and Terminology
1.2: Initial-Value Problems
1.3: Differential Equations as Mathematical Models
2.2: Separable Equations
2.3: Linear Equations
2.4: Exact Equations
2.5: Solutions by Substitutions
3.1: Linear Models
4.1: Preliminary Theory — Linear Equations
4.2: Reduction of Order
4.3: Homogeneous Linear Equations with Constant Coefficients
A homogeneous linear ordinary differential equation with constant coefficients is an ordinary differential equation in which coefficients are constants (i.e., not functions), all terms are linear, and the entire differential equation is equal to zero (i.e., it is homogeneous).
4.4: Undetermined Coefficients — Superposition Approach*
4.6: VARIATION OF PARAMETERS
Thanks for reading! Buy me coffee? https://ko-fi.com/jonngan