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Whereas any finite set of impedance data does not constrain the electrical conductivity σ (z) at a fixed level z in a 1D-model, the conductance function S (z2) as the depth-integrated conductivity from the surface to the depth z2 will be constrained. Assuming only the non-negativity of σ (z), it is shown that for a given depth z2 the models generating the lower and upper bound of S (z2) consist of a sequence of thin sheets. The determination of the positions of the thin sheets and their conductances leads to a system of nonlinear equations. As a limitation the present approach requires the existence of a model, which exactly fits the data. The structure of the extremal models as a function of z2 is discussed in examples with a small number of frequencies. Moreover, it is shown that any set of complex 1D impedances for M frequencies can be represented by a partial fraction expansion involving not more than 2M (positive) constants. For exactly 2M constants there are two complementary representations related to the lower and upper bound of S (z2). For the simple one-frequency case, a more general extremal problem is briefly considered, where the admitted conductivities are constrained by a priori bounds σ – (z) and σ + (z) such that σ – (z) ≤ σ (z) ≤ σ + (z). In this case, the extremal models for S (z2) consist of a sequence of sections with alternating conductivities σ – (z) and σ + (z). The sharpening of conductance bounds by incorporating a priori information is illustrated by an example.
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Anderssen, R.S., Seneta, E. (1972) A simple statistical estimation procedure for Monte Carlo inversion in geophysics. II: Efficiency and Hempel's paradox. Pure Appl. Geophys. 96:5-14
Barcilon, V. (1979) Ideal solution of an inverse normal mode problem with finite spectral data. Geophys. J. R. Astron. Soc. 56:399-408
Barcilon, V., Turchetti, G. (1979) On an inverse eigenvalue problem with truncated spectral data. In: Anger, G. (Ed.) Inverse and improperly posed problems in differential equations, pp. 25-34, Akademie-Verlag, Berlin
Brown, K.M. (1973) Computer oriented algorithms for solving systems of simultaneous nonlinear algebraic equations. Byrne, G.D., Hall, C.A. (Eds.) pp. 281-348. Academic Press, NewYork and London
Jones, A.G. (1980) Report on the COPROD study. Paper read at the Fifth Workshop on Electromagnetic Induction in the Earth and Moon, Istanbul, August 18-23
Lawson, C.L., Hanson, R.J. (1974) Solving least squares problems. Prentice-Hall, Englewood Cliffs
Oldenburg, D.W. (1983) Funnel functions in linear and non-linear appraisal. J. Geophys. Res. 88:7387-7398
Parker, R.L. (1972) Inverse theory with grossly inadequate data. Geophys. J. R. Astron. Soc. 29:123-138
Parker, R.L. (1974) Best bounds on density and depth from gravity data, Geophysics 39:644-649
Parker, R.L. (1975) The theory of ideal bodies for gravity interpretations. Geophys. J. R. Astron. Soc. 42:315-334
Parker, R.L. (1977) The Frechet derivative for the one-dimensional electromagnetic induction problem. Geophys. J. R. Astron. Soc. 49:543-547
Parker, R.L. (1980) The inverse problem of electromagnetic induction: existence and construction of solutions based on incomplete data. J. Geophys. Res. 85:4421-4428
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Parker, R.L., Whaler, K.A. (1981) Numerical methods for establishing solutions to the inverse problem of electromagnetic induction. J. Geophys. Res. 86:9574-9584
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Sabatier, P.C. (1979) On extremal solutions of Sturm-Liouville inverse problems. In: Anger, G. (Ed.) Inverse and improperly posed problems in differential equations, pp. 223-231. Akademie-Verlag, Berlin
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ICAS Mathematics test has five parts:
- Algebra and Patterns: Number problems related to patterns and relationship of numbers
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4 edition of Mathematics for Health Professionals found in the catalog.
Mathematics for Health Professionals
B. Louise Whisler
by Jones & Bartlett Publishers
|The Physical Object|
Biostatistics, Second Edition, is a user-friendly guide on biostatistics, which focuses on the proper use and interpretation of statistical methods. This textbook does not require extensive background in mathematics, making it user-friendly for all students in the public health sciences field. Engineering Mathematics: YouTube Workbook. Partial Differential Equations. Essentials of Statistics. Blast Into Math! Applied Statistics. Integration and differential equations. Elementary Algebra Exercise Book I. Principles of Insurance. Essential Engineering Mathematics. Mathematics for Computer Scientists. Introduction to Complex Numbers.
Avoids complex mathematics, making the book accessible to those from a non-quantitative background Online resources include additional questions, suggestions for further reading, links to useful websites and teaching slides An Introduction for Students and Health Professionals. Essential Epidemiology. An Introduction for Students and Health Book Edition: 3rd Edition. Whether you are preparing for a career in the health sciences or need a refresher on basic medical mathematics, Mathematics for Medical and Clinical Laboratory Professionals has what you need. Learn everything from basic algebra to advanced chemistry, statistics, and hematology all through the unique perspective of the health sciences.
Statistics for Health Care Professionals is an accessible guide to understanding statistics within health care practice. Focusing on quantitative approaches. The Pocket Field Guide for Disaster Health Professionals: Missioncraft in Disaster Relief is a disaster management toolkit containing briefing checklists, systems profiles, assessment forms, reporting templates, project management worksheets, and other tools required in the field. It also provides mission-critical technical reference information adapted from internationally .
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Math for Health Care Professionals is a comprehensive, foundational resource that is equally effective in the classroom or for self-study. It assumes no prior knowledge of mathematics or health care but merges the two topics into the capstone of a complete learning package, including a student workbook/5(10).
Actual health professionals also share how they use math in their careers. Exercises using drug dosages, intake and output, weights and measures, temperatures, IV drip rates, illustrations of syringes, prescriptions, medication labels, IV bags, and I and O charts give you hands-on practice with real-life health care skills requiring mathematics.5/5(1).
mathematics for health professionals Download mathematics for health professionals or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get mathematics for health professionals book now.
This site is like a library, Use search box in the widget to get ebook that you want. Mathematics for health professionals. Boston: Jones and Bartlett, © (OCoLC) Online version: Whisler, B.
Louise, Mathematics for health professionals. Boston: Jones and Bartlett, © (OCoLC) Document Type: Book: All Authors /. COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle.
All-You-Can-Learn Access with Cengage Unlimited. Cengage Unlimited is the first-of-its-kind digital subscription that gives students total and on-demand access to all the digital learning platforms, ebooks, online homework and study tools Cengage has to offer—in one place, for one price.
Students get unlimited access to a library of more t products for $ per Price: $ Book Description. This compendium of essential formulae, definitions, tables and general information provides the mathematical information required by engineering students, technicians, scientists and professionals in day-to-day engineering practice.
This book provides all the necessary information in a readable style that can be understood by anyone with even the most basic knowledge of mathematics Health Economics is ideal for all health professionals who are required to make policy decisions – including hospital managers, clinical directors and partners in family practices.
high school math. social sciences. literature and english. foreign languages. Mathematics for Public and Occupational Health Professionals by Ian Young is licensed under a Creative Commons Attribution-ShareAlike International License, except where otherwise noted. Please see the Acknowledgement Section of the book for the OER sources used in creating this book.
Math for Health Care Professionals is a comprehensive, foundational resource that is equally effective in the classroom or for self-study. It assumes no prior knowledge of mathematics or health care but merges the two topics into the capstone of a complete learning package, including a student workbook.
Goal: This is an introductory book on Mathematics for Medical and Health Science students to be taught at the first year level like Medical Physics book will contain enough materials at. The design of the mathematics program provides a quality education to all students.
This program is actively being revised and aligned to meet the standards of the Massachusetts Common Core State Standards. Each course offers a variety of strategies to provide students with opportunities to move toward meeting these standards.
Mathematics for Allied Health Sciences is a book for students majoring in Nursing and Radiology technology. This book includes an integrated review of the mathematical skills required in Allied. Applied Mathematics for Safety Professionals: Tips, Tools and Techniques to Solve Everyday Problems is a reference that safety and health professionals can turn to for time-saving solutions to complex problems.
Mathematical applications are included from a broad variety of fields. A library of equations from each field is enhanced by a large selection of practical examples with. The Neuropsychology of Mathematics explores how young children learn and acquire basic mathematical skills from a brain-based educational perspective.
It discusses the three primary ways in which numbers are formatted in the brain and examines the relationship between anxiety and mathematical performance during classroom learning situations. Martinson, Ida Marie is the author of 'Mathematics for health professionals' with ISBN and ISBN The major in biomathematics is a joint program with the department of biology.
It is an ideal major for students who have a strong interest in math and who plan to go to professional school in a health science field such as medicine, dentistry, or biostatistics. For more information, contact Mathematics Department.
Stress levels in health professionals have been shown to be high inmany countries and in most staff groups. This creates a personalcost to the individuals concerned, a financial cost to theorganisations in terms of absence, early retirement and complaints,and a health cost to patients in terms of the risk of poorerquality care that is received by patients from stressed.
Book of Math Terms Provides Help for Students, Professionals By Kirsten BairdThe Razorback Reporter An interdisciplinary team of four UA professors have published a book that features a list of mathematical terms that will guide beginners and experts through their academic studies, a professor in Curriculum and Instruction said.
The Language of. This important book offers an introduction to the theory and the varying types of assessment for health care professionals. The book includes information on such topics as Where have work based assessments come from?; Why do we have different parts to the same exam like MCQs and OSCEs?; How do colleges decide who has passed or not?; Why can people pick their .Find many great new & used options and get the best deals for Basic Mathematics for the Health-Related Professions by Lorraine J.
Doucette (, Paperback) at the best online prices at eBay! Free shipping for many products!Covers all the math essentials needed by today's allied health professionals. 14 units cover: pre-algebra basics, the metric system, reading drug labels, medicine cups, syringes, intravenous administration bags, parenteral dosages, basic intravenous administration, and Availability: Available.


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About the Author: Xtronic.org blog author. Electronics technician for the technical school of Brasilia - Brazil. Interested in electronics, circuits and technology in general.


Kenya Film Classification Board (KFCB) Chief Executive Officer, Dr Ezekiel Mutua presided over the annual 23rd National Mathematics Contest held at Mang’u High School, Kiambu County on June 22nd, 2019.
The National Mathematics contest seeks to generate young people’s interest in the subject and provide a platform for students to compete and socialize.
In his remarks, Dr. Mutua who addressed over 4000 students and teachers from all over the country, stressed on the values of hard work, resilience and integrity in the journey to success.
He lauded the organizers of the contest, noting that mathematics provides an opportunity for students to enjoy the fun in solving mathematical problems and appreciate the beauty of it.
The CEO added that Mathematics is an integral part of education and one of the big four academic disciplines namely: science, technology and engineering that is critical to financial literacy, and central to our daily lives.
Dr Mutua further told the students that Mathematics is more than numbers and formulas; it embraces logic and reasoning to resolve complicated and interlocking issues adding that many decisions made, in the public and private sectors alike rely partly on the use of mathematics.
The CEO urged the students to develop positive attitude in Mathematics if they want to succeed, adding that hard work will see them excel.
“I got a ‘Nine’ an equivalent of an ‘E’ in mathematics in my Fourth Form, but today I was the Chief Guest at a Mathematics competition. Failure is not the end. Life is mathematical and the greatest failure is to miss the formula. If you fail, go back and find where you missed the formula, where you miscalculated or misapplied the formula,” he said.
The main objective of the Mathematics contest is to popularize and demystify the subject through an interactive activity in a fun, relaxed environment.


NEW YORK, NY – There aren’t many people that would turn down a $1 million prize, but one math whiz felt like it was okay!
Our Russian lesson for the day is nyet, which means no. Not exactly the first thing that would come out of many people’s mouths when offered a $1 million prize, but it certainly came out of Grigory Perelman’s mouth. Perelman is a reclusive Russian mathematics genius. His lifestyle doesn’t make for stirring headlines, but he did when he decided to decline a lucrative math prize.
His decision was announced Thursday by the Clay Mathematics Institute in Cambridge, Mass., which had awarded Perelman its Millennium Prize. The math whiz successfully solved the Poincare (pwan-kah-RAY) conjecture, which deals with shapes that exist in four or more dimensions.
Jim Carlos, the institutes president, said Perelman’s decision was not a complete surprise, since he had declined some previous math prizes.
Carlson said Perelman had told him by telephone last week of his decision and gave no reason. The Russian mathematician did tell Interfax his reasoning, however.
“To put it short, the main reason is my disagreement with the organized mathematical community,” Perelman told Interfax. “I don’t like their decisions, I consider them unjust.”
The Associated Press attempted to reach Perelman for comments, a resident of St. Petersburg, but were unsuccessful.


I am currently a PhD student under supervision of Stefan Schreieder at Leibniz Universität Hannover. Since October 2023, I receive a doctoral scholarship from the Studienstiftung des deutschen Volkes.
I am working in algebraic geometry and my research interests are mainly algebraic cycles and rationality questions.
Institute of Algebraic Geometry
Leibniz University Hannover
Rationality questions and decomposition of the diagonal (pdf)
Advisors: Stefan Schreieder and Nebojsa Pavic
Schnitttheorie und enumerative Geometrie [in German] (pdf)
Advisor: Stefan Schreieder


Books for Students
The Great Ancestor Hunt
Lila Perl. Clarion Books, 1989.
A study of America's population expansion, which allows students
to work with place value and estimate distances on a map.
Coins and Currency
Brenda Ralph Lewis. Random House, 1993.
A manual of money that includes a history of coins and paper money,
how money is made, how to find and identify coins, and how to display a collection.
If You Made a Million
David M. Schwartz. Mulberry Books, 1994.
Describes how to use money earned to make purchases, pay off loans,
or accrue interest in a savings account.
The Kid's World Almanac of Amazing Facts About Numbers, Math & Money
Margery Facklam. World Almanac, 1992.
Shows the importance of numbers through their use in weights and
measures, and as money, signs, and symbols. A good resource for mathematics projects.
Math Fun with a Pocket Calculator
Rose Wyler and Mary Elting. Julian Messner, 1992.
A collection of calculator games and tricks. Recommended to help
students become familiar with a calculator.
Roman Numerals I to MM = Numerabilia romana uno ad duo mila
Arthur Geisert. Houghton Mifflin, 1996.
An introduction to Roman numerals.
The Step-by-Step Art of Paint Techniques
David Japp. World Publications, 1995.
One of the techniques covered in this art book is mural painting,
with color photographs following the initial drawing to the finished piece.
Books for Families
Lakshmi Hewavisenti. Gloucester Press, 1991.
This book of number games, puzzles, and projects helps students
recognize numbers as well as learn to add, subtract, and
Jelly Beans for Sale
Bruce McMillan. Scholastic, 1996.
This photo-essay shows how different combinations of pennies, nickels,
dimes, and quarters can buy varying amounts of jelly beans.
Math Fun with Money Puzzlers
Rose Wyler. Silver Burdett Press, 1992.
Explains the use of money in daily life, using puzzles, games, tricks,
and jokes. Useful for introducing a lesson.
Reference Books for Teachers
The Math Teacher's Book of Lists
Judith Muschla. Prentice Hall, 1995.
This collection of rules and lists is an information source for mathematics
teachers who at times seek an overall view of mathematics or want a new
twist on a specific topic.
Number Sense and Operations: Addenda Series, Grades KŠ6
Grace M. Burton, et al. National Council of Teachers of Mathematics, 1993.
This book focuses on the importance of understanding relationships
between numbers and the importance of representing numbers in various ways.
Multicultural Math: Hands-on Math Activities from Around the World
Claudia Zaslavsky. Scholastic, 1994.
This collection of reproducible worksheets and activities integrates number
systems from ancient cultures.
Math World Literature
Thailand In Pictures
Prepared by Geography Department
Lerner Publications, 1989.
Thailand Land of Smiles
Dillon Press, 1991.
Number Art: Thirteen 123s from Around the World
Leonard Evertt Fisher
Four Winds Press, 1982.
Hand Around the World: 365 Creative Ways to Build Cultural Awareness & Global Respect
Williamson Publishing, 1992.
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Calculus homework is a challenging course to take, even for the most gifted, enthusiastic math students. It’s filled with difficult terminology, complicated concepts, and excessive doses of memorization. It can be a real bear, especially if you’re not ready for it.
Calculus homework can be especially challenging when you’re studying for Calculus. It requires a tremendous amount of time, planning, and organization. Calculus homework can take up to three times as long as other assignments, meaning that you have to be extra careful with your time management. Fortunately, we’re here to help! We offer a multitude of resources to support you through your coursework and bring you success. So, if you’re ready to take the plunge and sign up for our quality homework help, let’s dive into it!
If you’re taking Calculus, you’re taking a two-year math course that builds on the math you’ve already learned, covering topics like limits, functions, and differentiation. Before you begin your Calculus homework and coursework, you’ll want to make sure you have a firm grasp on the fundamentals. This includes things like pre-calculus, algebra, and geometry.
If you’re not sure where to start, don’t stress it. Calculus is a challenging course, and we’re here to help you succeed in your Calculus homework!
If you’ve decided that Calculus is the course for you, congratulations! You’ve taken the first step towards success. Now, it’s time to get started on the road to Calculus mastery.
The first thing you’ll want to do is pick a Calculus course from one of our many offerings at Careers Experts. We offer free, online Calculus tutors who can help you get started on the right foot on your Calculus homework.
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If you’re feeling overwhelmed and you don’t know where to start, or you just want some extra help, we invite you to sign up for our newsletter.
We send out timely Calculus study tips, tricks, and hacks to help you succeed. You can also expect to get timely reminders about your upcoming Calculus homework assignments so you don’t forget.
And, don’t worry! We don’t spam or give away your email address. We only use it when you sign up for our mailing list.
Facing problems with a specific Calculus homework assignment? Don’t stress it. We’ve all been there. Just let us know what you need help with, and we’ll get right on it.
Your Calculus tutor will assist you with all your coursework, from pre-calculus through post-calculus. You can submit your assignments via our online tutoring platform, or you can email them to us if you prefer.
We also offer a dropbox option so you can share your files with us and we can review them offline. This is a great option for students who prefer to work at their own pace.
If you want to practice for exams or simply master the concepts, take advantage of our practice questions. They cover topics like functions, limits, vectors, and more.
These Calculus homework questions are designed to challenge you, push you to work out your problems, and help you solidify what you’ve learned.
If you’re struggling with a particular concept, our practice questions are a great way to test your knowledge and get ready for the exam.
If you’re ready to take your Calculus studies and Calculus homework to the next level, we invite you to try our quality homework help. We offer a plethora of resources to support you through your coursework and bring you success. So, if you’re ready to take the plunge and sign up for our quality homework help, let’s dive into it!


A few weeks ago, at the beginning of January, I was fortunate enough to attend the Joint Math Meetings (JMM) in Boston. The meetings are an annual math conference hosted by the American Mathematical Society (AMS) and the Mathematical Association of America (MAA) consisting of thousands of talks, mini-courses, poster sessions, and panels and are the biggest gathering of mathematicians in America each year. I attended this year for the first time with over 7,000 other mathematicians and can’t wait to go next year!
This past summer I participated in a Research Experience for Undergraduates (REU) at Central Michigan University and as part of our NSF grant we decided to attend the JMM. My 20 friends from the summer and our advisors flew in from all over the country, while I drove 2 short hours from Connecticut, to reunite in Boston. My research group was fortunate enough to give a talk and present a poster on our research during the meetings. Apart from our 10 minute talk and 2 hour poster session, we were free to go to panels, attend sessions, and visit exhibits; that’s where the real fun began.
It takes quite a bit of nerdiness to like math, and there was definitely plenty of that to go around at the meetings. Imagine 7,000 mathematicians walking around the Prudential center in Boston, taking over Starbucks, Dunkin Donuts, the Marriot, Sheraton, and many other stores and locations. I wonder what was running through the minds of baristas, hotel staff, and store owners as thousands of mathematicians milled about… At first I was intimidated but very quickly began to feel at home. The reality of it is that the math world is pretty small and the meetings become an annual family reunion of sorts. Even as an undergraduate I was reconnecting with ‘old’ math friends from my REU and alum of Stonehill’s math department. My math family has already begun to take form and I haven’t even gotten into graduate school yet!
Not only did I get to spend time with friends, but I learned and networked. A lot. We attended many interesting sessions including one titled “Benjamin Franklin’s Magic” and a session on Algebraic Topology. Imagine sitting through 15 minutes of “Homotopy Kac-Moody groups and infinite pseudoreflection groups”; we didn’t understand much of that lecture to say the least. A highlight of the meetings is always the Gibbs Lecture, which is an invited address geared towards sharing some of the greatest contributions mathematics has made to modern day thinking, was given by Professor Bradley Efron. Fun fact: Einstein gave the Gibbs Lecture at the JMM in 1934! I also had the opportunity to attend a graduate school fair where I met with representatives from several Ph.D. programs and heard about their programs while they got to meet me as well.
Overall the meetings were a phenomenal experience. I felt so at home among people who think, learn, and love the same thing I do. It is not every day that you can get a room full of hundreds of people laughing at some stereotypical corny math joke. I’ve found what I want to do when I grow up; I want to be a mathematician. Oh, and the best part: the meetings are in San Diego next year…sweet!!


Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis
Chaos is usually regarded as a distinct alternative to random effects such as environmental fluctuations or external disturbances. We argue that strict separation between chaotic and stochastic dynamics in ecological systems is unnecessary and misleading, and we present a more comprehensive approach for systems subject to stochastic perturbations. The defining property of chaos is sensitive dependence on initial conditions. Chaotic systems are "noise amplifiers" that magnify perturbations; nonchaotic systems are "noise mufflers" that dampen perturbations. We also present statistical methods for detecting chaos in time-series data, based on using nonlinear time-series modeling to estimate the Lyapunov exponent λ, which gives the average rate at which perturbation effects grow (λ > 0) or decay (λ < 0). These methods allow for dynamic noise and can detect low-dimensional chaos with realistic amounts of data. Results for natural and laboratory populations span the entire range from noise-dominated and strongly stable dynamics through weak chaos. The distribution of estimated Lyapunov exponents is concentrated near the transition between stable and chaotic dynamics. In such borderline cases the fluctuations in short-term Lyapunov exponents may be more informative than the average exponent λ for characterizing nonlinear dynamics.


Milford High School Graduation Requirements:
24 credits: 4 English, 4 Math (including Algebra II), 3 Science (including Biology), 3 Social Studies (including US History), 2 Foreign Language, 1 PE, 0.5 Health, 3.5 Electives, 3.0 in a Pathway.
You can check your transcript to see what classes you have completed on Home Access - go to Grades and then Transcripts.
Milford High School GPA Calculation:
For each final grade on your transcript, you must convert it to quality points and then find the average with all final grades from all courses.
Weighted quality points must be used with an honors, AP, or dual enrollment course.


This is a short summary of some of the terminology used in machine learning, with an emphasis on neural networks. I’ve put it together primarily to help my own understanding, phrasing it largely in non-mathematical terms. As such it may be of use to others who come from more of a programming than a mathematical background. Note that there are some overlapping or even slightly different meanings of some words or expressions depending on the context (e.g. “linear” and “dimension”), and I’ve tried to make this clear where appropriate. Disclaimer: It is by no means exhaustive or definitive.
Activation function Link to heading
This is the function which determines whether a unit should be activated or not, i.e. a “gate” between a unit’s input and output. It is computed by multiplying the input number(s) by the weight, i.e. the weighted sum of all of the inputs from the previous layer. At its simplest it could be a “step” function which outputs one number below the threshold and another above the threshold. In practice activation functions are non-linear (in the machine learning rather than statistical sense), i.e. a curved line rather than a straight line. One of the most common activation functions at the moment is the rectified linear unit. A sigmoid function might be used for binary classification. Given there may be a large number of activation functions (one for each unit in the network) and it may need to be computed a very large number of times, activation functions are designed for computational efficiency.
Back propagation Link to heading
Back propagation is an algorithm used to determine a network’s weights. It computes the gradient of the loss function with the respect to the weights, with the gradient determined by computing the derivatives or partial derivatives of the loss function, and does this each layer at a time iterating backwards.
Batch size Link to heading
The number of examples used in one iteration of training. Stochastic gradient descent will have a batch size of 1, while Mini-batch gradient descent will have more. In a Recurrent Neural Network, you’d typically have as large a batch size as your memory allows, given a larger batch size will work through the training examples more quickly and a small batch size can lead to irregularities in the rate that the loss decreases over the epochs, but some experimentation may be required to optimise batch size.
Bias unit Link to heading
In a neural network, a bias unit is a special unit typically present in each (non output) layer which is connected to the next layer but not the previous layer, and which stores the value of 1. It allows the activation function to be shifted to the left or to the right. It is sometimes called the intercept term in linear regression.
Classification Link to heading
In supervised learning, this is trying to predict a discreet value, e.g. true/false or sun/cloud/rain/snow. Contrast with Regression. A binary classification has 2 discreet values, while with more than 2 is a multiclass classification. You can think of classification as working with “labels”.
Convergence Link to heading
The process of getting very close to the correct answer.
Convolutional Neural Network Link to heading
A Convolutional Neural Network (CNN) is a network architecture commonly used for image processing. It uses convolutions, which are filters which pass over the image and apply transformations to a grid, to emphasise certain features in the image, e.g. vertical edges or horizontal edges. CNNs often use pooling, which is a way of compressing an image, e.g. max pooling preserves only the highest value in a grid. The network would typically train on the results of the convolutions, and the pooling, rather than the original input pixels.
Cost function Link to heading
See Loss function.
Derivative Link to heading
Slope of a line tangent to a curve (e.g. generated by a function). In neural networks, this is usually applied to the cost function to determine how correct or incorrect its output is - the steeper the curve, i.e. the bigger the gradient, the more incorrect it is.
Dimension Link to heading
A matrix is a two dimensional array, i.e. has rows and columns. A vector could be said to be a one dimensional array, i.e. just having a single column and multiple rows of data, or a single row and multiple columns. However, a vector’s dimension is long it is, i.e. how many elements it has. Similarly the dimension of a matrix often refers to the number of rows and columns (conventionally in that order), e.g. a 6 x 2 matrix has 6 rows and 2 columns. Sometimes dimensions are referenced in the shape.
Embedding Link to heading
Representing a higher dimension vector with a lower dimension representation, e.g. a single number representing a whole word or a category representing a specific book. Reducing dimensionality helps with training, and generated embeddings can help with visualisation. The term “vocabulary size” is used to indicate how many different mappings from number to character or word or part word there are.
Epoch Link to heading
One iteration of training, so that each example has been seen once. The term is most relevant where batch size is greater than 1, in which case it will be the number of examples divided by batch size.
Feature Link to heading
Information that is relevant to learning. An attribute of the subject. Features are like columns in a database, with distinct values for each row, e.g. separate features for houses could include the floor area and number of bedrooms. In the example of an image, it could be just the pixel data, but can also include metadata, such as tags describing the image.
Feature engineering Link to heading
More features are not necessarily better because some features may not help improve the results, and too many features may make learning impractical, so feature engineering is important for neural networks. A network may even have automated feature extraction, e.g. a layer for edge detection of an image. A useful rule of thumb when deciding which features to use is whether a human expert would be able to predict the results given the selected features, or at least say the features contain sufficient information to predict the results.
Feature scaling Link to heading
This is where the features are scaled to be of similar range to each other, typically in the -1 to +1 range. Feature scaling can ensure the learning gives a comparable amount of attention to all the different features, not just the features that have a range of large values. In gradient descent, this makes the contours less skewed or tight, which allows it to reach the global minimum more quickly. Feature scaling can be more important in some learning algorithms, e.g. SVMs.
Feature vector Link to heading
The combination of the features is called a feature vector. A feature vector is one row (or column) of input, and the dimension of the feature vector is the number of columns of that row (or rows of that column).
Function Link to heading
In its simplest form, a function is the method used to turn an input into an output. Machine learning is trying to find a function describing the relationship between data features x and classification or regression for y, i.e. y = f(x). A linear function is just a straight line. A quadratic function is a curve. A polynomial function is one with lots of ups and downs, i.e. a wiggly line. Note that many examples demonstrate with one or two features (dimensions) so it can be illustrated on a graph, but in practice it is often more dimensions that are more difficult to visualise in this way. See also Activation function, Optimisation function and Loss function.
Polynomial function Link to heading
A function where operations include non-negative integer exponents of variables, e.g. x-squared. A degree 2 polynomial has x-squared and generates a parabolic curve (one peak or trough), a degree 3 polynomial hàs x-squared and x-cubed and generates a cubic curve (one peak and one trough), etc.
Sigmoid function Link to heading
Also called logistic function. In contrast to a linear function which is a straight line, a sigmoid function starts off with a slow slope, has the largest slope in the middle, and finishes with a slow slope again.
Gradient descent Link to heading
Algorithm to find the local minimum of a function. It is an iterative process which determines the gradient at each step and proceedes in the direction of the steepest gradient. Step size is determined by the learning rate. The gradient is determined by calculating the derivative of the loss function. Also called batch gradient descent.
Stochastic gradient descent Link to heading
Stochastic gradient descent (SGD) uses individual training examples in each iteration to update parameters, in contrast to batch gradient descent which uses all the training examples in each iteration before updating parameters which can be computationally expensive with large datasets. This means progress on the path to the global minimum is quicker, although it is not as direct, and tends to converge near the global minimum rather than on it exactly (using a smaller learning rate can help, optionally with a learning rate that decreases over time). You would typically periodically check on progress twoards convergence by plotting the cost averaged over the last x training examples processed.
Mini-batch gradient descent Link to heading
Mini-batch gradient descent is between batch gradient descent and stochastic gradient descent, using b training examples in each iteration, where b is the mini-batch size. It can be faster than SGD if you use vectorisation to partially parallelise the derivative calculations (gradient computations).
Hyperparameters Link to heading
Configuration, e.g. learning rate, iterations, hidden layers, type of activation function. They help determine the final values of the parameters.
Inference Link to heading
Running an already-trained model to make predictions, without training, i.e. without adjusting the weights via back-propagation.
Learning rate Link to heading
In gradient descent, the learning rate is the size of each step taken. If it is too small it could take a long time to reach the target, and if it is too large it could miss the target.
Loss function Link to heading
The loss function determines the performance of a given model for given data, in the form of a single number denoting the difference between predicted values and expected values. The “score” from the loss function is simply known as the loss. The loss function is sometimes called the cost function, and sometimes represented as J(theta). It always has a term to calculate the error (like Mean Squared Error), and sometimes has a term to regularise the result. The derivative of the loss function is the gradient, which shows how quickly the loss (or cost) is improving. The lower the loss (or cost) the better. Mean Squared Error (MSE) and Mean Absolute Error (MAE) are the most common terms to calculate the error, with with MAE perhaps more suitable for time series because it doesn’t punish larger errors as much as MSE does, and there are other terms to calculate the error, e.g. Huber loss which is also less sensitive to outliers.
Matrix Link to heading
Rows and columns of numbers. Contrast with a vector which is a matrix with one column and many rows.
Network architecture Link to heading
The neural network architecture is the connectivity pattern between units, i.e. the number of layers (e.g. single layer, 2 layer etc.), the number of units in each layer, and the connection patterns between layers (e.g. fully connected layers, aka dense layers). It sometimes also includes the activation functions and learning methods. A single layer will just have the input layer and the output layer (the input layer is not counted hence this is called a single layer not a 2 layer), a 2 layer network will have the input layer, a hidden layer and the output layer, etc. A fully connected layer is one where each of the units in one layer are connected to each of the units in the next layer. The number of input units will be the same as the dimension of the feature vector, and the number of output units would match the number of “labels” required in classification. A reasonable default is one hidden layer, and if there are multiple hidden layers a reasonable approach is to have the same number of units in each hidden layer, with the number of units in a hidden layer the same or a small multiple of the number of number of units in the input layer.
Normalisation Link to heading
Optimisation function Link to heading
In a neural network, an optimisation function (or optimisation algorithm) would be used with back propagation to try to minimise the loss function as a function of the parameters. In other words, it uses the “score” from the loss function to work out how well it is doing and then makes adjustments to try to improve the “score” on the next iteration / epoch. Common optimisation functions are gradient descent, Adam and RMSProp (a benefit of Adam and RMSProp is that they automatically adapt the learning rate during training).
Overfitting Link to heading
The model is simply memorising the training data and recalling it, which means excellent results against data that has already been seen but generally much poorer results against data that has not been seen. It should learn generalisations which would apply to unseen data. It can happen when you have a lot of features and little training data. You can typically see this has happened when the validation loss is higher than the training loss (or the test loss is higher than the training loss if there is no validation set). To remedy, you can decrease the number of features (which might not be possible or desirable), regularise, or specifically with neural networks decrease network size, or increase the dropout hyperparameter, or get more training data. Overfitting is sometimes called “high variance”.
Parameters Link to heading
A parameter represents the weight learned for the connection between two units. Parameters are often represented by W and b, for the weights and biases respectively. You can calculate the number of parameters in a fully connected network by multiplying the number of units in a layer with the number of units in the next layer and adding all the results for all the layers together, and adding the bias units in the hidden and output layers. The parameters are also typically stored in a matrix for efficiency in calculating the forward propagation and back propagation, although sometimes have to be “unrolled” into vectors for advanced optimisation (i.e alternatives to gradient descent). See also Hyperparameters which are the configuration options excluding the weights.
Precision and recall, and F Score Link to heading
Precision is a measure of exactness (or quality), while recall is a measure of completeness (or quantity). Specificially, precision is the number of true positives divided by predicted positives (or true positives divided by true positives and false positives), while recall is true positives divided by actual positives (or true positives divided by true positives and false negatives). Therefore, perfect precision has no false positives (or no irrelevant results), and perfect recall has no false negatives (all the relevant results). When the classes are heavily skewed (e.g. when 99% of expected results are false and you could get reasonable error/accuracy by always predicting false), you can get a better insight into how well a learning algorithm is performing by looking at precision and recall rather than error and accuracy. To get a view of overall combined precision and recall, use the F Score (or F1 Score) which is 2 times ((precision times recall) divided by (pecision and recall)).
Rectified Linear Unit Link to heading
A Rectified Linear Unit (ReLU) outputs 0 if the input is negative or 0 and outputs the input if it is positive, i.e. it only returns x if x is greater than 0.
Recurrent Neural Network Link to heading
A Recurrent Neural Network (RNN) is a network architecture for sequences of data. Hidden layers from the previous run provide part of the input to the same hidden layer in the next run.
Regression Link to heading
In supervised learning, this is trying to predict a continuous output, e.g. price or time or age. Contrast with Classification, which is trying to predict discreet values. In broader statistical terms, regression is a technique for estimating the relationship among variables, and there are several types of regression, e.g. linear regression (estimates for continous output based on a function which generates a line, although not necessarily a straight line in statistical terms because e.g. polynomial regression is a type of linear regression using a polynomial function) and logistic regression (estimates for discreet values, using the “S-shaped” curve of a logistic function aka sigmoid function).
Regularisation Link to heading
Reduce the magnitude/values of parameters. In a neural network, this would often be performed in the loss function. Without regularisation a model might overfit, and with too much regularisation it could underfit. Not to be confused with Normalisation which adjusts the data rather than the prediction function.
Shape Link to heading
Shape in the context of an input layer is the original dimension of the feature, prior to flattening into a vector. In a Convolutional Neural Network used for image processing, a 28x28 pixel square grayscale input image could be represented by a 28 x 28 matrix of the grayscale values for each pixel and this would be flattened into a vector with an input shape of (28, 28) for the input layer, or a 28x28 pixel square RGB image could have the input shape (28, 28, 3) where 3 is the number of channels (i.e. R, G and B), or if input has a batch size of 4 a 28x28 RGB image would have an input shape of (4, 28, 28, 3). In a Recurrent Neural Network, input shape would be batch size, the number of timestamps, and series dimensionality (i.e. 1 for univariate and 2 or more for multivariate). Shape in the context of a tensor is the number of elements in each dimension, e.g. in a two-dimensional tensor the shape is the [number of rows, number of columns].
Support Vector Machine (SVM) Link to heading
A Support Vector Machine is a supervised learning algorithm. In simple terms, it is a way of finding the optimal line to separate features. Compared to a neural network, an SVM might be faster to train, and should always find global optima (although in practice neural networks tend not to suffer from the local optima problem).
Tensor Link to heading
A multidimensional array that can contain scalars (i.e. numbers) and/or vectors and/or matrices. A vector is a one-dimensional tensor and a matrix is a two-dimensional tensor. A picture, for example, can be represented by a 3 dimensional tensor, with fields for width, height and depth, or a series of pictures as a 4 dimensional vector. Grouping data in this way can be more computationally efficient.
Underfitting Link to heading
The model hasn’t learned much. You typically see this when the training loss and validation loss remain about the same over multiple epochs. To remedy, you could increase the network size and/or number of layers, or add features. Underfitting is sometimes called “high bias”.
Univariate and multivariate Link to heading
In relation to time series, a univariate time series has a single value at each time stamp (e.g. a chart showing birth rate), whereas multivariate has multiple values at each stamp (e.g. charts showing birth rate and death rate).
Validation Set vs Training Set vs Test Set Link to heading
There would normally be a Training Set, Validation Set and Test Set of data. Most would be Training e.g. 70%, then Validation e.g. 20%, then Test e.g. 10%. The Training Set is used for training, i.e. adjusting the weights. The Validation Set is used during training to calculate the accuracy and avoid overfitting, but does not adjust the weights (although may be used to tune hyperparameters). If the validation loss is higher (i.e. accuracy lower) than training loss you are likely overfitting, and if the validation loss is about the same as the training loss but neither are reducing you are likely underfitting. You’d normally choose a model with the lowest validation loss (noting that this is not necessarily the last checkpoint if there has been overfitting). The Test Set is used after training is complete to confirm the predictive accuracy. When you deploy a model to production, you may also use a Test Set of production data to measure the Domain Shift between production and test data, and continue sampling production data to measure the Drift (i.e. how performance is degrading as a result of data changing over time).
Vector Link to heading
A list of numbers. A vector can be seen of as a special type of matrix that has one row and multiple columns (sometimes called a “row vector”) or one column and multiple rows (a “column vector”). Pretty much all of AI/ML works on lists of numbers, so everything has to be converted to them - see also Embedding, Parameters, Shape, Matrix and Tensor. With images, after “flattening” from a width x height matrix, a number could represent the colour of a pixel with the length of the vector equal to the number of pixels wide by number of pixels long. Additional information is often added to the input vector - see Feature vector. In terms of implementation within the machine learning algorithms themselves, vectors are linear algebra vectors, with various properties relating to addition, multiplication etc., and mechanisms for measuring distances and determining the gradients for calculating errors and rate of change. For computational efficiency in machine learning, vectors are more likely to be used as part of a matrix or tensor.


Calculus: The Science of Fluxions
I go over a brief overview of the origins and main objectives of Calculus as a math science
The science of fluxions was Sir Isaac Newton’s terminology for the new field of science known today as calculus. Newton and German mathematician Leibnitz appear to have discovered the principles of calculus in about the same time; but Leibnitz published his work first. For years, there was friction between the two countries, England and Germany regarding which country was to take credit for the discovery of calculus. For the best Maths Tutor In Ireland company, call Ace Solution Books. In the final analysis, it appears that both men arrived at their findings at about the same time and independently of each other. Both men appear to have learned from Egyptian, Indian and ancient Greece sources.
The name calculus is derived from the Latin or Roman term meaning pebbles which were a type of counting stone. The term fluxion was Sir Isaac Newton’s term for the science of calculus. His book “Method of Fluxions was published in posthumously in 1736, although it was completed much earlier in 1671. Whether Leibnitz’s or Newton’s authorship of the science is accepted as being primary, the fact remains that the subject of calculus is the most powerful mathematical invention of modern times.
There are two major types of calculus, known as Infinitesimal Calculus and that part of the total which is called Differential Calculus. Both types are built on a foundation on analytic geometry and are related by the Fundamental Theorem of Calculus. In simple terms, the theorem states that the sum of infinitesimal changes over time or some other quantity will add up to the net change.
To use a living plant as an example, as it grows, you can see the difference in size, or the increase over a period of several days and you could measure the growth, using conventional means. However, if you were to measure the difference in size after only ten seconds, it would be much more difficult to determine. If you then were to attempt to determine not the amount of growth over the time period, but the rate of growth over those seconds, you would not be able to do that using algebraic terms.
The science of calculus allows you to determine the rate of change for infinitesimally small amounts. There are a number of functions which impact the rate of change for the above example of a growing plant; things such as amount of sunlight, water, the temperature and others. However, if all other things remain equal, the variable for the rate of growth is time. By using calculus these variables can be determined. Most of the work in calculus is done by graphing formulas in order to determine the slope of the rate of change.
Calculus can be used in any field of science where a mathematical model can be designed and an optimal solution is desired. This includes physical sciences, computer sciences, business, medicine, engineering, economics and statistics. Although in the past, calculus has been considered too complicated to study as an individual subject without the benefit of a teacher’s direction, internet courses are readily available and can be studied as a self-paced course of study.


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Statistics can be used everyday in a number of scenarios and situations. More often than not they have a tendency to visit undetected but studies have shown that an individual is uncovered close to five statistics each day typically, such as this one! A functional understanding of statistics can help students and adults understand data they are available across and interpret it properly.
Students frequently find statistics a little hard to learn which can be because of the profusion of graphs, charts and equations this subject involves. When you are accustomed to it, statistics becomes much simpler to know. Among the best methods to learn statistics is to use the concepts to everyday situations. Fortunately, statistics is a subject that students will discover lots of scenarios to apply recently learned concepts and ideas.
Studying statistics can be created simpler when students spend some time about them after each class. Studying the training when you’ve learned them can help you learn faster because the material is going to be simpler to recall. Practice around you are able to to be able to focus on various kinds of questions. Concentrate on areas you have difficulty solving. For instance, if you discover graphs confusing and therefore are not able to plot data in it, take more time solving problems which entail graphing.
Statistics is part of many courses attending college, like financial aspects, sociology, the sciences, and math. Try and discover the subject in addition to you are able to, not just in pass senior high school math, but that will help you with further studies too. Statistics tutoring is definitely an option you can look at if you discover the topic really tough. Locating a good tutor and scheduling regular tutoring sessions will truly help you to get ahead in stats.
Regular tutoring helps to ensure that you practice regularly. Additionally, you will be more prone to undergo training every day. Most statistics tutors offer homework help to be able to make sure your projects is true. Although some tutors are costly, there are also affordable tutoring services that offer great good value. Sites offers a lot of versatility in timing since students can schedule sessions every time they want and in advance too. There are a variety of tutors for every subject so you’ll also have an instructor open to solve your condition or answer your doubts.
Among the popular tutors providing to your specific needs, you should look for the economics tutor who would help you learn about economics in the right manner. They should handle your learning needs in the best manner to suit your pace and learning abilities.


In the first edition of Capital, Vol 1, Ch 1, Marx mentions that:
Both forms, the relative value-form of the one commodity, equivalent form of the other, are forms of exchange-value. Both are actually only vectors – determinations conditioned reciprocally by each other – of the same relative value-expression, but divided like poles between the two commodity-extremes which have been set equal. (Translation by Albert Dragstedt)
I have emphasized the word “vectors” because I had not noticed it before, but I had previously been struck by the necessity to translate the abstruse “Hegelian” discussion of value forms into the modern mathematical language of equivalence relations. This should go together with the modern treatment of proportions and differences as vectors to construct the rationals and the integers from “natural” (positive) magnitudes. Also connect with modern duality theory, convexity, optimum control, separating hyperplanes etc etc and refer back to Marx’s letter to Dr Kugelmann of July 11 1868 explaining that value is simply the particular social form taken by the necessary proportionalities between different kinds of labor.
Use of the term “vector” by Marx seemed anachronistic. Quick look at Hegel’s Philosophy of Nature Vol 1 did not shed any light. (But did have stuff about points and lines that may shed light on Marx’s discussion of swerves in his doctoral thesis on quantum mechanics).
Must check for reference in other versions and manuscripts. (Also related to “moments”, with inseparable magnitude and direction cf inseparale length and force in lever torque!)
Looked up “vectors” and came across “Giusto Bellavitis” credited with first exposition of what are now called vectors via “equipollence” (equivalence relation defining an affine space) in 1835. Apparently he also corresponded with Walras on this precise issue!
Here is a modern explanation, showing its superiority to the usual clumsiness.
Also mentioned in German biography of Grassmann. Must come back to this when studying Grassmann/Lawvere/univalent modal homotopy type theory!
Have downloaded “History of Vector Analysis” by Michael J. Crowe. Looks like Marx could have picked up from Leibniz/Ancient Greeks/Complex numbers (all of which he studied closely). Worth reading history carefully and perhaps tracking down in Marx.
Also downloaded several volumes on Grassmann!
Including German language biography by Hans-Joachim Petsche, which linked Marx and Bellavitis and Grassmann in a random google search. See notes 229 to 242 on p110. Might be worth getting rough translation (eg google) if not found in english volumes. (Since found similar in english version of biography by same author – did not look significant but check later when actually reading stuff on Grassmann).
Have previously downloaded “Polarity and analogy; two types of argumentation in early Greek thought” by G.E.R. Lloyd.
Also may relate to Douglas Hofstadter on “Fluid concepts and creative analogies” re concepts/categories/analogies and Freyd and Scedrov “Categories, Allegories”.
After all, panta rhei and everything is connected to everything else.
Meanwhile hopefully this gets it out of my head for a while.
Except reminded to include Galileo’s verbal version of kinematics that is vastly easier to understand as simple algebra than verbally via Bill Kerr at Strange Times. See “Changing Minds: Computers, Learning, and Literacy” by Andrea A. diSessa (chapter 1, especially p13):
THEOREM 1 If a moving particle, carried uniformly at constant speed, traverses two distances, then the time intervals required are to each other in the ratio of these distances.
THEOREM 2 If a moving particle traverses two distances in equal intervals of time, these distances will bear to each other the same ratio as their speeds. And conversely, if the distances are as the speeds, then the times are equal.
THEOREM 3 In the case of unequal speeds, the time intervals required to traverse a given space are to each other inversely as the speeds.
THEOREM 4 If two particles are carried with uniform motion, but each with a different speed, then the distances covered by them during unequal intervals of time bear to each other the compound ratio of the speeds and time intervals.
THEOREM 5 If two particles are moved at a uniform rate, but with unequal speeds, through unequal distances, then the ratio of the time intervals occupied will be the products of the distances by the inverse ratio of the speeds.
THEOREM 6 If two particles are carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed by the inverse ratio of the time intervals occupied.
A modern reader (after struggling past the language of ratios and inverse ratios) must surely get the impression that here there is much ado about very little. It seems like a pretentious and grandly overdone set of variations on the theme of “distance equals rate times time.” To make matters worse, the proofs of these theorems given by Galileo are hardly trivial, averaging almost a page of text. The first proof, indeed, is difficult enough that it took me about a half-dozen readings before I understood how it worked. (See the boxed text.)
In fact this is a set of variations on distance equals rate times time. Allow me to make this abundantly clear. Each of these theorems is about two motions, so we can write “distance equals rate times time” for each. Subscripts specify which motion the distance (d), rate (r), and time interval (t) belong to.
In these terms, we can state and prove each of Galileo’s theorems. Because Galileo uses ratios, first we divide equals by equals (the left and right sides of the equations above, respectively) and achieve:
THEOREM 1 In the case r1 = r2, the r terms cancel, leaving d1/d2 = t1/t2.
THEOREM 2 In the case t1 = t2, the t terms cancel, leaving d1/d2 = r1/r2. Conversely, if d1/d2 = r1/r2 then t1/t2 = 1 or t1 =t2.
THEOREM 3 In the case of d1 = d2, the d terms cancel, leaving (r1/r2)(t1/t2) = 1, or t1/t2 = r2/r1.
THEOREM 4 This is precisely our little ratio lemma, d1/d2 = (r1/r2)(t1/t2).
THEOREM 5 Solve the equation above for t1/t2; t1/t2 = (d1/d2)(r2/r1).
THEOREM 6 Solve for r1/r2; r1/r2 = (d1/d2)(t2/t1).
For direct contrast, I reproduce Galileo’s proof of theorem 1, which is one-sixth of the job we did with algebra, in box 1.
So now we’ve redone a significant piece of work by one of the great geniuses of Western science, with amazing ease. Solving problems is always easier after the first time around, but the difference here is almost mindboggling. What we did would constitute only an exercise for a ninth-grade mathematics student.
That, in fact, is the key. Galileo never had ninth-grade mathematics; he didn’t know algebra! There is not a single “=” in all of Galileo’s writing.
The fault is not with Galileo or with the education provided by his parents or with the schooling of the times. Algebra simply did not exist at that time. To be more precise, although solving for unknowns that participated in given relations with other numbers had been practiced for at least half a millennium, the modern notational system that allows writing equations as we know themand also the easy manipulations to solve themdid not exist. Fifty years after Galileo’s main work, René Descartes (1596-1650) would have a really good start on modern algebra. Later, by the end of the seventeenth century, algebra had stabilized to roughly the modern notation and manipulative practices, although it would be the twentieth century before algebra became a part of widespread technical literacy.
Theorem 1 and Galileo’s axioms refer to a particle being “carried”. Origins of “vector” is the Latin for “carrier” (also used for disease carriers like mosquitos and for gene carriers). Check whether Galileo’s original Latin/Italian uses the word vector. (Have downloaded original. Very quick look suggests he did not. Check again when actually reviewing the long boxed proofs (in english).
Consider doing parallel treatment of weight as defined by Euclid in terms of “heaviness” etc with axiomatization of both measurement and the concept itself as in “Foundations of measurement. Vol. 1. Additive and polynomial representations” by David H Krantz; R D Luce; Patrick Suppes; A Tvesky. Find simpler logical treatment of additive magnitudes by Suppes and the well known theorem about sub-magmas of reals (when Archimedean) mentioned in Birkhoff on Lattices. Also Tarski.
Could combine with historical treatment – weighing grain and gold with scales after long epochs of occasional gifts and exchanges with neighbouring communities etc etc. Better to follow Marx’s recommendation and start with “Contribution” before chapter 1 of Capital. Highlight role of arbitrage and merchants along with money.
Then discover “intrinsic” property is mass, not weight. Caricature “transformation problem” considering density when immersed in fluids more dense than air. Further frictional, viscous, aerodynamic, hydrodynamic and then relativistic complications then curvature of spacetime and whatever comes next when measuring dynamically instead of purely hypothetical “static equilibrium”.
Saw a development of money from barter expressed in modern notation. Could take a lot further with markov chains etc. Give actual proof that circulation expels a single money commodity. The extremely efficient algorithms for disjoint set data structure could illustrate how spontaneous arbitrage in exchange rapidly produces relatively stable partitioning into equivalents and selection of a money commodity as universal equivalent.
Also show full development of commodities naturally follows rather than precedes capital and wage labour where inputs and outputs both commodities and purpose is capital increasing itself.
But experience has shown that Marx’s method of presentation simply did not work. As Lenin said, decades later, none of the Second International understood it. (“Understanding” was a pretty low bar, need more for actually completing and developing it.)
Try opposite sequence. Start describing the appearance, including modern aggregate economic and related statistics and their changing distributions, history of regular crises, current situation. Explain current theories that don’t work. Need to get behind the appearance and extract essence of the dynamic. Comparison with other sciences (solar system etc). Then refer to established categories of value, money, affordance, demand, supply, input/output, activity, credit, yield curve etc.
Then offer (improved) Maksakovsky “pure” toy model as dynamic theory based on development of contradictions from reproduction schemes resolved cyclically in a spiral that both develops the level of technology and intensifies the contradictions of capitalism that prepare the way for revolutionary transition including concentration, centralization, globalization etc. (Ties in very well with the good bits of Stalinist “New Philosophy” in Leningrad textbook).
On the logical/mathematical side, hopefully some additional axioms similar to Galileo/Suppes/Tarski could bring out lattice normed bounded complete vector space of commodity bundles which are sufficient basis for lots of functional analysis and mathematial economics (eg to explain values “duality” as in linear programming etc (Kantorovich Stalin and Nobel prizes). Hopefully Grassmann’s Geometric Algebra (including Geometric Calculus) brings out “moments” and curved space better. See “New Foundations in Mathematics” by Sobczyck.
See also Kutateladze et al developing Kantorovich spaces etc from reals via forcing invertible infinitesimals with Boolean valued models. Hopefully can do similar with the more classical nullsquare infinitesimals in toposes.
See also Aliprantis for lots more duality theory with cones, positive operators etc in mainstream “General Equilibrium”.
Real dynamics of stochastic disequilibrium as in Marx and Maksakovsky much harder.
Delays central to Maksakovsky’s cycle. Differential inclusions, not equations (as in optimal control, differential games etc). “Variational Analysis” by Rockafellar and Wets based on set valued analysis. Path dependent stochastics as in “A Modern Theory of Random Variation” by Patrick Muldowney.
So studying “path dependent stochastic delay partial differential inclusion games” aka “chaotic evolution of recursively self-organizing complex systems” aka materialist dialectics. More likely to find useful stuff in mathematical biology literature than in economics literature.


4. Solving Equations Using Multiplication or Division: How can you use multiplication or division to solve equations? Online Resources 5. Solving Two-Step Equations: How can you use algebra tiles to solve a two-step equation? Resources Big Ideas Student Text Big Ideas Record & Practice Journal Big Ideas Resources by Chapter and Performance Tasks
This equation is not factorable, but there are two solutions to this equation: 1 + √2 and 1 −√2. Looking at the form of these solutions, weobtained these types of solutions thein previous section while using the square root property. If we can obtain a perfect square, then we can apply the square root property and solve as usual.
185. $3.00. PDF. Linear Equations Performance TaskIn this performance task, students take on the role as a member of the student activities committee with the task to plan the 8th grade dinner-dance. This task utilizes the concept of writing linear equations in slope-intercept form given the slope and y-intercept,
The core elements of performance required by this task are: • work with magic squares, calculating cell values • understand simple algebraic notation Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 15 1 1 2. Gives correct answer: 3x 2 2 3.
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Algebra 1 Key Topic: slope, linear equations, systems of linear equations, x-intercept, y-intercept Astronauts wear a spacesuit when they leave the pressurized vehicle to perform spacewalks. Some of the key features of the spacesuit are breathable air and heating and cooling control.
After allowing students to intuitively solve simple systems of linear equations like in the Counting Candy Sequel or the Tech Weight Sequel, you can use these tasks as a way to practice substitution and/or elimination as a way to solve algebraically. Task 1: Sticky Situation Act 1 - Introduce the Task. Show students this video.
equation of the form x = a, a = a, or a = b results (where a and b are different numbers). MGSE8.EE.7b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Performance Task: Student Copy 1 of 2
Tiered assessments allow students to show their understanding at varying degrees of challenge. Besides tests, we also offer tiered projects and performance assessments. Below, you'll find examples of tiered tests. We've worked to improve these over the years. Recently, we've worked to merge the three choices into a single test. Students can now choose green,…


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Chemists, working with only mortars and pestles, could not get very far unless they had mathematical models to explain what was happening “inside” of their elements of experience — an example of what could be termed mathematical learning the complete colored pencil book rapidshare. This volume contains the proceedings of Work Group 4: Theories of Mathematics, a subgroup of the Seventh International Congress on Mathematical Education held at Université Laval in Québec the complete colored pencil book rapidshare. Bringing together multiple perspectives on mathematical thinking, this volume presents elaborations on principles reflecting the progress made in the field over the past 20 years and represents starting points for understanding mathematical learning today the complete colored pencil book rapidshare. This volume will be of importance to educational researchers, math educators, graduate students of mathematical learning, and anyone interested in the enterprise of improving mathematical learning worldwide the complete colored pencil book rapidshare.
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In the September 20, 1950, ARP Magazine this Mathematic Bible Riddle was included.
Take the number of pieces of silver Joseph was sold for by his brother, add the number of pieces of silver Judas received as the price of betrayal; subtract the number of pence the Good Samaritan gave to the innkeeper; add the number of Jesus’ disciples; add the number of chapters in the tenth New Testament book and the answer will be the number of books in the Bible.
Try it! Let me know if you do!
3 thoughts on “On This Day”
Susan, I am dead certain that this old, 1950, ARP Magazine, mathematics ‘story problem’ would result in the number ’66’. However, most of the time I am happy just to get my checkbook balance ‘right’. The time that it takes for me to do the more essential things of life (like pay my bills) has kept me from confirming my suspicions about this ‘head teaser’. : )
Love this! And I got it right!!!
Terrific it works every time 😀 thanks.


Ever wonder which fraction problems need inverted and multiplied, or which ones call for finding the least common denominator before you can do anything else? Keeping track of the procedures used for adding, subtracting, multiplying and dividing these fractions can be tricky for grade schoolers (and their parents!). Here’s some help!
But first a plug! We love our Professor B Math! It’s a contextual way to learn all aspects of math from a practical viewpoint in the least amount of time possible, and at your own pace. In spite of being a straight A student, I never learned math this way in grade school…and consequently, I feel like it’s finally making sense to me these past few years of teaching Professor B’s methods to my children.
So my daughter is learning ALLLL about fractions and we got the coolest fraction “tips” in today’s lesson.
Here you go for the next time your child is stumped as to which procedure to use for which fractional operation…
Know the code:
f=fraction, MN=mixed number, LCD=least common denominator
When the first formula says “f+f = LCD then add” it’s a short hand way of saying: “fraction plus fraction equals finding the least common denominator, then adding the fractions”…
Okay…for the formulas:
- f+f=LCD then add
- f-f=LCD then subtract
- fxf=reduce (if possible), then multiply across
- f (divided by) f=invert and multiply
- MN+MN=LCD then add
- MN-MN=LCD then subtract
- MNxMN=set up fraction, then multiply fraction
- MN (divided by) MN=set up fraction then divide by fraction
If you drill your children on these formulas, and keep a “cheat sheet” taped to the inside of their mathbooks, eventually they won’t have to think very hard about which formula applies to their various fraction pursuits!
Gotta love it!
Professor B says whatever you do, if you learn your fractions backwards and forwards algebra will be a lot less intimidating!
Hope this helps some harried mom out there with her child’s homework!


I want to design a space in which the user is the architect of the space and constructs it using the help of others stepping in within the defined boundaries. Users act as anchor points and restructure the space dynamically as they move around. My explorations do not intend to be a game, or an interactive installation but merely a reinterpretation of how space and architecture is created/defined. In these diagrams, users are anchor points that form Delaunay triangulations.
In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulationDT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Connecting the centers of the circumcircles produces the Voronoi diagram.
In these diagrams, users act as points that form Delauney Triangulaltions. After a certain number of points, then a Voronoi is created and the users (with the help of each other) will then see the fruits of their labour: the construction of a 3D geometric shape.


See how much you can save over time. Use this calculator any time you need motivation.
About Savings Calculator
A savings calculator helps you determine the amount you need to save each month. You can enter your initial deposit, the amount that you will save each month, and the interest rate that you anticipate. Then, you can enter the amount of time that you expect to have the money in your account. Then, you can see how much your savings will be worth when it has accumulated to a certain amount. Using a savings calculator will help you plan your spending based on the estimated value of your account at the end of the period you chose.
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Using a savings calculator can help you plan the best way to deposit your money. With these tools, you can see how much you need to save each month and the interest will accumulate. It’s important to remember that interest in a savings account builds on itself, so if you don’t want to make any monthly deposits, you can just run the numbers. A small deposit every month will move you closer to your goal.
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When you’ve set your goals, you can use a savings calculator to find the best savings account. This will help you determine the amount to deposit each month to reach your savings goal. Some savings calculators will also show you how much money you need to save in a specific time frame to meet your goals. These calculations can help you choose the best savings account for your future. If you want to save more, set up an emergency fund for an unexpected expense.
A savings calculator can also help you set your goals. This tool can help you determine how much you need to save for your future. The amount you save will depend on your personal financial situation. Once you’ve set your goal, you can use a savings calculator to figure out the amount of money you need to save. It is important to remember that you should never borrow money to pay off a debt. It’s important to pay off your debts as early as possible.


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ExperienceI'm a graduate student, so I understand the difficulties learners go through when they face difficult subjects. As a tutor, I can provide guidance to help students become confident in their abilities to learn. I have worked with everyone from small children to graduate students, and I believe that everyone who has motivation can develop an understanding of any subject.
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Get professional results every time with this handy and easy-to-use tool for projects involving liquid or fluid measurements! Work in and convert between US, Imperial and metric units (Gallons, Quarts, Pints, Cups, Ounces, Tablespoons, Teaspoons, Liters and Milliliters). Use it at home, in the kitchen or in the shop for on-the-spot calculations – saving you time, money and frustration!
Also, you never have to look for a basic calculator again as long as you have your phone.
With the calculator you can:
•Add, subtract, multiply and divide and take percents of liquid volumes, for example 5gal + 3qt = 5.75gal or 3cup – 4oz = 2.5cup
•Enter compound units, for example (1cup 3Tbsp) x 2 + (4cup 5tsp) - 1qt = 2.4792cup.
•Mix US, Imperial and metric units in a calculation, for example 1L + 1gal(US) + 1 gal(Imp) = 9.3315L
•Convert a result to multiple units, for example 2 qt(US) = 0.5gal(US), 4pt(US), 8cup(US), 64oz(US), 128 Tbsp(US), 384tsp(US), 1.8927L, 1892.7059ml, 0.4163gal(Imp), 1.6653qt(Imp), 3.3307pt(Imp), 66.6139oz(Imp), 106.5823Tbsp(Imp), and 319.7468tsp(Imp).
• Enter long calculations, for example 2 x 6qt(US) x 4.5 – 2pt(US) – 24oz(US) + 2gal(US) = 60.25qt(US).
• Display a history of your work.
• Work with decimals, 7.5pt(US) + 6.3qt(US) = 20.1pt(US)
• Compare liquid volumes. Take the difference between two volumes to learn which one is bigger and by how much or take the ratio between two volumes to see which is bigger and by what multiplication factor. For example, 1gal(US)-1gal(Imp) = -0.201gal(US). The imperial gallon is larger than the US gallon! Try 1gal(Imp) ÷ 1gal(US) = 1.201. The imperial gallon is 1.201 times larger than a US gallon.
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•Even use it as a basic calculator, for example 12 x 12 = 144 or 5 x 5 x 4 x 6 + 11.23 = 611.23
It's incredibly easy to use, the display is easy to read and there's nothing like traveling light - A great addition to the toolbox of anyone from a home owners (compute fuel consumption, compute mileage or compare volumes in a grocery store), cooks or chefs(to scale food related recipes or convert recipes) or mechanical engineers/chemists (chemical volumes) /chemical engineers that need to easily convert, mix or manipulate volume measurements.
Liquid Calculator is a free trial software application from the System Maintenance subcategory, part of the System Utilities category.
The app is currently available in English and it was last updated on 2012-01-19. The program can be installed on Android.
Liquid Calculator (version 1.1) has a file size of 165.68 MB and is available for download from our website.
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