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More than 50% of the homework problems are new, and there are also a number of new sample problems. To help students build necessary visualization and problem-solving skills. Advanced Engineering Mathematics 9th Edition with Math Computer Guide Set (Hardcover) Published September 8th by John Wiley & Sons Hardcover, 0 pagesCited by: How is Chegg Study better than a printed Advanced Engineering Mathematics 8th Edition student solution manual from the bookstore.
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This note is a survey of observational astronomy across the electromagnetic spectrum. Topics covered includes: overview of current. Here is an unordered list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web.
I tried to select only the works in book formats, "real" books that are mainly in PDF format, so many well-known html-based mathematics web pages and online tutorials are left out.
Mathematics books Need help in math. Delve into mathematical models and concepts, limit value or engineering mathematics and find the answers to all your questions. It doesn't need to be that difficult. Our math books are for all study levels.
In addition to expanded explanations, the 10th edition includes new problems, updated figures and examples to help motivate students. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study.
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plus two others characteristics (cumulative frequency and cumulative relative frequency) thus including information about how they are sorted. Cumulative frequency of the i-th variant mi - is a number of values of a variable showing the frequency of variants less or equal the i-th variant.
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I am a theoretical and computational ecologist. My research program is motivated by the pervasive existence of context-dependency—results are only true under particular circumstances—in ecological systems, which fundamentally hinders generalization and prediction under uncertainty. To address this problem, my research centers on developing a theoretically rigorous and computationally feasible ‘common currency’, and using the common currency to synthesize empirical findings across study systems and scales.
I am currently a postdoc with Jonathan Levine at Princeton University. I was a postdoc co-supervised by Andrew Gonzalez at McGill University and Marie-Josée Fortin at University of Toronto from 2020-2022. I received a PhD in Civil and Environmental Engineering from MIT under the supervision of Serguei Saavedra, and a BS in Mathematics from Zhejiang University under the supervision of Yang-Yu Liu.
You can reach me via email: clsong.ecology AT gmail DOT com.
PhD in Civil and Environmental Engineering, 2016 - 2020
Massachusetts Institute of Technology
BSc in Mathematics, 2013 - 2016


Homeschool+™ Frequently Asked Questions
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Homeschool+ is a homeschool curriculum designed to support home educators in creating a customized learning environment for children ages 4 to 8. The curriculum includes fully adaptive math and reading programs and 19 online courses covering math, reading, science, language arts, social studies, art, Spanish, and more.
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Homeschool+ is perfect for homeschool families with children ages 4-to-8 years old and home educators looking for additional home instruction tools coupled with complete lesson plans, a lesson planner, and robust progress tracking. Learning experts designed the curriculum to be customizable to fit the needs of multiple homeschool styles and unique needs of your homeschool.
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I hate math. But life is all about numbers; especially a diabetic life. I guess my hatred for math could be one reason why I have such a difficult time getting through life and controlling my diabetes. I’m not bad at math, it actually isn’t all that difficult in a day-to-day perspective. I just hate doing it, especially all day, everyday, over and over again. Ugh!
For a diabetic, math comes up a million times a day. Counting carbs, protein, fat, and calories with every meal. Checking blood sugar at least four times a day. Determining insulin dosage based on those food counts and blood sugar readings at least four times per day.
Yeah, it’s a number game.
I was in bed trying to fall asleep the other night and I got to wondering just how many times I’ve pricked my finger and injected myself in my lifetime. Then the question evolved to how many doctors visits, tests, procedures, hospitalizations, surgeries, etc., I’ve had because of my diabetes.
I’ve been diabetic for 27 years. Let’s add up the numbers.
These are mostly estimates due to the fact that schedules and routines have changed year to year…and other factors.
Finger pricks: >40,000
Injections (not just insulin): >50,000
Doctor visits: >300
Lab Draws: >120
Tests (other than lab draws): >100
Procedures (not surgery): >30
And many, many more numbers to come!


intuitionism n : (philosophy) the doctrine that knowledge is acquired primarily by intuition
- This article is about Intuitionism in mathematics and philosophical logic. For other uses, see Ethical intuitionism.
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs.
Truth and proof
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. As the name suggests, in Brouwer's original intuitionism, the truth of a statement is taken to be equivalent to the mathematician being able to intuit the statement. The vagueness of the intuitionistic notion of truth often leads to misinterpretations about its meaning. Kleene formally defined intuitionistic truth from a realist position, however Brouwer would likely reject this formalization as meaningless, given his rejection of the realist/Platonist position. Intuitionistic truth therefore remains somewhat ill defined. Regardless of how it is interpreted, intuitionism does not equate the truth of a mathematical statement with its provability. However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he/she proves is in fact intuitionistically true. This gives rise to intuitionistic logic.
To claim an object with certain properties exists is, to an intuitionist, to claim to be able to construct a certain object with those properties. Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a constructive proof of existence. As such, intuitionism is a variety of mathematical constructivism; but it is not the only kind.
As well, to say A or B, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, A or not A, is disallowed since one can construct, via Gödel's incompleteness theorems, a mathematical statement that can be neither proven nor disproved.
The interpretation of negation is also different. In classical logic, the negation of a statement asserts that the statement is false; to an intuitionist, it means the statement is refutable (i.e., that there is a proof that there is no proof of it). The asymmetry between a positive and negative statement becomes apparent. If a statement P is provable, then it is certainly impossible to prove that there is no proof of P; however, just because there is no proof that there is no proof of P, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.
Intuitionistic logic substitutes justification for truth in its logical calculus. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has given philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett.
Intuitionism also rejects the abstraction of actual infinity; i.e., it does not consider as given objects infinite entities such as the set of all natural numbers or an arbitrary sequence of rational numbers. This requires the reconstruction of the foundations of set theory and calculus as constructivist set theory and constructivist analysis respectively.
History of IntuitionismIntuitionism's history can perhaps be traced to the nineteenth century. Cantor and his teacher Kronecker — a confirmed finitist — disagreed. Frege's effort to reduce all of mathematics to a logical formulation was a scientific breakthrough in the department of logic, and it greatly inspired the younger generation, including a youthful Bertrand Russell.
But Frege himself counted it as failure when a young Bertrand Russell sent Frege a letter about his hot-off-the-presses first volume, outlining the famous paradox now known as Russell's Paradox, that showed how one of Frege's rules of self-reference was self-contradictory. Frege, the story goes, plunged into depression and did not publish the second and third volumes of his work as he had planned. For more see Davis (2000) Chapters 3 and 4: Frege: From Breakthrough to Despair and Cantor: Detour through Infinity. See van Heijenoort for the original works and Heijenoort's commentary.
In the early twentieth century Brouwer represented the intuitionist position and Hilbert the formalist — see van Heijenoort. Kurt Gödel offered opinions referred to as Platonist. (see various sources re Gödel). Alan Turing considers: "non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive" (Turing (1939) Systems of Logic Based on Ordinals in Undecidable, p. 210) Later, Kleene brought forth a more rational consideration of intuitionism in his Introduction to Meta-mathematics (1952). For the view that there are no paradoxes in Cantorian set theory — thus calling into question the program of intuitionist mathematics, see Alejandro Garciadiego's now-classic Bertrand Russell and the Origins of the Set-Theoretic Paradoxes.
Branches of intuitionistic mathematics
- W. S. Anglin, Mathematics: A Concise history and Philosophy, Springer-Verlag, New York, 1994.
- Martin Davis (ed.) (1965), The Undecidable, Raven Press, Hewlett, NY. Compilation of original papers by Gödel, Church, Kleene, Turing, Rosser, and Post.
- Engines of Logic: Mathematicians and the origin of the Computer
- John W. Dawson Jr., Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley, MA, 1997.
- Less readable than Goldstein but, in Chapter III Excursis, Dawson gives an excellent "A Capsule History of the Development of Logic to 1928".
- Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Godel, Atlas Books, W.W. Norton, New York, 2005.
- In Chapter II Hilbert and the Formalists Goldstein gives further historical context. As a Platonist Gödel was reticent in the presence of the logical positivism of the Vienna Circle. She discusses Wittgenstein's impact and the impact of the formalists. Goldstein notes that the intuitionists were even more opposed to Platonism than Formalism.
- van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. The following papers appear in van Heijenoort:
- L.E.J. Brouwer, 1923, On the significance of the principle of excluded middle in mathematics, especially in function theory [reprinted with commentary, p. 334, van Heijenoort]
- Andrei Nikolaevich Kolmogorov, 1925, On the principle of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]
- L.E.J. Brouwer, 1927, On the domains of definitions of functions, [reprinted with commentary, p. 446, van Heijenoort]
- Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
- L.E.J. Brouwer, 1927(2), Intuitionistic reflections on formalism, [reprinted with commentary, p. 490, van Heijenoort]
- Jacques Herbrand, (1931b), "On the consistency of arithmetic", [reprinted with commentary, p. 618ff, van Heijenoort]
- From van Heijenoort's commentary it is unclear whether or not Herbrand was a true "intuitionist"; Gödel (1963) asserted that indeed "...Herbrand was an intuitionist". But van Heijenoort says Herbrand's conception was "on the whole much closer to that of Hilbert's word 'finitary' ('finit') that to "intuitionistic" as applied to Brouwer's doctrine".
- Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s
- Arend Heyting: Intuitionism: An Introduction
- Introduction to Meta-Mathematics
- In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.
- Constance Reid, Hilbert, Copernicus - Springer-Verlag, 1st edition 1970, 2nd edition 1996.
- Definitive biography of Hilbert places his "Program" in historical context together with the subsequent fighting, sometimes rancorous, between the Intuitionists and the Formalists.
- Paul Rosenbloom, The Elements of Mathematical Logic, Dover Publications Inc, Mineola, New York, 1950.
- In a style more of Principia Mathematica -- many symbols, some antique, some from German script. Very good discussions of intuitionism in the following locations: pages 51-58 in Section 4 Many Valued Logics, Modal Logics, Intuitionism; pages 69-73 Chapter III The Logic of Propostional Functions Section 1 Informal Introduction; and p. 146-151 Section 7 the Axiom of Choice.
- A. A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algorifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]
- A secondary reference for specialists: Markov opined that "The entire significance for mathematics of rendering more precise the concept of algorithm emerges, however, in connection with the problem of a constructive foundation for mathematics....[p. 3, italics added.] Markov believed that further applications of his work "merit a special book, which the author hopes to write in the future" (p. 3). Sadly, said work apparently never appeared.
intuitionism in Czech: Intuicionistická logika
intuitionism in German: Intuitionismus
intuitionism in Spanish: Intuicionismo
intuitionism in Esperanto: Intuiciismo
intuitionism in Croatian: Intuicionizam
intuitionism in Italian: Intuizionismo
intuitionism in Dutch: Intuïtionisme
intuitionism in Japanese: 数学的直観主義
intuitionism in Polish: Intuicjonizm (matematyka)
intuitionism in Portuguese: Intuicionismo
intuitionism in Russian: Интуиционизм
intuitionism in Turkish: Sezgici Matematik
intuitionism in Chinese: 数学直觉主义


Elliptic Curve Cryptography (ECC)
Elliptic Curve Cryptography (ECC) is a key-based technique for encrypting data famous for being smaller, faster, and more efficient than incumbents. ECC focuses on pairs of public and private keys for decryption and encryption of web traffic. ECC is frequently discussed in the context of the Rivest–Shamir–Adleman (RSA) cryptographic algorithm.
There are two main different types of encryptions - symmetric encryption, which uses one key to both encrypt and decrypt (e.g. AES), and asymmetric encryption, which uses two different keys (e.g. RSA). These are often called a public and private key, where the private key is not to be disclosed.
RSA uses integer factorization cryptography based on algebraic number theory, while elliptic curve cryptography (ECC) uses integer factorization cryptography based on elliptic curves.
Elliptic Curve Cryptography is a choice for public-key-cryptography, based on elliptic curves over finite fields.
ECC is used as the cryptographic key algorithm in Bitcoin because it potentially can save ~90% of the resources used by a similar RSA system.
RSA vs. ECC
ECC and RSA both generate a public and private key and allow two parties to communicate securely. One advantage to ECC however, is that a 256-bit key in ECC offers about the same security as a 3072-bit key using RSA. ECC allows resource-constrained systems like smartphones, embedded computers, and cryptocurrency networks to use ~10% of the storage space and bandwidth required by RSA.
Based on the idea that the key that anyone uses to encrypt data can be made public while the key that is used to decrypt your data can be kept private. As such, these systems are known as public key cryptographic systems. The first, and still most widely used of these systems, is known as RSA — named after the initials of the three men who first publicly described the algorithm: Ron Rivest, Adi Shamir and Leonard Adleman.
When both the RSA algorithm and the Diffie-Hellman key exchange algorithm were introduced, these new algorithms were revolutionary because they represented the first viable cryptographic schemes where security was based on the theory of numbers. It was the first to enable secure communication between two parties without a shared secret.
Cryptography went from being about securely transporting secret codebooks around the world to being able to have secure communication between any two parties without worrying about someone listening in on the key exchange.
After the introduction of RSA and Diffie-Hellman, researchers explored other mathematics-based cryptographic solutions looking for other algorithms beyond factoring that would serve as good Trapdoor Functions.
In 1985, cryptographic algorithms were proposed based on an esoteric branch of mathematics called elliptic curves.
An elliptic curve is the set of points that satisfy a specific mathematical equation, something that looks a bit like the Lululemon logo tipped on its side.
There are other representations of elliptic curves, but technically an elliptic curve is the set points satisfying an equation in two variables with degree two in one of the variables and three in the other. An elliptic curve has some properties that make it a good setting for cryptography.
Neal Koblitz and Victor Miller independently co-discovered elliptic-curve cryptography which is part of the mathematics that allows encrypted communication on the internet today.
In an interview with All About Circuits (AAC) in 2019, Dr. Koblitz said:
"What changed my feeling about number theory was the invention of RSA cryptography (Rivest-Shamir-Adleman) in about 1977. That was the first important application of number theory to computer security."
"The idea of elliptic-curve cryptography came in 1984. I, along with several other people received a pre-print, a rather preliminary-version, of an algorithm that Hendrik Lenstra developed to factor large integer numbers. If this algorithm was sufficiently fast, it could be a threat to RSA cryptography. "
Elliptic Curve Cryptography (ECC)
December 12, 2023
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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!


Dial Gauge Angle Finder
This Dial gauge angle finder can be used to measure the angle of the surface of an object with respect to the horizon. The weighted needle always points up, perpendicular to the horizon, while the side of the angle finder is held against the surface to be measured. The angle of the solar panels aimed at the sun is measured with the angle finder.
- Dial gauge angle measurements
- Measures angle from the horizon
- Measures 0 to 90 degrees
- Accuracy of 0.5 degrees
Make STEM Fun & Easy!
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A car accelerates uniformly from rest to 20 m/s in 5 seconds. Determine the acceleration of the car and the distance traveled.
First, you calculate the acceleration by subtracting the initial velocity from the final velocity and then dividing by the time.
vf - vi = 20 - 0 = 20 m/s
a = 20 / t = 20 / 5 = 4 m/s^2
Next, you calculate the displacement by multiplying the initial velocity by the time and adding that to the product of one-half times the acceleration times the time squared.
d = (vi * t) + (.5 * a * t^2)
d = (0 * 5) + (.5 * 4 * 5^2) = 50 m


Our topic this term is called ‘Toys’. We will be using a variety of fiction texts as our stimulus: Benjamin Boo, Brown Paper Bear, Here Comes Traction Man and Toys in Space. The children will be writing labels and captions, story writing and producing instructions and information pages related to these books and to toys they’ve looked at or made.
Our topic will be linked to the books we are reading and using in our literacy lessons.
In art we will be making junk model aliens and ‘cuddly’ alien toys. In science we will be looking at materials that float and sink; we will be discovering the best materials to use for a mini parachute; we will also be looking at friction and gravity. In history the children will be finding out how the materials used in toys have changed over time; the children will also make old fashioned peg dolls (with a super hero twist!) and thread spinners. We will be taking part in a toy workshop where the children will have the opportunities to play with Victorian and Edwardian toys.
This term the children will be consolidating the core number concepts learned in Reception. This includes counting on from any number, finding one more or one less than a number to 20 and solving one-step addition and subtraction questions. We will also begin learn how to count in multiples of 2, 5 and 10 (we will use counting songs from YouTube).
We will continue to learn about - and explore – shapes. The children will revise the names and properties of common 2D shapes; they will also learn about 3D shapes and their properties. In addition the children will learn how to describe direction, movement and position.
Your child will need to wear the following kit which should be kept in school throughout this half term: light blue t-shirt, black shorts and plimsolls or trainers. The t-shirt will be provided by the school.
PE days are as follows:
1H - Tuesday morning
1N - Wednesday morning
1C - Thursday Morning
1M (VR) - Tuesday Afternoon
1W (VR) - Wednesday Afternoon
Home reading books will be changed on your child's PE day.


A normal calculator, often referred to as a basic calculator or pocket calculator, is a simple device designed primarily for performing fundamental arithmetic operations. Here are the key characteristics of a normal calculator:
- Arithmetic Functions: A normal calculator can handle basic arithmetic functions, including addition, subtraction, multiplication, and division. It's ideal for everyday calculations like balancing your checkbook, calculating tips, or solving simple math problems.
- Limited Functions: Normal calculators typically have a limited set of functions and buttons. You'll find keys for numbers, basic operations (+, -, *, /), and possibly a percentage key.
- Compact and Portable: These calculators are compact, lightweight, and easy to carry around. They are battery-powered and fit easily in a pocket or purse.
- Affordable: Normal calculators are inexpensive and readily available. They are suitable for everyday use and are widely used in schools and offices.
A scientific calculator is a more advanced and versatile tool designed for complex mathematical and scientific calculations. Here's what sets it apart from a normal calculator:
- Advanced Functions: Scientific calculators are equipped with a wide range of functions beyond basic arithmetic. They can handle trigonometric functions (sin, cos, tan), logarithms, exponentiation, square roots, and more. These functions are essential for students and professionals in fields like mathematics, engineering, and science.
- Multi-line Display: Scientific calculators typically feature a multi-line display that allows users to see both the input and the result simultaneously. This is incredibly useful for tracking complex calculations.
- Constants and Variables: Scientific calculators often provide the ability to work with constants and variables, making them indispensable for solving algebraic equations and scientific problems.
- Programmable Features: Some high-end scientific calculators offer programmable functions, allowing users to create custom calculations and formulas. This is particularly useful in research and engineering.
- Notation Modes: Scientific calculators may support different notation modes, including degrees and radians, to accommodate various mathematical conventions.
When to Use Each Calculator:
Normal Calculator: Use a normal calculator for basic everyday math, such as addition, subtraction, multiplication, and division. It's perfect for quick calculations where advanced functions aren't necessary.
Scientific Calculator: Opt for a scientific calculator when you need to perform more complex mathematical or scientific calculations. These are invaluable tools for students, engineers, scientists, and professionals working with advanced mathematics.
In summary, normal calculators and scientific calculators serve different purposes, with the latter offering an array of advanced functions tailored to scientific and mathematical applications. Choosing the right calculator depends on your specific needs, so be sure to consider the complexity of your calculations before making your selection. Whether you need a basic calculator for simple math or a scientific calculator for advanced calculations, having the right tool on hand can make a world of difference in your work or studies.


Trading in the Global Sports Betting Market
If bets were placed on the toss of a coin, the probability of heads is 50 %, the probability of tails is 50%. Regarding bookmaker decimal odds, 50% probability is expressed as odds of 2.0. Which means that if bets were continuously placed on either heads or tails at odds of 2.0, over the long term the bettors’ bankroll would break even.
However, if odds of 2.2 were available from the market, the odds of 2.2 are miss-priced as they are greater than 2, which means a real odds edge is available. The correctly priced odds are 2 which is a probability of 50%, odds of 2.2 is a probability of 45.5%, which means a real odds edge of +4.5% (50% – 45.5%) exists.


Mathematical Modeling Engineer
Unravel Complexity and Drive Efficiency with Our Mathematical Modeling Engineer
In the ever-evolving landscape of engineering, one tool has emerged as a driving force behind innovation and efficiency – mathematical modeling. At the intersection of abstract mathematical concepts and real-world engineering challenges, our Mathematical Modeling Engineer operates, providing solutions that illuminate complex problems and revolutionize decision-making processes.
Who is a Mathematical Modeling Engineer?
A Mathematical Modeling Engineer employs mathematical methods and theories to develop models that mirror real-world engineering systems, scenarios, or phenomena. This unique approach transcends empirical observation, unveiling intricate dynamics, predicting outcomes, and informing strategic decisions.
How Our Mathematical Modeling Engineer Can Benefit Your Business:
- Reveal Complex Physical Problems in Engineering: With the power of mathematical modeling, complex engineering challenges are converted into solvable mathematical problems. By analyzing these models, our engineer can unravel intricate dynamics, discover underpinning principles, and generate solutions that might have remained obscured in a purely empirical approach.
- Reduce Troubleshooting Costs: In engineering, troubleshooting can be a lengthy and expensive process. Mathematical modeling offers an efficient alternative. By simulating various scenarios or system configurations, our Mathematical Modeling Engineer can identify potential problems before they occur, saving time and reducing troubleshooting costs.
- Accelerate R&D Processes: Mathematical models enable experimentation that would be too costly, time-consuming, or even impossible to conduct in the real world. By testing hypotheses and exploring system behavior in the mathematical realm, our engineer can expedite the R&D process, leading to faster innovation and lower costs.
Why Choose Our Mathematical Modeling Engineer?
Expertise Across Domains:
Our Mathematical Modeling Engineer brings a solid foundation in mathematics and a deep understanding of various engineering disciplines. This combination enables them to develop reliable and versatile models for a wide range of applications.
Advanced Tools and Techniques:
Equipped with state-of-the-art mathematical software and advanced modeling techniques, our Mathematical Modeling Engineer delivers robust, accurate models that pave the way for efficient problem-solving and effective decision-making.
Holistic Engineering Solutions:
As part of our comprehensive suite of engineering services, our Mathematical Modeling Engineer can collaborate with other specialists in our team, ensuring that you receive holistic solutions tailored to your unique engineering needs.
With extensive experience in applying mathematical modeling to real-world engineering problems, our engineer understands the nuances and complexities of various sectors, providing insights and solutions that are practically viable and impactful.
Our Mathematical Modeling Engineer can help your business navigate complexity, improve efficiency, and accelerate innovation. Reach out to us today to discover how our Mathematical Modeling Engineer can transform your engineering challenges into strategic advantages.
Case Study 1: Optimizing Production in Automotive Manufacturing
Background: An automotive manufacturer was seeking to optimize their production line to improve throughput and reduce costs.
Solution: Our Mathematical Modeling Engineer developed a model simulating the entire production process, taking into account various factors such as machine speeds, setup times, worker schedules, and raw material availability. By manipulating these variables within the model, they identified an optimal configuration that increased throughput and minimized costs.
Result: The company implemented the suggested changes and saw a 15% increase in production throughput and a significant reduction in operational costs.
Case Study 2: Risk Mitigation in Renewable Energy Project
Background: A renewable energy company wanted to assess the financial viability and potential risks of a large-scale wind farm project.
Solution: Our Mathematical Modeling Engineer built a comprehensive model incorporating factors like projected wind speeds, turbine efficiency, maintenance costs, and potential downtime. They were able to predict energy output, identify potential risks, and calculate projected returns.
Result: Armed with this information, the company could make informed decisions about the project, balancing potential risks and returns. The project has since been successful, with performance closely matching the predictions from the mathematical model.
Case Study 3: Reducing R&D Costs in Medical Devices
Background: A medical device company was developing a new product but wanted to reduce the time and costs associated with R&D.
Solution: Our Mathematical Modeling Engineer created a model simulating the functionality of the device, allowing for extensive testing and refinement in a virtual environment. This approach significantly reduced the need for physical prototyping and testing.
Result: The company was able to accelerate the development process, reduce R&D costs, and bring the product to market faster than traditionally possible.
These case studies demonstrate how our Mathematical Modeling Engineer can provide actionable insights across various sectors, driving efficiency, mitigating risks, and accelerating innovation.


Get a jump start on back to school shopping and score this Texas Instruments 2-Line Scientific Calculator for only $8.88 (Reg. $17). If you have a high school student, you probably need this.
- Robust, professional grade scientific calculator. Logs and antilogs
- It has 2-line display shows entry and calculated result at same time
- Easily handles 1 and 2 variable statistical calculations and three angle modes (degrees, radians, and grads) and scientific and engineering notation modes
- It has 1-year limited warranty
- The front of the calculator is black, the back cover is in fact a dark blue-gray


04/28/2011: JUMPIN' MATH JAM FREE has just been released! It is a free ad supported trial version of JUMPIN' MATH JAM, but includes only addition and multiplication. Give it a try!
04/12/2011:Get a preview of JUMPIN' MATH JAM on YOUTUBE! You can learn how the game works, discover features, and determine if this app is right for you!
04/09/2011: JUMPIN' MATH JAM USERS: Thank you very much for your positive feedback!
v1.1 Has just been released in response to your requests for new features! Check it out on the Android Market!
APPTITUDE DEVELOPERS is pleased to announce the launch of its first mobile application for the Android platform, JUMPIN' MATH JAM!
JUMPIN' MATH JAM is a really fun way to practice addition, subtraction, multiplication, and division facts as you (a frog) try to beat the turtle in a race to the next level. It's a fun game that's more entertaining than math flash cards, and can be quite addictive! For 5th graders and older, the "random" mode is recommended as it takes you through all operations up to the 12's tables. For earlier elementary students, focusing on just one operation at a time can help reinforce skills. As you go through the levels, the game moves faster and faster! JUMPIN' MATH JAM can be found in the Android Marketplace.


A North Carolina man's love for math ending up winning him more than just good grades in school — he was the lucky winner of a nearly $200,000 lottery jackpot.
Jonathan Ruby, of Raleigh, recently tried his luck in the Cash 5 lottery games, purchased a $1 ticket from the Han-Dee Hugo's on Hillsborough Street for the Nov. 28 drawing, according to a release from the NC Education Lottery. The lover of math used a well known number — pi, or 3.1415 — to choose his ticket, a move that ended up winning him a third of the $578,823 jackpot, splitting the prize with two other lucky winners.
"I've always been an extremely big math person," said Ruby. "I picked my numbers based on pi."
Ruby told lottery officials that the lucky sequence has shown up in different ways throughout his life, so he took it as a sign to get his ticket.
"I kept seeing that number so my karma told me to use it," he said. "I even lived at a 314 address as a child."
Ruby had a good feeling about his ticket, sure that he had won something, but it wasn't until later that he realized just how lucky he was.
"I thought I won a dollar or two," Ruby said. "I was calm until I got home and checked the numbers and then I got very excited."
Ruby claimed his $192,941 jackpot at lottery headquarters on Monday (May 9), bringing home a total of $137,012 after taxes. When asked what he plans to do with his prize, the 64-year-old bartender told lottery officials he plans to pay some bill and save the rest for retirement.


Mr. Luis was my mathematics teacher throughout my matriculation at the Oprah Winfrey Leadership Academy for Girls (OWLAG). I entered OWLAG from amongst the most under-resourced primary schools of my admission class. This meant that I was both at a relative academic disadvantage while being newly inducted to the level of academic rigour of a private South African high school. With all the odds stacked against me, I graduated high school as one the top students in mathematics and valedictorian of my class, and Mr. Luis was undoubtedly paramount in my ability to achieve this feat.
Throughout my time at OWLAG, Mr. Luis was committed to excellence in education and had the unshakeable belief in any students’ ability to grasp mathematics given a teacher who is skilled at playing to their students’ strengths while cognisant of areas that pose a challenge for them. He embodied this commitment and belief through relentless and tireless help and support, going as far as giving up his weekday afternoons and Saturdays spending countless hours giving extra review classes and helping me and my class keep up with the increased pace in mathematics he introduced because he envisioned us performing at the level of some of South Africa’s best schools, despite our backgrounds.
Almost ten years later, and having graduated salutatorian of my undergraduate class pursuing a BS in Biology at Spelman College in Atlanta; earned a MSc in Medical Anthropology at the University of Oxford in the UK and currently pursuing a medical degree at Stanford University in California, Mr. Luis remains one of the best teachers I have ever had. His love for mathematics (and teaching) remains unmatched. He always finds innovative ways of teaching and making mathematics instinctive. I know I speak for many of my OWLAG peers when I say, he created an engaging and fun learning environment that made an infamously hard subject rather enjoyable. Whether it was to travel the world or be the next world’s best mathematicians, we always left his classes feeling incredibly inspired.
Past Student Of John Luis


In the Maths department we aim to embed the whole school curriculum intent, through and beyond our curriculum. We do these in line with the three principles across the school.
|Globally Diverse||Ambitious for the Future||Inquisitive Learners|
In Maths we aim to raise awareness around the following financial topics through the mathematics students learn:
In Maths we also aim to raise awareness of how mathematics can be used to mislead, Teachers identify this through:
Students are introduced to famous mathematicians around the world and their contributions to the mathematics we study today.
In Maths we strive to ensure that all students are numerate by the end of Key Stage 3. We aim to end the phrase ‘I can’t do maths’ by developing confidence and instilling a growth mindset in all our students.
We regularly raise awareness of different professions relating to maths and identifying possible ways where students will require maths in future careers.
We regularly set homework to promote self study and organisation.
We hold regular revision sessions to support independent study. Pupils also have the opportunity to purchase revision resources to help support them in their independent studies.
The mathematics department provides opportunities for students to explore their knowledge and face challenges. Previous opportunities provided include UKMT team challenges, the British Museum, Faraday Challenge and Thorpe Park.
At KS5 pupils are taught the applications and interpretations IB mathematics course; this encourages students to further develop their understanding through investigations and applying these to real life. Students currently have the opportunity to study this at both standard and higher level
There is a large focus in the mathematics curriculum towards problem solving. Staff work with students to develop the resilience and critical thinking skills to break down larger problems into smaller more manageable ones. These are often also taught through real life applications
In all Key Stages students are introduced to different careers in mathematics in lessons and as part of careers week within the school.
The mathematics department are passionate about the subject they teach and want all pupils to share this passion.
Our curriculum develops problem solving skills and provides students with the opportunity to make connections. They will develop their mathematical communication both verbally and in written work. Students will develop resilience, independence and take ownership of their learning through rich open tasks and research based learning.
We want to ensure that students take pride in their work. Presentation is key to showing good mathematics with logic and organised thought. Students will critically evaluate their own work and develop their previous learning through encouragement of reflections, reviewing and correcting.
We want pupils to leave KS4 with the skills to continue studying maths at KS5 and beyond.
Our schemes of learning in all Key Stages develops mastery of skills and will develop links to other subjects. Students will develop a greater depth to their knowledge and be able to apply this to various contexts. They will also better develop their vocabulary with subject specific words.
The mathematics curriculum is a spiralling curriculum allowing students to revisit and build on their knowledge developing the idea that we can always further our knowledge and understanding of different topics. Especially as they mature and their prioritise for studies change depending on their choices for their future.
In key stage 3, we have a mastery curriculum, allowing students to develop both fluency and understanding. They build on their knowledge, developing the idea that we can always further our knowledge and understanding of different topics. We use rich open tasks, developing resilience, independence and ownership of learning. Clear written communication and presentation is promoted which students can use in all aspects of their learning. During ‘Maths Week’, students have the opportunity for self-discovery, particularly in Year 8 when they investigate Pythagoras’ Theorem.
We use rich and open ended tasks which allow students to discover their own insights and apply to the real world. This helps with resilience and problem solving too. We regularly set homework to promote self study and organisation.
The Mathematics department provides opportunities for students to explore their knowledge and face challenges. Previous opportunities provided include UKMT team challenges, the Maths Feast, the British Museum, Bletchley Park, Faraday Challenge and Thorpe Park.
We have an annual ‘Maths Week’ which presents maths in real life and problem solving situations to help students make links between the subject and their everyday life. In addition, Year 7 considers estimating as it is a vital skill in budgeting and financial mathematical fluency.
We teach about best buys through ratio in Year 8, so students can understand value for money. Class discussions give students the opportunity to build listening skills and mutual respect. Investigations ask students to share responsibility with others and work collaboratively. Students are asked to reflect upon their own learning, to further understand and develop their learning journey.
In key stage 4 students study the GCSE Mathematics course, we have designed our curriculum so topics are interwoven and further develop mastery, problem solving and resilience, leading to independent learners who take ownership of their own learning. Students also have the opportunity to make greater connections between subjects and develop their own learning. Statistics GCSE for our most able students gives the opportunity for critical thinking and real life applications for their learning by forcing justification behind answers, spotting misleading information and understanding/arguing different points of view. We teach financial maths such as compound interest, depreciation, loans and mortgages. Through work on representing and analysing data, students begin to think critically about information that is presented to them as well as being exposed to situations where data may be misleading.
Some students also pursue GCSE Statistics which allows students to become better critical learners, looking further than the initial data or information to understand the situation. We continue setting regular homework to promote self study and organisation.
In key stage 5 students can study IB Mathematics Applications and Interpretation, through this course we continue to develop students' independence of study. The introduction of IB mathematics gives more opportunities for application of mathematics and real life scenarios. The Internal Assessment stretches students beyond the outlined curriculum, giving opportunities for personal investigation. The course also encourages students to further develop their understanding through investigations and applying these to real life.
We continue to develop students’ independence with the introduction of the internal assessment. Firm deadlines are introduced which compel students to manage their time. Students are given the opportunity to teach each other, becoming responsible for others' learning as well as their own. In this scenario, students use their presentation and pupil speaking skills as well as ensuring they are prepared in a timely manner.


Darko Odic’s research examines how children acquire abstract cognitive and perceptual representations, including those of number, confidence, and mathematical thought. By examining individual and developmental differences in, for example, how children’s earliest intuitions about number inform their formal math abilities, he seeks to understand why children sometimes learn complex ideas with ease, and sometimes struggle.
My plans for the fellowship period
To learn, we must make mistakes. But mistakes are only useful when they are noticed, evaluated, and corrected. When learning mathematics, a topic particularly challenging for many children around the world, “error monitoring” is especially important: math problems often require one answer in a sea of (literally) infinite possibilities. Unlike learning about gravity through building blocks, math problems do not physically fall down when a mistake is made; unlike learning social skills, math problems don’t flash with moments of anger or joy; unlike reading, math mistakes do not lead to meaningless strings of words.
During the Fellowship period, I will be investigating one potential mechanism for how children catch errors in mathematics: by using their intuitive (but approximate) sense of number. We hypothesize that young children use their intuitive number system to arrive at a probabilistic prediction of the likely answer for a simple equation and therefore catch when they are likely to have made a mistake. We have recently shown that differences in children’s number sense correlate with detecting math mistakes made by others. Over the next 2-4 years my lab plans to: (1) use pupillometric measures to measure the degree of surprise children experience when catching mistakes in mathematics; (2) temporarily enhance or impair children’s intuitive number sense and observe the effects this has on their error monitoring; and (3) train children to actively make predictions about the outcomes of math problems to increase the chance of catching and learning from mistakes. I aim to provide both a mechanistic understanding of how core cognition contributes to formal concepts and to characterize why some children are more successful at learning about math compared to others.
How will my work change children’s and youth’s lives?
My research focuses on children’s early mathematical abilities, which have consistently been shown to predict later school and job success, income level, happiness, and personal health. It will help identify mechanisms that allow children to be more independent, effective, and self-efficacious learners, ultimately informing educators on how children can learn complex topics with minimal external feedback. The current proposal, therefore, helps identify the conditions and skills that children should ideally maximize to deepen their development and ability to contribute to society. And, because the intuitive number sense is present in every child from birth, this work also has important universal implications, and seems to promote education initiatives across traditional cultural and economic divides.


Use this calculator to compute how much you would need to have invested in order to withdraw a specified amount each year over the course of a specified period of time. For example, if you want to be able to withdraw $500 during each month of your expected 20-year retirement, this calculator will tell you that if you expect to earn a 10% interest rate you will need to have $51,812.30 saved up by the time you retire. This is often referred to as "Present Value of an Annuity" analysis.
This is a hypothetical example used for illustrative purposes only and does not represent the return of any specific investment. Actual rates of return will vary over time; particularly for long-term investments.
To compute the Present Value of an Annuity, fill in the first three text boxes and then click the "compute" button.


In this episode of the Oxfordshire Teacher Training podcast, Matthew Coatsworth discusses the wonderful work of the NCETM – the National Centre for Excellence in the Teaching of Mathematics – with NCETM Professional Development Leads and Oxfordshire Teacher Training Mentors Claire Shorrock and Crispin Hoad. Claire and Crispin outline some of the key work of the NCETM, the BBO Maths Hub, Teaching for Maths Mastery and the five big ideas, as well as how to cater for the wide variation of mathematical ability within classes and how to adapt teaching effectively.
Please note that this episode was recorded remotely and unfortunately the sound quality from Claire’s device is a little distorted. We hope it does not affect your listening too much!
Links to resources mentioned in the podcast:
(Bucks, Berks and Oxon Maths Hub)
(KS3 Subject Audits)


A First Course in Probability 9th Edition by Sheldon Ross, ISBN-13: 978-0321794772
[PDF eBook eTextbook]
- Publisher: Pearson; 9th edition (December 21, 2012)
- Language: English
- 480 pages
- ISBN-10: 032179477X
- ISBN-13: 978-0321794772
This book is intended as an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and the sciences (including computer science, biology, the social sciences, and management science) who possess the prerequisite knowledge of elementary calculus. It attempts to present not only the mathematics of probability theory, but also, through numerous examples, the many diverse possible applications of this subject.
Content and Course Planning
Chapter 1 presents the basic principles of combinatorial analysis, which are most useful in computing probabilities.
Chapter 2 handles the axioms of probability theory and shows how they can be applied to compute various probabilities of interest.
Chapter 3 deals with the extremely important subjects of conditional probability and independence of events. By a series of examples, we illustrate how conditional probabilities come into play not only when some partial information is available, but also as a tool to enable us to compute probabilities more easily, even when no partial information is present. This extremely important technique of obtaining probabilities by “conditioning” reappears in Chapter 7, where we use it to obtain expectations.
The concept of random variables is introduced in Chapters 4, 5, and 6. Discrete random variables are dealt with in Chapter 4, continuous random variables in Chapter 5, and jointly distributed random variables in Chapter 6. The important concepts of the expected value and the variance of a random variable are introduced in Chapters 4 and 5, and these quantities are then determined for many of the common types of random variables.
Additional properties of the expected value are considered in Chapter 7. Many examples illustrating the usefulness of the result that the expected value of a sum of random variables is equal to the sum of their expected values are presented. Sections on conditional expectation, including its use in prediction, and on momentgenerating functions are contained in this chapter. In addition, the final section introduces the multivariate normal distribution and presents a simple proof concerning the joint distribution of the sample mean and sample variance of a sample from a normal distribution.
Chapter 8 presents the major theoretical results of probability theory. In particular, we prove the strong law of large numbers and the central limit theorem. Our proof of the strong law is a relatively simple one that assumes that the random variables have a finite fourth moment, and our proof of the central limit theorem assumes Levy’s continuity theorem. This chapter also presents such probability inequalities as Markov’s inequality, Chebyshev’s inequality, and Chernoff bounds. The final section of Chapter 8 gives a bound on the error involved when a probability concerning a sum of independent Bernoulli random variables is approximated by the corresponding probability of a Poisson random variable having the same expected value.
Chapter 9 presents some additional topics, such as Markov chains, the Poisson process, and an introduction to information and coding theory, and Chapter 10 considers simulation.
As in the previous edition, three sets of exercises are given at the end of each chapter. They are designated as Problems, Theoretical Exercises, and Self-Test Problems and Exercises. This last set of exercises, for which complete solutions appear in Solutions to Self-Test Problems and Exercises, is designed to help students test their comprehension and study for exams.
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I am a native Slapoutian and a third generation graduate of Holtville High School. I am married to Casey Antley who is also a graduate of Holtville High School. We have a daughter, Scarlett, and three large fur babies. My husband and I both take a lot of pride in our school and our community.
I graduated from Holtville in 2009 and went on to AUM where I graduated in 2014 with my degree in Secondary Education - Mathematics. Recently, I returned to AUM where I completed my Masters in Math Education. After college, I spent a year teaching 7th grade math at Eclectic Middle School before returning "home" to HHS in 2016. I have been teaching math here at HHS ever since. I love math and I hope to pass on my love for the subject to my students.
Please feel free to contact me at anytime if there is anything I can do to help your student succeed!


Summer Curriculum Correspondence
It is already that time of the year to start thinking about the summer. It is vital that each student retains and even improves academic skills during the summer weeks. The jump to third grade is a very significant one.
If your student participates in the Summer Correspondence program, each week for 6 weeks during the summer, your student will receive a packet to help him/her maintain grade level skills. If they finish all of the packets and return the completed cover sheet when school restarts in August, they earn a recognition.
If you would like your student to participate in this summer's Curriculum Correspondence program, please send in six (6) number 10 (that's legal sized) envelopes. Each envelope must have postage affixed. Each envelope must have your mailing address and your student's name. Each envelope should have your mailing address as the return address. Turn the envelopes in to your child's teacher.
We are starting Unit 7, Arrays, Partitioned Rectangles, and Equal Shares, this week. Please see the informational letter sent home on Monday for more specific info.
In this unit, students should be able to arrange items in rectangular arrays and partition rectangles into equal shares. Use the paper Square-Inch Tiles sent home Monday for practice.
Students should be able to:
• easily identify and know the difference between columns and rows
• measure a rectangle and draw lines to partition it into square centimeters or square inches
• partition a rectangular array into halves, thirds, and fourths
• partition a rectangular array by measuring in square centimeters or square inches
• fold a circle to make equal halves and fourths
• fold a rectangle into equal halves, thirds, and fourths in different ways
• solve word problems involving lengths
• use a number line diagram to add and subtract within 100
• calculate the perimeter (distance around) of a shape


By Haripriya Mehta, Co-founder of MehtA+
When the CEO of Math Kangaroo USA, Ms. Joanna asked if MehtA+ would be interested in sponsoring 3000+ digital kits for the 2021 Math Kangaroo competition, without a moment of hesitation, I said yes.
My brother (and the other co-founder of MehtA+) and I had participated in Math Kangaroo since elementary school. I had placed in the top 10 nationally twice and my brother received 1st place internationally twice in high school with a perfect score besides placing top 10 nationally several other times! And while winning and receiving an award, a certificate and a medal were exciting, there was much greater joy in learning and strengthening our mathematical and logic skills as we prepared for and took the Math Kangaroo test. The skills we learned during Math Kangaroo preparation helped me succeed at MIT and my brother at Stanford where we majored in electrical engineering and computer science.
Even so, we wanted our digital kits to be special for the top 10 winners in grades 1–5, which they would be receiving in addition to a certificate from Math Kangaroo. After all, it had been a difficult year with the ongoing pandemic and we wanted to create something that students would be looking forward to. We didn’t have any budget because all of the instructors in our organization are current university students or recent alumni. But, inspired by Math Kangaroo’s volunteers such as Ms. Grazyna, a longtime volunteer of my hometown’s Math Kangaroo chapter, who have dedicated their lives to facilitating this math competition, MehtA+ instructors were absolutely ready to volunteer their time to design a memorable prize!
Designing a prize was definitely a challenging feat. MehtA+’s target demographic for camps, workshops, tutoring and consulting services are middle and high school students so we were not sure what elementary-aged students would like. We also did not want it to be a completely digital prize because it is often more exciting to play with a physical prize. More importantly, we had heard from a lot of grownups that they did not want their students to be looking at their computer screens any more than they needed to, since students had been looking at screens non-stop throughout the virtual school year.
Then, one fine morning, as I was cleaning my room, inspiration struck. A little kangaroo stuffed animal fell out of my bookshelf. This was a kangaroo I had won in Math Kangaroo as an elementary student. I remember how much I had cherished it and kept it ever since. This is what the elementary students would love. MehtA+ instructors could teach elementary school students how to create their own paper model of a kangaroo, while learning principles of mechanical engineering and design.
I enjoyed the MehtA+ Digital Kit. It was fun to assemble and is a great decoration for my study area!
I hurriedly called our Product Lead and instructor, Ms. Marwa who had completed her undergraduate studies in mechanical engineering at MIT, if she could design such a kit. She gladly agreed despite her busy schedule — she was reaching out to labs for her PhD program in Mechanical Engineering at MIT, which she was to begin a couple of months later. She was also fasting as it was Ramadan!
Ms. Marwa first took the Math Kangaroo logo and vectorized it. She exported the file to DXF format so she could create a 3-D model of the kangaroo by extruding the image in CAD software. She then created the unfolded pattern of the 3-D model. However, it turned out that the design of the 3-D model was larger than A4 paper (normal printer paper)! A design larger than A4 paper would mess with the integrity of the paper structure because there would be individual cutouts split between multiple papers rather than having one continuous pattern. Ms. Marwa spent several more hours trying to modify the structure so that it would fit on an A4 paper and it was easy for students to assemble.
It was important for us that the students could create the kangaroo using materials easily accessible to them. The kit was designed in such a way that students could print out the design on printer paper or cardstock and only needed household items to create it — scissors, Q-tips, and glue/tape was sufficient.
Ms. Marwa also recorded videos that taught students how to fold the kangaroo cutout as well as how to create similar cutouts using software that mechanical engineers use — TinkerCad and Pepakura. We figured that this was a great opportunity for younger students to strengthen their motor skills and practice cutting and coloring. For older students, this was a great way to understand the applications of mathematical concepts that they had learned in preparation for the Math Kangaroo.
Growing up, none of our instructors had this kind of exposure to the applications of the theory we learned in school. While there were many 3-D cutouts of different animals found on the web, no one explained to the students how these cutouts were made using mechanical engineering software. We always strive to introduce new concepts to students, while they are having fun!
The kit was a huge success. 2nd grader Gideon B. from RSM Winchester created, colored and sent us a picture of his beautiful 3-D paper kangaroo. One 4th grade student remarked, “I enjoyed the MehtA+ Digital Kit. It was fun to assemble and is a great decoration for my study area!”
We are excited to announce that we will be sponsoring Math Kangaroo USA again in 2022. For students in grades 1–12, please don’t forget to register for Math Kangaroo by December 15, 2021. Who knows, you might be receiving one of our prizes next year!


The Diffie-Hellman system
The Diffie-Hellman algorithm is a public key algorithm. It was invented in 1976 by Whitfield Diffie and Martin Hellman (US patent 4,200,770). It allows two parties, commonly called Alice and Bob, to agree on a key that they can use to encrypt messages they want to send to each other. They can to this even when an eavesdropper (Eve) listens in on their entire conversation.
The security of this algorithm depends on the assumption that it is easy to raise a number to a certain power, but difficult to compute which power was used given the number and the outcome.
The Diffie-Hellman system allows Alice and Bob to agree on a key even when Eve is listening to everything they say to each other. Alice and Bob need to agree on a prime number p, which they can do by simply sending it to each other. Eve is allowed to learn this number p. In practice the number p is often simply advertised somewhere public.
Given a prime number p, it is possible to come up with a number g (the so-called generator) with a very interesting property. Every number between 1 and p-1 can be written as a power of g when calculating modulo p. For example, using p = 5 the generator is 2, because
- 20 = 1
- 21 = 2
- 22 = 4
- 23 = 3 (because 8 = 3 mod 5)
Alice and Bob agree in the same way on a generator g for the numbers between 1 and p-1.
The public key
The numbers p and g serve as the public key.
Alice and Bob both choose random numbers (a and b respectively). Alice then computes ga and Bob computes gb. They exchange their results.
The key that Alice and Bob now agree on is simply ga*b. This is all very easy to compute. Alice knows a and gb, and Bob knows b and ga, and
Alice and Bob can use the key ga*b to encrypt messages with any secret key algorithm.
The security of the Diffie-Hellman system depends on the assumption that it is easy to raise a number to a certain power, but difficult to compute which power was used given the number and the outcome. For example, it's easy to compute 210 = 1024, but more difficult to determine that 1024 is the 10th power of 2.
Eve knows ga and gb, but since she does not know a or b itself, she cannot compute the key in a reasonable amount of time. To do that, she has to calculate the g-logarithm of ga (which mathematicians write as logg(ga)). And calculating this logarithm takes a long time.


Confidence Intervals and
Two basic uses of inferential statistics are possible:
a)interval estimation Ð so-called "confidence intervals"
b)point estimation Ð so-called "hypothesis testing"
Interval estimation ("Confidence Intervals") and point estimation ("Hypothesis Testing") are two different ways of expressing the same information.
Using Confidence Intervals we make statements like the following:
- the probability that the population mean (µ, or the 'true' value of the population mean) lies between 19 and 23 is 0.80;
- the probability that the population correlation value (r, or the 'true' value of the correlation between these two variables) lies between 0.20 and 0.24 is 0.60;
Using Hypothesis testing we say:
- the probability that our sample mean comes from a population with a mean of 21 is greater than 0.95
- the probability that our two sample means come from the one populations is less than 5% (i.e.,0.05).
- the probability that our sample comes from a population in which the true correlation is zero is 0.60.


To the Williams Community,
I am very sad to report the death this morning of a true stalwart of our faculty, Professor of Mathematics Ollie Beaver, who taught as recently as this semester despite advancing illness.
Ollie earned her Ph.D., at UMASS Amherst, when that was quite a difficult path for a young mother, and in 1979 joined what was then a much smaller and male-dominated math department than the one she helped it become. From the start she was especially dedicated to women and minority students in math and science. She help found, and for many years directed, the Summer Science Program that has given countless Williams students a strong footing in those subjects. She often mentored those students throughout their four years.
Meanwhile, she pursued her research in measure and probability theory. She was the second person ever to win the Louise Hay Award from the Association for Women in Mathematics, and the phenomenal growth in our math department is attributable in no small part to her.
A college citizen of the highest order, she served also as department chair, as the Gaudino Scholar, and for many years as chair of the Winter Study Committee. In fact, the list of her committee assignments is impressively long. She loved all this work, and set about doing it effectively and without fanfare.
After such a painfully swift loss, our thoughts and hearts will be for a long time with her family, especially her husband Don, professor of history of science.


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Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Eight International Conference
New Trends in the Applications of Differential Equations in Sciences (NTADES 2021)
6-9 September 2021, Sts. Constantine and Helena (Bulgaria)
The conference will be held in the Joliot-Curie International House of Scientists, St. Constantine and Helena, Bulgaria.
Last modified: 2021-01-13


Estimating Soybean Yield Potential
Harvest will be quickly upon us, and yield estimates will be occurring across the countryside. Crop yield potential is normally estimated by determining the number of plants in an area, the number of reproductive units, and the weight of those reproductive units.
Potential Yield = plants/acre x seeds/plant x seed weight
Yield potential estimates for most row crops have three yield components, but how those yield component numbers are determined are different for each crop. Before estimating soybean yield potential, a word of caution; estimating soybean yield potential can be tricky. Soybean yield components can vary greatly by soybean product, different parts of the field, and from plant to plant. It is not advised to conduct a yield estimate during early reproductive stages as the plant has not fully determined the number of seeds that it may produce. Soybean yield potential estimates should only occur after the R6 (full seed) growth stage. Yield estimates should be conducted in at least 5 to 10 random places in the field and these random places should be representative of the plants in that area. By sampling in multiple places, a more uniform yield estimate for the entire field can be obtained.
Step 1: Determine number of plants per acre.
- Take time to calculate the number of harvestable plants per acre, as this number could have changed from when stand counts were determined earlier in the season.
- Determine the row width and measure the corresponding row length to achieve 1/1000th of an acre.
- Count the number of plants in the row for 1/1000th of an acre and multiple by 1000, to estimate the number of plants per acre.
- For example, in 30-inch rows, measure 17 feet 5 inches in row length and count the number of plants in that row. If 114 plants were counted in the row, multiply by 1000 to have an estimated 114,000 harvestable plants per acre.
Step 2: Estimate the number of pods per plant.
- To estimate pods per plant, count the number of pods on each plant for 10 consecutive plants in one row.
- Divide the total number of pods by 10 to determine the average number of pods per plant.
- For example, if a total of 350 pods were counted, divide by 10 to reach an average of 35 pods per plant.
Step 3: Estimate number of seeds per pod.
- When determining the number of pods per plant, look at those pods from the 10 consecutive plants from Step 2.
- If there is about an even mix of three and two seeded pods, use the general 2.5 seeds per pod value.
- Typically, a value of 2.5 seeds per pod is used with healthy soybean plants when there is an average growing season. However, in stressful growing conditions, this number may be lower. This is when it would be advised to count the number of three seeded, two seeded, and one seeded pods to determine if a lower value of 2 or 1.5 seeds per pod should be used.
Step 4: Estimate the weight of the seed
- Seed weight is the most difficult yield component to estimate correctly.
- Seed weight can vary by seed product and growing conditions.
- When there is ample rain in August, larger seeds are typically produced which correspond to larger seed weights.
- Stressful conditions of hot and dry usually produce plants with small seeds and lower seed weight.
- A general ‘rule of thumb’ factor for seed weight is 2500 seeds per pound. However, a better indicator is typically what the original seed size was from the seed bag tag at planting. The seed weight factor can be adjusted if the final yield estimate is unusually too high or low.
Step 5: Estimate the final yield (bushel/acre)
Make sure to set up the equation correctly to have the correct units for each portion of the equation. For the equation, the weight of one bushel of soybean (60 pounds) is needed. Using the numbers in the examples above, the hypothetical soybean potential yield estimate from one spot in the field would be:
Potential yield (bu/acre) = (plants/acre) x (pods/plant) x (seeds/pod) ÷ (seeds/lb) ÷ (lb/bu)
bu/acre = 114,000 plants/acre x 35 pods/plant x 2.5 seeds/pod ÷ 2500 seeds/lb ÷ 60 lbs/bu = 66.5 bu/acre
These steps should be repeated at multiple locations across the field to get a representative average of estimated yield potential for the field. A reminder that yield potential estimates are only as good as the numbers used in the equation. Yield estimates made closer to harvest are more reliable than yield estimates taken earlier in the season when pod and seed development are occurring. Contact your local Channel Seedsmen concerning questions and assistance in estimating soybean yield potential.
Knott, C. and Lee, C. 2018. Cultural practices. Chapter 4. A Comprehensive Guide to Soybean Management in Kentucky. ID-249. University of Kentucky College of Agriculture Food and Environment Cooperative Extension Service.
Source verified 6/22/23. 1110_274488