Art in Number image: Black Ice, Creative Commons, Pexels.com

## SquareCirclez Mathematics, learning, computing, travel - and whatever...

• How To Solve Linear Congruences
by Kathleen Knowles on January 29, 2021 at 5:38 pm

Numbers are congruent if they have a property that the difference between them is integrally divisible by a number (an integer). The number is called the modulus, and the statement is treated as congruent to the modulo. Mathematically, this can be expressed as b = c (mod m) Generally, a linear congruence is a problem of findingRelated posts:How To Solve Three Variable Linear Equations An equation between two variables is known as a Linear...Newton’s Method interactive graph I added a new interactive graph that helps explain...Solving Equations With The Addition Method There are two ways of solving an equation: the addition...Solving For Y in Terms of X Using Fractions The first step to solving any mathematical equation is to...

• Calculating Probability with Mean and Deviation
by Kathleen Knowles on January 29, 2021 at 5:37 pm

Calculating probability with mean and deviation depends on the type of distribution you'll base your calculations on. Here, we'll be dealing with typically distributed data. If you have data with a mean μ and standard deviation σ, you can create models of this data using typical distribution. We can find the probability within this data based on that meanRelated posts:New z-Table interactive graph Investigate the meaning of the z-table using this interactive...Solving Probability with Multiple Events What is the probability of two events happening? My friend...Probability And the probability of this happening is......Calculating Exponential Decay with a Variable In the Exponent Exponents are used to represent any function that changes rapidly. If...

• Solving Trigonometric Equations and Identities
by Kathleen Knowles on December 21, 2020 at 7:23 pm

It is said that spies and other nefarious characters will carry many passports, enabling them to claim a different identity instantly. Despite the many identities, we know that all these passports are aliases of the same person. Trigonometric identities are different ways of representing the same expression. They are used to solve a trigonometric equation when appliedRelated posts:Solving Systems of Equations by Using Elimination In mathematics, an equation is a statement where two mathematical...Solving Quadratic Equations by Factoring The term "quadratic" traces from the Latin word "quad," which...Solving Absolute Value Equations Absolute value for most of us is a brief unit...Solving Equations With The Addition Method There are two ways of solving an equation: the addition...

• Understanding Shapes with Oblique Angles
by Kathleen Knowles on October 27, 2020 at 7:43 pm

Oblique shapes are extraordinarily common, but many people don’t know what they are. Oblique shapes are shapes made of oblique angles. Oblique angles are: Acute angles: angles that are 0-90 degrees Obtuse angles: angles that are 90-180 degrees The sides that form the angles of an oblique shape are never perpendicular. In simple terms, theseRelated posts:Calculating Polygon Angles and Sides Lengths A polygon is any closed plane figure. It comes from the...Random triangles What is the probability that a randomly chosen triangle is...Determining if a Plane Figure Can Be Regular Faced with the task of teaching your kids the geometry you...How do you find exact values for the sine of all angles? Can you find exact values for the sines of all...

• Calculating Weight Using Different Gravity Loads
by Kathleen Knowles on October 14, 2020 at 7:31 pm

One can define gravity as a universal force that acts between two objects. It tends to pull objects towards the center of the earth. Each body in the universe possesses a particular amount of matter. This is known as mass, which is defined as the amount of matter contained in a substance. Anything that occupiesRelated posts:Calculating Mass From Force and Weight We've all heard the term “mass” in school before. But...Calculating Acceleration Due To Gravity on a Plane Ever wondered why, when a body is thrown upwards, it...Calculating Acceleration with Force and Mass As is usually the case in mathematics and physics, formulas...Determining Velocity with Time and Change in Acceleration Every object experiencing an acceleration must have a velocity. This...

## Recent Questions - Mathematics Stack Exchange most recent 30 from math.stackexchange.com

• Aymptotics of $\sum_{i=0}^n e^{-i^2/n + i^a}$ as $n \to \infty$
by ARG on February 26, 2021 at 6:26 pm

To what function is $s(n) = \sum_{i=0}^n e^{-i^2/n + i^a}$ with $a \in ]0,1]$ asymptotic (as $n \to \infty$)? or what are relatively tight lower and upper bounds? Here is what I tried so far: go over to a "Riemann sum" $$n \sum_{i=0}^n \tfrac{1}{n} e^{ -n (i/n)^2 + n^a (i/n)^a} \approx n \int_0^1 e^{ -n x^2 + n^a x^a}$$ when $a = 1$ this integral is actually easy to evaluate (it's asymptotic to $\sqrt{\pi} \cdot e^{n/4}/\sqrt{n}$) and gives a very tight bound for the asymptotics of the sum. When $a<1$, I used the following estimates: $-n x^2 + n^a x^a >0 \iff 0< x < n^{(a-1)/(2-a)}$. The maximum of the exponent happens when $-2nx + an^ax^{a-1} =0 \iff x = (\tfrac{a}{2})^{1/(2-a)}n^{(a-1)/(2-a)}$. This gives a maximal value of the exponent of the integrand $$-n (\tfrac{a}{2})^{2/(2-a)}n^{(2a-2)/(2-a)} + n^a (\tfrac{a}{2})^{a/(2-a)}n^{(a^2-a)/(2-a)} = \bigg( -(\tfrac{a}{2})^{2/(2-a)} + (\tfrac{a}{2})^{a/(2-a)} \bigg)n^{a/(2-a)} = ( 1-\tfrac{a}{2}) (\tfrac{a}{2})^{a/(2-a)} n^{a/(2-a)}$$ Up to a mistake of $1$, one can ignore the part of the area of the integral which is the region of the unit square. There remains at most a rectangle with dimensions $n^{(a-1)/(2-a)} \times \mathrm{exp}\bigg( ( 1-\tfrac{a}{2}) (\tfrac{a}{2})^{a/(2-a)} n^{a/(2-a)} \bigg)$. Hence $s(n) \leq n^{1/(2-a)} \mathrm{exp}\bigg( ( 1-\tfrac{a}{2}) (\tfrac{a}{2})^{a/(2-a)} n^{a/(2-a)} \bigg)$ (for large $n$). The lower bound [for $n$ large] $s(n) \geq \mathrm{exp}\bigg( ( 1-\tfrac{a}{2}) (\tfrac{a}{2})^{a/(2-a)} n^{a/(2-a)} \bigg)$ is also not too hard to get (just restrict the sum to the a few values of $i$ so that $i/n \approx (\tfrac{a}{2})^{1/(2-a)}n^{(a-1)/(2-a)}$). Numerics indicate that $s(n) \approx K n^{1/2} \mathrm{exp}\bigg( ( 1-\tfrac{a}{2}) (\tfrac{a}{2})^{a/(2-a)} n^{a/(2-a)} \bigg)$ for some $K \in [1,2]$

• Prove that $\sum_{n=-\infty}^{\infty}\frac{1}{(x+\pi n)^2}=\frac{1}{sin^2x}$
by Ozod on February 26, 2021 at 6:24 pm

$$\sum_{n=-\infty}^{\infty}\frac{1}{(x+\pi n)^2}=\frac{1}{sin^2x}$$ Marko Riedel Dec 30 '18 at 14:46 Show 5 more comments 1 Answer order by votes Up vote 3 Down vote Accepted With the quoted proof being unsatisfactory we try again. With the goal of evaluating $$\sum_{n=-\infty}^\infty \frac{1}{(u+n)^2}$$ where $u$ is not an integer we study the function $$f(z) = \frac{1}{(u+z)^2} \pi\cot(\pi z).$$ which has the property that with $S$ being our sum, $$S = \sum_n \mathrm{Res}_{z=n} f(z) = \sum_n \frac{1}{(u+n)^2}.$$ 😓😓😓😓 please help prove that

• Why isn't the determinant function zero?
by user9343456 on February 26, 2021 at 6:24 pm

Let $P_1,\ldots,P_n$ be $n$ dimensional column vectors. The determinant function $D(P_1,\ldots,P_n)$ is a multilinear, alternating real function of these vectors. Since it's alternating, if $P_1,\ldots,P_n$ are not linearly independent, then $D$ evaluates to zero. So far so good. Let $E_1,\ldots,E_n$ be the standard basis column vectors ($n$-dimensional). Let the $i$-th entry in $P_j$ be $p^i_j$. My question is: due to multilinearity can't I write $D(P_1,\ldots,P_n)$ as $$D(p^1_1E_1,\ldots,p^1_mE_1)+D(p^2_1E_2,\ldots,p^2_mE_2)+\ldots+D(p^m_1E_m,\ldots,p^m_mE_m)$$ and say that all terms are zero, since in each term, the arguments are not independent? So the entire thing must be zero? I'm not able to figure out where I'm going wrong.

• Question in Hatcher proposition 1.26 proof
by love_sodam on February 26, 2021 at 6:22 pm

(a) If $Y$ is obtained from $X$ by attaching 2-cells as described above, then the inclusion $X\hookrightarrow Y$ induces a surjection $\pi_1(X,x_0)\to\pi_1(Y,x_0)$ whose kernel is $N$.Rather than writing down the meaning 'describe above' and '$N$' in the statement, I think it's better to look p.49 of Hatcher's algebraic topology here.The question is I can't understand the red line in the image. I understand why they're trying to choose such $\delta_\alpha$, but why such an element exists in $\pi_1(A\cap B,z_0)$? The map $\pi_1(A\cap B)\to \pi_1(A)$ is not surjective. Could you explain this?

• Hi! I'm having a problem with the integral $\displaystyle\int_{n}^{n+1}x^2 dx$
by Algoak on February 26, 2021 at 6:22 pm

Basically, i noticed that when $n\subset \mathbb{N}$, $\displaystyle\int_{n}^{n+1}{x^2}dx$ gives a prime number divided by 3 or a multiplication of prime numbers. Here's a table i made.is this something already known or a "function that generates prime numbers"? I don't know much about math and I don't have a great academic background so I'm asking this here.

## Surrey Mathematics Research Blog The blog on research in mathematics at the University of Surrey

• Alessandro Torrielli gives virtual seminars to universities in Germany and Italy
by Tom Bridges on February 26, 2021 at 4:22 pm

Alessandro Torrielli has given two virtual seminars this month. The first, on 18 February, was a talk in the Emmy Noether Seminar at the Institute for Theoretical Physics at the University of Leipzig in Germany. The title of the talk was “Massless integrable scattering in the AdS/CFT correspondence” (link here). The second talk, on 24

• Paper of Bin Cheng and Thomas O’Neill on Monge-Ampère equation published in JFA
by Tom Bridges on February 26, 2021 at 9:52 am

The paper “Interior estimates for Monge-Ampère equation in terms of modulus of continuity” co-authored by Bin Cheng and Thomas O’Neill has been published in the Journal of Functional Analysis. The paper forms part of the PhD thesis of Thomas. The published version is available for download here and the final form arXiv version is available

• Paper of Juan Miguel Nieto García, Alessandro Torrielli, and Leander Wyss published in JGP
by Tom Bridges on February 25, 2021 at 6:18 pm

The paper “Boosts superalgebras based on centrally-extended su(1|1)^2” co-authored by Juan Miguel Nieto García, Alessandro Torrielli, and Leander Wyss has been accepted for publication in the Journal of Geometry and Physics. The final form arXiv version can be found here, and the screenshot below shows Figure 1 from the paper.

• Two grants awarded to the Data Group by the Surrey Institute for Advanced Studies
by Tom Bridges on February 24, 2021 at 9:03 am

An IAS Fellowship grant has been awarded to the Data Group. The co-investigators are Stefan Klus and Naratip Santitissadeekorn, with support from David Lloyd. The Fellowship grant funds a visit by Jason Bramburger (Seattle). An IAS Workshop grant has also been awarded with PI Payel Das (Physics, Surrey), with support from Masanori Hanada and David

• Two new papers published by Stefan Klus
by Tom Bridges on February 22, 2021 at 11:45 am

Stefan Klus has had two new papers published in the latter part of 2020. The paper “Kernel based approximation of the Koopman generator and Schrodinger operator“, co-authored with Feliks Nuske (Paderborn) and Boumediene Hamzi (Imperial College) was published in the journal Entropy, and chosen by the journal for a cover story (link here). The second

## Wolfram Blog » Mathematics News, views, and ideas from the front lines at Wolfram Research.

• 3D-Printed Jewelry Made with the Wolfram Language Showcases the Beauty of Mathematics
by Christopher Hanusa on February 15, 2021 at 8:09 pm

h2.bookpost{display:block;} img.bookpost{padding-top:20px} I enjoy turning mathematical concepts into wearable pieces of art. That’s the idea behind my business, Hanusa Design. I make unique products that feature striking designs inspired by the beauty and precision of mathematics. These pieces are created using the range of functionality in the Wolfram Language. Just in time for Valentine’s Day, we recently launched Spikey [...]

• Step-by-Step Math Tools in Wolfram|Alpha Help Your Chemistry Course Prep
by Becky Song on February 12, 2021 at 8:51 pm

Math is one of the main things that deters students from wanting to learn more about chemistry. Being a chemical engineering student, I understand this, especially for students who just have to get chemistry out of the way as a general education requirement. Essentially, step-by-step solutions are like your own on-demand math tutor: in addition [...]

• How We Navigated a Hybrid Remote Learning Environment Using Wolfram Technology
by Timothy Newlin on January 14, 2021 at 6:00 pm

The past year of learning ushered in a variety of new experiences for instructors and students alike, and the United States Military Academy at West Point was no exception. In addition to masks in the classroom, reduced class sizes to allow for social distancing, rigorous testing and tracing efforts, and precautionary remote video classes, we [...]

• New Wolfram Language Books on Wolfram|Alpha, Calculus, Applied Engineering and System Modeler
by Paige Bremner on October 29, 2020 at 3:24 pm

h2.bookpost{display:block;} img.bookpost{padding-top:20px} The pandemic has postponed or canceled a lot of things this year, but luckily learning isn’t one of them. Check out these picks for new Wolfram Language books that will help you explore new software, calculus, engineering and more from the comfort of home. Hands-on Start to Wolfram|Alpha Notebook Edition New from Wolfram [...]

• Learn Linear Algebra in Five Hours Today with the Wolfram Language!
by Devendra Kapadia on August 14, 2020 at 1:44 pm

Linear algebra is probably the easiest and the most useful branch of modern mathematics. Indeed, topics such as matrices and linear equations are often taught in middle or high school. On the other hand, concepts and techniques from linear algebra underlie cutting-edge disciplines such as data science and quantum computation. And in the field of [...]

• New Wolfram Books: Releases from Wolfram Media and Others Featuring the Wolfram Language
by Amy Simpson on July 2, 2020 at 6:11 pm

h2.bookpost{display:block;} img.bookpost{padding-top:20px} The first half of 2020 has brought with it another exciting batch of publications. Wolfram Media has released Conrad Wolfram’s The Math(s) Fix. Keep an eye out for the upcoming third edition of Hands-on Start to Wolfram Mathematica later in 2020. The Math(s) Fix The Math(s) Fix: An Education Blueprint for the AI [...]

• New 12.1 Dataset Interactive Controls and Formatting Options
by Christopher Carlson on June 23, 2020 at 2:08 pm

In his blog post announcing the launch of Mathematica Version 12.1, Stephen Wolfram mentioned the extensive updates to Dataset that we undertook to make it easier to explore, understand and present your data. Here is how the updated Dataset works and how you can use it to gain deeper insight into your data. New Interactive [...]

• Using Integer Optimization to Build and Solve Sudoku Games with the Wolfram Language
by Paritosh Mokhasi on June 2, 2020 at 1:40 pm

Sudoku is a popular game that pushes the player’s analytical, mathematical and mental abilities. Solving sudoku problems has long been discussed on Wolfram Community, and there has been some fantastic code presented to solve sudoku problems. To add to that discussion, I will demonstrate several features that are new to Mathematica Version 12.1, including how [...]

• From Sine to Heun: 5 New Functions for Mathematics and Physics in the Wolfram Language
by Tigran Ishkhanyan on May 6, 2020 at 5:02 pm

Mathematica was initially built to be a universal solver of different mathematical tasks for everything from school-level algebraic equations to complicated problems in real scientific projects. During the past 30 years of development, over 250 mathematical functions have been implemented in the system, and in the recent release of Version 12.1 of the Wolfram Language, [...]

• 非線形偏微分方程式への有限要素法の適用
by Koji Maruyama on April 29, 2020 at 9:19 pm

Mathematica 12 has powerful functionality for solving partial differential equations (PDEs) both symbolically and numerically. This article focuses on, among other things, the finite element method (FEM)–based solver for nonlinear PDEs that has been newly implemented in Version 12. After briefly reviewing basic syntax of the Wolfram Language for PDEs, including how to designate Dirichlet [...]