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

• 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...

• Calculating Acceleration Due To Gravity on a Plane
by Kathleen Knowles on October 8, 2020 at 3:40 pm

Ever wondered why, when a body is thrown upwards, it comes back down at an increased speed? It is due to the acceleration caused by gravity. Near the earth's surface, there is almost no gravitational force experienced, but it varies at large distances from the earth. Gravity is a force that is experienced between twoRelated posts:Determining Velocity with Time and Change in Acceleration Every object experiencing an acceleration must have a velocity. This...Calculating Acceleration with Force and Mass As is usually the case in mathematics and physics, formulas...Finding Dimensional Formula For Acceleration Physical quantities are used to quantify the properties of a...Calculating Mass From Force and Weight We've all heard the term “mass” in school before. But...

• Calculating Mass From Force and Weight
by Kathleen Knowles on October 6, 2020 at 8:24 pm

We've all heard the term “mass” in school before. But what actually is mass? And how can we calculate it if we know the force and weight of an object? Well, I’m glad you asked. To calculate mass, you need to know the force of gravity that's acting on the object, and its weight. AndRelated posts: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...Calculating Exponential Decay with a Variable In the Exponent Exponents are used to represent any function that changes rapidly. If...Finding Dimensional Formula For Acceleration Physical quantities are used to quantify the properties of a...

• Separating Variables in Differential Equations
by Kathleen Knowles on October 6, 2020 at 8:23 pm

Separation of Variables, widely known as the Fourier Method, refers to any method used to solve ordinary and partial differential equations. To apply the separation of variables in solving differential equations, you must move each variable to the equation's other side. Get to Understand How to Separate Variables in Differential Equations As indicated in theRelated posts:Solving Systems of Equations by Using Elimination In mathematics, an equation is a statement where two mathematical...Factoring Polynomials in Algebraic Equations A polynomial is a mathematical expression containing variables in 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...

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

• Eigenspaces of $A^TA$ has dimension 1
by No name on November 28, 2020 at 1:36 am

Given any matrix A, I'm rather sure that all eigenspaces of $A^TA$, which are $\ker(A^TA-\lambda I)$ with $\lambda$ being any eigenvalue of $A^TA$, has dimension 1. But I'm stuck to prove this. Any suggestions? Thank you in advance.

• Formula for number of permutations
by tangor on November 28, 2020 at 1:35 am

How can I determine a formula for number of permutations of set {1, ...,2n} under condition that for any k the numbers: 2k and 2k-1 are not adjacent? I think that the formula can be a sum. Can anyone explain me this? Thank you for any answer or a clue.

• Is my answer ok? (Exercise 6 on p.79 in "Analysis on Manifolds" by James R. Munkres.)
by tchappy ha on November 28, 2020 at 1:33 am

I am reading "Analysis on Manifolds" by James R. Munkres. There is the following exercise (exercise 6) on p.79 in this book.Let $f:\mathbb{R}^{k+n}\to\mathbb{R}^n$ be of class $C^1$; suppose that $f(\mathbf{a})=\mathbf{0}$ and that $Df(\mathbf{a})$ has rank $n$. Show that if $\mathbf{c}$ is a point of $\mathbb{R}^n$ sufficiently close to $\mathbf{0}$, then the equation $f(\mathbf{x})=\mathbf{c}$ has a solution.My answer is here:For $\mathbf{x}\in\mathbb{R}^{k+n}$ and $\mathbf{y}\in\mathbb{R}^n$, we define $g(\mathbf{x},\mathbf{y}):=f(\mathbf{x})-\mathbf{y}$. Then, $g:\mathbb{R}^{k+2n}\to\mathbb{R}^n$ is of class $C^1$.$g(\mathbf{a},\mathbf{0})=\mathbf{0}$.$Dg(\mathbf{a},\mathbf{0})=\begin{bmatrix}\frac{\partial f}{\partial(x_1,\dots,x_{k+n})}(\mathbf{a})&-I_n\end{bmatrix}$. Since $Df(\mathbf{a})$ has rank $n$, $$\det\frac{\partial g}{\partial(x_{i_1},\dots,x_{i_n})}(\mathbf{a},\mathbf{0})=\det\frac{\partial f}{\partial(x_{i_1},\dots,x_{i_n})}(\mathbf{a})\ne0$$ for some $\{i_1,\dots,i_n\}\subset\{1,\dots,k+n\}$. So, by the implicit function theorem, if $\mathbf{c}$ is a point of $\mathbb{R}^n$ sufficiently close to $\mathbf{0}$, then the equation $g(\mathbf{x},\mathbf{c})=0$ has a solution. So $f(\mathbf{x})=\mathbf{c}$ has a solution.

• Rewriting this Reimann sum as a definite integral
by Future Math person on November 28, 2020 at 1:33 am

Can someone help me rewrite this Reimann sum as a definite integral?$\displaystyle \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}\left(-7+\frac{14i}{n} \right)^9\sin\left(4+ \left(-7+\frac{14i}{n} \right)^8 \right)$$\Delta x=\frac{1}{n} so this means b-a=1. If I rewrite the \frac{14i}{n} as 14\frac{i}{n}, then this means I could say that a=0 and b=1 so this means my definite integral is:$$\int_{0}^{1}(-7+14x)^9\sin(4+(-7+14x)^8)dx$$but I am not sure if this is correct or not. If this IS correct, how would I do this integral with no integration technique other than u-substitution or manipulation? I can't seem to do it otherwise. • Condition for normality of a linear operator by Ted Baker on November 28, 2020 at 1:29 am For finite-dimensional Hermitian inner product space$V$and$T \in \mathcal{L}(V)$, prove that if for all$v \in V$,$||T^* v|| \leq |Tv||$, then$T$is normal.My first confusion is that a theorem in Axler states that$T$is normal if and only if$|Tv| = |T^* v|$. The$\leq\$ does not seem needed here. Do I only need to rule out, by a proof by contradiction, that the inequality is strict? If so, hints on how to approach this would be appreciated.

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

• Jacob Brooks passes PhD viva
by Tom Bridges on November 26, 2020 at 5:05 pm

Congratulations to Jacob Brooks for passing his PhD viva on Thursday 26 November! Jacob’s thesis is entitled “Inhomogeneous semi-linear wave equations“. The External Examiner was Prof Dr Peter van Heijster of Wageningen University & Research (WUR) in the Netherlands, and the Internal was Tom Bridges. The viva was chaired by Jonathan Deane. The project was

• Alessandro Torrielli gives a virtual seminar in the Integrability, Dualities and Deformations Series
by Tom Bridges on November 18, 2020 at 4:15 pm

Alessandro Torrielli gave a virtual seminar today (Wednesday 18 November) in the Integrability, Dualities and Deformations Seminar Series. The title of his talk is “How one massless TBA was exactly solved“. In the talk, he reviewed the exact solution of the massless relativistic Thermodynamic Bethe Ansatz which describes the non-trivial BMN limit of massless right-right

• Philip Aston part of a team awarded UKRI funding for physiological sensor research
by Tom Bridges on November 17, 2020 at 10:38 am

A multidisciplinary team led by Prof Christian Heiss (FHMS) and including Prof Philip Aston, Dr David Birch (FEPS, Engineering), Prof Simon Skene (FHMS), and Dr Radu Sporea (FEPS, EE), have been awarded a £50k Healthy Ageing Catalyst Award, granted by UKRI, to facilitate a one-year research project on creating a low-cost prototype device for monitoring

• Paper of Sergey Zelik on Lieb-Thirring inequalities on manifolds published by JFA
by Tom Bridges on November 11, 2020 at 6:26 pm

The paper “Lieb–Thirring constant on the sphere and on the torus” co-authored by A. Ilyin, A. Laptev, and S. Zelik, has been published in the Journal of Functional Analysis. The paper proves, on the 2-sphere and on the 2-torus, the Lieb–Thirring inequalities with improved constants, for orthonormal families of scalar and vector functions. A link

• Paper of Jan Gutowski on multi-form modified Dirac operators published in Journal of Geometry and Physics
by Tom Bridges on November 3, 2020 at 1:14 pm

The paper “Eigenvalue estimates for multi-form modified Dirac operators“, co-authored by Jan Gutowski and George Papadopoulos (King’s College London), has been published in the Journal of Geometry and Physics. The paper gives estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford

## Featured Blog Posts - Data Science Central

• Human Decision-making in a Big Data World
by Bill Schmarzo on November 27, 2020 at 9:56 pm

This blog was originally written in 2012, and am republishing it because with the increased use of black box AI / ML models to power key operational decisions, these human decision-making traps need to be thoroughly and holistically addressed during the analytics definition stage to avoid the dangers of unintended consequences. Organizations are looking to integrate big data and advanced analytics into their business operations in order to become more…

• The Four Stages of Robotic Process Automation (RPA)
by Amit Dua on November 27, 2020 at 10:21 am

“Robotic process automation is not a physical [or] mechanical robot,” says Chris Huff, chief strategy officer at Kofax. In fact, there is no presence or involvement of any robots in the automation software, as its name says. RPA or Robotic Process Automation is an amalgamation of three factors:Robotic –…

• DSC Thursday News, 26 Nov 2020
by Kurt Cagle on November 26, 2020 at 4:41 am

DSC Thursday News, 26 Nov 2020…

• New Features In Angular 10.1.0 You Cannot Miss!
by James Smith on November 25, 2020 at 10:38 am

Why developers prefer Google's Angular app development for their projects? Keep reading to find out! Mobile app development is gaining popularity all over the world. Businesses are trying to incorporate this tech into their process in order to bridge the gap between the users and…

• Artificial Intelligence- The Technology Growth Driver for Banking Services
by Rishabh Sinha on November 24, 2020 at 12:30 pm

Across industries and enterprises, AI has been the key growth driver, and this is mainly due to the services offered under this technology umbrella. Every industry is now witnessing enhancements in business processes, costs, and efficiency due to AI's introduction in their field. Depending on the industry, AI offers something unique and has helped the key decision-makers to make fast and accurate decisions based on various implementation solutions.…

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

• 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 [...]

• Finally We May Have a Path to the Fundamental Theory of Physics… and It’s Beautiful
by Stephen Wolfram on April 14, 2020 at 2:07 pm

div.purplestripe { max-width:620px; background-color: #E6E6FA; padding-left:10px; margin-bottom:10px; } #blog .post_content .purplestripe a,#blog .post_content .purplestripe a:link,#blog .post_content .purplestripe a:visited { font-family:"Fira Sans",Arial,Sans Serif; font-size:11pt; color:#3d137d;} Website: Wolfram Physics Project Technical Intro: A Class of Models with the Potential to Represent Fundamental Physics How We Got Here: The Backstory of the Wolfram Physics Project I Never Expected [...]

• Hitting All the Marks: Exploring New Bounds for Sparse Rulers and a Wolfram Language Proof
by Ed Pegg Jr on February 12, 2020 at 6:00 pm

The sparse ruler problem has been famously worked on by Paul Erdős, Marcel J. E. Golay, John Leech, Alfréd Rényi, László Rédei and Solomon W. Golomb, among many others. The problem is this: what is the smallest subset of so that the unsigned pairwise differences of give all values from 1 to ? One way [...]

• An Intriguing Identity: Connecting Distinct and Complete Integer Partitions
by George Beck on January 9, 2020 at 2:59 pm

Number theory is a very old subject that in modern times has branched into various large areas. One of these is additive number theory, with problems like this: when is a prime the sum of two squares? Primes are part of the more classical area now called multiplicative number theory, so as this problem of [...]