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

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

• Explaining Trigonometric Ratios: Sin
by Kathleen Knowles on April 19, 2021 at 5:56 pm

Trigonometric ratios are the functions relating to a right-angled triangle. As everyone knows, a triangle has three sides: the hypotenuse (the longest side), the perpendicular (side opposite to the angle), and the base (side adjacent to the angle). Relationship Between Sine and Other Trigonometric Ratios Trigonometric ratios express the sides of a right-angled triangle. There are sixRelated posts:Explaining Trigonometric Ratios: cos Trigonometry examines the relationship between the sides of a triangle,...Solving Trigonometric Equations and Identities It is said that spies and other nefarious characters will...Calculating Polygon Angles and Sides Lengths A polygon is any closed plane figure. It comes from the...Unit Circle: an Introduction The unit circle ties together 3 great strands in mathematics:...

• What is a Shifting Function
by Kathleen Knowles on April 3, 2021 at 2:04 am

Most people have seen some basic graphs before. Graphs are pictorial representations of data and values along axes. By understanding basic graphs and how to apply translations to them, you'll realize that each new graph is a variation of the old one. It is not a completely different graph than you've ever seen before. UnderstandingRelated posts:How to find the equation of a logarithm function from its graph? This article explains how to determine the equation of...New applet: What does b do in a quadratic function? y = ax2 + bx + c is a parabola....How to find the equation of a quadratic function from its graph A reader asked how to find the equation of a...How to draw y^2 = x - 2? How do we draw parabolas that are moved around the...

• Heron’s Formula Explained
by Kathleen Knowles on April 3, 2021 at 1:56 am

The perimeter of a triangle is its three sides added together. The area of a triangle is quite interesting. It will usually be expressed as area = (1/2) x base x height, A = (1/2)b*h, where height is measured as a vertical line from the base to the opposite vertex. This formula makes calculating theRelated posts:Quadratic formula by completing the square - easier method Here's an easier way to derive the Quadratic Formula using...How to Use the Distance Formula on a Coordinate Plane Let's say you want to measure the length of any...Calculating Polygon Angles and Sides Lengths A polygon is any closed plane figure. It comes from the...Understanding the Discriminant in a Quadratic Formula A quadratic equation in algebra is an equation in which...

• Fundamental Rules of Exponents
by Kathleen Knowles on April 3, 2021 at 1:50 am

Knowing the fundamental rules of exponents will help you simplify mathematical equations and statements so you can arrive at a solution with relative ease. As with everything, it may take some practice to get the hang of working with exponents. But if you take your time, don't allow yourself to be overwhelmed by variables, and memorize each rule,Related posts:How To Multiply X with Different Exponents Do you still remember the concept of variables and exponents?...How To Solve Negative Polynomials A polynomial is a mathematical expression made of variables and...Solving Basic Algebra Using Steps and Revisions Algebra is a type of math that focuses on solving expressions...Stumbling blocks in math - the way it is written and explained What does integral exponent in math really mean? A student...

• Explaining Trigonometric Ratios: cos
by Kathleen Knowles on April 3, 2021 at 1:47 am

Trigonometry examines the relationship between the sides of a triangle, more specifically, right triangles. A right triangle has a 90° angle. The equations and ratios that describe the relationship between the sides of a triangle and its angles are trigonometric functions. In this particular article, we're going to explain one specific ratio: "cos" or cosine.Related posts:Solving Trigonometric Equations and Identities It is said that spies and other nefarious characters will...Unit Circle: an Introduction The unit circle ties together 3 great strands in mathematics:...Calculating Polygon Angles and Sides Lengths A polygon is any closed plane figure. It comes from the...Finding The Angle Line Using the Y-Axis To find the angle line using the y-axis, you'll need...

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

• Function with a vertical and a horizontal asymptote
by mov0021 on May 17, 2021 at 1:15 am

I’ve been doing research on a function and I obtained a plot of the behaviour of the system. It has the following shape:But I’m having a shortage of imagination to think of a known function that shows the following features:All functions are bounded with the $x = 1$ asymptote and another one, except for the uppermost one. The uppermost function has a maximum at $\left(W(\frac{1}{e}), 1+\frac{1}{W\left(1/e\right)}\right)$, which is roughly $(0.278465, 4.59112)$, as seen on the plot. The uppermost one has got the $x = 1$ asymptote and the $y = 0$ asymptote.What bothers me the most is having two asymptotes, which is a must for purposes of the research. I appreciate any input that can shed some light on my research.

• Extrema of function defined on a circle not located at origo
by vault on May 17, 2021 at 1:14 am

Find the global maxima and minima of the function $$f(x,y) = \sqrt{(x-2)^2 + y-1^2} + 4x - x^2$$ on the circle $$(x-2)^2 + y-1^2 \leq 1$$. I found $f_x$ and $f_y$ and used them to find $x=2, \sqrt{(x-2)^2 + y-1^2} = \frac 1 2$ and $y=1$, giving the points $(\frac 5 2 , 1)$ and $(\frac 3 2 , 1)$ and they provide the singularity $(2,1)$. The last thing I need to check are the boundaries. Here I set $$(x-2)^2 + y-1^2 = 1$$ into $f(x,y)$, finding $g(x) = 1 + 4x - x^2$. I am not able to continue this, finding the last solutions around the boundary.

• I got mixed up by the use of triple bar vs equals sign in linear congruences. Is it worth asking the professor for a regrade?
by William Minnis on May 17, 2021 at 1:10 am

On an exam I recently took, I was asked the following question: What are the number of solutions to the equation: $ax = b$ (mod $m$) when gcd($b,m$) = 1 and gcd($a,m$) = $d$ > 1 I know this is entirely sematic, but I was confused when I saw them use = instead of ≡ in the equation. For this reason, I interpreted the right-hand side to be a specific value $(b\%m)$ and stated there was one solution. In reality, there are no solutions as the expression ax + km is always a multiple of d and b is not. What I really want to know is if my interpretation was valid at all, and if it is worth asking the professor for a regrade in the context of ≡ vs = or if I just take the L.

• lim x---> 0 [[x]] sin (x)=0 use squeeze theorem to prove.
by Robanjeet Singh on May 17, 2021 at 1:08 am

How can you use the Squeeze theorem to prove something that limits DNE? for example one of my problems says to find the limit x approaching 0 to the floor function since the limit of that function does not exists I'm confused? Problem lim x---> 0 [[x]] sin (x)=0 use squeeze theorem to prove.

• Are the given sum of abolute value functions increasing?
by Seyhmus Güngören on May 17, 2021 at 1:08 am

We have the set $\{0,\cdots,2^K-1\}$ in binary representation, e.g. $\{00,01,10,11\}$ for $K=2$. Then each $0$ is replaced by $b_k$ and each $1$ is replaced by $1-b_k$, This gives a set e.g. $$B=\{b_1b_2,b_1(1-b_2),(1-b_1)b_2,(1-b_1)(1-b_2)\}$$ The same process is done in reverse order for the $a_k$s i.e., by replacing each $0$ by $1-a_k$ and each $1$ by $a_k$. The result is then $$A=\{(1-a_1)(1-a_2),(1-a_1)a_2,a_1(1-a_2),a_1a_2\}$$ Then we find the absolute difference sum of these two sets: $$f(a_1,b_1,a_2,b_2)=|b_1b_2-(1-a_1)(1-a_2)|+|b_1(1-b_2)-(1-a_1)a_2|+|(1-b_1)b_2-a_1(1-a_2)|+|(1-b_1)(1-b_2)-a_1a_2|$$ Here each $(a_k,b_k)$ is on a convex continuous function connecting $(0,1)$ to $(1,0)$, where $1>a_K>\cdots>a_1>0$ and $0<b_K<\cdots<b_1<1$.Given any set of pairs $(a_k,b_k)$ for $k\in\{1,\ldots,K\}$, the function $f$ is non-increasing (most probably decreasing) in every $b_k$.Example: $(a_1,b_1)=(0.1,0.2)$ and $(a_2,b_2)=(0.2,0.1)$. Then if we take $b_1$ as variable we expect that $f(0.1,b_1,0.2,0.1)$ is decreasing for $0<b_1<0.9$ (due to convexity). I verified that this is true in Mathematica. If we take $b_2$ as variable we expect that $f(0.1,0.2,0.2,b_2)$ is decreasing for $0<b_2<0.1$ (due to convexity). This is also true according to Mathematica. I guess this should be general for any example. I dont have any idea how to prove it, however.

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

• Paper of Tommaso Macrelli and Martin Wolf published in Physical Review Letters
by Tom Bridges on May 13, 2021 at 9:29 am

The paper “Becchi-Rouet-Stora-Tyutin-Lagrangian double copy of Yang-Mills theory“, co-authored by Leron Borsten (Heriot-Watt), Jurčo Branislav (Charles U, Prague), Hyungrok Kim (Heriot-Watt), Tommaso Macrelli, Christian Saemann (Heriot-Watt), and Martin Wolf, has been published this week (12 May) in Physical Review Letters. The paper is published gold open access and is available here. The following two diagrams

• Dan Hill speaks in the Leeds Nonlinear Dynamics Seminar
by Tom Bridges on May 12, 2021 at 8:06 am

Dan Hill spoke this week (Tuesday 11 May) in the Leeds Applied Nonlinear Dynamics (LAND) seminar series. The talk is titled “Making mountains out of Magnets: Localised patterns on the surface of a ferrofluid“, and will be on the existence of localised radial and cellular patterns with an application to the ferrofluid experiment. A link

• Paper of Masanori Hanada on gauge/gravity duality published in Physical Review D
by Tom Bridges on May 9, 2021 at 9:49 am

The paper “Bulk geometry in gauge/gravity duality and color degrees of freedom“, with sole author Masanori Hanada, has been published this week (6 May 2021) by Physical Review D (link here). The paper contains a breakthrough in gauge/gravity duality. Historically, gauge/gravity duality claims superstring theory is equivalent to certain non-gravitational quantum theories. Although there is

• Paper of Tom O’Neill and Bin Cheng to appear in the Journal of Differential Equations
by Tom Bridges on April 30, 2021 at 2:52 pm

The paper “An interior a priori estimate for solutions to Monge-Ampère equations with right-hand side close to one” co-authored by Tom O’Neill and Bin Cheng has been accepted for publication in the Journal of Differential equations. A link to the final form arXiv version is here. The image below shows the main theorem proved in

• Carina Dunlop and Naratip Santitissadeekorn give keynote webinars in the Surrey CMCB series
by Tom Bridges on April 29, 2021 at 8:14 am

The Surrey Centre for Mathematical and Computational Biology (CMCB) Seminar this week, on Thursday 29 April, will host keynote talks by Carina Dunlop (on “Integrating cell mechanics and cell mechanosensing“) and Naratip Santitissadeekorn (on “Uncertainty quantification for a time-series of count data and influence network“). A link to the seminar page with abstracts is here.

## Featured Blog Posts - Data Science Central

• What Makes Power BI the Most Powerful Data Visualization Tool
by Imenso Software on May 14, 2021 at 7:20 am

Nowadays, businesses have to rely on data in unprecedented ways. In fact, businesses hailing from various disciplines use massive amounts of data on a daily basis. They gather data from several sources, offline and online. However, it is also important to compile and process that data and analyze it using apt software solutions. That is why Data…

• Tech-Driven Transformation of the Legal Sector
by Pamela Hobbs on May 14, 2021 at 7:00 am

Legal Tech refers to the technology used in the legal sector. It has significantly transformed how attorneys and other legal professionals perform their duties. Moreover, it has brought a lot of opportunities for law offices (solo legal practices, law firms, and corporate/government legal departments) by digitally transforming legal operations, helping them meet client…

• The Secret behind Train and Test Split in Machine Learning Process
by Shanthababu P on May 14, 2021 at 6:30 am

What is Data Science and Machine Learning? Data ScienceData Science is a broader concept and multidisciplinary. Data science is a general process and method that analyze and manipulate data. Data science…

• 10 Email Marketing Tools For You To Consider
by Elena Osipova on May 13, 2021 at 4:21 pm

Email Marketing can be challenging. I learnt this lesson from my experience in the digital marketing sphere and being a support representative at an email software company. Why? There are a number of reasons. They come in different forms and from various places and refer…

• The Coming College Crisis
by Kurt A Cagle on May 13, 2021 at 3:30 pm

This article is also available as a podcast on Spotify. and was originally published on The Cagle Report. Education, especially college education, is facing an existential crisis. Partially due to demographic factors, and in part due to decisions…

## Mathematics – Wolfram Blog News, Views and Insights from Wolfram

• Is Your Function Continuous? Squaring Away the New Function Properties in the Wolfram Language
by Devendra Kapadia on March 30, 2021 at 8:17 pm

The Wolfram Language has several hundred built-in functions, ranging from sine to Heun. As a user, you can extend this collection in infinitely many ways by applying arithmetic operations and function composition. This could lead you to defining expressions of bewildering complexity, such as the following: &#10005 f = SinhIntegral[ LogisticSigmoid[ ScorerHi[Tanh[AiryAi[HermiteH[-(1/2), x] - x

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

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 earrings in

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

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 Media and

• 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

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 Age is

• 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