Done (suggested Feb. 2016, done March 2016) I propose to deprecate abstract-algebra. It seems redundant with other tags, very broad, and the usage seems inconsistent. I feel it is comparable to the meanwhile removed topology and the deprecated geometry. (Explication on terminology: To depreca...

For various reasons it is often useful to be able to find questions which have some given tag as a standalone tag. (For example, when discussing whether some tag is useful for this site or not. Or when burnination of some tags is discussed.) Is it possible somehow find those questions which only...

Let $n>3$ be a positive integer.We denote the symmetric group of $n$ elements by $S_n$ and the identity mapping by $id$. For every $f\in S_n$, $f(1,2,\ldots,n)=(a_1,a_2,\ldots,a_n)$, denote $a_n$ by $m(f)$. For any positive integer $1\leq k\leq n-2$, define $f_k\in S_n$ as follow: $f_k:(1,\ldot...

We use $N^+$ to denote the set of positive integers.For any finite array $A:(a_1,b_1),...,(a_k,b_k)$,where every $(a_i,b_i)\in N^+\times N^+$,we call $A$ is good if for every $i\in \{a_1,b_1,...,a_k,b_k\}$,$i$ appears exactly two times in $a_1,b_1,...,a_k,b_k$.For example,$A_1:(1,1)$,$A_2:(1,2),(...

I'm finding programming various combinatorial searches (connected to mathematical music theory) in a general purpose computer language tedious, so I'd like pointers to computer platforms/environment purpose-built to enable such searches. If the answer has the form "Maple can do that" or "Mathe...

suppose we have a map $f:\mathbb{Z}\longrightarrow\mathbb{C}[t^{\pm}]$ with property that $f(i)=-f(-i)$. The algebra $\mathcal{T}=\mathbb{C}[t^{\pm}][L^{\pm},M^{\pm}]/[LM=tML]$ acts on $f$ by $(Lf)(i)=f(i+1),(Mf)(i)=t^if(i)$. Should the annihilating ideal of $f$ generated by annihilators in sy...

How to compute the value of [G(1,x)+G(2,x)+G(3,x)+....+G(x,x)] efficiently? When x can be as large as million.G = greatest common divisor.

I need to choose $k$ pairs of numbers out of first $n$ natural numbers such that the elements in each pair are $l$ distance apart. For example, if $n = 10, k = 3$ and $l = 2$, $\{(1,3),(4,6),(7,9)\}$ is a valid choice. How many such valid choices are there? I worked out for $l = 1$ and it seems t...

i am early undergraduate looking for a good textbook gor algebra, i don't want a too wordy book@@

How to prove following for $n\geq0$ ? $$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$ Where, at any point $\vec{r}$, the $\vec{\Omega}$ can be described by the polar angle $\theta$ measured with respect to the z axis and an azimuth...

'Twas the night before Christmas and under the tree Was a heap of new balls, stacked tight as can be. The balls so gleaming, they reflect all light rays, Which bounce in the stack every which way. When, what to my wondering mind does occur: A question of interest; I hope you concur! From each poi...

I have posted this question here without answer. Maybe I can get some light here. Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where $|l_i|$ is the length of $l_i$. Let $\alpha_1,...,\alpha_{n-1}$ be $n-1$ angles such that $\alpha_i>...

Test Polygon: Consider the following polygon as attached. Let the known parameter be as follows: •Member to member connectivity, i.e. it is known that A – B, X – E, F – B, etc. for all the members. •All the space co-ordinates (x, y, z) for each co-ordinate is known. Note that we do not kno...

Imagine I have two $n$ x $m$ assemblies of $P = (n*m)$ unit square cells on the plane, $(c_{(a,1)}, ..., c_{(a,P)}) \in A$ and $(c_{(b,1)}, ..., c_{(b,P)}) \in B$, where every cell, $c_k$, in a particular assembly must have a distinct color/value. Here, $c_{(a,i)} \neq c_{(a,j)}$ & $c_{(b,i)} \n...

You have three points. A,B and C. They define a circle segment that starts at A, goes through B and ends at C. Find the smallest bounding box that encompases the circle segment. Here is a picture: https://docs.google.com/drawings/d/14YwCO0UeMzu-rTLmqULg7HDU5XPYZvSBQdyfu0l71Fs/edit I started cre...

Hi, How to calculate number of circles in side a square, if we know the side of the square and the circles all are equal size. Thanks.

Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^1$-orbibundle over $M/S^1$. What's the proper way to define the Euler class of this $S^1$-orbibu...

Let $S$ be a set of 2D points $(x,y)$ with positive real coordinates, i.e. $x,y>0$. An 2D rectangle $R$ is called an ${Origin-Rectangle}$ if it is decided by the origin $(0,0)$ and another point $(x,y)$ with $x,y>0$. Denote $S_R$ as the subset of points in $S$ covered by $R$, i.e. $S_R = S\cap ...

So we have such situation: In this illustration, the first quadrilateral is shown on the Image Plane and the second quadrilateral is shown on the World Plane. [1] In my particular case the Image Plane has 3 quadrilaterals - projections of real world squares, which, as we know, have same size...

We have a grid with red squares on it. Meaning we have an array of 3 squares (with angles == 90 deg) which as we know have same size, lying on the same plane and with same rotation relative to the plane they are lying on, and are not situated on same line on plane. We have a projection of the s...

Eric Broug in his book Islamic Geometric Patterns gives straightedge and compass construction of some simpler patterns. It is clear his techniques will provide constructions for many Islamic patterns. Looking at formal constructibility, the Wikipedia pages gives Gauss' result that 7, 9, 11, 13,...

Some of the classical triangle centers can be expressed as solutions to minimization problems: Given a triangle $A_1, A_2, A_3$ define $d_i, i=1,2,3$ to be the distance of a given point $P$ to $A_i$, and $f_q$ as the sum of the $q$-th power of these distances:$f_q = \sum_{i=1}^3 d_i^q$. I'm looki...

At the risk of posting too low-level a question... Please consider two tori, with tube radii $r_1$ and $r_2$, and center-of-the-hole to center-of-the-tube radii $c_1$ and $c_2$. I'd like to find an analytical expression for the overlap volume as a function of the distance between the hole cent...

The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray originating at the point will cross before escaping. If we use projective duality, then regression ...

The $H$-representation of a convex polytope $S$, is just a set of linear inequalities corresponding to the intersection of halfspaces: $S = ( x | Ax\leq b )$. One could also represent a convex polytope as the convex-hull of its vertices, called the $V$-representation: $S = conv(a_1,..., a_m)$. ...

How many different rectangles (in terms of area) can fit in a 20-unit-wide square? The rectangles can be squares, and their dimensions are integers.

Consider any n points on the circumference of a circle. Draw a straight line link between each pair of points with a link weight consisting of the cosine of the angle the link subtends at the centre. It seems that If the convex hull of the point set contains the centre of the circle, then some p...

By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller circles uncovered. Is it possible to rearrange the 19 circles to accommodate a twentieth circle of ...

I want to pick a random direction in n-dimensional space. How can I do this? The reason I want to do this is to pick a neighbor for hill climbing optimization.

Say we have a space of dimension $D$. Say we have a $D$-cube of side $l$ centered at the origin and inside it we have a point $P\in \mathbb{R}^D$ and a collection of $D-1$ angles $\phi_1, \phi_2, \ldots \phi_{D-1}$. Then, say we have a line $r$ that is defined by the point $P$ and the angles. Is...

I am not a specialist in maths, so I thank you very much for any help you can give me. Consider two circles C1, C2. Q1: Find the points that are in the intersection of C1 and C2, this is easy ! Q2: Find two points p1 and p2, such that (p1 \in C1) and (p2 \in C2), and (distance(p1, p2)= D). Is ...

Greetings, We have a horn-shaped 3d body, which is represented as a list of vertices and faces. Each face is a triangle represented by 3 vertices. The body is positioned along the Z-axis (height). We would like to make several cuts at certain heights. Each cut (a plane perpendicular to the Z- ax...

Say we have $n$-gons $P$ and $Q$. Is there any necessary condition for $Q = f(P)$, for some linear transformation $f : \mathbb{R}^2 \to \mathbb{R}^2$? Sorry if this is too elementary / general.

Would the volume of an ellipsoid continuously increase if one keeps adding radii along new dimensions? What is the volume of ellipsoid with infinite dimensions?

There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.

Given two points A and B on the surface of the hyperboloid x^2+y^2-z^2=1. How to find the shortest distance between them along the surface?

I have polygon chains similar to the following... http://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Self_crossed_polygonal_chain.svg/220px-Self_crossed_polygonal_chain.svg.png ...given the chain in the image, how would I go about calculating a chain that defines the same shape but withou...

It is possible to explain me the 18th proof of the irrationality of square root of 2 from the following site? http://www.cut-the-knot.org/proofs/sq_root.shtml

i am early undergraduate looking for a good textbook gor algebra, i don't want a too wordy book@@

TL;DR: I suggest to blacklist the deprecated tags that still exist on some questions, in order to prevent them from being added to new questions. There are a few tags which are deprecated and should not be used at all, but still exist on the site.1 Ideally such tags should be completely removed,...

Done (suggested Feb. 2016, done March 2016) I propose to deprecate abstract-algebra. It seems redundant with other tags, very broad, and the usage seems inconsistent. I feel it is comparable to the meanwhile removed topology and the deprecated geometry. (Explication on terminology: To depreca...

What is the exact relationship hyper-complex and quaternionic Kahler manifolds? From Wikipedia we get that hyper-Kahler manifolds are both hyper-complex and quaternionic Kahler. Thus, the two families have a non-empty intersection. But this is all I can conclude. Moreover, quaternionic projectiv...