# All Questions

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### Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1$ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
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### Birational equivalent and isomorphic represenation of a subalgebra

here I am again with another exercise which gives me a hard time. Let $A$ be the subalgebra of $\mathbb{C}[t]$ of all polynomials $f(t)$ such that $f(1) = f(-1)$. Let X be an alebraic set such ...
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### Dynamic programming approach for multidimensional problem

I use a dynamic programming approach to optimize the behaviour of individuals playing a game.I have one strategy matrix that describes the behaviour of individuals in situation 1, which depends on ...
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### sum of perpendicular distances from the sides of a triangle.

I am trying to solve a problem and got stuck in the following:- P, A’, C’ are respectively points on the sides AC, CB, and AB of ⊿ABC. PA’ and PC’ are the perpendiculars to the sides of the ...
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### Im stuck on (b)

This is about changing fractions into a mixed expression. So I have to do divide them. But I don't know why (b) has to leave spaces
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### Is it possible to have a sphere $S^m$ equidistant to sphere $S^n$ in $R^k$?

Is it possible to place a sphere $S^m$ and another sphere $S^n$ in Euclidean $k$-dimensional space $R^k$ in such a way that the distance from any point of the first sphere to any point of the second ...
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### How to simplify the kronecker product of four product

Suppose that $A$ and $B$ are $N\times N$ matrices, and $I$ is a $m\times m$ identity matrix, then here comes the kronecker product $$K_1 = (I\otimes A)\otimes(I\otimes B).$$ I now wonder how can we ...
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### Is $(x^2,xy)$ a primary ideal in $k[x,y]$ for $k$ a field?

In Example of Page 52 in Atiyah's Introduction to Commutative Algebra $\mathfrak a = (x^2,xy)$ is not a primary ideal in $A = k[x,y]$ where $k$ is a field. I think, for any $z \in \mathfrak a$, ...
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### Lines and planes-recursive formula

A family of $n$ lines is drawn in the plane such that each pair of lines cross and no $3$ dinstinct lines have a point in common Let $r(n)$ denote the number of regions into ...
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### Euler's Equation [duplicate]

I'm new around here, I'm fourteen, and I am in Ninth Grade. Can somebody tell me what Euler's equation exactly is, and why it's important, and what we can use it for? The whole $e^{i\pi} = -1$ thing. ...
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### Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
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### which functions can be obtained as a composition of a continuous function with itself?

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
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### Tricky question about binomial expansions.

State the binomial expansion of $(1+x)^n$ So I can do this $$(1+x)^n=\sum_{i=0}^{n} {n\choose i}x^i$$ Then given $n=2k$ is even. Derive an expression for $$\sum_{i=0}^{2k} (-1)^i{2k\choose i}$$ ...
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### How to find curvature between two lines in each four quadrants?

everyone plz help me by providing formula to calculate curvature of two lines in each quadrant. as this line in picture it could be in different direction too. so i have to calculate curve in every ...
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### Functional Minimization of Exponential Decay

I would like to find a function $f$ that minimizes the functional: $$\ln(f(x))f(x)-\frac1x$$ over some range of $x > 0$. Is this a good application for functional calculus and the Euler-Lagrange ...
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### Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$

So elements of this ring look like $$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ If I want to find the multiplicative inverse element for the above general element, then i'm trying to find $x,y,z\in\mathbb{Q}$ such ...
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### finding the overall loss or profit percentage

A person sells two shirts for 880 each . he gets a 10% profit on one whereas 20% loss on the other. Find the overall profit or loss percentage I didn't understand how to find the ...
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### maple allvalues function?

I have two simultaneous equations that I have solved (for two parameters) using Maple. The solutions are themselves complicated expressions of yet another parameter. I had to use the allvalues'' ...
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### Find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not.

I am trying to find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not. Have you got any ideas of easy examples? Thank you!
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### The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
Let $X$ be a right-continuous Feller Dynkin process. For $r>0$ we define the $\{\mathcal{F}_t\}_t$ stopping time (which is called escape time) $$\eta_r=\inf\{t\geq 0: \|X_t -X_0\|\geq r\}$$ We have ...
We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: ...