# All Questions

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### If $f(x)=\frac{\sin(\pi x)}{\pi\sin(x)}$ and $f'(x_0)=0$, then evaluate $(f(x_0))^2(1+(\pi^2-1)\sin^2x_0)$

The questions reads: Let $f(x)=\dfrac{\sin(\pi x)}{\pi\sin(x)}$, $x\in(0,\pi)$ and let $x_0 \in (0,\pi)$ be such that $f'(x_0)=0$. Evaluate $$(f(x_0))^2(1+(\pi^2-1)\sin^2x_0)$$ I have used ...
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### Normals PO,PB and PC are drawn to the parabola $y^2=100x$ from the point $p=(a,0)$

Normals $PO,PB, PC$ are drawn to the parabola $y^2=100x$ from the point $p=(a,0).$ If the triangle $OBC$ is equilateral, then a possible value of $a$ is:
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### wasserstein metric with order or similar measure which considers order

I have come across the Wasserstein metric, in which the order is not considered. Suppose that I have $$X=[100 203 103 204 105] \\ Y=[100 103 203 204 105]$$ According to Wasserstein, the distance ...
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### Integrating this definite integral from 0 to infinity using contour integration

I am trying to derive sterling's approximation(image 2). I derived this integral from the gamma function. I need to compute the following it using contour integration. I need the answer in the ...
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### If $\frac{t^2-x^2}{(t^2+x^2)^2}+\frac{(1-x)^2-t^2}{((1-x)^2+t^2)^2}=0$ for real $t$ and $0<x<1$, prove that $x=1/2$

$$\frac{t^2-x^2}{(t^2+x^2)^2}+\frac{(1-x)^2-t^2}{((1-x)^2+t^2)^2}=0$$ where $0<x<1$ and $t\in\mathbb{R}$. Prove that $x=1/2$. It is evident that $x=1/2$ satisfies the above equation. Please ...
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### Consider the set $A=\{X\in\mathcal P(\mathbb Z),X=\{k,k+2\} \}$. Show that $A$ is countable infinite.

Consider the set $A=\{X\in\mathcal P(\mathbb Z),X=\{k,k+2\} \}$. Show that $A$ is countable infinite. Hello everyone. I am a little confused on this one. Can I get some help? Thank you. This is what ...
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### Binary operation ab+a defined on Q. Is it a group?

A binary operation ∗ is defined on Q such that a∗b=ab+a. Is it a group? I think that it's associative and the identity element is 0, but what about the inverse element? Am I right to think that for ...
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### Best book Self teaching calculus at A level?

I am from Pakistan where I have completed the A level mathematics in my first year of a level. I plan on taking the further mathematics Course this year. I am looking for a good book to self teach my ...
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### What assumptions on $f: X \to Y$ such that $\Delta: X \to X \times_Y X$ is a closed immersion?

Let $X, Y$ be separated schemes and $f: X \to Y$ a morphism of schemes. Is it then always the case that $f$ is separated (i.e. $\Delta: X \to X \times_Y X$ is a closed immersion)?
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### Minimum squared curvature shape

[Related but different: Which shape does an elastic rod take as its ends are getting closer? - different functional] I'm looking for a shape that elastic beam makes when its ends are brought into ...
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### Why is it difficult to find the Global Optimum?

When studying Calculus I learnt that it it possible to take the derivative of a function to find its minimum and maximum points. I then wondered what happens if there are more than one minimum and ...
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### What is the formula to find the number of vertices and edges of a cubic lattice of n dimensions?

When $n = 1$, it is a basic cube with $8$ vertices and $12$ edges. When $n = 2$, $v = 8$, $e = 12$ we have: What is the formula to find vertices and edges for other values of $n$?
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### An alternate motivation 1988 IMO question #6 (the infamous one)

This is an especially famous problem that you can check out an alternative asking and full solutions to here and, more formally, here. This post is not asking for solutions or full fledged proofs; ...
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### Expected value of number of different cards

I have this question: From a deck of $10$ cards, Tom draws two cards and places them back in the deck. Now Jerry draws two cards from the deck. Let $X$ be the number of cards that was chosen ...
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### Distributive property for convergent infinite sum?

When is $$c\sum_{k=1}^∞a_{k}=\sum_{k=1}^∞ca_{k}$$ where $c$ is a constant and the sums converge. 1) Does this always hold for convergent infinite sums? If so why? (What is the proof?) 2) If it ...
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### On circumcircle, incircle, trillian theorem, power of a point and additional constructions in $\triangle ABC$

The problem was at this deleted question originally. Given: 1) $\triangle ABC$ -- an arbitrary triangle 2) with circumcircle $\omega$ centered at $O$ 3) and incenter $I$. 4) Let $D$ be the ...
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### Understanding the Poisson bracket for this system in $so(4)$

I have the following system of ODEs: \begin{align*} \dot{x}_1 &= x_2x_3A_{32} + x_5 x_6 A_{65}\\ \dot{x}_2 &= x_1 x_3 A_{13} + x_4 x_6 A_{46}\\ \dot{x}_3 &= x_1 x_2 A_{21} + ...
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### What differential equations do these solutions belong to?

Find the differential equation whose solution is given by $y=Ae^{2x}+Be^{-2x}$ and $y=c(x-c)^2$ where $c$ is a constant. The first one is easy. It is $y’’-4y=0$ but I’m having trouble with the second ...
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### Comparing similar vectors and detecting expressions in n-dimensional space

I have more of a methodological problem than an exact mathematical formula to solve. At first I'll describe the environment: I have a large set of delay-vectors (e.g. for a chip/cell) characterized ...
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### Convex formulation of the smallest distance to a point outside of a polyhedron

Consider a polyhedron $S$ whose set of extreme points (vertices) is $\{v_1, v_2,\dots,v_k\}$. Given a point $y \notin S$, we would like to find the point with the smallest distance to $y$. Provide a ...
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### Proving Lorentz Transformation identities

Given the defining property of Lorentz transformation (which is linear) $\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma}$, prove the following identities (i) ...
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### Deriving a contradiction from two cases

Assuming that we have a true mathematical case (A). The negation of A) is equivalent to the following statements: (1) There exist a polynomial function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that ...
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### Volume between sphere $x^2+y^2+z^2 = 2$ and paraboloid $z= x^2+y^2$

Express the volume of region $D$ upper bounded by the sphere $x^2+y^2+z^2=2$ and the paraboloid $z=x^2+y^2$. a- Cartesian Coordinates b- Cylindrical Surface c- Spherical coordinates If you help me ...