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1answer
30 views

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 ...
0
votes
1answer
8 views

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:
1
vote
2answers
19 views

Banach Algebra: $r_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2$

I am trying to see that if we have a complex Banach algebra with unity, we will have that $$r_\sigma(x)=\|x\| \iff \|x^2\|=\|x\|^2.$$ I was able to do the first implication: Since we know that $\|x^2|...
3
votes
1answer
25 views

Is an integral domain $\frac{\mathbb{C}[x,y]}{<x^4+x^3y+y^4>}$?

Is an integral domain $\frac{\mathbb{C}[x,y]}{<x^4+x^3y+y^4>}$ ? Where $\mathbb{C}[x,y]$ is a commutative ring of polynomials over $\mathbb{C}.$ I know the fact that for any field, $\mathbb F[...
0
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0answers
13 views

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 ...
-1
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0answers
14 views

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 ...
0
votes
1answer
39 views

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 ...
0
votes
2answers
13 views

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 ...
0
votes
2answers
35 views

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 ...
0
votes
1answer
16 views

Tensor products and linear maps

Let $R$ be a commutative ring (with unity) and let $M,M',N,N'$ be $R$-modules. I know that there is a standard linear map $$\varphi:\,Hom_R(M,M')\oplus Hom_R(N,N')\longrightarrow Hom_R(M\otimes_R N,\, ...
0
votes
1answer
14 views

one function for 3 scenarios

Just wondering, is there a function that depends and a small number of parameters and can model these 3 scenarios (depending on the chosen parameters): The first is a constant, the second a log-...
0
votes
2answers
24 views

Prove a set is totally disconnected

Let $I=[0,1]$ be a set with the topology induced from the usual topology of $\mathbb{R}$ and $D \subset I$ an open and dense subset. Prove that $I-D$ is totally disconnected.
0
votes
1answer
26 views

integrate $e^{\frac{x-y}{x+y}} dydx$

Integrate $e^{\frac{x-y}{x+y}} dydx$ over the set $D=\{(x,y):x\geq 0,y\geq 0,1\leq x+y\leq 2\}$. I tried to do substitution $u=x-y$ and $v=x+y$ so i know that $ 1\leq v\leq 2$ but i couldn't figure ...
2
votes
0answers
15 views

Application of Girsanov's theorem

Given a Brownian motion $W(t)$ under $P$ and the stochastic process $\hat{W}(t) := W(t) - \int_{0}^{t}\theta(s)ds$ which is a Brownian motion under an equivalent measure $Q \sim P$. The stochastic ...
3
votes
0answers
17 views

example of a physical phenomenon

can anyone give an example of a physical system(phenomena) that often arises in the field of engineering research whose states-space has to be taken as a complete separable metric space(I precisely do ...
1
vote
2answers
26 views

Lagrange's theorem to prove $b^{p-1}=1$

My problem: (a) Let $p$ be a prime and let $b$ be a nonzero element of the field $Z_p$. Show that $b^{p-1}=1$. Hint: Lagrange. (b) Use (a) to prove that if $p$ is a prime and $a$ is an integer then $...
0
votes
1answer
23 views

How can I solve the system of ODEs $x'=-5x+10y, y'=-4x+7y, z'=z^2-2z+1$?

How can I solve the following system of differential equations? $$\begin{aligned} x' &= -5x + 10y\\ y' &= -4x + 7y\\ z' &= z^2 -2z + 1\end{aligned}$$ I have to give a function $f(x)$, ...
0
votes
1answer
31 views

Find number of $n\in\{1,2,\dotsc,1000\}$ s.t. $\exists x\in\mathbb{R}^+$ where $x^2+\lfloor x^2\rfloor=n$.

Find number of $n\in\{1,2,\dotsc,1000\}$ such that $\exists x\in\mathbb{R}^+$ where $x^2+\lfloor x^2\rfloor=n$. My approach : As $\lfloor x^2\rfloor$ and $n$ are integers, then $x^2$ is a (positive) ...
0
votes
0answers
10 views

Measurable variables

Consider $\pi$ that is measurable variable with respect to the $\sigma-\text{algebra}$ $\mathcal{F}$. Is this equivalent to write $\pi\in\mathcal{F}$? By using the concept of measurability it means ...
0
votes
0answers
9 views

Heat equation in a rectangle

I'm trying to solve the following problem. Consider the heat equation in the rectangle $[0,\pi] \times [0,\pi]$ with Dirichlet boundary conditions. Calling $\Omega:=(0,\pi) \times (0,\pi)$ we have ...
0
votes
0answers
12 views

How does $\lim\limits_{Re(s)\rightarrow -\infty} Li_s(e^{w})$ become $\Gamma(1-s)(-w)^{s-1}$?

I am trying to prove the following property of polylogarithm. $$\lim\limits_{Re(s)\rightarrow -\infty} Li_s(e^{w}) = \Gamma(1-s)(-w)^{s-1} \text{ for } -\pi < \Im(w) < \pi.$$ As per Wikipedia's "...
2
votes
0answers
27 views

Chern class in Bott-Tu

I've met some trouble in understanding Chern class. I first touch the Chern class in classyfying space of characteristic class. For $\pi:E\rightarrow X$ and we have such relationship:$$Vect^n_{\Bbb C}(...
0
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0answers
6 views

Compute gradient of 2D complex valued function using DFT methods (spectral gradient)?

I'm looking to see if it is possible to compute the gradient of a complex function using FFT methods. I am able to do this for a 1D real valued function with the DFT using th following theory. $$F_k=\...
0
votes
1answer
19 views

Benford's law – formula

My I ask what does the "k" represent in Benford's law? The formula I'm struggling with is in here – enter image description here Thanks in advance.
0
votes
2answers
15 views

gcd and divisibility in ℤ

I'm studying the divisibility properties in ℤ, there's a passage in my manual I find difficult to grasp. I don't understand why it is consequent that $d_1$ divides $d_2$ in the following statement: ...
2
votes
0answers
11 views

Criteria for a distribution function to uniquely determine a measure on $(\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2)$

I know that for a function $F:\mathbb{R}\to[0,1]$ with the below criteria, $F$ is a distribution function for a probability measure $\mu_F$. Indeed, we can define $\mu_F=F(b)-F(a)$ for all $a,b\in\...
0
votes
0answers
10 views

Weak formulation incompressible Euler equations

A divergence-free vector field $v \in L_{loc}^2(\mathbb{R}^n_{x}\times \mathbb{R}_{t},\mathbb{R}^n)$ is said to be a weak solution of the incompressible Euler equation $$\frac{\partial v}{\partial t} ...
2
votes
1answer
32 views

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 ...
0
votes
0answers
19 views

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)?
0
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0answers
6 views

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 ...
0
votes
0answers
21 views

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 ...
-1
votes
0answers
13 views

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 $n$?
0
votes
1answer
55 views

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; ...
1
vote
1answer
26 views

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 ...
0
votes
2answers
20 views

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 ...
1
vote
1answer
21 views

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 ...
4
votes
0answers
24 views

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} + ...
0
votes
2answers
31 views

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 ...
-1
votes
0answers
8 views

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 ...
0
votes
2answers
31 views

Why can the equal sign $(=)$ be included in the below inequality?

Let $\left(X_n\right)_{n\geq1}$ be a sequence of random variables defined on a given fixed probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$. Consider that $\mathbb{P}\...
0
votes
0answers
41 views

Why do some use “$\;\stackrel{\text{def}}{=}\;$” for definitions?

Why do some use the equality symbol "$\;\stackrel{\text{def}}{=}\;$" for some definitions? For instance, $$\varepsilon \;\stackrel{\text{def}}{=}\; \frac{\Delta(L)}{L}$$
-9
votes
0answers
33 views

Dude - you should'nt miss it - need some suggestion bout improving your probability skill… [closed]

trust me, i had been trying to solve this damn problem for the whole day - and finally, nice. i gave up. so the problem is: how much 3*3 magic square are there...? seems simple, but for a math noob ...
0
votes
2answers
41 views

Show that $X + Y$ and $|X − Y |$ are uncorrelated

Be $X$ and $Y$ Independent Bernoulli-distributed random variables with parameter $p= \frac{1}{2}$. Show that $X + Y$ and $|X − Y |$ are uncorrelated. So I have to show $cov(X + Y,|X − Y |)= 0$ $cov(...
1
vote
1answer
12 views

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 ...
0
votes
0answers
16 views

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) ...
0
votes
0answers
29 views

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 ...
0
votes
1answer
42 views

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 ...
0
votes
1answer
23 views

Simplifying a logical expression

I'm looking for a way to simplify this logical expression: ((x == y) and (x > 0 or z > 0)) or ((x != y) and (x > 0 and y > 0 and z > 0)) All ...
0
votes
1answer
24 views

Positiveness of a continuous function on an interval

Consider a continuous function $f:D\subset \mathbb{R}\rightarrow \mathbb{R}$. Let $I \subset D$ be an interval. $f$ has a property that for each pair $x_{1},~x_{2}\in I$ with $x_{2} \ge x_{1}$, if $f(...
0
votes
1answer
24 views

Magnetic field induction due to charged radially symmetrically shaped body rotating about an axis

I had been solving questions based on finding magnetic field induction at the center of geometrical figures, with charge uniformly distributed on their surface with density $\sigma$, rotating about ...

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