1
vote
1answer
14 views

Why boundary of a locally closed set is nowhere dense?

Let $X$ is locally closed , i.e. exist open $U$ S.t. $X=\overline{X} \cap U $ , and $bd (X) = \overline{X} \setminus \mathring{X} $. How can I show that $ bd(X) $ is nowhere dense? I read topics ...
1
vote
0answers
8 views

Specific requirements for Runge's theorem to hold

This question is exercise 8.2 in Conway's Functions of One Complex Variable I. It states: Let $\mathbb{D}\subset\mathbb{C}$ be the open unit disk, and let $K=\{z\in\mathbb{D}: \frac{1}{4}\leq ...
1
vote
0answers
11 views

Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
1
vote
0answers
9 views

Boxcar average algorithm of the specified width.

Ok, I need to write a java algorithm which simulates the SMOOTH function written in IDL where the SMOOTH function is given by $$ R_i = \begin{cases} \displaystyle \frac{1}{w} \sum_{j = ...
1
vote
1answer
10 views

Distributing infinite supply of $n$ distinct objects into $k$ identical urns

I have $n$ distinct objects, namely {$n_{1\le i \le n}$} with an infinite supply of each of them, and I have $k$ identical, indistinguishable urns to place the objects in. Each urn will contain ...
0
votes
0answers
15 views

Growing slower than exponential?

Consider $2^{cn}+a(n)$ with $c$ being an exponential growth rate. Am I right that $$ 2^{cn}+a(n)\sim 2^{cn} $$ only, if $a(n)$ grows slower than exponential? And if yes.. when does $a(n)$ grow ...
3
votes
2answers
71 views

Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
0
votes
1answer
16 views

Instalments ( simple interest)

The price of a T.V set worth Rs. 20,000 is to be paid in 20 instalments of Rs. 1000 each. If the rate of interest be 6% per annum, and the first instalments be paid at the time of purchase, then the ...
-1
votes
0answers
13 views

Simple interest ( Instalments)

A sum of Rs 10 is lent to be returned in 11 monthly instalments of Re 1 each, interest being simple. The rate of interest is? 100/11 % 10 % 11 % 240/11 % What I understand is this - Rs 1 for 11 ...
2
votes
1answer
59 views

Theoretical and computational results of ODE differ a lot! Why?

Hello. I have a problem in that my theoretical and practical(computational) calculations differ a lot. in 1 order of magnitude actually. Impulse I needed to jump over the saddle point in one ...
3
votes
1answer
115 views

What is the name of this paradox?

What is the name of the mathematical paradox which is arises from the following? If we imagine a point on a two-dimensional coordinate system (line graph), which moves from the positive part of the ...
1
vote
2answers
31 views

definite integral of a complex function

I wonder if there is a way to evaluate this definite integral... $$\frac{2}{\pi}(\ln (2) + \int_{0}^{\infty}({\sqrt{\frac{1}{t^{4}} - \frac{4e^{-4t}}{(1 - e^{-4t})^{2}}}} - \frac{1}{t^2 + 1})dt)$$ ...
0
votes
0answers
6 views

Principal value methods for fourier laplace etc.

I recently saw this here: $$\int_{-\infty}^{\infty} \frac{1}{\omega^2} e^{j\omega t} d\omega$$ and I was unable to understand how such an integral could be computed. I want to learn about this ...
1
vote
0answers
9 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
0
votes
1answer
13 views

Is this legitim to see polynomial growth?

Let $(a_n)$ be any monotoninally in creasing sequence where $a_n$ is a real number dependent on $n$. Of course it is $a_n\leq a_n^2$ for all $n\geq 1$. Does this means that any increasing sequence ...
11
votes
2answers
83 views

$\forall x,y>0, x^x+y^y \geq x^y + y^x$

Prove that $\forall x,y>0, x^x+y^y \geq x^y + y^x$ A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an ...
17
votes
4answers
3k views

1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?

A hypothetical example: You have a 1/1000 chance of being hit by a bus when crossing the street. However, if you perform the action of crossing the street 1000 times, then your chance of being ...
5
votes
1answer
42 views

Try to generalize a problem in Hatcher: finite vs. infinite CW-complexes

While solving a problem in Hatcher I got this doubt in my mind, In the 2nd chapter (Homology) Hatcher asked us to prove the following question... If $X$ is a finite dimensional CW-complex then, ...
0
votes
3answers
18 views

Proof of the construction of Dirac Delta

The Dirac Delta function pops up in a wide variety of applications, especially in applications that require Laplace and Fourier transforms. But my question is: what's the proof that the distribution ...
2
votes
0answers
10 views

Open embedding of affine toric varieties implies Cone is face of the other

let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion ...
-2
votes
6answers
58 views

What must be added to $(x^3-3x^2+4x-13)$ to obtain a polynomial which is exactly divisible by $(x-3)$? [on hold]

Please explain with details. I am not able to understand this question with examples. Please describe. than you everyone for your supportive questions . :)
-1
votes
0answers
24 views

Poisson Kernel - Complex Analysis

This is a problem from Ahlfors, Complex Analysis, pag. 171 #5. "Show that the mean value formula $u(z_0)=\frac{1}{2 \pi} \int_0^{2 \pi} u(z_0 +re^{i \theta} ) d \theta$ remains valid for $u=log ...
1
vote
3answers
37 views

Is it possible to make a dataset given these following mean, range, and standard variance?

Given that the: $Mean = 30$ Range $= X_n - X_1 = 10$ $S^2 = Variance = 40$ is it possible to construct a dataset with those values? So, to be honest I have no idea how to properly approach this ...
0
votes
1answer
19 views

using matrix with cos/sin etc.

I need to check if the equation is linear independent so: $$ \alpha x^2 \cos x + \beta x + \gamma \sin x = 0 $$ I got 3 equations of it: $$\beta \pi/2 + \gamma = 0$$ $$\alpha \pi^2(-1) + \beta \pi = ...
1
vote
0answers
32 views

Pullback map distributes over wedge product (proof)

To prove that the pullback map distributes with the wedge product is it first best to prove that it distributes over the tensor product and then use the relation $$dx^{\mu_{1}}\wedge\cdots\wedge ...
2
votes
1answer
38 views

Can someone help explain a proof from Feller Vol1 III.5?

One will need a copy of Feller's text (3rd edition) to answer this question. The proof I'm having difficulty with is Theorem 1, pages 84-85. When he discusses the r=1 case, he says ... "To the ...
0
votes
0answers
13 views

Behavior of eigenvalues of certain matrices

I am trying to analyze the behavior of the 2 highest eigenvalues of matrices of this form : Symmetric $n*n$ matrices that contains only : $1/k$ (for fixed k), -1,1 and 0. My hope is to find some ...
1
vote
0answers
32 views

Question about your function,

I'm Xavier Vigan, a physical oceanographer. I've been using your $f(x)=\dfrac 12 \times \left(X+C-\sqrt{S+(X-C)^2}\right)$ function to calibrate quantile vs quantile plots. Because of the shape of ...
5
votes
2answers
68 views

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the ...
0
votes
0answers
14 views

Whether the definition of fractional derivative is correct or not?

The fractional derivative is defined as $D^\alpha f(x)=\Gamma(1+\alpha)\lim_{h\rightarrow 0^+}\frac{f(x+h)-f(x)}{h^\alpha}$.
0
votes
2answers
19 views

If I have a matrix M=[A,B;0,C], how do I prove that rank(A)+rank(C)<=rank(M)?

. . . . . . . A . . B . . . . . . . 0 0 0 . . . 0 . 0 . C . 0 0 0 . . . If I have a matrix $M$ as displayed in the text above ($A$ ...
1
vote
2answers
485 views

Rotating the gradient

Suppose I have a triangle T in 3dimensional space and i want to rotate it in arbitrary ways. The coordinates for T are given by $f: T_R \in \mathbb{R}^2 \rightarrow T \in \mathbb{R}^3 $ where $T_R$ is ...
2
votes
1answer
30 views

Proof of convergence for the proximal point algorithm

I'm trying to come up with a super simple proof of convergence on the proximal point algorithm, which uses the iteration scheme $x^{i+1} = \mathbf{prox}_{tf}(x^i)$ where $\mathbf{prox}_{tf}(z) = ...
0
votes
2answers
19 views

$\Bbb{Z}_{2}(\alpha)$ as splitting field

i have problems with an exercise: let $\alpha$ be a root of the polynomial $X^{3}+X^{2}+1$ in $\Bbb{Z}_{2}$. Prove that $\Bbb{Z}_{2}(\alpha)$ is the splitting field of this polynomial over ...
0
votes
1answer
44 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
1
vote
0answers
14 views

Gradient w.r.t. boundary conditions in PDE

I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation: $\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$ Here ...
1
vote
1answer
32 views

In $S_{3}$ what is the group generated by $(123)$?

In $S_{3}$ what is the group generated by $(123)$? Is there a way to find the elements of the group generated by $(123)$?
0
votes
0answers
10 views

Translation:Bayes Classificator -> precise math?

I want to understand the most simple form of the Bayes classificator (see here) but I want to understand it in a really precise, clean, mathematical way. Math description of the setting: Let us ...
1
vote
3answers
54 views

Why is this equality involving factorials true?

$$ (n +1)! -1 +(n +1)(n +1)! = (n +2)! -1 $$ Can someone explain me how in the world is this true? :D Thanks (yes I'm trying to understand induction).
1
vote
1answer
33 views

Closed immersions and complete linear systems

Let $X$ be a local complete intersection subscheme in $\mathbb{p}^n$ for some integer $n>0$. Denote by $i:X \to \mathbb{P}^n$ the induced closed immersion, ...
4
votes
4answers
75 views

In every set of $14$ integers there are two that their difference is divisible by $13$

Prove that in every set of $14$ integers there are two that their difference is divisible by $13$ The proof goes like this, there are $13$ remainders by dividing by $13$, there are $14$ numbers ...
0
votes
1answer
35 views

Let $X: U(0,3 \pi)$ - Uniform on $(0,3 \pi)$ Find the distribution of Y and the expectation of Y if Y is:

$$Y=\begin{cases} -\sin X , x \in(0, \pi] \\- \frac{1}{2} , X \in[\pi, \frac{3 \pi}{2}]\\ \cos X, X \in [\frac{5 \pi}{2}, \frac{11 \pi}{4}] \\ \frac{3}{4}, X \in (\frac{11 \pi }{4}, 3 \pi) ...
2
votes
1answer
45 views

What is the use and motivation for this particular concept in permutations?

Say you have the permutation $(54231)$ element of $S_5$ Now you drop say the "4" and then re-rank the remnant permutation on the other elements. Then you are left with, $(4231)$ element of $S_4$ ...
0
votes
1answer
20 views

Take the outcome of a draw in ELO formula

Is there any way to get the probability of a draw outcome using ELO formula as it only gives the Win probability ELO formula is given by $E = \frac{1}{1+10^\frac{d}{a}}$ where d is the difference in ...
1
vote
0answers
11 views

Dimension of the restricted representation

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
0
votes
1answer
17 views

a question on the extension of an operator?.

It is known that $C_0^{\infty}(\Omega)$ is dense in $W_0^{1,p}(\Omega)$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$. Let $T:C_0^{\infty}(\Omega)\rightarrow\mathbb{R}$ be a continuous linear ...
4
votes
1answer
32 views
0
votes
0answers
20 views

maximum area under the function, with length constraint

Suppose I have a function $f$, such that $f(0) = f(1) = 0$. Given the length $l$ of the curve between $0$ and $1$, which function maximizes the area under the curve? I know that if $l \leq \pi/2$ the ...
1
vote
2answers
60 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
0
votes
1answer
22 views

Laplace transform of $H(-t)$

How to compute the Laplace transform of $H(-t)$, where $H$ is the Heaviside step function? Does it exist? Basically, I want to compute the Laplace transform of $e^{2t}H(-t)$. I know how to compute ...

15 30 50 per page