1
vote
1answer
34 views

Completeness and Separability of $C[0,\infty]$

Let $C[0,\infty]$ be the space of all continuous functions on $[0, \infty ]$ with metric $$ \phi(\omega_1, \omega_2) = \sum^{\infty}_{n=1} \frac{1}{2^n}\max_{0{\leq} t {\leq} ...
0
votes
1answer
23 views

How should I approach this Conditional Probability Problem?

Can anyone give a hint on how to begin this problem? Suppose $Y = X^2 + W$ where $W$ is Gaussian $N(0, 1)$ noise. Then derive an expression for $P(Y\mid X)$. I know about Bayes' Rule but I'm not ...
1
vote
2answers
20 views

How many 2m-permutations, consisting only of cycles of even length?

How many 2m-permutations, consisting only of cycles of even length? I have found this formula: $$Q_2(n) =((2n − 1)!!)^2$$ but how it can be proven?
0
votes
1answer
22 views

How can we show that 3-dimensional matching $\le_p$ exact cover?

In exact cover, we're given some universe of objects and subsets on those objects, and we want to know if a set of the subsets can cover the whole universe such that all selected subsets are pairwise ...
0
votes
0answers
12 views

Finding the dual of this linear program?

I'm having trouble figuring out the dual to this linear program. When I put it into a online solver I'm not getting the same answer. Heres the Primal: Max $2x_1 - x_2$ $2x_1 + 3x_2 - ...
0
votes
1answer
18 views

Cesàro summable sequences

During some homeworks the following question came into my mind (it is not part of the homeworks): Let $(a_k)_{k \in \mathbb{N}}$ be a Cesàro summable sequence in $\mathbb{C}$ and let $a := \lim_{n ...
0
votes
0answers
24 views

What does the positive minimal radius mean for an open cover?

The formal definition seems a bit confusing, any more words or pictures to explain it? Definition. Let $U$ be an open cover of $(M,d)$. We say that $U$ has positive minimal radius when there ...
2
votes
0answers
12 views

Question on perpetual American put

Define $u(x):=\sup_{\tau \in T_{0,\infty}}E[e^{-r\tau}(K-S_{\tau})_{+}1_{\tau<\infty}$]. $T_{0,\infty}$ the set of stopping times taking values in $[0,\infty)$ and ...
2
votes
2answers
15 views

Verify one of DeMorgan’s Laws for sets

Verify one of DeMorgan’s Laws for sets: $$\bigcap \{S\setminus U:U \in \mathcal U\} = S \setminus \bigcup \{U :U \in \mathcal U\}.$$ Can anyonw show me how to do this? a little confused, thanks
0
votes
3answers
25 views

Partial fraction decomposition of rational functions

How can I start this with Partial fractions? $$ \frac{10}{(s^2-4)(s^2+4)}+\frac{1}{s^2+4}$$ I was thinking of something like: ...
0
votes
1answer
20 views

Simplifying the polynomial for integration

Hi, I am trying to find the length of the function above from $x = 1$ to $x = 2$. I applied the length formula but I was not able to simplify it past $\sqrt{x^6 + \frac 12 + \frac 1{16x^6}}$. Can ...
0
votes
5answers
48 views

The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$

If $ a_0 ,a_1 ,\ldots,a_k $ are real numbers such that $$ a_0 + a_1 + \cdots + a_k = 0$$ Find $$ \lim_{n \to \infty } (a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k} )$$ I just ...
17
votes
2answers
208 views

Prove $\gamma_1\left(\frac34\right)-\gamma_1\left(\frac14\right)=\pi\,\left(\gamma+4\ln2+3\ln\pi-4\ln\Gamma\left(\frac14\right)\right)$

Please help me to prove this identity: $$\gamma_1\left(\frac{3}{4}\right)-\gamma_1\left(\frac{1}{4}\right)=\pi\,\left(\gamma+4\ln2+3\ln\pi-4\ln\Gamma\left(\frac{1}{4}\right)\right),$$ where ...
-3
votes
0answers
29 views

Algorithm to numerically solve this system of three polynomial equations of degree $6$ [on hold]

Mathematica Nsolve gave all $6$ solutions without an initial guess, whereas sympy Nsolve gave one solution closest to the ...
-1
votes
0answers
18 views

Is there a generic, or international, naming of the Wyatt Earp Effect?

Since succumbing to the Wyatt Earp Effect (that is, entertaining the false belief that probability applies to the past) is a well-known logical fallacy, and therefore known in other languages, it ...
0
votes
1answer
35 views

Minimum and maximum number of edges graph with 25 vertices and 6 connected components can have

Let G be a simple graph with 25 vertices and 6 connected components. Find (i) the minimum number of edges that G can have. (ii) the maximum number of edges that G can have. What I know: The maximum ...
1
vote
0answers
8 views

Finding the tangent space of a subgroup

My professor set the following question and I have an answer, though would like someone with more experience to cast a critical eye over the details as I don't necessarily trust my result! Define the ...
0
votes
2answers
12 views

Stationary probability in an M/M/$1$ queue with a lazy server

Customers arrive to a single server queue according to a Poisson process with rate $\lambda$. Each customer requires Exponential($\mu$) service time. In the beginning when there are $0$ ...
1
vote
3answers
21 views

expected value of a random palindrome

If you choose a 6-digit palindrome at random, what is the expected value for the number? All possible palindromes are equally likely to be chosen. Beginning number must be NONZERO, so numbers like ...
1
vote
1answer
25 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow ...
0
votes
0answers
8 views

Half-integer Bessel function evaluated at one

Let $K_\alpha$ denote the modified Bessel function of the second kind and order $\alpha$. We have $$ K_{1/2}(1) = 2e\sqrt{\pi /2},\\ K_{3/2}(1) = 7e\sqrt{\pi /2},\\ K_{5/2}(1) = 37e\sqrt{\pi /2},\\ ...
0
votes
1answer
42 views

Solve that equation [on hold]

Solve equations - $\sqrt{\frac{x+7}{x+1}}+8=2x^2+\sqrt{2x-1}$ $\sqrt{\frac{6}{3-x}}+\sqrt{\frac{8}{2-x}}=6$ $\sqrt[3]{2x+1}-\sqrt[3]{3x^2+x-1}=\sqrt{x}-1$
0
votes
0answers
7 views

binomial quantile function

Let $Q(\epsilon, n, p)$ be the $\epsilon$-quantile of a binomially distributed random variable with $n$ trials and success probability $p$. I am interested in the following question: Fix $0 < ...
0
votes
1answer
27 views

Prove $O(n)$ is compact

I have to prove $O(n)$ is compact, I know if I can prove it bounded and closed in $\mathbb{R^{n\times n}}$, I will be done. But how to check boundedness and closed ness. For closedness I would like to ...
1
vote
1answer
715 views

Find equations of the ellipses given conditions on the directrices, foci, and vertices

The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation: with distance between directrices $27$, and between foci $3$; with a focus at $(-\sqrt{13},0)$ ...
0
votes
0answers
10 views

Maple Cumulative Distribution Error help

I'm trying to write the cumulative distribution function into maple but I seem to get an error. The code is: with(Statistics): N:=RandomVariable(Normal(0,1)): CDF(N,t,inert=true); I ...
1
vote
2answers
33 views

An example of a not locally compact space in $\mathbb R^2$

Are the two subspaces $X$ and $\operatorname{cl}(X)$ of Euclidean space $\mathbb R^2$ locally compact? $$X = \{(x,\sin 1/x) \mid 0 < x \le 4\}\cup\{(x,\sin 1/x) \mid -4 \le x \lt 0\} \cup ...
1
vote
0answers
31 views

Zeroes of a complex valued function

How to find the zeroes of $f_r(z)=ze^{r-z}-1$ for any $r>0$ and real in say the unit disc? Or in any other domain? I understand we have to use Rouche's theorem somehow but I am not sure what is ...
4
votes
0answers
37 views

Duals of representations of affine group schemes, in particular $\mathrm{GL}_n$

Duals of representations of affine group schemes Let $R$ be a commutative ring. If $G$ is a group and $V$ is a dualizable i.e. finitely generated projective $R$-module on which $G$ acts, then it is ...
1
vote
1answer
27 views

Angular bracket operation in differential geometry?

There is an angular bracket operation in geometry, which looks like $$\langle X,Y\rangle$$ where $X$ and $Y$ are apparently $(0,1)$ tensors. It appears for instance in the answer to the following ...
2
votes
1answer
24 views

Triple integration using spherical coordinates

Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere $$x^2+y^2+z^2 ≤ 2$$ cut off by the plane z=1 and restricted to the first octant. Note: In ...
3
votes
2answers
59 views

Are these two definitions of the natural numbers equivalent?

If we consider two definitions of the natural numbers: Definition 1 $N$ is the set that satisfies all of: There is an element $0$ in $N$. For each element $n$ in $N$, there is the successor of ...
2
votes
1answer
43 views

Simplest Solution for a Round Table Q

Company of 3 Turks, 3 British and 3 French sit at a round table. What is the probability that no two countrymans sitting next to each other? All the people are different, but sitting orders different ...
0
votes
2answers
48 views

Modular equations

$$ x^{13}\equiv4\pmod{101}\\x\equiv5^{5^{5^{5}}}\pmod{47\cdot27} $$ Equations are separate. How should I approach these? Both has something to do with Euler's theorem, I believe, but all my attempts ...
0
votes
0answers
28 views

Automorphisms of symmetric groups

For $ X $ a finite set, $\operatorname {card} X\not=2$, we have $\operatorname {Inn}\operatorname {Sym} X\cong\operatorname {Sym} X $. Indeed, we have the action of $ \operatorname {Sym} X $ on itself ...
1
vote
1answer
40 views

Lebesgue Dominated Convergence: Alternative Proof?

For dominated decreasing positive functions one has: $$\int f\mathrm{d}\mu<\infty:\quad\int(f-s_n)\mathrm{d}\mu\downarrow 0\quad(0\leq s_n\uparrow f)$$ Is there an alternative to Lebesgue directly ...
0
votes
2answers
57 views

Given three plots of trigonometric functions, find an expression for each function [on hold]

Considering the three graphs in figure 1 showing trigonometric functions, $f(x)$, $g(x)$ and $h(x)$. Using these graphs, write the expression for each function.
0
votes
2answers
16 views

Finding a measurable function with an independent uniform distribution

Suppose $X,Y,U$ are random variables on some probability space such that $U$ is independent of $(X,Y)$. Prove there exists a measurable function $f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ ...
0
votes
1answer
15 views

Joint PDF Correlation

In the problem I am given $f(x,y)=2,\ 0 < x < y,\ 0 < y <1$. I'm trying to find the correlation $\rho$ which I know is equal to $$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$ ...
1
vote
0answers
42 views

“Research level” differential equation from “stochastic backpropagation” paper

This is ultimately a differential equation, but to motivate this and give background, I am trying to solve for $B$ from Rezende et al.'s "Stochastic Backpropagation and Approximate Inference in Deep ...
0
votes
1answer
22 views

limit of state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
1
vote
3answers
76 views

Bochner Integral: Integrability

Attention This question has been slightly modified!! Problem Given a measure space $\Omega$ and a Banach space $E$. Consider Bochner measurable functions $S_n\to F$. Then integrability is given ...
3
votes
1answer
91 views

Does the special Pell equation $X^2-dY^2=Z^2$ have a simple general parameterization?

In Carmichael's Diophantine Analysis ($\S8$), he notes that the equation $$X^2-dY^2=Z^2 \qquad(\dagger)$$ has a two-parameter solution $$x=m^2+dn^2, \quad y=2mn, \quad z=m^2-dn^2. \qquad(\star)$$ He ...
1
vote
1answer
335 views

A ship is being pulled by two tugboats problems

A ship is being pulled by two tugboats. The larger tugboat exerts a force 25% greater than the smaller tugboat and at an angle of 20 degrees west of north. In what direction should the smaller tugboat ...
1
vote
1answer
25 views

What's the meaning of the length of glide reflection?

Problem 4 of this sheet states: Let $\Gamma$ be a discontinuous, fixed point free group of glide reflections and translations. Let $g$ be a glide reflection of minimal length in $\Gamma$, and let ...
0
votes
1answer
27 views

Unital maps taking values in abelian C*-algebras

It is known that a bounded linear functional $f$ on a unital C*-algebra $A$ is positive if and only if $f(I)\geqslant 0$. Is the same true for bounded linear operators $T\colon A\to C(X)$ with $T(I) = ...
0
votes
1answer
51 views

What is the error in this fake-proof of the complex number i? [duplicate]

The error is from the 3rd step to the 4th step. But why is this an error? Can't $i$ be interchangeable with $\sqrt{-1}$? $-1 = i\cdot i = \sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$
1
vote
2answers
64 views

Finding a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$

To find a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$ knowing that $u(x,y)=1$ if $xy=1.$ I thought it may be useful to do the change $v=\log x, w=\frac{1}{2}\log ...
1
vote
1answer
27 views

A problem in locally compact Hausdorff space

I am trying to solve the following problem. Let $X$ be locally compact Hausdorff and $Y$ be Hausdorff. (a) If $f: X \to Y$ is continuous and open map then show that $f(X)$ is locally compact. (b) ...
2
votes
1answer
75 views

Definition of a Elliptic curve

I've seen two different definitions of an elliptic curve. 1) The first one being that it is a nonsingular projective curve of genus 1. 2) The other definition nonsingular projective curve of ...

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