2
votes
0answers
8 views

Self-duality in a lattice

Is there any self-dual lattice $(X,\le)$ such that there is not any self-duality $f:X\to X$ such that $f\circ f = 1_X$?
-4
votes
1answer
143 views

Questions about a real orthogonal $3 \times 3$ matrix

Let $A$ be an orthogonal $3 \times 3$ matrix with real entries. Pick out the true statements: a. The determinant of $A$ is a rational number. b. $d(Ax,Ay) = d(x, y)$ for any two vectors $x, y \in ...
2
votes
1answer
30 views

Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
-3
votes
0answers
26 views

Subset Lp space. [on hold]

Let S closed vector space subset $L^1$($\mu$), where $\mu(X) < \infty$. Assume $f \in S \Rightarrow f \in L^p(\mu)$, for some $p>1$. Prove that $\exists p>1$ so that $S \subset L^p(\mu)$?
2
votes
2answers
39 views

Calculate the inequality

$a,b,c$ are the sides of a triangle.Then show that $$(a+b+c)^3 > 27(a+b-c)(b+c-a)(c+a-b)$$ Also give the case where equality holds i.e. $$(a+b+c)^3=27(a+b-c)(b+c-a)(c+a-b)$$ I tried triangle ...
1
vote
0answers
13 views

Maximum likelihood estimate vs likelihood ratio tests?

Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a ...
2
votes
0answers
10 views

Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...
0
votes
0answers
11 views

Quantile as solution to minimization problem

I'm studying basics of quantile regression now and I have trouble prooving that $\tau-$th quantile of real-valued random variable $Y$ is a solution to the following minimization problem (in the ...
1
vote
1answer
15 views

Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
0
votes
0answers
8 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
0
votes
1answer
27 views

Why $f(p)=(1+u^p)^{1/p}$ is always decreasing when $u>0$ and $p>1$?

I tried differentiating it and get a whole mess: $$f'(p)=(u^p+1)^{1/p}\left(\frac{u^p\log{u}}{p(u^p+1)}-\frac{\log{(u^p+1)}}{p^2}\right)$$ And I don't know how to prove that this is always negative ...
1
vote
2answers
65 views

If T is an infinite subset of $\mathbb{N}$ show that there is a 1-1 mapping of T onto $\mathbb{N}$ [duplicate]

Possible Duplicate: An infinite subset of a countable set is countable Infinite subset of Denumerable set is denumerable? Elementary set theory homework proofs Like the title says: If T ...
3
votes
2answers
51 views

Improper integral of a cosine

I'm trying to follow some equations in an electrical engineering paper that I'm reading. I'll spare you the details, but at one point I come across: $$\lim_{ T \rightarrow \infty }\int_{-T/2}^{T/2} ...
0
votes
1answer
8 views

Orthonormal basis $L^2(a,a+2\pi)$

Let $$\mathcal{B}=\{\frac{1}{\sqrt{2\pi}},\frac{cos x}{\sqrt{\pi}},\frac{sin x}{\sqrt{\pi}},\frac{cos2x}{\sqrt{\pi}},\frac{sin 2x}{\sqrt{\pi}},\dots\}$$. This is an orthonormal basis of ...
6
votes
1answer
77 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
1
vote
4answers
3k views

Formula for number of lines you can draw through $n$ points

So I've got a homework question I'm stuck on. It's asking me to develop a formula that when given $n$ points, it gives the number of straight lines that can be drawn through those points. For ...
0
votes
4answers
54 views

How to prove {$a_n$ } is increasing where $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{ 2+ a_n}$ [duplicate]

I already found out that this sequence is bounded above and $a_n <2 \forall n \in \mathbb Z_+ $ I think I'm missing a point as I can't think of a way to prove that the sequence is increasing.
1
vote
0answers
33 views
+50

Quotient of Poincare dodecahedral space-example of spherical orbifold

Let $\mathcal{O}$ the orbifold with underlying space $S^3$ and singular locus the trefoil knot with local groups of order five. Then how can we see that it is the quotient space of the Poincare ...
0
votes
3answers
67 views

Circle in a complex plane.

Let $C$ be a circle in the complex plane, and let $x$ be a fixed, non-zero complex number. Prove that $\{xz : z \in C\}$ is also a circle. I would really appreciate any help that would get me ...
5
votes
3answers
132 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...
0
votes
0answers
6 views

Lower bound for a relative of the central binomial coeff

The central binomial coefficients $\binom{2m}{m}$ have g.f. $\frac{1}{\sqrt{1-4z}}$ and lower bound $\frac{4^m}{\sqrt{4m}} \le \binom{2m}{m}$. I'm interested in a related integer series $$T(2m, m) = ...
1
vote
2answers
130 views

NumberLine Word-Problem

In the figure below C is the mid-point of segment AE and AB < DE. Which is more (Ans A) A)BC B)CD I know its going to be like $B-A < E-D$ . How will i ...
2
votes
3answers
28 views

if the third vertex lies on the line 4x+3y-12=0, find the area of the triangle

the vertices of the base of an isosceles triangle are $(-1,-2)$ and $(1,4)$ . if the third vertex lies on the line $4x+3y-12=0$, find the area of the triangle.
0
votes
1answer
87 views

Function is continuous or not

If $f$ is a function defined on $[0, 1]$ for which $f'(\frac{1}{2})$ exists but$f'(\frac{1}{2})$ is not an element in $[0,1]$ then f is discontinuous at $x=\frac{1}{2}$. I need to find the ...
1
vote
2answers
71 views

Retrieve the initial cubic Bézier curve subdivided in two Bézier curves

I have a cubic Bezier curve subdivided to two cubic Bezier: Assuming that "t_cut" is the t value where this initial Bezier is cut: example of function subdivision(BezierCurve initialCurve, ...
1
vote
4answers
66 views

Integration of $x/(x^2+1)$ from $-\infty$ to $\infty$

I am trying to find the area of this graph $\int_{-\infty}^\infty\frac{x}{x^2 + 1}$ The question first asks to use the u-substitution method to calculate the integral incorrectly by evaluating ...
0
votes
0answers
12 views

Integral inequality with gamma function

I have some trouble with paper I'm reading. The goal is this: let $s=\frac{1}{2}+\frac{1}{\log n}+it$. $M$ is a function such that $M(s)=O(\log^{3}(N(|t|+2)))$. Define $$U(s)=\frac{1}{2\pi ...
0
votes
1answer
24 views

Is there a name for a property defined in terms of open sets?

We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
0
votes
1answer
13 views

Hypotenuse and angle ratio relationship

In triangle ABC $\angle BAC=90$, $\angle ABC$:$\angle ACB $=1:2 and AC = 4cm. Calculate the length of BC. I tried this by constructing an equilateral triangle as in the figure. I am interested in ...
2
votes
4answers
251 views

How do you work out $\sqrt[4]{16^3}$ without a calculator.

$$\sqrt[4]{16^3}$$ I just don't know what to do when I get to $4096$. The original equation was $16^{3/4}$.
2
votes
1answer
52 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
1
vote
1answer
17 views

Elementary Matrix and row of zeros

If you have the following matrix can $k$ be any number? \begin{pmatrix} 1 & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & 1 \end{pmatrix} So this is obviously an assignment question, ...
3
votes
2answers
98 views

Find the following limit: $L=\lim_{n\to \infty}\frac{\left(2\sqrt[n]{n}-1\right)^n}{n^2}$

Find the following limit: $$L=\lim_{n\to \infty}\frac{\left(2\sqrt[n]{n}-1\right)^n}{n^2}$$ I think use Taylor's expansion give $\left(2\sqrt[n]{n}-1\right)^n$ or there is a workaround, but I do not ...
3
votes
3answers
115 views

Math Puzzle - Rock Problem

A child is arranging rocks in layers. He can arrange the rocks, in such way that, any layer has lesser rocks than its base layer. Given n rocks, In how many ways can the child arrange the rocks?
0
votes
1answer
13 views

Change of basis of linear map

Suppose T: $\mathbb{R}^{2}$$\rightarrow\mathbb{R}$$^{2}$ is linear and has matrix $\begin{pmatrix}4&9\\1&1\end{pmatrix}$ with respect to the standard basis of $\mathbb{R}$$^{2}$. What is the ...
3
votes
2answers
68 views

Equality of real numbers whose squares sum to 1

I am helping a friend with homework and got stuck on the following problem. It doesn't seem like it should be that hard, I don't know why I'm having a hard time: For $a,b\geq0$, if $a^2+b^2=1$, then ...
2
votes
1answer
36 views

A certain two-step subgroup of a nilpotent group

Let $\Gamma$ be a finitely-generated, torsion-free, nilpotent group, of nilpotency class $n\ge 2$. Is there an $N \lhd \Gamma$, such that (i) $N$ is two-step nilpotent, (ii) $\Gamma / N$ is torsion ...
3
votes
2answers
43 views

How do I find the inverse of a $\cos^2 \theta$?

This was originally a physics question, but the math is what is throwing my brain into loops. Basically, I need to find $\theta$: $$ \frac{7}{8}= \cos^2(\theta) $$
1
vote
0answers
7 views

Intersections on General Nonsingular Projective Varieties

Let $X$ be a nonsingular, integral projective variety of dimension at least 2 over $k$ algebraically closed. Let $Y$ and $Z$ be two codimension 1 subschemes (effective Weil divisors) of $X$. Must they ...
3
votes
2answers
57 views

determining $n$ in a given sequence $\frac{1+3+5+…+(2n-1)}{2+4+6+…+(2n)} =\frac{2011}{2012} $

Given that: $$\frac{1+3+5+...+(2n-1)}{2+4+6+...+(2n)} =\frac{2011}{2012} $$ Determine $n$. The memorandum says the answer is 2011 but how is that so? Where did I go wrong?
0
votes
0answers
9 views

limits changed F(g(x))=F(lim G(x))

lim x tends to 0(F(G(x))=F(lim x tends to 0(G(x))) I have seen this step in a derivation of a result which is not the point of interest here. The book wrote the reason for it was that it is when F ...
4
votes
1answer
520 views

Simplify $\frac{9}{2}(1 + \sqrt 5)\sqrt{10 - 2\sqrt 5} + 9\sqrt{5 + 2\sqrt 5}$

Simplify $\displaystyle{\frac{9}{2}(1 + \sqrt 5)\sqrt{10 - 2\sqrt 5} + 9\sqrt{5 + 2\sqrt 5}}$. I get this when I was doing another Q, but I don't know how to further simplify it. Can anyone help me, ...
4
votes
4answers
140 views

Solving $a(x+2)=\pi-cy$ for $x$, arrived at an answer different from the one in the book

In an algebra review book, one exercise asked to solve for $x$: $$a(x+2)=\pi-cy$$ I arrived at the following: $$x=\frac{\pi-cy}{a}-2$$ The book stated the correct answer is: $$x=\frac{\pi-cy-2a}{a}$$ ...
4
votes
1answer
150 views

Vector calculus and Frenet-Serret equations

I have shown the first two equality and I am working on the showing the 1st equals the 3rd. \begin{alignat*}{4} \frac{1}{\rho}\hat{\mathbf{{n}}} &= \frac{d\hat{\mathbf{{u}}}}{ds} &{}= ...
3
votes
1answer
14 views

General lists of techniques to prove whether a set is a generator of a matrix group

It seems like a rather common problem in group theory, at least in undergraduate maths, to check whether a set is a generator of a group. The question is usually as follow: Given a group $G$, and a ...
1
vote
2answers
53 views

How many times can transitivity property be applied

Can transitivity property be applied for infinite number of times for a certain problem??
0
votes
0answers
5 views

Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
-1
votes
1answer
28 views

Order of center of character

I am working on a course in representation theory and I've got completely stuck on some exercises regarding $|G:Z(\chi)|$ where $G$ is a finite group with an irreducible representation $\theta : ...
1
vote
1answer
35 views

Identity criterion for Gateaux deriative

I'm trying to prove that if $f,g$ are two functionals on a Banach space $X$ that have the same Gateaux derivative for all $x \in X$, then $f-g = \text{constant}$. I can show that under the hypotheses, ...
1
vote
1answer
25 views

Algebraic Integers in $\mathbb{Q}(\sqrt{m})$ and Norms on them

I'm having a problem with a section of Niven's book the Theory Of Numbers. I am trying to show: If an integer $\alpha \in \mathbb{Q}(\sqrt{m})$ is neither zero nor a unit, prove that ...

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