2
votes
1answer
64 views

What is $\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}$

What is $$\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}$$? What I did I know that \begin{align*} &\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}\\ &=\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}+\sum_{i=1}^m\sum_{...
1
vote
1answer
88 views

Suggested Reading for Combinatorics

What are some suggestions for texts on introductory combinatorics and its applications? I would prefer if the applications would be to other branches of mathematics rather than outside of mathematics. ...
0
votes
2answers
2k views

what is the cardinality of a Null set?

Does the cardinality of a Null set is same as the cardinality of a set containing single element? If a set A contains Null set as its subset, then the null set is taken into account to calculate the ...
1
vote
1answer
37 views

How to recognize this sequence pattern?

What's the pattern for the following sequence -1,-1,1,1,-1,-1,1,1... And what is it's series? I tried (-1)^(n^3+1) and many things with no successful results.. And i didn't find any similar question ...
1
vote
2answers
113 views

Submersion surjective on the complex projective space $\mathbb{C}P^1$.

If $S^3=\{ (z_1,z_2)\in\mathbb{C}^2\mid \vert z_1\vert^2+\vert z_2\vert^2=1\}$ and $\pi:S³\rightarrow\mathbb{C}P^1$ for $(z_1,z_2)\mapsto [(z_1,z_2)]$ since $[(z_1,z_2)]=\{ (w_1,w_2)\in\mathbb{C}^2-\...
2
votes
1answer
90 views

Given a particular triangle that has been constructed, I want to prove that one of the angles must be $> 45$ degrees. [duplicate]

Suppose you are given an acute triangle $XYZ$ with the following properties: At $\angle XZY$, the $\angle$ bisector is drawn and extended all the way to $XY$. Lets call the point where it intersects ...
19
votes
3answers
698 views

Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show ...
3
votes
1answer
85 views

$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$

While solving $y'=x^2-e^y$ I'm stuck on the last step that requires to evaluate this integral. $$\displaystyle\int e^{\frac{x^3}{3}} \mathrm{d}x$$ I don't know how to approach it. I know that it ...
0
votes
1answer
29 views

Solving for all coefficients of powers of x given a equation

Given a equation of following: $$ xa_{1}+x^2a_{2}+x^3a_{3}+x^4a_{4}+x^5a_{5}+......=0 $$ $\forall x>0$ I want to prove that $a_i=0 \forall i\ge1$ I am not sure if this is true.
0
votes
0answers
475 views

What exactly is a true value of a parameter?

I am currently studying the properties of the Maximum Likelihood Estimator. One of these properties being the asymptotic normality, I found the following equation: $$\sqrt{n}(\hat{\theta} - \theta_{0}...
2
votes
2answers
58 views

Max/Min Problem

For $x$, $y$, and $z$ positive real numbers, find $\frac{z}{x}$ such that $(x,y,z)$ achieves the maximum value of $$\sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}} + \sqrt{\frac{2z+3x}{6x+...
1
vote
1answer
45 views

Inverse of the matrix product $\boldsymbol{A} \cdot\boldsymbol{S} \cdot \boldsymbol{A}^{T}$

If I have an $n\times n$ symmetric matrix $\boldsymbol{S}$ and a $m\times n$ matrix $\boldsymbol{A}$ is there any relation between $(\boldsymbol{A} \cdot\boldsymbol{S} \cdot \boldsymbol{A}^{T})^{-1}$ ...
3
votes
0answers
55 views

Iterative equation

I have an equation that I want to try and solve iteratively. I don't have any background in numerical analysis so unsure as to how to go about it. Any help would be greatly appreciated. My equation is:...
0
votes
1answer
43 views

Integral of $\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$.

Can some one help me with the integral $$\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$$ According to my exercise I should be able to get $0$. Please help me .
1
vote
2answers
44 views

Inequality of sides of triangle [duplicate]

If a,b,c are the sides of a triangle ABC then prove $$ a/(b+c) + b/(a+c) + c/(a+b) < 2 $$I tried to solve this by triangle inequality but i was not able to get to the solution.
1
vote
1answer
80 views

Why test problems in convex optimization are mostly random?

Very often people who compare performance of different algorithms in convex optimization use randomly generated data. For instance, this often happens in compressed sensing and signal processing. Is ...
0
votes
1answer
71 views

Real analysis. Uniformly continuous

Suppose $$f:\mathbb R\to\mathbb C$$ is continuous and $f(x)=0$ for all $|x|>1$. Show $f$ is uniformly continuous on $\mathbb R$. This is not homework. I'm trying to study for a test. I ...
1
vote
2answers
42 views

Compactness of a sum of spheres in $\mathbb{R}^n$

Let $S(a,t)=\{x\in\mathbb{R}^n : d_e(a,x)=t\}$ be $n$-dimensional sphere in $(\mathbb{R}^n,\cal{T}_e)$ (natural, euclidean topology). Then for $A\subset\mathbb{R}^n$ and $r:A\longrightarrow (0,\infty)$...
1
vote
2answers
33 views

Proving a Triangle inequality

If $ a^2 + b^2 > 5c^2 $ in a triangle ABC then show c is the smallest side.I tried to solve this by cosine rule but i was not able to find the answer
18
votes
3answers
603 views

Closed form of $\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$

I have homework to evaluate this integral $$I=\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$$ Here is what I have done so far. I tried integration by parts using $u=\tanh(x)\,\tanh(2x)$ and ...
1
vote
1answer
34 views

Inequality with absolute value and a parameter inside it

I've been stuck on this problem for quite a while even though it seemed trivial to me at first. Basically, I have this: $$\lvert ax+4 \rvert>\frac1x$$ It is quite easy to conclude that only $x&...
0
votes
1answer
44 views

Suppose $x$ and $y$ are in the same orbit. Show that $G_x$ and $G_y$ are conjugate subgroups.

Let $X$ be a $G$-set and $x,y \in X$. Let $G_x$, $G_y$ be the stabilizers of $x$ and $y$ respectively. Suppose $x$ and $y$ are in the same orbit. Show that $G_x$ and $G_y$ are conjugate subgroups. ...
2
votes
3answers
15 views

Independence of two r.v.

Let $\{ X_n, n\ge \}$ be iid with $P[X_1=1]=p=1-P[X_1=0]$. What is the probability that the pattern $1,0,1$ appears infinitely often? \ Hint: Let \begin{align*} A_k=[X_k=1,X_{k+1}=0, X_{k+2}=1] \end{...
0
votes
1answer
40 views

search for a theorem related to $\sum\limits_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu$

I need to use the following inequality: $$\sum_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu\tag{1}$$ what is the name of such a theorem?
0
votes
2answers
40 views

$X_i = \{ (x, y) \in \mathbb R^2 : x^2 + y^2 = i^2\}$. Show that the collection of sets $X_i$ for $i \in I$ form a partition of $\mathbb R^2$

$X_i = \{ (x, y) \in \mathbb R^2 : x^2 + y^2 = i^2\}$. Show that the collection of sets $X_i$ for $i \in I$ form a partition of $\mathbb R^2$. Do we use bijection? Is there any other way to solve it ...
1
vote
1answer
58 views

characteristic function of complement of non-Borel set

Let $C$ be the Cantor set and $M \subseteq C$ a subset, $M \notin \mathcal B(\mathbb R)$. Obviously the characteristic function of $M, \chi_M:[0,1] \rightarrow \mathbb R$, is not (Borel-) measurable. ...
1
vote
1answer
27 views

CHECK: Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$.

Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$. Hint: When working this problem, I found that in showing the two sets equal I had to be extremely careful. Try not to make big jumps. At one ...
1
vote
1answer
239 views

Find simple path with the largest sum of weights

In the longest path problem, given a weighted graph G and a starting node v0, find the simple path starting with v0 with the largest sum of weights. Show that the longest path problem is NP-hard in ...
3
votes
1answer
100 views

Conditional expectation on exchangeable variables

Let $X=(X_1,X_2,...,X_n)$ be an exchangeable $n$-dimensional vector of random variables with values in $\mathbb{R}$, let $\mathfrak{G}$ be the $\sigma$-algebra of permutation-invariant (symmetric) ...
2
votes
2answers
101 views

Could we calculate pi using an iterative series

I know that, as a hobbyist mathematician, this is generally a term we can use to express pi \begin{equation*} \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} +...
0
votes
2answers
90 views

Cardinality with Cartesian Cross Product problem

A and B are finite sets Prove that |AxB| = |A||B|. I need a solution/hint. I suspect that the answer has to do with the fact that the we can say that |A| = |B| and then from that say = |AxB|. I ...
0
votes
2answers
167 views

Minimum value of a positive definite binary quadratic form along integers

Is there a formula for the least non-zero value of $$f(x,y):=ax^2+bxy+cy^2$$ as $x,y$ assume integer values? Here $a,b,c$ are integers with $a,d>0$ and $b^2-4ac<0$.
1
vote
1answer
29 views

Show that if $\lim_{x\rightarrow\infty}f(x)=a \ \text{and} \ \lim_{x\rightarrow\infty}f'(x)=b$ then $b=0$

Consider: $$f:[c,\infty] \rightarrow\mathbb{R}$$ Show that if:$$\lim_{x\rightarrow\infty}f(x)=a \ \text{and} \ \lim_{x\rightarrow\infty}f'(x)=b$$ Then: $$b=0$$
0
votes
1answer
128 views

What does it mean for an integral to “vanish”?

I had a question; What does it mean for an integral to vanish in complex analysis? There is supposedly something, which says if the integral "vanishes," the sum of the residues is 0. But what does ...
3
votes
1answer
41 views

Find $n$ such that $(\mathbb{Z}/n\mathbb{Z})^\times \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$.

In the ring $(\mathbb{Z}_n, +, \times)$, we know that only $\phi(n)$ of the elements have multiplicative inverses which form the multiplicative group $\mathbb{Z}_n^\times$. However, since $\phi(n)$ ...
1
vote
2answers
40 views

Is this approach wrong?

Where have i gone wrong? $\frac{dq}{dt}=1000\cdot 2^{t}$ $dq=1000\cdot 2^{t}\cdot dt$ $\int\ln dq=\int\ln2000t\cdot dt$ $\ln\int dq=\int\ln2000t\cdot dt$ $\ln(q)+c_1=t\cdot\ln(2000t)-2000t+c_2$ ...
1
vote
1answer
78 views

What permutations of matrix entries do row and column transpositions generate?

Let $M$ be a square matrix. By transposing rows and columns, can we get any permutation of the entries of $M?$ If we can't, which permutations are generated?
1
vote
2answers
45 views

Cauchy-Schwarz inequality with zero angle?

Cauchy-Schwarz Inequality: If $\textbf{u}$ and $\textbf{v}$ are vectors in a real inner product space $V$, then $$|\left\langle\textbf{u},\textbf{v}\right\rangle|\leq||\textbf{u}||\ ||\textbf{v}||...
3
votes
1answer
83 views

Topological form of Martin's Axiom

I'm currently studying consequences of Martin's Axiom: Martin's Axiom (MA): Suppose that $\left\langle P, \leq \right\rangle$ is a ccc partially ordered set and $\{D_\alpha\}_{\alpha < \lambda}$ ...
0
votes
0answers
30 views

How to prove that $abd = abcd + abc'd$ for all general occassions

It is true for example that $abd = abcd + abc'd$. Each of the terms on the right part of the equation contains all the used letters. Is there anyway to prove that any term is equal to the sum of the ...
2
votes
4answers
73 views

I need Arctan but only Arctan2 is supplied

I'm a new programmer and I'm programming the projectile of a missile using the equation $\theta = \arctan(v^2\pm\sqrt{v^4-g(gx^2+2yv^2)}/gx)$ where $g$: the gravitational acceleration—usually taken ...
0
votes
0answers
50 views

If a function is continuously derivable is it also continuously differentiable?

I'm doing an computer science online test and my professor is putting loads of trick questions in in. I'm wondering whether this is also one. I have a question for which we stated at the lectures ...
0
votes
1answer
80 views

Eigenfunctions of a second derivative operator

Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\alpha >0$ is a positive real ...
2
votes
1answer
57 views

Calculating limit-help needed (difference of cubes)

I'm stuck trying to find this limit: $$\lim_{x\to \infty} (\sqrt[3]{(x-2)^2(x-1)}-x)$$ Thanks in advance!
9
votes
0answers
147 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
0
votes
1answer
27 views

Sequence limit sophisticated

$$\frac{3+(-1)^n+9\sqrt{n}-7n^5-2[\sqrt[3]{n}]n}{(3n-1)(n-2)(2n-3)(n-4)(4n-5)+2^{-n}}$$ What is the easiest method to calculate this sequences limit when $n\rightarrow \infty$
0
votes
1answer
48 views

Set of orthogonal matrix over $\Bbb{R}$: Closed, convex, open?

Reading my course on topology we haven't answering this exercise: Show that the set of orthogonal matrix over $\Bbb{R}$ is closed. Is it convex? Open For the fact is closed I wrote $\mathcal{O}...
2
votes
3answers
90 views

Proving that there exist 2 distinct vertices $u,v$ in $G$ such that $d(u) = d(v)$

I have difficulties understanding the proof given below showing that there exist 2 distinct vertices $u,v$ in $G$ such that $d(u) = d(v)$ where $G$ is a non-trivial graph. Proof: It's clear that $0 \...
5
votes
1answer
76 views

If $f(x)$ is close to $0$ then necessarily $x$ is close to $\ker f$?

Suppose that $X$ is a real Banach space and $f:X \to \mathbb{R}$ is a continuous linear functional. Is it true that for any $\varepsilon>0$ there is a $\delta>0$ such that for any $x \in X$ we ...
1
vote
2answers
44 views

Tricky problem on function equation

How do you evaluate $$q(t)=A\cdot (2000t)^t\cdot e^{-2000t}$$ when $q(0)=4000$? I just can't get around $(2000\cdot0)^0$.

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