1
vote
1answer
42 views

To find a $y_n$

I need to prove this theorem for positive series: $\forall {x_n}$ such that $\displaystyle\sum_{i=1}^{\infty} x_n$ diverges, $\exists y_n$ such that: 1) $\displaystyle\sum_{i=1}^{\infty} y_n$ ...
1
vote
1answer
28 views

Poisson actions defined in terms of coactions.

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, ...
5
votes
2answers
150 views

Forty poor thieves

Forty thieves have 4000 gold coins to split between them. A group of five thieves is $poor$ if together they have less than or equal to 500 gold coins. Let N be the minimum number of poor groups of ...
0
votes
1answer
28 views

Transpose solution to non-homogeneous system

Let $A$ is an nxn real matrix and $A^t$ its transpose. Is it the case that if $Ax=0$ has a non-trivial solution that $A^tx=0$ has a non-trivial solution? My guess is yes, mainly because the rank of $...
0
votes
5answers
147 views

Prove that $\displaystyle \lim_{x \to 0} \dfrac{|x|}{x}$ does not exist

I thought absolute values were positive? Why is the there a negative $x$ in example $7$ in the attached picture. Can someone explain?
3
votes
3answers
52 views

Question about limits $\lim_{x\to\infty}\frac{x-2}{e^{1/x}\cdot x}$

How to calculate this: $$\lim_{x\to\infty}\frac{x-2}{e^{1/x}\cdot x}$$
1
vote
1answer
79 views

Attractors of a Random Boolean Network?

I need some direction on the topic of Random boolean networks (NK-boolean networks or Kauffman automata). I know some of the results like if K=1 the systems settles down to fixed points, if K=2 it ...
0
votes
3answers
957 views

Let A and B be sets. Prove that A = B iff the power set of A is equal to the power set of B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! Let A and B be sets. Prove that A = B iff the power set of A is equal to the power set of B. Assume that A and B are ...
2
votes
0answers
129 views

Bound for variance of maximum of normal random variables

Suppose that $(X_1,\ldots,X_n)=\mathbf{X}\sim N(\mathbf{0},\Sigma)$ is an $n$-dimensional normal random vector. I want to show the bound $$ \text{Var}\left(\max_{i\leq n} X_i\right)\leq \max_{i\leq n} ...
0
votes
2answers
120 views

How should a programmer store and solve simultaneous algebraic equations?

I need to store and solve simultaneous algebraic equations (no trig, no calc, no logs) as part of a larger program. I am not yet committed to a particular language, so long as it's a free one. For ...
0
votes
0answers
34 views

Just what is the importance of operators that produces an eigenvalue?

For some operators, there is a well known eigenvalue associated with it, for example the energy operator in quantum mechanics $i\hbar \partial_t$, this is very important indeed and gives us physical ...
5
votes
4answers
106 views

Smallest prime that divides $n^2+5n+23$

Find the smallest prime that divides $n^2+5n+23$ for some integer $n$. I thought taking all primes less than 29 one by one and then solving the equations and then some manipulation. (Like $n^2+5n+23=...
1
vote
0answers
41 views

Dimension of space of linear maps between infinite dimensional vector spaces

Let $F$ be a field, and suppose $V$ and $W$ are vector spaces over $F$. What is the dimension (meaning cardinality of any basis) of the space of linear maps from $V$ to $W$? I hope there is an answer ...
7
votes
1answer
165 views

Is The Statement $b^n\equiv 1\pmod n$ equivalent to “$x\mapsto b^x-x\pmod n$ is a bijection”?

Suppose that $n$ is a natural number and $b$ is one coprime to it such that $b^n\equiv 1\pmod n$. Does it follow that, if $b^x-x\equiv b^y-y\pmod n$, then $x\equiv y\pmod n$? This is inspired by the ...
2
votes
1answer
36 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
1
vote
1answer
33 views

measure convergence

Let $(X,\mathcal{F},\mu)$ be a finite measurable space. Define $$d(f,g) = \int_X \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}\mu(dx) $$ Proof that $d(f_n,f)\to0 \Leftrightarrow\ f_n$ converge in measure to $f$ ...
2
votes
1answer
78 views

Iterated function

Let $$f(x)=x−\frac{1}{x}$$ Find the number of real solutions to $f(f(f(f(x))))=1$. Do I evaluate it completely, or is there some other way. After third composition it got nasty, so I left it.
0
votes
2answers
27 views

Evaluating integral convergence

I have an the integral $$\int_{-\infty}^{6} xe^{\frac{x}{2}}\; dx$$ I know that this integral is convergent but I can not find how to evaluate its' convergence other than finding the limit of the ...
0
votes
0answers
29 views

What is the range of this complex function?

The function is: $f(z) = 2x^2+(1-x^2)(1+i)$ on the unit disk $|z|\leq1$. I simplified this down and got that $Re(f(z)) = x^2+1$ and $Im(f(z)) = 1 - x^2$, which then led me to think that the answer is:...
1
vote
1answer
29 views

Suppose that $A$ is a $2\times2$ matrix with singular values $\sigma_1 = \sigma_2 = \sigma > 0$,Let u,v be any pair of orthonormal vectors in $\Re^2$

Suppose that $A$ is a $2\times 2$ matrix with singular values $\sigma_1 = \sigma_2 = \sigma > 0$. a. Show that if $z\in\Re^2$ with $\Vert z\Vert _2=1$, then $\Vert Az\Vert _2 = \sigma$. b. Let $...
1
vote
0answers
27 views

Shouldn't big oh definition be if and if only if, not just if? [duplicate]

This is from Discrete Mathematics and its Applications Shouldn't the if in that definition be an if and only if? Say we know that $n^2$ is in O($n^2$). Then from one side of the if and only if, we ...
0
votes
1answer
36 views

Trying to prove that if two matrices with the same Eigenvectors are summed, the result has the same eigenvectors

This is a proof I am tyring to work out: A and B are square matrices, of same size. I am trying to show that if the eigenvector V of both and A and B, then v is also an eigenvector of $ M = c_1 A + ...
1
vote
1answer
48 views

Show that the right cancellation law holds in $S$

Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z = y$. (This $z$ may depend on $x$ and $y$....
1
vote
0answers
97 views

Prove $H \cap K$ is itself a subgroup of $G$ if $H$, $K$ are subgroups of $G$

Let $G$ be a group. If $H$, $K$ are subgroups of $G$, then prove that $H \cap K$ is itself a subgroup of $G$. What I need: I am hoping you wonderful people could read this over and make sure it is ...
0
votes
1answer
21 views

Suppose $s_n$ is a sequence such that lim $s_n = 5$.

This is a two part problem, and if anyone can help explain them, I would appreciate it. (a) State the definition of what it means to say that lim $s_n = 5$. (b) Prove that there exists a real number ...
5
votes
2answers
335 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: t\in(0,\pi)\},\...
1
vote
1answer
46 views

Prove that f is unique, and express f(x) in closed form.

Let $f : [−1, 1] \to R$ be a continuous function such that (i) $f(x) = \frac {2−x^2}{2}f(\frac {x^2}{2-x^2}) $ for every $x$ in [−1, 1], (ii) $f(0) = 1$, and (iii) $\lim_{x\to 1^{-}}\...
1
vote
3answers
368 views

Solve the differential equation $y'-xy^2 = 2xy$

I get it to the form $\left | \dfrac{y}{y+2} \right |=e^{x^2}e^{2C}$ but I'm not sure how to get rid of the absolute value and then solve for y. I've heard the absolute value can be ignored in ...
1
vote
1answer
42 views

Determine whether $\sum_{t=2}^n\frac{\log t}{\Omega(t)}\sim n\log n$

Determine whether $$\sum_{t=2}^n\dfrac{\log t}{\Omega(t)}\sim n\log n$$ Let $r=n^{1/\Omega(n)}$, where $n$ is a positive integer and $\Omega(n)$ is the total number of prime factors of $n$. If $r$ is ...
1
vote
3answers
40 views

Why is $\int$ $1\over(a^2+b^2)^{3/2}$ $da$ $ = $ $ a\over b^2\sqrt {(a^2+b^2)}$

$$\int\frac{da}{(a^{2}+b^{2})^{3/2}} =\frac{a}{b^{2}\sqrt{(a^{2}+b^{2})}}.$$ Found this in the solution to a problem in my physics textbook, and left clueless.
0
votes
3answers
208 views

For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B. ...
2
votes
0answers
43 views

Finding the Legendre transform of an “entropy type” functional

I want to find the Legendre Transform of $$T(f) = \int_{\mathbb{R}^2} f \log \left(\frac{f}g{}\right) \, dx$$ on a set $H_M = \{ f: f \ge 0 \text{ and } \int_{\mathbb{R}^2} f =M\}$, where g is some ...
10
votes
1answer
1k views

Why are people interested in solving the Navier-Stokes equations if people can find an good approximate solution?

Why are people interested in solving the Navier-Stokes equations if people can find a good approximate solution? Also especially when people have supercomputers?
2
votes
1answer
43 views

Prove that $X_n/\lambda_n \to$ 1 in probability for $X_n \sim \text{Pois}(\lambda_n)$

Let $X_n \sim \text{Pois}(\lambda_n)$, where $\lambda_n \to \infty$ as $n \to \infty$. Prove that $X_n/\lambda_n \to 1$ in probability Should I solve this problem using chebychev's inequality? I'm ...
-3
votes
2answers
67 views

Smallest rectangle satisfying certain conditions

What is the least possible area of a rectangle whose sides and diagonals all have positive integral lengths? I tried picking random numbers but don't have any strategy for this
2
votes
1answer
75 views

when a crossed product group is inner amenable

Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not. For example, when $\alpha$ is trivial, $G=H\...
2
votes
4answers
144 views

Area of triangle bounded by line and degenerate “crossed lines” conic

The question is Show that the two lines given by $$(A^2 - 3B^2)x^2 + 8ABxy +(B^2 - 3A^2)y^2=0$$ and the line given by $$Ax+By+C=0$$ determine an equilateral triangle of area $$\frac{C^2}{\...
1
vote
1answer
275 views

A linear program for maximizing a fraction

Given $\lambda_1,\ldots,\lambda_n \geq 0$ and an $n\times n$ matrix $A$, I wish to maximize the ratio $$ \frac{\lambda_1x_1 + \cdots + \lambda_nx_n}{x_1+\cdots+x_n}, $$ where $x_1,\ldots,x_n \geq 0$ ...
0
votes
1answer
43 views

Sum of exponentials

Question: Does there exists $t\in \mathbb{R}$ such that the sum below holds true? $$\sum_{n=1}^N a_ne^{ib_nt}=0$$ where $a_n\in\mathbb{C}$ and $b_n\in\mathbb{R}$.
0
votes
1answer
20 views

Shifted DE Differential Equation

I just want to know what kind of a shift the $-15$ causes in this equation: $P' = 0.08P(1-\frac{P}{1000})-15$
1
vote
0answers
40 views

Solving $q = \sin\left(\frac{a}{2}\right)*\cos(B_x)$ for $B_x$

I'm a bit rusty and am having trouble using Trig Identities to solve for $B_x$. Can someone show me how to do this? $$q = \sin\left(\frac{a}{2}\right)*\cos(B_x)$$ I want to solve for $B_x$ ...
0
votes
1answer
51 views

How to solve this question with Itô lemma?

Let $$M(t) = \int_{0}^t Y (u)dB(u).$$ where $$E \left[ \int_{0}^t Y^2(u)du\right] < \infty.$$ Use Itô’s rule to find the differential $dQ$ of the process $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ ...
1
vote
3answers
178 views

Contour Integration of Line Segments

I am trying to use contour integration to find the integral of: $$ \int_\gamma ydz $$ where we have the union of line segments from $0$ to $i$ and then to $i+2$. I simply do not understand how to ...
10
votes
3answers
255 views

Number Theory : What are the last three digits of $9^{9^{9^9}}?$

I was doing some basic Number Theory problems and came across this problem and was all thumbs : Find the last three digits of $9^{9^{9^9}}$ How would I go about solving this problem? I am a ...
-2
votes
1answer
84 views

Addition Problem with Missing Digits

In the addition problem shown each $\ast$ denotes a missing digit and the $\ast$'s are not necessarily identical. What final four digit sum will result from the proper restoration of the missing ...
0
votes
1answer
61 views

Bilinear forms in ${\Bbb R}^3$

Consider $V={\Bbb R}^3$ and bilinear forms $B_k:V\times V\to {\Bbb R}$ $(k=1,2,3)$. Use the hairy ball theorem to show that one can not have $B_1,B_2,B_3$ such that $$ B_1(x,y)^2+B_2(x,y)^2+B_3(x,y)^...
1
vote
0answers
26 views

I'm interested in the solution set satisfying the equation $\log_{10} p\times\log_{10} q=\log_{10} r$

The equation interested in is $\log_{10} p\times\log_{10} q=\log_{10} r$ where $p,q,r\in\mathbb N$ are natural numbers. Here, I want not to consider some trivial solutions that make any one of ...
0
votes
2answers
73 views

Let $ k $ belong to the naturals. Prove that $3k^2+2$ is never a perfect square

I'm been struggling with this proof for a couple of hours. I originally thought I could prove it by contradiction and let some $n^2=3k^2+2 $to prove there is a contradiction, but it got me nowhere. ...
0
votes
2answers
26 views

Possibilities for Unbounded Sequences

Let $(s_n)_1^\infty$ be an unbounded sequence in $\mathbb{R}$. Some possibilities for $s_n$ are: $\lim s_n = +\infty$ $\lim s_n = -\infty$ $s_n$ oscillates between large and negatively large numbers....
0
votes
1answer
30 views

Find the matrix A in the following

$$\left[ \begin{array}{cc} 2 & -6&-4 \\ -2 & 7& 3 \\ 3&-8&-6 \end{array} \right]^{-1} +2 \cdot A= \left[ \begin{array}{cc} 7 & 2&5 \\ 6 & -10& -7 \\ 5&-8&...

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