4
votes
4answers
1k views

Let $∑_{n=0}^∞c_n z^n $ be a representation for the function $\frac{1}{1-z-z^2 }$. Find the coefficient $c_n$

Let $∑_{n=0}^∞c_n z^n $ be a power series representation for the function $\frac{1}{1-z-z^2 }$. Find the coefficient $c_n$ and radius of convergence of the series. Clearly this is a power series with ...
3
votes
1answer
85 views

the infinite sum of symmetric random variables is also symmetric

Definition. Let $(\Omega, {\mathcal F}, \mathbb{P})$ be a probability space and $X$ a random variable in $\Omega$. $X$ is said to be ${\mathbf symmetric}$ (about $0$) if $X$ and $-X$ are equal in law....
1
vote
2answers
68 views

Analytic hypersurface as union of irreducibles

Let $X$ be a complex manifold. Then any analytic subvariety $V$ of codimension 1 (that is, any analytic hypersurface) can be expressed uniquely as the union of irreducible analytic hypersurfaces $V=...
3
votes
4answers
290 views

Parabolic sine approximation

Problem Find a parabola ($f(x)=ax^2+bx+c$) that approximate the function sine the best on interval [0,$\pi$]. The distance between two solutions is calculated this way (in relation to scalar product):...
0
votes
2answers
77 views

Determining the correctness of a formal proof

Is the following formal proof, proving $\forall A\forall B \forall C[A+C=B+C\Longrightarrow A=B]$ correct?? Proof 1) $a+c=b+c$.............................................................Hypothesis ...
1
vote
2answers
83 views

Help solving a not so difficult limit?

Giving that $(a_{n})$ is an arithmetic series of ratio $r$ , why is $$ \lim_{n\to\infty}\frac{1}{\frac{\sqrt{a_{n+1}}}{\sqrt{r}}\arctan \frac{\sqrt{r}}{\sqrt{a_{n+1}}}}\cdot \frac{1}{1 + \frac{\sqrt{...
4
votes
1answer
40 views

If $f( \log_2 x)-f( \log_3 x) \le \log_5 x$ then $\log_5 \left( \frac{3}{2} \right) \int_0^1 f(x)dx \le 2$

Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is continuous in $x_0=0$, and Riemann integrable on $[0,1]$. If $f(0)=0$ and $f( \log_2 x)-f( \log_3 x) \le \log_5 x, \forall x>0$, prove that $...
1
vote
1answer
58 views

How to prove existence of solutions to the system of Diophantine linear equations

Let say that we have a linear system $\mathbf{Ax} = \mathbf{b}$, where $\mathbf{A} \in \mathbb{Z}^{m,n}$, $m \neq n$, $\mathbf{A}$ is a full row rank matrix: $rank(\mathbf{A}) = m$ and $\mathbf{b} \in ...
1
vote
2answers
85 views

Problem from I.N.Herstein (Linear transformation)

Let $A=(a_{ij})$ be such that for each i, \begin{equation*} \sum_{j} a_{ij}=1. \end{equation*} Prove that $1$ is characteristic root of A. Generalisation: Let $A=(a_{ij})$ be such that for each i, \...
3
votes
2answers
999 views

how to find coordinates of a point perpendicular to a line?

The point P is at the foot of the perpendicular from the point a(0,3) to the line $y=3x$ 1) find the equation of the line AP and find the coordinates of P I have found the equation of the line which ...
2
votes
1answer
49 views

Compact metric implies uncountable w*-dense set

I am reading a proof of the following: Let $X$ be a separable Banach space. The Szlenk index is countable iff $X*$ is separable. In the proof of => it uses the following: If $X^*$ is not separable, ...
5
votes
0answers
176 views

resizing rectangle within triangle

Imagine I have a parking lot that changes in width and length and in number of levels, and all of the levels need to be visible to a cctv camera at a fixed position, and I would want the camera to see ...
2
votes
1answer
93 views

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ ...
0
votes
1answer
67 views

Limits of the use of trigonometric substitution in integration

I'm trying to understand exactly when trigonometric substitution can be used. Here's the whole detailed process and the approach I'm currently taking. I'm mainly curious about whether the ...
1
vote
1answer
90 views

Is there a novel way to integrate this without using complex numbers?

I've been reading a post on Quora about lesser known techniques of integration and I'm just curious if there's also a novel way to integrate this type of integral without resorting to complex analysis....
1
vote
0answers
67 views

Elements in commutator subgroup

We know that $|G| < \infty$ and $|G:Z(G)|^2 < |G'|$. Show that $G'$ contains elements which are not commutators. ($G'$ is a commutator subgroup of G) Can I get any hints?
1
vote
0answers
59 views

Why can't the number $\pi$ (or any transcendental number) be definable?

Why can't the number $\pi$ (or any transcendental number) be definable without parameters in the model structure $(\mathbb{R}=(\textrm{real field}), <, +, .)$?
1
vote
1answer
63 views

Calculus Question (Maclaurin's theorem)

It's given that $y=\frac{1}{1+\sin (2x)}$, show that when $x=0$, $\frac{d^2y}{dx^2}=8$ Using Maclaurin's theorem, find the first three terms for $y$. Evaluate $\lim_{x\rightarrow \infty }\frac{y}{x^2+...
0
votes
1answer
41 views

Integrable variable $X$ , integral of the expected value of $|X|$ over a small set $A$

Let $(\Omega, \Sigma, P)$ be our probability space. Prove that a random variable $X$ is integrable, that is $\mathbb{E}(X) < \infty$ $\iff$ $$\forall \varepsilon >0 \exists \delta>0 : \...
1
vote
2answers
172 views

Does the arithmetic mean minimize the sum of absolute values of deviations? [duplicate]

We have $x_1,x_2,\ldots,x_n \in \mathbb{R}$. I conjecture there to be a number $M \in \mathbb{R}$ such that for any $i=1,2,\ldots,n$ the quantity $$|x_i - M|$$ is as small as possible. How do you go ...
4
votes
1answer
101 views

Limit $\lim_{x \to +\infty}\left(x^\frac{7}{6}-x^\frac{6}{7}\cdot \ln^2( x) \right)$ using L'Hôpital's rule.

$$\lim_{x \to +\infty}\left(x^\frac{7}{6}-x^\frac{6}{7}\cdot \ln^2( x) \right)$$ I can not decide the limit. I understand that it is necessary to apply L'Hôpital's rule, when there will be a fraction. ...
1
vote
1answer
44 views

Diffeomorphism in Lie Group

$G$ is a Lie group and consider $L_{g}: G \rightarrow G$ ($L_g(h)=gh$). What i need to show that $L_{g}$ is diffeomorphism. Is it something obvious? Can someone explain it to me?
3
votes
3answers
115 views

Epsilon Delta Proof of Limit

Let $f : (−\infty,0) → \mathbb{R}$ be the function given by $f(x) = \frac{x}{|x|}$. Use the $\epsilon -\delta$ definition of a $\lim\limits_{x \to 0^-} f(x) = -1.$ Workings: Informal Thinking: We ...
2
votes
1answer
58 views

Relation between the density function (measure theory) and density (physics)

I was reading some notes on Ergodic Theory and there is this sentence: Suppose we distribute mass on $X$ according to the mass density $fd\mu$, $f \in L^1(\mu)$,$ f \geq 0$, and then apply $T$ ...
2
votes
1answer
34 views

$\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation

I have a homework as follows : Prove that $\int_0^\pi g(x)\sin(nx) dx$ coverges to $0$ for any function $g$ is bounded variation on $[0,1]$. my attempt: for any bounded variation function $g$, $$\...
2
votes
1answer
27 views

there exsit postive integer $x,y$ such $p\mid(x^2+y^2+n)$ [duplicate]

For any give the postive integer $n$,and for any give prime number $p$. show that there exsit postive integer $x,y$ such $$p\mid(x^2+y^2+n)$$ My approach is the following: Assmue that $n=1,p=2$,...
4
votes
3answers
93 views

Compute $\sum_{n=0}^\infty \frac {\sin ((2n+1)\phi)}{2n+1}$ for $0<\phi<\pi$

Compute $\sum_{n=0}^\infty \frac {\sin ((2n+1)\phi)}{2n+1}$ for $0<\phi<\pi$ I just want to make sure I did this one correctly. Can I do this $$\sum_{n=0}^\infty \frac {\sin ((2n+1)\phi)}{2n+...
0
votes
1answer
75 views

What is the linear series $|mL|$?

I am studying complex geometry and I am trying to find out what is the definition of the linear series $|mL|,$ where $L$ be a line bundle over a compact Kahler manifold $X^n.$ In particular, I know ...
3
votes
1answer
234 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
1
vote
0answers
19 views

If there is an $\epsilon_x >0$ such that $cl(B(x,ϵ_x))$ is compact for each $x \in X$, then $X$ is complete. [duplicate]

If there is an $\epsilon_x >0$ such that $cl(B(x,ϵ_x))$ is compact for each $x \in X$, then $X$ is complete. We first take a Cauchy sequence $(x_n)$ in $X$ and since it is bounded we get an $x$ ...
0
votes
0answers
72 views

Ring homomorphisms on the set of rationals that coincide on integers

Let $R$ be a ring and let $f, g: \mathbb Q \to R$ be two ring homomorphisms such that $f|_{\mathbb Z}=g|_{\mathbb Z}.$ Then $f=g.$ I was trying to prove the above mentioned statement. According to ...
1
vote
0answers
107 views

Trying to prove Tietze extension theorem

I am trying to prove Tietze extension theorem for metric spaces that is " If $X$ is a metric space , $F$ is a closed set in $X$ and $f:F \to [0,1]$ is a continuous function , then there is a ...
2
votes
1answer
33 views

Find $\lim_{n \to \infty} n^2 (x_n-x_{n+p}) $

Let $f:\mathbb{R} \to \mathbb{R}$ with $f(x)=e^{x^2}$ and $F$ a primitive of $f$ for which $F(0)=0$. a) Prove that for all $n \in \mathbb{N}$ there is an unique $x_n$ such that $F(x_n)=\frac{1}{n}$. ...
1
vote
1answer
234 views

L,R,H,D,J relations on a completely simple semi group represented my a rees matrix

I am trying to tackle the following semigroup question. I can't see why my answer is wrong but I haven't used the fact the semigroup is COMPLETELY simple anywhere so I think there must be an error ...
3
votes
1answer
50 views

Link between tetrahedral numbers and combinatorics problem?

So I was trying to figure out a combinatorics problem involving the number of unique paths between two coordinates (can't move backwards such as from (1,1) to (0,1)) and I got stuck. I decided to draw ...
0
votes
0answers
42 views

A symmetric matrix is diagonalized by a matrix of its orthonormal eigenvectors

A proof of this statement is given in appendix A of this document. However, the author proves that every symmetric matrix has orthogonal eigenvectors. After the proof, it reads: By the second ...
0
votes
1answer
135 views

Proving some property of a Formal Logic Language [duplicate]

I am stuck at this problem: Let $\Sigma = \{\lnot,\lor,\land,\rightarrow,\leftrightarrow,(,),P_1,...,P_n\}$ be an alphabet. Now let's define the set of logical expressions $\mathscr{L} \subseteq \...
-1
votes
1answer
39 views

Perform the following block multiplications

i understand this process. but i do not understand that how may i calculate this. kindly tell me.
0
votes
2answers
85 views

Prove that there are infinitely many composite numbers in 10^n-9

Consider the sequence {1, 91, 991, 9991, 99991, ...} Prove that infinitely many of its elements are composite numbers.
1
vote
2answers
57 views

Any sequence in a compact metric space has an accumulation point but not necesseraly a limit?

Would it be correct to claim that $\forall X$ compact metric space and $\forall \{x_n\} \subset X$ sequence in it, $\{x_n\}$ has an accumulation point in $X$, but not necessarily a limit in it?
1
vote
2answers
91 views

A Number Theory problem (GCD) [duplicate]

Prove that $$gcd(2^{2^m}+1, 2^{2^n}+1)=1$$ if $m, n$ are positive integers such that $m \neq n$. A Hints to solve the problem is also given in the book as follows: Let $m>n$. Then $2^m=2^{n}2^{m-...
1
vote
1answer
82 views

Can you generate math problems that are solveable?

If you take Linear Programming, it problems are formulated like this: You know that Cabinet X costs 10 cents per unit, requires 6 square feet of floor space, and holds 8 cubic feet of files. ...
1
vote
3answers
59 views

how to solve inequation involving modulus

how to solve $$ǀx^2 + 3xǀ + x^2-2 ≥ 0 ?$$ I got stuck in the above problem. What would be the classic process to solve these type of problems. Also, if u have some fast processes then please explain ...
0
votes
1answer
214 views

Proof of Lallement’s Lemma

i am a bit weak with my congruence manipulation when it comes to semigroup theory. Could you possibly check my proof and give me constructive criticism on it? Lemma: Let $S$ be a regular semigroup ...
2
votes
1answer
146 views

Can we rewrite $Tr(ab^2)$ as $Tr(f(a)b)$?

Can we rewrite $Tr(ab^2)$ as $Tr(f(a)b)$ where $Tr:F_{2^{kn}}\rightarrow F_{2^{k}} $ is trace map, $k \neq 1$, $f$ is a function just depends to $a$.
0
votes
1answer
72 views

Mixture Algebra word problem

Brand X sells 21 oz. bags of mixed nuts that contain 29% peanuts. To make their product they combine Brand A mixed nuts which contain 35% peanuts and Brand B mixed nuts which contain 25% peanuts. How ...
0
votes
1answer
19 views

logarithmic limit in R^n

How does one prove that $$\lim_{|x|\to 0} \frac{ \log(1+|x|^2)}{|x|^2} = 1$$ when $x$ is a vector in $\mathbb{R}^n$, without using the multivariate Taylor expansion?
5
votes
1answer
144 views

How to find $\int {1\over x^2+x+1}\,dx$?

I have this indefinite integral to evaluate: $$\int {1\over x^2+x+1}\,dx$$ I thought it should be solved with IBP but it wouldn't work. I would appreciate any help or hint as for what identities I ...
0
votes
1answer
32 views

Show that $\left|\sum_{k=1}^\infty \frac{1}{k} \sin \left(\frac{x}{k+1}\right)\right| \leq |x|$ for all $x$

Let $$ f(x) = \sum_{k=1}^\infty \frac{1}{k} \sin \left(\frac{x}{k+1}\right). $$ I need to show that $|f(x)| \leq |x|$ for all $x\in\mathbb{R}$. Here's what I have so far: $$ \left|\sum_{k=1}^\infty \...
5
votes
1answer
100 views

Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that "....

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