1
vote
0answers
9 views

Does a closed form exist for the following summation?

How do i calculate this sum? Wolfram alpha can't calculate it, but the sum surely converges.. $$S=\dfrac{\sin x}{2}+\dfrac{2\sin {2x}}{5}+\dfrac{3\sin {3x}}{10}+\dfrac{4\sin {4x}}{17}......\infty$$
0
votes
0answers
8 views
0
votes
1answer
10 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, ...
0
votes
1answer
11 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
0
votes
1answer
11 views

Finding first few terms in power series expansion of general solution

I need to find the first four nonzero terms in the power series expansion for the general solution to the differential equation $y''-x^2y=0$. My work thus far: ...
0
votes
0answers
12 views

How to prove the value of the following infinite sum?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}*\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ It is obviously true for all $x\in \mathbb N_0$
0
votes
0answers
8 views

Least squares approximation: Legendre polynomial

Find the best quadratic least squares approximation to $f(x)=e^x$ on $[-1,1]$ with respect to the inner product $\langle f(x),g(x) \rangle = \displaystyle\int_{-1}^1 f(x)g(x)dx$. I cannot figure out ...
0
votes
0answers
8 views

What is the 'normal form' or 'canonical form'?

I learned 'Jordan canonical form' in linear algebra. But some question asked me about canonical form or normal form , not Jordan canonical form. Please help me.
0
votes
1answer
7 views

What is the meaning of the notation $cf(add(\mathcal I))$?

I am trying to prover the following claim: Assume that $\mathcal I$ is an ideal over some infinite set $X$. Then, $cf(add(\mathcal I))=add(\mathcal I)$. Doe anyone know what is the meaning of the ...
1
vote
4answers
26 views

$\lim_{n \rightarrow \infty} f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$

Let $$f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$$ Let M be the set of x s.t. $\lim_{n \rightarrow \infty} f_n(x)$ exists. For each $x \in M$ let $f(x) = \lim_{n \rightarrow ...
0
votes
0answers
6 views

Is the (type)class `Functor` itself a functor?

Simple yes or no question, but one that's hard to google/search for due to repetition of terms: Since Functor is the set (ok, class) of all types for which an ...
1
vote
1answer
16 views

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets? (or equivalently, that $\mathbb R$ contains no more then $\mathfrak c$ $G_\sigma$ sets? The more general ...
0
votes
0answers
8 views

On the integrability of vector fields

Let $X$ and $Y$ be a vector field on $M$ and satisfies $[X,Y]=X$. If $X$ and $Y$ are pointwise linearly independent for some point $p$, then there is a sub manifold $N$ of $M$ such that $T_xN$ is ...
2
votes
0answers
9 views

Graph planarity and electronic circuit boards

In another MSE question, I found the following definition for 2-layer circuit board decomposition of a graph: A circuit board is defined as a pair of planar graphs with vertices identified, i.e. ...
0
votes
0answers
16 views

Term by term integration

Let $D \subset \mathbb{R}^{d} $ be open. For $u,v \in C_{0}^{\infty}(D)$, we define \begin{eqnarray*} \mathcal{A}(u,v)=\sum_{i,j=1}^{d} \int_{D} \frac{\partial u(x) }{\partial x_{i}}\frac{\partial ...
1
vote
2answers
21 views

Eigenvalues of $A:\;A^2 +2A=0$

Let $A_n$ be square matrix where $n \geq 2$ and $A^2 +2A=0$. Then A is singular A is nonsingular 0 and -2 are eigenvalues of A either 0, or -2 is not an eigenvalue of A (1)-(4) are ...
1
vote
1answer
43 views

Why can one take the power of $e$ directly?

The definition of Euler's constant to the power $x$, $e^x$, is $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + {...}$$ And of course, we have the ...
0
votes
2answers
15 views

Gambling question: multiply quotes.

Reading various betting forum I came across different threads claiming betting multiple is worse than betting on single events. Could you explain why? [Clairification for the ones not familiar with ...
0
votes
0answers
28 views

I have questions on where you stood around my age in terms of math.

first of all I don't know if this is the right place to ask, but I feel a lot of really math-knowledgable individuals roam this site, so I gave it a go. So, basically my question is, am I not really ...
3
votes
1answer
17 views

C([0,1]) is not weakly sequentially complete.

I'm studying Functional Analysis by myself. For a counterexample of every Banach space is not weakly sequentially complete, I was suggested to check C([0,1]) is not weakly sequentially complete. For ...
0
votes
0answers
8 views

Shannon entropy and mutual information

What is the relation between Mutual information and Shannon entropy of two random variables x and y? In other words, What is the relation between MI(x,y) and H(x), H(y) ?
0
votes
0answers
9 views

A further question on reparametrization.

Hatcher contains the following paragraph: Define a reparametrization of a path $f$ to be composition $f\psi$ where $\psi:I\to I$ is any continuous map such that $\psi(0)=0$ and $\psi(1)=1$. ...
0
votes
1answer
12 views

Straight Lines co ordinate geometry

At what angle with the line x+y=4, a line through (1,2) be drawn so that the distance between the point of intersection of the lines and the point (1,2) is 6/(root 3)?
0
votes
1answer
13 views

Stuck with deriving tangent line at a point

I have been asked to find the tangent to the given curve at the point indicated. This is what I know: We find the slope $m$ at the specific point on the graph by ...
0
votes
0answers
12 views

von Neumann Algebras and measures

I read that any abelian von Neumann algebra ist isomorphic to $L^\infty(X,\mu)$ for some $X$ and $\mu$. This seems to be reasons, to consider any von Neumann Algebra as non-communitative measurable ...
0
votes
2answers
26 views

Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ (corrected inequation)

Prove that Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ algebraically or geometrically. $n\sin\frac{2\pi}{n}-n\sin\frac{\pi}{n}$ means the area of a regular n-gon + the area ...
0
votes
1answer
18 views

Notation question: (X,Y) and (Y,X) identically distributed?

Given that $X$ and $Y$ are random variables with finite expectation (say, 1-dimensional), what does it mean to say that $(X,Y)$ and $(Y,X)$ are both identically distributed? From what I can see the ...
1
vote
0answers
15 views

boundedness of $\{\int_E(g f_n)\}$ implies boundedness of $\{f_n\}$

I need some help on this problem: let $E$ be a measureable set, $1 \le p < \infty$ and $q$ is the conjugate of $p$. suppose that $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g \in ...
0
votes
0answers
11 views

conjunctive normal form

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: y= ax+b y= -ax+b y=b So, please How can I represented those ...
1
vote
1answer
23 views

Continuity of series $\sum_{n=0}^\infty \frac {x^n sin(nx)} {n!}$?

Let $$ S(x) = \sum_{n=0}^\infty \frac {x^n sin(nx)} {n!}~~,~~ S_k(x) = \sum_{n=0}^k \frac {x^n sin(nx)} {n!}$$ $$ \left |S(x) - S_k(x) \right| = \left | \sum_{n=k}^\infty \frac {x^n sin(nx)} {n!} ...
1
vote
1answer
18 views

optimization word problem

You want to construct a rectangular dumpster which uses a thicker steel for the bottom of the container than for the walls. If the top will be covered by plastic doors which will cost 50 dollars ...
0
votes
2answers
37 views

What does $a = b$ mean when $a, b \in S$

If we have a set $S$ and $a, b \in S$. What does the expression $a = b$ mean in this context? Does it strictly mean that $a$ and $b$ refer to the same element of $S$. Or maybe they are different ...
4
votes
1answer
100 views

Is $2013^{2014}+2014^{2015}+2015^{2013}+1$ a prime? (usage of a computer not allowed)

Prove or disprove: $$2013^{2014}+2014^{2015}+2015^{2013}+1$$ is a prime number, without using a computer. I tried to transform the expression $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, but couldn't reach ...
-1
votes
3answers
37 views

What is the Cadinality of $\mathbb{Z}$ free product of $\mathbb{Z}$??

i want to know cadinality of $\mathbb{Z}$*$\mathbb{Z}$. Is it countable? or uncountable?
0
votes
2answers
12 views

Algebraic Multiplicity of Eigenvalues for a Linear Mapping

I am stuck on the following problem: For the following linear mapping, $L(\bar{x})=\bar{x}-2 \frac{\bar{x} \centerdot \bar{n}}{||\bar{n}||^{2}} \bar{n}$ where $\bar{n} \in \mathbb{R}^{n}$, find the ...
1
vote
0answers
12 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
3
votes
1answer
29 views

Is there a self-homeomorphism of the 2-sphere with exactly 3 fixed points?

I don't believe so, but I'm not sure how to prove it. The Lefschetz-Hopf theorem says in this case that the sum of the fixed point indices is 0 or 2 (since our map is a self-homeomorphism). My ...
1
vote
0answers
24 views

Finding average time to run a marathon

Here is a problem that I am working on. I have tried to put it in simplified and analogical terms here and it may sound a bit absurd but I am desperate for a solution. 100 runners start a 5 day ...
2
votes
0answers
13 views

Does there exist Latin square critical sets for which deleting any entry results in arbitrarily many completions?

For those familiar with Latin squares terminology, I'll get straight to the point: Q: For all $N \geq 2$, does there exists a critical set $C$ (for a Latin square of any finite order) such that ...
-2
votes
1answer
40 views

Calculus question about finding the length [on hold]

A rectangular plot of ground has two adjacent sides along highways 40 and 60. In the plot is a small lake, one end of which is 256ft from highway 40 and 108 ft from highway 60. Find the length of the ...
0
votes
2answers
19 views

Kernel, row space and orthogonality

The set of solutions of $Ax=0$ i.e. kernel or null space of $A$ is perpendicular to each row of A. But why is the kernel of $A$ perpendicular to the row space of it? In other words why is it ...
0
votes
0answers
20 views

Calculate length of radial intersecting a rectangle

In a rectangle like below, I need to calculate the length of any radial, from the center of the rectangle to where it intersects with the edge of the rectangle. Further, the angle of the radial is ...
0
votes
0answers
7 views

Asking one example of unbounded joint density

For $d\geq2$, let $X_{i}=\left\{Y_{i-1},Y_{i-2},...,Y_{i-d} \right\}$, and assume the sequence $\left\{X_i \right\}$ is strictly stationary. Let $f_{j}(x_{0},x_j)$ denote the joint density of ...
1
vote
0answers
64 views

Compute $\int_1^e \frac{dx}{x(x+(\ln x)^2)}$

My friend asked me how to integrate the following: $$\int_1^e \frac{dx}{x(x+(\ln x)^2)}$$ How am I going to solve this?Any help is greatly appreciated. Thanks.
1
vote
2answers
38 views

Particular solution of RE: $u_{n+1} - 2u_n = n^22^n$

Find the particular solution of recorrence equation $u_{n+1} - 2u_n = n^22^n$. I am developing a practical method using operators $E$ e $\Delta$, defined by $E(u_n) = u_{n+1}$ and $\Delta(u_n) = ...
0
votes
2answers
40 views

Linear algebra proof

Let $W$ be a subspace of $\mathbb{R}^n$. Let $\vec{v}_1 ,\vec{v}_2 \in \mathbb{R}^n$. Suppose that $\vec{p}_1$ is the projection of $\vec{v}_1$ onto $W$ and $\vec{p}_2$ is the projection of ...
9
votes
1answer
205 views

Apparent Paradox in the Idea of Random Numbers

This question is a bit less than rigorous, but it's only because I don't know how to formulate it rigorously. Suppose there was some machine, or function, or whatever that could output a random ...
1
vote
1answer
16 views

Finding the convergence radius of a complex laurent series

Find the maximal ring where the following series converges: $$\sum_{n=1}^\infty\frac{3^n+2^n}{(z-5)^n}+\sum_{n=0}^{\infty}\frac{n^2}{20^n}(z-5)^{2n}$$ I think that taking the minimum between the ...
1
vote
1answer
22 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
1
vote
2answers
33 views

Any well-ordered set must be inductive?

(Goldrei, Classic Set Theory, Exercise 3.17) This exercise asks to show that well-ordered set $X$ is inductive ($\varnothing \in X$ and for every $x \in X$, $x^{+} = x \cup \{x\} \in X$). In other ...

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