# All Questions

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### 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.. $$S=\dfrac{\sin x}{2}+\dfrac{2\sin {2x}}{5}+\dfrac{3\sin {3x}}{10}+\dfrac{4\sin {4x}}{17}......\infty$$
0answers
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### A train having 200 m length going speed of 60 km/hr. The Train crossed the bridge within 1 minute? Then what is the bridge length?

Can anyone give the idea about how to solve this type of problems?
1answer
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### 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 ...
1answer
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### 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 ...
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### 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 ...
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### 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. ...
0answers
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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...
2answers
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### 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 ...
0answers
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### 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 ...
1answer
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### 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 ...
0answers
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### 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) ?
0answers
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### 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 $\psi(1)=1$. ...
1answer
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### 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)?
1answer
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### 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 ...
0answers
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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...
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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 $g \in ... 0answers 11 views ### conjunctive normal form in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: y= ax+b y= -ax+b y=b So, please How can I represented those ... 1answer 23 views ### Continuity of series$\sum_{n=0}^\infty \frac {x^n sin(nx)} {n!}$? Let $$S(x) = \sum_{n=0}^\infty \frac {x^n sin(nx)} {n!}~~,~~ S_k(x) = \sum_{n=0}^k \frac {x^n sin(nx)} {n!}$$ $$\left |S(x) - S_k(x) \right| = \left | \sum_{n=k}^\infty \frac {x^n sin(nx)} {n!} ... 1answer 18 views ### optimization word problem You want to construct a rectangular dumpster which uses a thicker steel for the bottom of the container than for the walls. If the top will be covered by plastic doors which will cost 50 dollars ... 2answers 37 views ### What does a = b mean when a, b \in S If we have a set S and a, b \in S. What does the expression a = b mean in this context? Does it strictly mean that a and b refer to the same element of S. Or maybe they are different ... 1answer 100 views ### Is 2013^{2014}+2014^{2015}+2015^{2013}+1 a prime? (usage of a computer not allowed) Prove or disprove:$$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 ... 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? 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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)}$$How am I going to solve this?Any help is greatly appreciated. Thanks. 2answers 38 views ### Particular solution of RE: u_{n+1} - 2u_n = n^22^n Find the particular solution of recorrence equation u_{n+1} - 2u_n = n^22^n. I am developing a practical method using operators E e \Delta, defined by E(u_n) = u_{n+1} and \Delta(u_n) = ... 2answers 40 views ### Linear algebra proof Let W be a subspace of \mathbb{R}^n. Let \vec{v}_1 ,\vec{v}_2 \in \mathbb{R}^n. Suppose that \vec{p}_1 is the projection of \vec{v}_1 onto W and \vec{p}_2 is the projection of ... 1answer 205 views ### Apparent Paradox in the Idea of Random Numbers This question is a bit less than rigorous, but it's only because I don't know how to formulate it rigorously. Suppose there was some machine, or function, or whatever that could output a random ... 1answer 16 views ### Finding the convergence radius of a complex laurent series Find the maximal ring where the following series converges:$$\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 ... 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 ... 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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