1
vote
3answers
75 views

Show these two are the same (statistics)

Show that $$\sum_{i=0}^n(x_i-\bar x)^2=\sum_{i=0}^n(x_i-\bar x)x_i$$ This is what I have done. Expanded the square $$\sum_{i=0}^n(x_i-\bar x)^2=\sum_{i=0}^n(x_i-\bar x)\times\sum_{i=0}^n(x_i-\bar ...
1
vote
1answer
92 views

A closed form for $\sum_{k = 1}^{\infty} k^{-k}$? [duplicate]

Is there any closed form for this expression? $$\displaystyle\sum_{k = 1}^{\infty} k^{-k}$$ I got this while playing with my scientific calculator, so I really have no idea about how I could find one. ...
3
votes
0answers
44 views

Combinatorial proof of $\sum\limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} =4^n$ [duplicate]

$$\sum_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} = 4^n$$ Is there a combinatorial proof of above identity, without any arithmetic transformation? Thanks...
0
votes
1answer
90 views

Rolle's theorem for a proof [duplicate]

Consider the continuous functions $f : \mathbb R \to \mathbb R, f(x) = 1 + e^x \cos(x)$ and $g : \mathbb R \to \mathbb R, g(x) = 1 + e^x \sin(x)$. Using Rolle's Theorem, prove that between any two ...
0
votes
2answers
34 views

on visualising arithmetic with roots ansd radicals

Is there a visual way to simplify $4\sqrt{12}+4\sqrt{27}$? I know the answer is $20\sqrt{3}$, but I want to geometrically explain it to a 14 year old. Is there also a way to geometricaly interpret ...
1
vote
2answers
650 views

Calculus - Find critical points

I need help finding the critical points of this function: $f(x)=x-2 \sin x $ I found $f'(x)=1-2 \cos x $ and $f''(x)=2\sin x$ I know the next step is to set $f'(x)=0$ but when I do that I get ...
1
vote
0answers
58 views

Gauss and Stocks teory

Given $\phi\in C^1(R)$, and we define the curve and surface $\gamma=${$(x,y):y=\phi(x),0\le x\le 1$} $S=${$(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1$} a.I need to prove that $A(S)=2\pi\int_\gamma ...
0
votes
1answer
67 views

Multiplication of negative numbers is always positive [duplicate]

Show that Multiplication of negative numbers is always positive eg. (-1)*(-1)=1
0
votes
3answers
50 views

Determine if the given integral is convergent

$$\int_0^{\pi/2}{\log x\over x^a}\,\mathrm dx,\quad a<1$$ I tried solving using the $\mu-test$. so if I consider $\mu=1$ then $\lim\limits_{x\rightarrow 0} {x\log x\over x^a}$ Solving further, I ...
2
votes
1answer
129 views

A doubt about the isomorphism between an integral domain and its field of quotients.

I am currently reading the proof of the fact that An integral domain can be imbedded into a field. Let $a,b,c,d\in D$, where $D$ is an integral domain. $(a,b)=(c,d)$ iff $ad=bc$ ...
1
vote
2answers
219 views

Proving non-existence of isometry

We call two metric spaces $(X,d), (Y,d')$ isometric if there are inverse functions $f: X \to Y$, $g: Y \to X$ with $d(x,x') = d'(f(x), f(x'))$ and $d'(y,y') = d(g(y), g(y'))$. An example of ...
0
votes
1answer
96 views

can the gradient of a harmonic function =0 at some interior point of a manifolds with two ends?

M is a complete noncompact Riemannian manifold with two ends. There exists a nonconstant bounded harmonic function f defined on the whole M. Then is it possible that $|\nabla f|=0$ at some interior ...
0
votes
1answer
150 views

Zariski tangent space.

I'd like to investigate in two examples of Zariski tangent space: Let $X$ be $Spec(\mathbb{C}[x,y]/(y-x^2))$ with structure sheaf $\mathcal{O}_X$. Let $p=(x-a,y-a^2)$, How can calculate $T_pX$? We ...
0
votes
1answer
128 views

Two questions on endomorphisms

The first question is: Let $f$ be an endomorfism of an $n$-dimensional vector space $V$ with $n$ different eigenvalues. How does one show that $V$ has precisely $2^n$ subspaces which are ...
0
votes
0answers
34 views

Diagonalizing operator over $L^2(\mathbb{T})$

I've been asked to diagonalise an operator on $L^2(\mathbb{T})$, given by $Tf(z) = f(z^{-1}$). I know that I'm expected to find a $U$ such that $TU = UM_f$, where $M_f$ is the multiplication operator, ...
1
vote
5answers
347 views

Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
3
votes
3answers
4k views

Prove that every positive semidefinite matrix has nonnegative eigenvalues

There is a theorem that every positive semidefinite matrix only has eigenvalues $\ge0$. How can i prove this theorem?
2
votes
1answer
200 views

Geometric Intuition of Gaussian Curvature

Curvature of a curve at a point can be understood as how rapidly the curve tries to move away from the tangent of the curve at that point. And for curved surfaces we have defined the Gaussian ...
3
votes
1answer
88 views

Show that 2S = S for all infinite sets

I am a little ashamed to ask such a simple question here, but how can I prove that for any infinite set, 2S (two copies of the same set) has the same cardinality as S? I can do this for the naturals ...
6
votes
1answer
2k views

Is the sum of factorials of first $n$ natural numbers ever a perfect cube?

If $S_n = 1! + 2! + 3! + \dots + n!$, is there any term in $S_n$ which is a perfect cube or out of $S_1$, $S_2$, $S_3$, $\dots S_n$ is there any term which is a perfect cube, where $n$ is any natural ...
0
votes
1answer
105 views

Summation of reciprocal of Product of Factorials.

How can this summation be evaluated: $${∑ {1\over {a_1!a_2!....a_m!}}}$$ Where $$a_1+a_2+.....+a_m=n$$ Also $a_i !=n $ and $m<n$.
2
votes
2answers
76 views

Minimisation of a distance sum

I have a list $L$ of $N$ numbers, and I want to choose $k$ numbers $\{x_1,x_2, \ldots,x_k\} \subseteq L$ in such a way value $S$ of the those K numbers is minimum. $$ S = \sum_{0< i < j <= k} ...
0
votes
1answer
219 views

Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$.

Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$ for some $q\in \mathbb{Z}$. Any ideas on how to start? Do I use a proof by contraposition? Also what's the definition of ...
2
votes
1answer
149 views

Visualizing Markov and Chebyshev inequalities

I am helping a class on introductory probability covering Markov and Chebyshev's inequalities. I would like to give the students a nice visualization for why they are true or at least to show what ...
1
vote
0answers
36 views

Sum with chebyshev function

I want to compute the value of the following sum . $$\sum_{n=1}^{+\infty}(\psi(2n+1)/(2n)-\psi(2n)/(2n)).$$ $\psi$ is the second chebyshev function. It is defined in Mathematica as $$\psi(x) = ...
3
votes
4answers
95 views

Irrationality of a unique positive root of $\sin{x} = x^2$

The equation $\sin{x} = x^2$ has a unique positive real root. I wonder if there is any standard technique how to show that this number is irrational (rational), preferably a technique which works also ...
0
votes
1answer
60 views

prove uniqueness of a measure

If $(X,\mathcal{M},\mu)$ is a measure space and $\mathcal{\overline{M}}:=\{E\cup F:E\in\mathcal{M}\text{ and }F\subset N\text{ for some }N\in \mathcal{N}\}$ is a completion of $\mathcal{M}$ with ...
1
vote
2answers
174 views

bridgeless graph

I need to prove that every graph containing only even vertices is bridgeless. I understand that an even vertex is one with an even degree. Therefore an even vertex is one which is connected to an ...
1
vote
3answers
858 views

Integral points on a circle

Given radius $r$ which is an integer and center $(0,0)$, find the number of integral points on the circumference of the circle.
-1
votes
3answers
145 views

How to find $\liminf f_n $

Let $$f_n = \chi_{[n, n+1]} $$ We know $$\int\limits_{\mathbb{R}} f_n dm = 1 $$for all $n$. However, I am a little unsure whether $\liminf f_n (= \lim f_n) = 0$ or $1$. Can someone explain me the ...
2
votes
3answers
268 views

A question about the proof of the fact that contractible spaces are simply connected

In greeberg's algebraic topology, the following fact is used in the proof that contractible spaces are simply connected without justification: Let $p:\mathbb{I}\rightarrow X$ be a continuous function ...
5
votes
3answers
99 views

To what divisors $a$ of $n$ can Euler's Theorem multiplied by $a$ be generalized, i.e. when is $a^{\phi(n)+1}\equiv a \pmod n$?

Euler's Theorem $$a^{\phi(n)}\equiv 1\pmod n,$$ which is valid only iff $a$ and $n$ are coprime, can be "generalized" a bit to $$a^{\phi(n)+1}\equiv a\pmod n, (*)$$ where some zero-divisors of $n$ are ...
2
votes
1answer
118 views

How to extract the indeterminates from a set of polynomial?

I am a biologist and I am facing a huge problem. I would like to extract the indeterminates of a set of polynomials, for example, I have: $f_{1} = \{\\x_{3}^{2} + x_{1}*x_{2} + x_{1} + x_{1}*x_{3},\\ ...
3
votes
3answers
419 views

Set membership and inclusion confusion

I've recently started studying discrete mathematics for my computer science degree, and I have a doubt regarding membership and inclusion. I'll explain with an example. Consider the set: $ A = \{1, ...
4
votes
3answers
155 views

Needing help picturing the group idea.

I have a question; X and Y are subgroups of a group G. If $|X|, |Y| < \infty$, show that $|XY| = \frac{|X||Y|}{|X \cap Y|}$ but I can't really picture what it is talking about to even get ...
1
vote
2answers
64 views

Integration by substitution in a special case, “inverting” the interval

By letting $y=a-x$, show that $$\int_0^{a}f(x)\,dx = \int_0^a f(a-x)\,dx $$ Can someone help me to solve this question .This is only the first part of the question. We have to use this proof to do the ...
-2
votes
1answer
656 views

Analytical solution to Poisson’s equation 1D

I need some help in finding u(x) analytically where equation and the boundary conditions are satisfied
2
votes
1answer
163 views

Cauchy Schwarz inequality true with positive semi-definite inner product space

In one of exercises in a linear algebra book I have been asked to prove the following : "Suppose we modify the inner product definition such that $\langle u,u\rangle=0$ need not imply $u=0$." I have ...
1
vote
2answers
115 views

Partial functions - where can I learn more about this (heuristic, informal) system of conventions?

Is there a name for the following (heuristic, informal) system of conventions for dealing with partial functions and undefined expressions? I'd like to know whether it has any undesirable quirks that ...
3
votes
1answer
136 views

Singular values in SVD

I have recently started reading about SVD. If factorization of a matrix $A$ is required, we calculate the eigenvectors of $AA^T$ and $A^TA$ and they become the column vectors of $U$ and $V$ matrices ...
1
vote
2answers
221 views

3D geometry cube; find a distance

Let A1B1C1D1A2B2C2D2 be a cube with A1B1C1D1 being the bottom face and A2B2C2D2 the top face. Given that A1A2 is of length 1 what's the distance between D2A1 and A2B1.
-1
votes
2answers
79 views

Map from $\mathbb {R}$ to $\mathbb {R^2}$

Is there a way to construct an injective function that map from $\mathbb{R}$ to $\mathbb{R^2}$? If yes, please give me an example. Thank you!
1
vote
3answers
50 views

Total Percent increase

If the price of an item is increase by 8% from 2005 to 2006. then from 2006 to 2007 it is also increased by 8%. what is the total percent increase in price of an item from 2005 to 2007.?
1
vote
0answers
71 views

inverse of integral transform

given a general integral transform $$ g(x)= \int_{0}^{\infty}dyf(y)K(xy) $$ for a general formula of the kernel $ K(xy) $ is there an inverse of the Integral transform to obtain $ f(x) $ from above ...
2
votes
0answers
50 views

Finding a sufficient condition for a set to be finitely decomposable into open sets..

Let us call a set in a topological space finitely decomposable set (FDS) iff it can be rewritten using the standard set operations $\cup$ and $\sim$ and only finitely many open sets. I'm looking for ...
0
votes
1answer
134 views

inclusion exclusion principle basic question

Hello I have found a question about exclusion principle and I have love that you will help me with that question. Prove that for each 201 number from[1,300] we can find that there is always two ...
5
votes
1answer
156 views

Distance from point to vertices of convex hull

let $P = \{p_1, \ldots, p_k\}$ be any $k$ points on the unit $n$-sphere in $\mathbb{R}^{n+1}$ and $p_0 = 0$ the origin. Furthermore, let $CH(P\cup \{p_0\})$ denote the (possibly degenerate) convex ...
1
vote
2answers
107 views

How to prove that $x/y$ is continuous in R

$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$. Prove (formally) that $f$ is continuous. I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of ...
-1
votes
1answer
29 views

Complex linear system help?

Having been away for the lesson this was done in, I have no clue how to do this homework question, and there is nothing in the notes about it. The question is to solve this linear system: ...
1
vote
1answer
107 views

If $A$ Hurwitz, $(A+A^*)$ is Hurwitz?

If I have $A$ Hurwitz matrix, is $(A+A^*)$, with $A*$ the transpose of $A$, still Hurwitz? Any reference or proof? Because if $(A+A^*)$ is still Hurwitz I can say that it is even negative definite ...

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