3
votes
2answers
391 views

Coordinates and distance in higher dimensional spherical and hyperbolic space

For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the ...
2
votes
4answers
239 views

Combinatorial proof help

I have to prove the identity using a combinatorial proof: $\displaystyle\sum\limits_{k=0}^n 2^k \binom{n}{k} = 3^n$ I think this should be my combinatorial proof: We want to form a committee of $k$ ...
1
vote
1answer
187 views

Pattern Recognition with basic probability

I am a college student and have Pattern Recognition as a course this semester and I am really struggling with it, reasons could be difference in teaching style of the instructor and different type of ...
2
votes
0answers
301 views

Inverse function theorem for matrices and vectors

I am trying to figure out why a result invoked in a proof might be proved. Basically, the following equation holds \begin{equation*} ...
8
votes
3answers
294 views

Prove Inequality $\frac{\left(1-\alpha\right)\left(1+{\alpha}^{k}\right)}{\left(1+\alpha\right)\left(1-{\alpha}^{k}\right)}\geqslant\frac{1}{k}$

I would like to prove the following inequality: $$ \frac{\left(1 - \alpha \right )\left(1 + {\alpha}^{k} \right )}{\left(1 + \alpha \right )\left(1 - {\alpha}^{k} \right )} \geqslant \frac{1}{k} \ ...
0
votes
1answer
514 views

Orthonormalization and polynomial vector spaces

I am studying the Gram-Schmidt process of orthonormalizing a basis of a given vector space. Suppose you have a vector space $W = \{ w_1,w_2...w_n\}$. Our goal is to find a set $U = \{u_1,u_2 ... ...
2
votes
2answers
2k views

Rate of increase in the area of a square

I really do not understand how to do these problems, so many weird math tricks and rules and I am getting caught up on at least a dozen in this problem. Anyways I am supposed to find: Each side of ...
5
votes
2answers
871 views

Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$

How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in ...
1
vote
4answers
3k views

Velocity word problem

I know how to find velocity but I just can't make sense of this problem. If a rock is thrown verically upward from the surface of mars with velocity 15m/s, its height after t seconds is $h=15t - ...
0
votes
2answers
106 views

What (formal) language does this describe? And, how do I prove it's regular?

I have this problem that I can't seem to be able to wrap my head around, and I was wondering if there was someone here that could help me understand it. Let $L_1$ be a regular language over $\{a, b, ...
9
votes
3answers
498 views

Proper, smooth action with singular orbit space

This is a problem from Lee, Smooth Manifolds (Problem 9.4). It's not an homework problem, I'm just systematically solving every problem of that book, and I got stuck on this one. Usually I try not to ...
0
votes
1answer
149 views

$cl(C_c(\Omega))$ is a subset of $C_0(\Omega)$

I am trying to show that $cl(C_c(\Omega))$ is dense in $C_0(\Omega)$ where $C_c(\Omega)$ is the set of compactly supported continuous functions $f: \Omega \rightarrow R$ and $C_0(\Omega)$ is the set ...
1
vote
2answers
94 views

The vectorial ODE $ D\left( X \right) = AX$, with $A$ a constant coefficient matrix

Let $ X(t) $ a vector function in $ R^n $, and let A be an $ n \times n $ matrix with constant coefficients. Let us use $ D(X) $ to denote the derivative with respect to $t$ of the function $X$ (this ...
2
votes
1answer
206 views

If $X$ is regular then so is $C(X,X)$

I can't find a proof of this result, can anyone help me? Let $X$ be a topological space and $C(X,X)$ the space of all continuous functions from $X$ to itself. Suppose $X$ is regular, then $C(X,X)$, ...
0
votes
3answers
120 views

$\dfrac{-1}{n} < x$ for all $n\in \mathbb{N}$ $\Rightarrow x\geq 0$

I want to show $\dfrac{-1}{n} < x$ for all $n\in \mathbb{N}$ $\Rightarrow x\geq 0$ And I'am not allowed to use limits. Any ideas? I have tried to use contraposition, but with no luck.
5
votes
1answer
82 views

Independent statements that cannot be weakened

Let $T$ be a theory and let $\phi,\psi$ be statements that are independent of $T$. Say that $\psi$ is a $T$-weakening of $\phi$ if $T$ proves $\phi \Rightarrow \psi$ but cannot prove $\psi \Rightarrow ...
3
votes
2answers
2k views

Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT

Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ...
14
votes
2answers
272 views

A question on partitions of n

Let $P$ be the set of partitions of n. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : ...
11
votes
1answer
408 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
3
votes
2answers
344 views

Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
3
votes
0answers
70 views

The number of local rings $R$ such that $R^\ast$ is cyclic of order $n$

For $n>0$, let $c_n$ be the number of local rings $R$ such that $R^\ast$ is cyclic of order $n$. Note that $c_1 =1$. (A local ring $R$ such that $R^\ast = \{1\}$ has precisely two elements. See Is ...
2
votes
2answers
185 views

$S^n$ homeomorphic to $I^n/\partial(I^n)$

Let $S^n$ the unit sphere in $\mathbb{R}^{n+1}$ and $I=[0,1]$ the unit interval. By using polar and spherical coordinates, we can show that $$S^1\text{ is homeomorphic to }I/\partial I,\text{ and}$$ ...
6
votes
0answers
151 views

Invertibility of Toeplitz operator in $\ell_1$

Suppose we have a Toeplitz operator $$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & ...
4
votes
2answers
173 views

Basic Question on Gradients

I am having trouble understanding how the gradient of a scalar field is the direction along the $x$-$y$ plane that yields the maximum inclination. Sure it takes into account the partial derivatives ...
3
votes
2answers
322 views

Primes and proofs

1) Are there infinitely many primes of the form $a_n$? if $p_1 = 2 < p_2 = 3 <\cdots$ is the sequence of primes then are there infinitely many $n$ for which $p_1p_2\dots p_n + 1$ is prime? For ...
4
votes
1answer
356 views

Question about representing the Dual Space

In Fulton and Harris' Representation Theory, right at the beginning when they introduce representations, they note The dual $V^{\ast} = \mbox{Hom}(V,{\mathbb C})$ of $V$ is also a representation, ...
0
votes
1answer
43 views

The proportion between distinct labels in a multiset and the total amount of labels

Say we have a (multi)set $\alpha$ of $n$ balls, each of them is labeled with a number in $\{1,\ldots,m\}$ (where $m<n$ ). Denote by $d$ the amount of distinct labels in $\alpha$. Is it true that ...
0
votes
0answers
61 views

Pointed bundles as short exact sequences

Let $\pi: E \to B$ be a pointed continuous surjection, and let $F = \pi^{-1}(b_0)$ be the fiber over the basepoint (base fiber for short). Then $$* \to F \to E \stackrel{\pi}{\to} B \to *$$ is a short ...
5
votes
2answers
149 views

Can I use $x \in \{a, b, c\}$ to mean that $a, b$ and $c$ are valid solutions?

When solving an equation, can I use the notation $x \in \{a, b, c\}$ to mean that $x=a$, $x=b$ and $x=c$ are all possible solutions to the equation?
0
votes
2answers
366 views

Discrete Math - Calculating the number of different functions possible

Consider $f:\{1,\dots,n\} \to \{1,\dots,m\}$ with $m > n$. Let $\operatorname{Im}(f) = \{f(x)|x \in \{1,\dots,n\}\}$. a.) What is the probability that a random function will be a bijection when ...
2
votes
1answer
112 views

Is a Fréchet differentiable map between complex Banach spaces locally given by a “power series”?

Let $X,Y$ be Banach spaces over $\mathbb{C}$ and let $U \subset X$ be open. If $f:U \to Y$ is Fréchet differentiable at every point of $U$, can we locally expand $f$ as a "power series"? To be more ...
2
votes
4answers
353 views

How can I verify the Pythagorean Trig Identity using approximations for $\sin x$ and $\cos x$ derived from their infinite series representations?

How can I verify the Pythagorean Trig Identity using approximations for $\sin x$ and $\cos x$ derived from their infinite series representations? I can see that the infinite series of $\sin x$ and ...
2
votes
2answers
414 views

Is there a comprehensive algebra exercises site online?

I'm trying to find a place that has guided walkthroughs/answer guides for intermediate algebra, college algebra, and precalc. Maybe trig, also. Any ideas?
1
vote
1answer
371 views

A plane Geometry Problem

The triangle $ABC$ has $CA=CB$, circumcenter $O$ and incenter $I$. The point $D$ on $BC$ is such that $DO$ is perpendicular $BI$. Show that $DI$ is parallel to $AC$.
1
vote
2answers
252 views

Do Ambivalent Axioms have a place in Mathematics?

I can't think of any examples of ambivalent axioms in mathematics (two ideas are ambivalent if there are sometimes conflicts between them), so please let me humor you with a strange example. Suppose ...
5
votes
0answers
92 views

Showing that $[\mathbb C(z):\mathbb C(f_0)]=$topological degree of $f_0\in\mathbb C(z)$ as a function from the Riemann sphere to itself

McKean & Moll offer the following sketch of a proof. Let $P(x)=a_0(x)-f_0b_0(x)$ in $\mathbb C(f_0)[x]$ where $f_0(z)=\dfrac{a_0(z)}{b_0(z)}$ is in $\mathbb C(z)$. Then evidently, $\deg P=\#$ ...
2
votes
1answer
276 views

Linear Fractional Transformations

Let $p(x)= \pi A_\alpha \pi^{-1}(x) = y$, where $$A_\alpha = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\\ \end{pmatrix}$$ and $\pi:S\to ...
6
votes
1answer
219 views

Can you deduce Neumann boundary data from Dirichlet boundary data?

Say for the following problem, suppose boundary of $\Omega$ is $C^{1,1}$: $$ \left\{ \begin{aligned} -\Delta \phi &= \mathrm{div} \,\vec{u}\quad \text{ in } \Omega \\ \phi&=0 \quad \text{ ...
2
votes
1answer
193 views

A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$

I am going over some counterexamples for the the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. In particular I have been trying to understand what happens if you remove the various ...
5
votes
0answers
211 views

Can the 57-cell be made in vZome without strut crossings?

Here's the 57-cell in vZome with lots of strut crossings: Is it possible to construct the 57-cell in vZome without any strut crossings? That is, 57 nodes, 171 struts, in the 57-cell / Perkel graph ...
2
votes
5answers
520 views

Integrating $\int \frac{1}{1+e^x} dx$

I wish to integrate $$\int_{-a}^a \frac{dx}{1+e^x}.$$ By symmetry, the above is equal to $$\int_{-a}^a \frac{dx}{1+e^{-x}}$$ Now multiply by $e^x/e^x$ to get $$\int_{-a}^a \frac{e^x}{1+e^x} dx$$ ...
2
votes
1answer
190 views

Maximal ideals in a circular discrete convolution algebra

Let $G$ = $\mathbb{Z_n}$ and let $A$ = $\ell^1(G)$ with convolution, over $\mathbb{C}$. Let $A_{\mathbb{R}}$ denote the subring of real valued functions in $A$, so $A_{\mathbb{R}}$ is an algebra over ...
4
votes
2answers
257 views

If you know the convergent sequences, how do you know the open sets?

I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes.... Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
0
votes
2answers
124 views

Question on Triangles

In a right triangle, the length of hypotenuse is $c$. The centers of three circles of radius $c/5$ are found at its vertices. Find the radius of the fourth circle which touches the three given ...
0
votes
1answer
92 views

Does this sum have an upper bound?

If we have an infinite sequence of positive numbers whose sum is $$ S = \sum_{i=1}^\infty a_n $$ and $$ \lim_{n \to \infty} a_n = 0 $$ Can we draw conclusion that $S$ has an constant upper ...
0
votes
0answers
110 views

If $Rm$ is free, how do you show $m \otimes n \neq 0$?

This is a follow up to the post Showing $ m\otimes n$ is free given $m,n$ are free. I am curious about conditions that would eliminate the possibility for a counterexample in the case that was ...
4
votes
1answer
706 views

Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it

I am trying to find a bound for variance of an arbitrary distribution $f_Y$ given a bound of a Kullback-Leiber divergence from a zero-mean Gaussian to $f_Y$, as I've explained in this related ...
3
votes
2answers
5k views

Proving formula for product of first n odd numbers

I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$): $$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$ How ...
4
votes
1answer
192 views

Open ball over the real numbers

In my book, it says that any open ball $B(a,r)$ over the real numbers is equal to the open interval $(a-r,a+r)$. I wonder how I can prove that this is true, only using the metric axioms. If the ...
1
vote
1answer
533 views

Need help with Lagrange Multipliers

I need to maximize $U = BM$ with constraits: $6B +3M = 60$, $B>0$ and $M>0$. The Lagrange function is $L=U + \lambda (6B+3M-60) + KB + HM$. So $$\partial_{\lambda}L= 6B+3M-60=0$$ ...

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