# All Questions

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### 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 ...
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### 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 $k$ ...
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### 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 ...
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### 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*} ...
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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} \ ... 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 ... ... 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 ... 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 ... 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 - ... 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, ... 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 ... 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 ... 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 ... 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), ... 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. 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 ... 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 ... 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 : ... 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 ... 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 ... 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 ... 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}$$... 0answers 151 views ### Invertibility of Toeplitz operator in \ell_1 Suppose we have a Toeplitz operator$$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & ...
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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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 $AC$.
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### 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 ...
### 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, $\deg P=\#$ ...