1
vote
3answers
81 views

If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$

If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$ Prove that $\tan A+\tan B= 8ab/((a^2+b^2)^2-4a^2)$
0
votes
0answers
339 views

Standard Deviation Around an Arbitrary Mean

I'm collecting data from x and y axis offset from origin of the impact points of rounds I've shot at a target, and I've calculated my standard deviations in the x and y directions as $ \sigma_x $ and $...
0
votes
1answer
41 views

Is the optimization problem right?

If we want optimize the following problem $$ \min_x \{a(x)+c(x)\} $$ and we have $$ a = \min_y b(y) $$ then, could we directly optimize the following problem? $$ \min_x \{b(x)+c(x)\} $$
-1
votes
1answer
161 views

Real Analysis vs. Functional Analysis version of Arzela-Ascoli Theorem

Consider a collection of functions defined on a compact set such that they are uniformly bounded and equicontinuous. Then 1) Real analysis version: Every sequence in that collection will contain a ...
4
votes
0answers
259 views

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification)

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification) Proof: Let G be generated by c, b, where $c^2 = b^2 = 1$. ...
4
votes
0answers
40 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
0
votes
1answer
58 views

Infinite dimensional transpose?

I know that if $L$ is a linear transformation from $V$ to $W$ where $V,W$ are finite dimensional, then we can conclude that the dimension of image (rank) of $L$ is same as that of its transpose, i.e., ...
0
votes
1answer
94 views

Existence of a holomorphic function such that f(z)^2 = z^2-1

Does there exist a holomorphic function $f:\mathbb{C} \setminus [-1,1] \rightarrow \mathbb{C}$ such that $f(z)^{2} = z^{2}-1$ holds for all z in the domain? I figured that the answer should be NO. ...
1
vote
3answers
116 views

Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$

Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$. Ok I'm finally understanding what each of these ...
2
votes
3answers
41 views

Time spent to sort $10^7$ records with insertion sort

I am stuck with my revision for the upcoming test. The question asks" An implementation of insertion sort spent 1 second to sort a list of ${10^6}$ records. How many seconds it will spend to sort ${...
0
votes
1answer
50 views

How does derivative of a squared function work?

I was working through a physics problem related to magnetic flux, but was confused at the math the solution uses. I understand up till the last line: $ c=1.65-.12t\\ A=c^2/4\pi\\ \Phi_B=BA=(\frac{B}{...
5
votes
1answer
92 views

Why it is true for rapid decreasing function $g$ that: $\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\leq A_{l,k}(1+|y|)^{l}$

If $g$ is of rapid decrease, that is $\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x)|<\infty$, then we have: $$\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\...
0
votes
1answer
99 views

On the existence of a positive fundamental period

A function $f$ has period $t$ if for all $x$ in the domain it is true that $f(x+t) = f(x)$. A function is called periodic if it has (at least one) period. Take any periodic function $f$ -periodicity ...
2
votes
2answers
162 views

Factoring numbers of the form $11111111$

Why $11111111$ is divisible by $73$? How can we get all the prime factors? It is clear that it is divisible by $11$. Is there any formulae for $1111...11$ ($n$ times)? Give me some idea. Thanks in ...
1
vote
1answer
83 views

Find all conformal automorphisms

Let G be a non-empty simply connected and bounded domain in $\mathbb{C}$ and let $a, b \in G $ with $a \neq b$. Find all conformal automorphisms of G such that a dn b are two fixed points. Moreover, ...
7
votes
1answer
85 views

Is there a proof of $\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$ using induction?

Can someone prove (or disprove) this equality? $$\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$$ where the value of $x$ can vary. This is a pattern found in derivatives and stuff but I'm not ...
0
votes
2answers
46 views

a question about abstract algebra, a question related to permutation

Given $X=\{1,2,......n\}$, let us call a permutation $p$ of $X$ an adjacency if it is a transposition of the form $(i,i+1)$ for $i\le n-1$. Prove that $(i,j)$ is a product of an odd number of ...
0
votes
1answer
109 views

Proof of why the partition function Z in probabilistic graphical models (PGM) is NP-complete

I was wondering if someone knew why computing the partition function for probabilistic graphical models is NP-Hard? I would like to see a full blown rigorous proof, however, I am as happy to get a ...
3
votes
1answer
146 views

Corollaries of Green-Tao Theorem?

there is already a good thread which discusses some corollaries of the Green-Tao Theorem, here: Constructing arithmetic progressions The question I was wondering about is of a similar flavor but isn'...
1
vote
1answer
44 views

How to show using Cauchy integral formula?

Let f(z) be an entire function such that $\;|f(z)| \le M|z|^n\;$ for any $\;z ∈ \Bbb C\; , \;\text{and}\;M >0\;$ is a fixed number. Using Cauchy integral formula, show that for any $$\;k\ge n+1\;...
0
votes
1answer
43 views

Calculation of the Fourier transform of $x/(x^2+1)^2$ using the properties of the transform

I am trying to calculate the Fourier transform of $$f(x)=\frac{x}{(x^2+1)^2}$$ using the property of Fourier transform. So I am trying to use$$\widehat{g_1(x)g_2(x)}=\frac{1}{2\pi}\widehat g_1 *\...
0
votes
0answers
26 views

Suffix string starting at $i$

$S$ is the string of characters:TACGCGGT$ For string S and each of the positions $i=1,2,\dots,9$ write down the suffix string starting at position $i$. What is ...
1
vote
1answer
92 views

Matrices similar only to themselves

Find all matrices similar only to themselves, i.e., $PTP^{-1}=T$ for any invertible $P$. My attempt: $PT = TP$. Am I going about this correctly? If so, how do I find all matrices that are ...
0
votes
3answers
77 views

Let x be any non zero real number. Show that $x^8-x^5-\frac{1}{x}+\frac{1}{x^4}\ge0$

Let x be any non zero real number.Show that $x^8-x^5-\frac{1}{x}+\frac{1}{x^4}\ge0$ This question is from the Regional Olympiad Materials.This is trivial proof, but I am stuck designing the ...
2
votes
2answers
80 views

Proving a Combinatorial Theorem

The Theorem My Problem I don't really understand how the $RHS$ counts the number of final positions for a $1$. I understand how summing all of these cases would be the same as counting all the ...
1
vote
0answers
171 views

Calculating centre of mass of a cylindrical wedge

I'm sort of stuck with a problem. Don't know if I'm applying the theory in a wrong way. Here's the problem: Let $S$ be the solid enclosed by the cylinder $y^2+z^2=9$ and the planes $x=0$, $y=3x$ and $...
0
votes
1answer
140 views

Probability that no two teams in a tournament win the same number of games

Six teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $\dfrac{1}{2}$ probability of winning any game it plays. Find the probability ...
0
votes
2answers
46 views

Inner product on the k-tensor space

This is homework so no answers please. The problem is "Given inner product vector space V, define an inner product on $T^{k}(V)$ by declaring the standard basis $\{e^{*}_{i_{1}}\otimes...\otimes e^{*...
1
vote
0answers
130 views

Splitting of short exact sequence of sheaves

Let $X$ be a smooth projective variety over a field, say $k$. Consider the short exact sequence of $k$-modules, $$0 \to A_1 \to A \to A_2 \to 0$$ where $A$ and $A_2$ are $k$-algebras. Since these can ...
1
vote
1answer
40 views

How do I perform taylor expansion of the following?

Taylor expansion about $(x,y)$ of $f(x + a,\; y + k\; f(x + b,\; y + c))$ I do not understand what happens to the second $f$ inside. The inspiration for this question is Runge-Kutta methods.
14
votes
1answer
232 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
0
votes
1answer
54 views

On Prime and Maximal Ideals in a Commutative Ring with Unity

Let $R$ be a commutative ring with $1 \neq 0$, $I$ and $P$ are ideals of $R$. If $P$ is prime and $I \cap P \neq 0$, does it follows that either $I \subseteq P$ or $I$ is also a prime ideal ...
12
votes
1answer
386 views

Limit cycle of dynamical system $x'=xy^2-x-y$, $y'=y^3+x-y$

Consider a planar ODE system $z'=f(z)$ with $z=(x,y)$, $$ f(x,y)=(xy^2-x-y,y^3+x-y). $$ Using polar coordinates, one can get $$ r'=r(r^2\sin^2\theta-1),\quad \theta'=1. $$ With Mathematica one can ...
2
votes
1answer
91 views

Method for solving $\frac{dy}{dx} = 2xy+2$

I tried using the integrating factor method, since the equation is first order linear: suppose $R$ is the integrating factor. We have $\log(R) = \displaystyle \int -2x dx = -x^2+c_0$ so $R = c_1 e^{...
2
votes
1answer
69 views

How to calculate the residue of the fourier transform?

I have been struggling calculating the Fourier transform of $f(x)=\frac{x}{(x^2+1)^2}$. I tried to calculate $f(t)=\int\frac{x}{(x^2+1)^2}e^{-ixt}\,dx$ directly by integration by parts, but it is not ...
5
votes
2answers
133 views

Can the Kahler differentials of a “good” local ring R be free of rank not equal to dim(R)?

Let $R$ be a local ring containing a field isomorphic to its residue field $k$. Assume $R$ is a localization of a finitely-generated $k$-algebra. Can $\Omega_{R/k}$ be free of rank $r\neq\dim{R}$? ...
1
vote
1answer
63 views

Determine whether each function is one-to-one, onto, or both

$g:\mathbb Z \times \mathbb Z$ where $g$ is defined by $g(x)=x-1$ My guess is that this is onto and one-to-one. But is the correct interpretation of this problem that $g$ is a function of an ...
1
vote
1answer
499 views

Find the inverse of a Trig Matrix

I don't have a clue of what's going on. We haven't learn this in class so I need all the help possible. The more detailed of an explanation, the better. Thanks in advance. The only info I have is that ...
3
votes
1answer
145 views

Product measures and $\sigma-$ finite measures

Problem similar to folland chapter 2 problem 51. The actual problem in Folland mentions that $X,Y$ are not necessarily $\sigma-$finite. Then how can I use Fubini-Tonelli theorem?
0
votes
0answers
361 views

understanding and visualizing the span of sets

I've been researching for a while and trying to wrap my head around spanning of vector spaces completely (by visualizing them in R3) before moving on to Linear Independence, Basis' and anything else ...
1
vote
0answers
35 views

lines that only intersect a curve at 2 points

I have some data points that define a curve and what i need to find is the slope of the lines definedby line 1 = p1&p2 line 2 = p1&p3 line 3 = p1&p4 . . . line29= p1&p30 line30= p2&...
-1
votes
1answer
82 views

Is the following statement true ? If $L$ is a decidable language and $L' \subseteq \; L$, then $L'$ is also decidable ? Prove your answer is correct [closed]

Is the following statement true ? If $L$ is a decidable language and $L' \subseteq \; L$, then $L'$ is also decidable ? Prove your answer is correct I can't figure out this question. Any tips ?
3
votes
2answers
88 views

Derivative of integral question

Given $f$ is function with continous derivatives, how do I obtain $f(x)$ in terms of $x$ from the equation below? Thanks in advance. $$ f(x)=\lim_{t\to 0} \frac{1}{2t} \int_{x-t}^{x+t} s f'(s) ds $$
1
vote
1answer
157 views

How to FIND the limit of a sequence with epsilon definition?

So I have this sequence in hand: $x_n=\frac{\sqrt{n}}{n+1}$, and I can intuitively see that its limit as $n\to\infty$ is 0, and I can verify it with the $\varepsilon$ definition of the limit. But I am ...
2
votes
1answer
86 views

The limit of the powers of an primitive non-negative matrix over its spectral radius

Let $A$ be a non-negative primitive matrix. Then $$\lim_{n\to\infty}\left[\frac{A}{\rho(A)}\right]^n=xy^T,$$ where $x, y$ are the Perron roots of $A$ and $A^T$ respectively, they satisfy $x^Ty=1$. ...
0
votes
2answers
43 views

Is there a simpler way to rewrite this binomial chain?

Consider some binomial chain that looks like this: $$\binom{N}{k_1}\binom{N-k_1}{k_2}\binom{N-k_1-k_2}{k_3}\binom{N-k_1-k_2-k_3}{k_4} \cdots \binom{N-k_1-k_2-\cdots -k_{t-1}}{k_t}$$ Where all ...
4
votes
1answer
180 views

The spectral radius of $A$ and its transpose

Let $A$ be a non-negative irreducible $n\times n$ matrix. Then the function $$f(t)=\rho(tA+(1-t)A^T)$$ is increasing on $[0,1/2]$, and is decreasing on $[1/2,1]$. Here are the notations. $A$ is non-...
1
vote
1answer
101 views

Best approximation for a closed set in a finite dimensional normed space

First of all I'd like to mention that it is a part of my home work so I'd like if you won't give the answer itself, but try to guide me into it. I've been losing my mind for the last couple of hours ...
2
votes
1answer
117 views

Segment ordered density conjecture.

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset \overline{...
1
vote
1answer
46 views

Iterated uniform random variables

Define a sequence of r.v.'s {$X_n$}$_{n\ge 1}$ iteratively, such that $X_1\sim\text{Unif}(0,1]$ and $X_{n+1}\sim\text{Unif}(0,X_n]$. Could someone please explain why this is equivalent to: Let a ...

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