3
votes
1answer
146 views

determinant of cosine coordinates

Let $n\geq3$. How to find the determinant of: $$ \begin{pmatrix} cos(\alpha_1 -\beta _1) & cos(\alpha_1 -\beta _2) & \cdots & cos(\alpha_1 -\beta _n)\\ cos(\alpha_2 -\beta _1) & ...
0
votes
0answers
51 views

the orthogonal complement of the set of all polynomials of odd degree

What is the orthogonal complement in $L^2([ -π, π])$ of the set of all polynomials of odd degree? It is the set of all functions in $L^2([-π, π])$ such that $f(x)=f(-x)$ almost everywhere. But I can ...
0
votes
2answers
108 views

If two sets (including A) are mutually exclusive, then A is the union of the sets.

In Mathematical statistics with applications, problem 2.5 they ask: Show that $(A\cap B)$ and $(A\cap \bar{B} )$ are mutually exclusive, and therefore that $A$ is the union of two mutually ...
3
votes
4answers
75 views

If $f'(z_0)\neq 0$ then $f$ is one to one on some open disk $D_r(z_0)$ [duplicate]

This is what I am trying to prove Let $D\subset\mathbb{C}$ be open and $f$ be analytic in $D$. If there is $z_0\in D$ such that $f'(z_0)\neq 0$ then there exists $D_r(z_0)\subset D$ and $f$ is ...
3
votes
4answers
942 views

Prove that limit inferior is same as limit superior for a convergent sequence

I was reading the book "Understanding Analysis" by Stephen Abbott on my own. I came across the following problem. Let $(a_n)$ be a convergent sequence. Let $y_n$=sup{$a_k:k\geq n$}. Then lim sup ...
1
vote
2answers
138 views

Simplification a trigonometric equation

$$16 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \cos\frac{8 \pi}{15} \cos\frac{14 \pi}{15}$$ $$=4\times 2 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \times2 \cos\frac{8 \pi}{15} \cos\frac{14 ...
3
votes
0answers
44 views

What are coquantales?

A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product. What are examples of non-diagonal comonoids in ...
2
votes
0answers
146 views

best book for practicing unsolved problems in differential equation and linear algebra

I started reading differential equation and linear algebra. Can anyone provide the link/book name where I may get many questions to practice. Generally, in the end of book only few problems are there. ...
4
votes
2answers
123 views

Inequality involving $\frac{\sin x}{x}$

Can anybody explain me, why the following inequality is true? $$\sum_{k=0}^{\infty} \int_{k \pi + \frac{\pi}{4}}^{(k+1)\pi-\frac{\pi}{4}} \left| \frac{\sin \xi}{\xi} \right| \, \text{d} \xi \geq ...
4
votes
0answers
104 views

$\lim_{x\to0}\frac{\sin x-x}{x^3}$ without de l'Hospital's Rule? [duplicate]

I know, how to calculate $$ \lim_{x\to0}\frac{\cos x-1}{x^2} $$ without differential calculus. Calculating $$ \lim_{x\to0}\frac{\sin x-x}{x^3} $$ using de l'Hospital's rule or Taylor expansion is also ...
5
votes
1answer
127 views

Projective resolution of $k$ over $R=k[x,y]/(xy)$

I want to prove that $\operatorname{Tor}_{n}^{R}(k,k)=k\oplus k,\,\,\forall n\ge 1$. I found the projective resolution $$ R^4\stackrel{d_2} \longrightarrow R^3\stackrel{d_1} \longrightarrow ...
1
vote
2answers
186 views

How to express the equations as the Square root Like?

$$2\sin \frac{\pi}{16}= \sqrt{2-\sqrt{2+\sqrt{2}}}$$ What law is need to be applied here? Do I have to make the $\frac{\pi}{16}$ in a form that will be give us $\sqrt{2}$ like sin 45 degree?
1
vote
1answer
76 views

Endomorphisms of a ring

Let $R$ be a ring with identity and let $R^n=P⊕P'$ be a direct sum decomposition with right $R$-modules as its components. We take $e\in\operatorname{End}(R^n_R)$ as the projection of $R^n$ onto $P$, ...
1
vote
1answer
183 views

Evaluating a surface integral using symmetry

Let $S$ denote the surface of the cylinder $x^2+y^2=4,-2\le z\le2$ and consider the surface integral $$\int_{S}(z-x^2-y^2)\,dS$$ How can one use geometry and symmetry to evaluate the integral without ...
0
votes
0answers
73 views

How to construct this modular Galois representation?

I want to know how to construct this modular Galois representation: The $\bmod p$ representation is semi-simple. For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup ...
1
vote
1answer
90 views

How to find $\max|f(z)|$ in complex analysis?

The $M-L$ estimation lemma inequality states: $$\left |\int_\Gamma f(z) dz\right| < ML(\Gamma)$$ Where $M = \max|f(z)|$ and $L(\Gamma)$ is the arc length of $\Gamma$. Here: Wikipedia: ...
7
votes
2answers
164 views

Problem Solving Question With Polynomials

For any polynomial $p$ with real coefficients, let $$ S(p):= \{x\in \mathbb{R} \mid p(x) \in \mathbb{Z}\} $$ Prove that if $p$, $q$ are two polynomials such that $S(p) = S(q)$, then either ...
3
votes
1answer
154 views

Sufficient condition for $M$ to have constant curvature

I decided to keep my original question. However, I'm having trouble only in a part of it (check NOTE) Let's consider a Riemannian manifold $(M,g)$, with the Levi-Civita connection $\nabla$. I would ...
0
votes
1answer
39 views

Subgroups isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5} $

Let $A=\mathbb{Z}_{360}\oplus\mathbb{Z}_{150}\oplus\mathbb{Z}_{75}\oplus\mathbb{Z}_{3}$. I need to calculate the number of subgroups of $A$ which is isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$ ...
0
votes
1answer
63 views

Proof of the Curtis-Hedlund Theorem: Why is there a function $\mu\colon A^S\to A$ such that $\tau(x)(1_G)=\mu(x_{|S})$ for all $x\in A^G$?

Here is the Curtis-Hedlund Theorem and its proof [the sets $V(\cdot,\cdot)$ used in this proof are explained below.]: My problem is I am not sure that I have understand that correctly. So I ...
1
vote
1answer
180 views

How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
30
votes
1answer
1k views

Xmas Combinatorics 2014

Inspired the various** algebraic X'mas greetings sent to me over the festive period, I thought I would try to devise one of my own. $$\Large ...
2
votes
1answer
448 views

Nilpotent Matrices properties

Let $N \in M_n(\mathbb{F})$ be nilpotent. Prove that for any $1 \leq k \in \mathbb{N}$ a matrix $B \in M_n(\mathbb{F})$ exists such that $B^k=I+N$ I have no idea how to het started here. We've just ...
2
votes
1answer
78 views

What is the algebraic role of the mathematical constant $\gamma$?

Mathematical constants $\pi$, $e$, $i$ have a lot of algebraic roles. They appear as identity elements, idempotents, invariant elements etc against various operations and sets. This is illustrated by ...
1
vote
1answer
47 views

Prove that $L(Im(L))=Im(L)$

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$. Let $L:V\rightarrow V$ be a linear mapping such that $Im(L)\cap Ker(L)=\{0\}$ Show that $L(Im(L))=Im(L)$ I know how to prove ...
3
votes
0answers
71 views

Homological algebra and Grothendieck Topologies

I have recently became familiar with the theory of Grothendieck Topologies and Cech cohomology for sheaves over a site. It seems that many of homological concepts in algebra, can be formulated in ...
2
votes
1answer
69 views

Build a bijection $f: \mathbb{Q} \to \mathbb{Q}\setminus[0,1].$

Build a bijection $f: \mathbb{Q} \to \mathbb{Q}\setminus[0,1].$ What about $f(x)=x+1$ if $x>0$ and $f(x)=x-1$ for $x<0?$
2
votes
1answer
68 views

$P$ is normal complement in $G$ if and only if $H$ has normal subgroup $Q$ which $H=P\times Q$.

suppose that $G$ is finite group and $P$ is aabelian $p$-sylow subgroup of $G$ and $H=N_{G}(P)$. show that $P$ is normal complement in $G$ if and only if $H$ has normal subgroup $Q$ which $H=P\times ...
3
votes
2answers
133 views

Replacing $\sin(z)$ with $1 - e^{2iz}$

I have seen many integral evaluations within logs where they change the sine to: $$\sin(z) \rightarrow 1 - e^{2iz}$$ Such as here: Contour integral evaluation. I dont understand how those ...
2
votes
2answers
69 views

Build a bijection $\mathbb{R} \to \mathbb{R}\setminus \mathbb{N}$ [duplicate]

Build a bijection $f: \mathbb{R} \to \mathbb{R}\setminus \mathbb{N}$. What about $f(x)=\pi \cdot x?$
0
votes
1answer
34 views

Absolute value of a complex number

Is it valid :$$ abs(a+ib)/abs(c+id)=abs[(a+ib)/(c+id)]$$ ? and also $$abs(a+ib)^m*abs(c+id)^n=abs[(a+ib)^m*(c+id)^n]$$ when i stands for imaginary number. I were trying to multiply by complex ...
4
votes
2answers
86 views

Cyclic inequality for $n$ numbers

$a,b,c,x_1,x_2,x_3,...,x_n>0, a+b+c=1,\displaystyle \prod_{i=1}^n x_i=1 $ . Prove that $$(ax_1^2+bx_1+c)...(ax_n^2+bx_n+c)\geq1$$. I've tried just writing out as a product using the product sign ...
2
votes
1answer
90 views

Path properties of Brownian Motion: relation between its maximum and hitting time

Let $B(t)$ be a Brownian motion. $$T_a=\inf\{t>0,B(t)=a\}$$ $$M(t)=\max_{0\le s\le t} B(s)$$ There is a statement in Durrett's textbook (3rd last line in page 318, 4th edition): ...
-1
votes
1answer
246 views

20% of X is Y, then X is? [closed]

My math is trouble ! Question: Number of 1030 is 20% submission of of a number. what is that number ? That number is bigger that 1030. Let me explain in programming language: 1030 = 20% (X) - X , ...
2
votes
2answers
3k views

Nspire cx CAS - Laplace inverse fails

I'm trying to calculate that easy integral but I get undef. When I replaced $\infty$ with $1000$, I got the right answer. ($e^{-1000}$ is zero roughly). Although this calculator knows that ...
2
votes
0answers
174 views

Equipartition of energy

Let $u$ solve the initial-value problem or the wave equation in one dimension: $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = h & \text{on } ...
0
votes
0answers
35 views

Finding a function in a differential equation

$$\left [ (m+1)f(x)+\frac{\mathrm{d}(\xi(x) f(x))}{\mathrm{d}x} \right ]y^{m}+\left [ (n+1)g(x)+\frac{\mathrm{d}(\xi(x) g(x))}{\mathrm{d}x} \right ]y^{n}=0$$ Find $\xi(x)$ I tried to do the ...
5
votes
1answer
56 views

What is the connection with quadratic map

While reading Prof. Tao's Wordpress blog. I noticed he mentioned a different function $\displaystyle\Lambda_2(n):= \sum_{d|n}\mu(d)\log^2(n/d)\ldots(\ast)$ and said that this function vanishes ...
3
votes
3answers
94 views

Number of Solutions of $y^2-6y+2x^2+8x=367$? [closed]

Find the number of solutions in integers to the equation $$y^2-6y+2x^2+8x=367$$ How should I go about solving this? Thanks!
1
vote
1answer
125 views

Infinite number of poles in a domain of a meromorphic function?

Let $U\in \mathbb{C}$ be an open set and $f$ be meromorphic. Let $\gamma$ be a simple closed path in $U$ and let $D$ be the interior of $\gamma$ in $U$. Also suppose there are no zeros on the trace of ...
0
votes
2answers
74 views

Do all the properties of exponents work for every real exponent? [closed]

I am writing an essay, but i doubt at typing that the properties of exponents works for every Real, I know it works for every Rational number. $$\forall a,n,m\in\mathbb R$$ or $$\forall a\in\mathbb ...
3
votes
0answers
78 views

mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
0
votes
1answer
36 views

Finding the residue of the improper integral $\frac{1}{z^4+4}$

$$f(z) = \frac{1}{z^4+4}$$ the roots of this are: $z^2=\pm i\sqrt{2} \implies z=\pm\sqrt{i\sqrt{2}}$ and $z=\pm i\sqrt{i\sqrt{2}}$ i.e. $$f(z) = \frac{1}{(z\pm\sqrt{i\sqrt{2}})(\pm ...
3
votes
2answers
157 views

Need help with proof of $SO(3)$ is path connected

I am working on an exercise in Tapp's matrix groups for undergraduates. It is a proof that $SO(3)$ is path-connected. $SO(3)$ is the group $$ SO(3) = \{A \in O(n)\mid \det A = 1 \}$$ where $O(n)$ is ...
1
vote
1answer
160 views

Maximum and minimum average question

During a cricket match, India playing against NZ scored in the following manner: Partnership | Runs scored $1$st wicket$~~$ | $112$ $2$nd wicket | $58$ $3$rd wicket$~$ | $72$ $4$th wicket$~$ | $9$ ...
1
vote
1answer
95 views

Strange Sigma Notation

How do I interpret this form of sigma notation? Do e1 and e2 take on all combinations of 1 and -1? If they do, what's the point? They just get multiplied inside the sum! FYI, this comes from equation ...
0
votes
0answers
171 views

Identifying the types of singularities, poles and branch points on different functions

Someone please confirm that what i'm doing below is correct, -Thanks. $$sin(1-z^{-1}) \tag{1}$$ $sin(1-z^{-1})=\frac{z-1}{z}-\frac{(z-1)^3}{3!z^3}+\frac{(z-1)^5}{5!z^5}$ -has only an essential ...
1
vote
2answers
191 views

Stokes' Rule for an initial-value problem

Assume $u$ solves the initial-value problem $$\begin{cases}u_{tt}-\Delta u = 0 & \text{in } \mathbb{R}^n \times (0,\infty) \\ u = 0, u_t = h & \text{on }\mathbb{R}^n \times \{t=0\}. ...
5
votes
1answer
87 views

Evaluate $\int_{-2}^{-1}\frac{\text{d}x}{\sqrt{-x^2-6x}}$.

Problem statement [from Charlie Marshak's Math GRE Prep Problems]: Evaluate $\displaystyle \int\limits_{-2}^{-1}\dfrac{\text{d}x}{\sqrt{-x^2-6x}}$. My work: notice that $$\begin{align} -x^2-6x ...
0
votes
1answer
63 views

Why do we have $u_n=\bigl(\frac{n}{n+1}\bigr)^{n^2}=\exp(-n+o(n) )$

Why do we have $u_n=\left(\dfrac{n}{n+1}\right)^{n^2}=\exp\left(-n+o\left(n\right) \right)$ My attempts : $\ln\left(1+\dfrac 1n\right)=\dfrac 1n-\dfrac1{2n^2}+o\left(\dfrac{1}{n^2}\right)$ but I ...

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