0
votes
1answer
67 views

Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$

I need your help: Calculate the Radius of convergence of the following: $$ \sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$ Im new to this subject, so I'd appreciate it if you can add explanations to ...
4
votes
2answers
179 views

Does $f_{n}\left(x\right)=x\arctan\left(nx\right)$ converge uniformly?

Does $f_{n}\left(x\right)=x\arctan\left(nx\right)$ converge uniformly when $x\in\left(0,\infty\right)$? I know that the limit function is $f\left(x\right)=\frac{\pi}{2}x$. I think it does converge ...
5
votes
4answers
90 views

Multiplying the long polynomials for $e^x$ and $e^y$ does not give me the long polynomial for $e^{x+y}$

As an alternative to normal rules for powers giving $e^xe^y=e^{(x+y)}$ I am multiplying the long polynomial of the taylor series of $e^x$ and $e^y$. I only take the first three terms: $$ ...
4
votes
2answers
47 views

Unsure about changing order of integration

I want to change the order of integration of the following $$\int_{1}^{+ \infty} \left (\int_{1}^{\sqrt{y}} x^3e^{-xy} dx \right) dy.$$ I get the bounds $1 \le y \le x^2$ and $1 \le x \le \infty$, ...
3
votes
0answers
57 views

About Jordan volume

I have a multivariate calculus homework, and I don't understand completely the definition of Jordan volume. Here's my question: Be $R \subset \mathbb{R}^2$ with Jordan area and for each $h>0$ be ...
2
votes
1answer
76 views

What is an equation for cosine function graphed below?

my answer ? Please correct me if am wrong and explain your reasoning $y\,\,\, = \,\,\,5\cos {3 \over 2}\left( {x\,\, + \,\,{\pi \over 3}} \right)\,\, - \,\,2$
4
votes
1answer
149 views

Integral formula for $\frac{1}{\Gamma(z)}$

Let $c>0$. How to prove that for any complex number $z$, $$\frac{1}{\Gamma(z)}=\frac{1}{2\pi}\int_{-\infty}^\infty (c+it)^{-z}e^{c+it}\,dt?$$ where $\Gamma(z)$ is the Gamma function.
4
votes
2answers
95 views

Why can't you find all antiderivatives by integrating a power series?

if $f(x) = \sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n$ why can't you do the following to find a general solution $F(x) \equiv \int f(x)dx$ $F(x) = \int ...
1
vote
1answer
32 views

Proper Notation for Predicate Variable Not In Universe?

Let's say I have the simple statement Q(x): x + 1 > 2x. For a universe of all integers, I can easily compute any truth value. If my universe of discourse changes to, say, x | x < 1, what is the ...
1
vote
0answers
138 views

Confusion related to optimization of log(det(X)) function

I have this confusion related to optimization of the log(det(X)) function. I didn't get how it implicitly maintains the constraint of X being positive definite. For eg if I have a matrix ...
2
votes
0answers
31 views

Does the sum over the normed differential of the prime power function equal $2\log2\pi$?

Let $p\in \Bbb P$ a prime and a prime power function: $$\xi_p(x) = p^x$$ with $x \in \Bbb R^+_0$ hence: $$\xi'_p = \frac{d}{dx}\xi_p=\xi_p \log p$$ Taking into account E. Muñoz García and R. ...
9
votes
1answer
793 views

Structure of p-adic units

I am trying to understand the structure of the $p$-adic units. I know that we can write $$\mathbb{Z}_p^\times \cong \mu_{p-1} \times 1 + p\mathbb{Z}_p,$$ where $\mu_n$ are the $n$th roots of unity in ...
1
vote
0answers
25 views

Problem on reflections - clarification needed

Suppose that $R_1,...,R_p$, $S_1, . . . , S_q$ are all reflections across planes that contain $\bf0$. Show that if $R_1 · · · R_p = S_1 · · · S_q$ (denoting composition of reflections) then $(−1)^p = ...
2
votes
2answers
81 views

If $X=[x_{ij}]_{n \times n}$ then how prove $X^n=0$

Let $n\in \mathbb N$ and $A_1,A_2,..,A_n$ be arbitrary sets. Now define $X=[x_{ij}]_{n \times n}$ where $$x_{ij}= \begin{cases} 1 , & \text{$A_i$$\subsetneq$}A_j \\ 0 , & \text{otherwise} \\ ...
-4
votes
1answer
140 views

How to find $p$ when $ ({\frac{1}{2}})^p + ({\frac{1}{4}})^p + ({\frac{1}{8}})^p - 1 = 0. $ [duplicate]

Kindly mention solution-techniques along with solution
0
votes
1answer
91 views

How to get $e^{\sqrt{\log (x)}} \leq e^{log(x)}=x \leq x^n$?

Hi i was browsing through various asynptotic questions and got stuck in the mid due to the following daubt in the answer given in the link: Prove that $e^{\sqrt{\log x }}=O(x^n)$. How beni got: ...
3
votes
2answers
132 views

Two equal functions on a topological space

Can anybody help please help me, I have to answer this problem in topology: "Let $f$ and $g$ be continuous functions from the topological space $T$ into $\mathbb{R}$, with the usual topology. Show ...
1
vote
1answer
45 views

Orthogonality on a Matrix Ring

" A real square matrix is orthogonal if and only if its columns form an orthonormal basis of $\mathbb{R}^n$." Im looking for a generalization of that fact in Matrix Rings. If $A \in M_n(R)$ is matrix ...
2
votes
2answers
255 views

Subsequential Limits

I'm working through Rudin's PoMA at the moment, and I've been learning about subsequential limits. However, I'm somewhat confused and I have a question, which is more conceptual than an actual ...
0
votes
1answer
40 views

Expansion of $(z-1)^2 / (z-2)(z^2+1)$ in $z = 2$

I have to expand $$ f(z) := \frac { (z-1)^2 }{(z-2)(z^2+1)} $$ around $z = 2$. I wanted to write $f(z) = \frac 1 {z-2} g(z)$ and then expand $g(z)$. Some hints would help a lot :)
0
votes
1answer
96 views

Entropy Inequality

I have very hard time to prove the following inequality or to show a contradiction. $H(X_1,X_2,X_3) + H(X_1,X_2,X_4)+ H(X_1,X_3,X_4) + H(X_2,X_3,X_4) \leq 3(H(X_1,X_2) + H(X_3,X_4))$ The problem ...
2
votes
1answer
64 views

How to solve differential equation of form $y'+\frac{p(x)}{y}=q(x)$

How can I go about solving a differential equation of the form $y'+\frac{p(x)}{y}=q(x)$. Thanks in advance.
2
votes
3answers
99 views

About a condition for a continuous mapping to be open.

The text (Foundations of General Topology, by Pervin, Second edition) says a (continuous) mapping $f$ of $X$ into $X^*$ is open iff $f(i(E))\subseteq i^*(f(E))$ for every $E\subseteq X$. EDIT: ...
5
votes
2answers
86 views

When does $A^p$ is a diagonal $2\times 2$ matrix imply that $A$ is a diagonal matrix

Let $A \in\mathrm{SL}(2,\mathbb{C})$ and $p > 1$ be a natural number. Under which conditions the following statement is true? $A^p$ is a diagonal matrix implies $A$ is a diagonal matrix For ...
1
vote
3answers
70 views

homomorphisms induce isomorphism

Assume that we have the following commutative diagram with groups and homomorphisms where $b$ and $c$ are injective homomorphisms $\begin{array}[c]{ccc} A&\stackrel{a}{\rightarrow}&B\\ ...
1
vote
2answers
473 views

Prove that $U$ is a self adjoint unitary operator

Let $W$ be the finite dimensional subspace of an inner product space $V$ and $V=W\oplus W^\perp $. Define $U:V \rightarrow V$ by $U(v_1+v_2)=v_1-v_2$ where $v_1\in W$ and $v_2 \in W^\perp$. Prove that ...
20
votes
0answers
607 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
62
votes
26answers
4k views

Is there a great mathematical example for a 12-year-old?

I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets. Why? Because the maths department at her school is outrageously good, and set ...
2
votes
0answers
100 views

First time dealing with limits with complex numbers in it.

I am solving the following problem. Investigate the behavior (convergence of divergence) of $\Sigma a_n$ if $$a_n = \frac{1}{1+z^n}, \quad \text{ for } z \in \Bbb C.$$ First of all, I am ...
1
vote
2answers
43 views

Probability question help me please?

A company produces monitors.30 % of the monitors do not work.What is the probability that in a box with 15 monitors. a) 3 monitors dont work b) at least 6 monitors dont work c) less than 10 monitors ...
-1
votes
2answers
98 views

Which of the following is an equation for the graph shown?

Which of the following is an equation for the graph shown? I think the answer is c for because I think we know hte ampiltude is 3 and and it has 4 periods? please correct me if I'am wrong.
0
votes
1answer
57 views

Group theory, function from group to another

I have two groups A) 0 1 2 3 4 5 6 A B C D E F G B) 0 1 2 3 4 5 6 G A B C D E F What is the function to map from A to B? so that ...
0
votes
2answers
1k views

The graph of a periodic function is shown below. Determine the amplitude.

The graph of a periodic function is shown below. Determine the amplitude. I think the answer is A. If i'am wrong could explain why and your reasoning to the correct answers thanks david.
1
vote
1answer
32 views

If $\mid \lambda_i\mid=1$ and $\mu_i^2=\lambda_i$, then $\mid \mu_i\mid=1$?

If $|\lambda_i|=1$ and $\mu_i^2=\lambda_i$, then $|\mu_i|=1$? $|\mu_i|=|\sqrt\lambda_i|=\sqrt |\lambda_i|=1$. Is that possible?
3
votes
2answers
102 views

What stops me from making this conclusion?

Suppose I want to find $\sin^6x+\cos^6x$. What stops me from saying that $\sin^2t=\sin^6x$, and $\cos^2t=\cos^6x$? Of course this is wrong because $\sin^2t+\cos^2t=1$ and $\sin^6x+\cos^6x$ does not ...
2
votes
2answers
215 views

Definition of $C^k$ boundary

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition. I need this 'cause I'm not understanding what the Evan's book said. ...
11
votes
3answers
372 views

Irrationality of $\sqrt 2$ using induction

I came upon this exercise in a textbook. I know that $\frac{n}{b} \ne \sqrt{2} $ for all $b \gt 0$ and $n \le N_0$. How can I then show that $\frac{N_0 + 1}{b} \ne \sqrt{2}$ for all $b \gt 0$?
4
votes
2answers
460 views

When an infinite union of countable sets is uncountable?

If I have $\kappa$ countable sets, when their union is not countable? only if $\kappa$ is uncountable? Using AC make differences?
3
votes
5answers
208 views

Difficulties with partial integration

I have asked several questions on the site regarding this topic already, but I can't seem to grasp this at all. Consider the following example: $$ h(x) = e^{2x} \sin x$$ We have to find the ...
2
votes
1answer
86 views

A small question about $M/\partial M$, where $M$ is a manifold.

I have a question about topology. Suppose $M$ is a space with boundary $\partial M$, is that possible to obtain a space with the quotient topology $M/\partial M$? If this is possible, does ...
6
votes
1answer
295 views

equilateral triangle and trisection

This is problem 1.9.5 from Geometry Revisited by Coxeter and Greitzer: If two lines through one vertex of an equilateral triangle divide the semicircle drawn outward on the opposite side into three ...
2
votes
1answer
123 views

Infinite Fibonacci word is not periodic

How can one prove that the infinite Fibonacci word is not periodic? PS. Maybe, beginning from some position.
9
votes
2answers
397 views

weak* separable question

(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
2
votes
0answers
75 views

T'wo level full factorial design question

I need to prove the following identities for a factorial design experiment of the form $2^k$. $\overline{Y}(AB+)-\overline{Y}(AB-)=0.5$[$A(B+)-A(B-)$] $A(B+)=A+AB$ $A(B-)=A-AB$ ...
3
votes
3answers
257 views

Simplifying a quotient of complex numbers

Given the equation I am supposed to simplify : $$\frac{(7 - 4i)}{(5 + 3i)}$$ I conclude that I should first multiply both the numerator and denominator by $(5 - 3i)$ (note : or by $7 + 4i$ but ...
4
votes
1answer
590 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq ...
10
votes
2answers
3k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
2
votes
2answers
253 views

Determine the general solution for $2\cos 2x−5\cos x+2=0$

Determine the general solution for $2\cos 2x−5\cos x +2=0$ my answer I got was : $1.05+n\pi, 4.19+n\pi$
1
vote
0answers
38 views

System of second order lineal differential equations

I have the following system: $x'' = \alpha^2 y - x $ $y'' = x- y $ I have no idea how can I start. Please give me some hint.
3
votes
2answers
122 views

Change of a variable in a generating function

Assuming I have a generating function $$\sum_n c(m,n,k)x^n=\left(x\frac{1-x^m}{1-x~~~}\right)^k$$ (mentioned in this answer where $c$ represents the number of compositions of $n$ to $k$ parts of ...

15 30 50 per page