0
votes
1answer
7 views

Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
0
votes
0answers
5 views

Volume integral $\int_V y \frac{dy}{dt} + z\frac{dz}{dt} dV$

How do I solve $$\int_V y \frac{dy}{dt} + z\frac{dz}{dt} dV$$ Where $V$ is the volume of a hollow sphere. Usually I would use spherical coordinates, but I don't know how to express $\frac{dy}{dt} $ ...
0
votes
0answers
8 views

prove that matrix is diagonal by matrix rank and eigenvalue rank

$A$ is matrix $9\times9$ with rank of $5$, there is rank$(A-3I)=5$, the matrix has another eigenvalue of 5. I need to prove that $A$ is diagonal and find the similar diagonal matrix of $A$. I'm stuck, ...
0
votes
1answer
9 views

Can a point $z$ which belongs to a closed set be a limit point of an open set which is disjoint from the closed set in topological space $X$?

Say $X$ be a topological space, and $U$ and $V$ are open and closed sets respectively. Furthermore, $U$ and $V$ are disjoint. Now there is a point $z \in V$. Is it possible for the point $z$ to be a ...
1
vote
0answers
6 views

How much faster will a task be in percentage terms?

This isn't necessarily a difficult question, but it's something I'm trying to set straight. The task I have in mind is mowing my lawn. Let's say I have a 16" lawnmower. I decide I'd rather have a ...
0
votes
0answers
6 views

Information Theory - Uniquely Decipherable code problem

Hi I'm having trouble with parts bii) and c) of the following problem. For bii) I feel I might need to apply Markov's inequality but I'm really not sure. Thanks!
-1
votes
0answers
7 views

Sum of reciprocal of $p \log p$ where $p$ is a prime [duplicate]

Does $$\sum_{p\in Primes} \frac{1}{p \log p}$$ converge? Notice $$\sum_{n\in \mathbb{N}} \frac{1}{n \log^{1+\epsilon} n}$$ converges for all $\epsilon>0$, so one can wonder is the primes sparse ...
2
votes
0answers
16 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
0
votes
0answers
7 views

Generalizing the Remainder Factor Theorem

Today, I spent most of my time developing a systematic procedure for finding remainder polynomial when higher degree polynomials are divided by some polynomial of degree $\leq$ the degree of the ...
0
votes
3answers
23 views

How to evaluate $\lim\limits_{x\to1}\frac{x^{m+1}-x^{n+1}+x^n-mx+m-1}{(x-1)^2}$?

I used substitution $t=x-1 \Rightarrow t\rightarrow 0$ After this, the limit is: $$\lim\limits_{t\to0}\frac{(t+1)^{m+1}-(t+1)^{n+1}+(t+1)^n-m(t+1)+m-1}{t^2},m,n \in \mathbb{N}$$ Then I used ...
0
votes
1answer
10 views

Injective linear endomorphism of hilbert space is bijective?

Is it true that an injective continuous endomorphism of a hilbert space is bijective? If not, are there conditions that imply this? I know this would follow from the rank nullity theorem in finite ...
3
votes
0answers
10 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
0
votes
0answers
12 views

Existence of analytic function on a unit disc $\triangle$

Let $\triangle$ be the open unit disc. Then can there be analytic functions with the property (1) $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=3/4$ 2) $f(\frac{3}{4})= -\frac{3}{4}$ and ...
0
votes
1answer
10 views

Equivalence: Injective function from natural numbers to a set $X$, and injective but not surjective function from $X$ to $X$

How do I go about proving the equivalence of these statements? (1) There is an injective function $f: \mathbb{N} \rightarrow X$ (2) There is an injective but not surjective function $g:X \rightarrow ...
1
vote
1answer
5 views

Basis for row space of matrix: REF vs. RREF.

When finding a basis for the row space of a matrix, I reduce the matrix to row echelon form, and find the rows that have pivots in them. Does it matter wether you use the echelon or the reduced ...
0
votes
1answer
17 views

How to find $g(x)$, if: $f(x)=(x+1)/x$, and $f(g(x))=x$?

I know that the answer is that $g(x)=1/(x-1)$ But how to come to that answer remains a mystery to me Please give me some good advice Thanks in advance :)
1
vote
0answers
33 views

Roots of $f(x) = x^3+x^2-2x-1$

Show: $a_1=2\cos(\frac{2\pi}{7})$ is a root of $f$. [Edited here] $a_2 = a_1^2-2$ is a root of $f$. $a_3 = a_1^3-3a_1$ is a root of $f$. The first one is easy, since $e^{\frac{2\pi i}{7}}$ is a ...
-2
votes
0answers
19 views

$f'(x) \equiv 0 \pmod{p}$ with $\deg f < p$ implies $f(x) \equiv c \pmod{p}$

Let $f(x)$ be a rational function in $\mathbb{F}_{p}(x)$. Suppose for some prime $p$ we had $f'(x) \equiv 0 \pmod{p}$ and $\deg f < p$. Why must $f(x) \equiv c \pmod{p}$ for some constant $c$?
0
votes
0answers
4 views

An MCQ involving Rayleigh - Ritz method for the functional $I(y) = \int_{0}^{1}(\frac{1}{2}(y^{'})^2 - y)dx$

Let $y_\text{app}$ be polynomial approximation, involving only one coordinate function, for the functional $$I(y) = \int_0^1 \left(\frac{1}{2}(y^{'})^2 - y\right) \, dx ;\quad y(0) = 0,\ y(1) = 0$$ ...
0
votes
1answer
9 views

$A$ be a non-empty closed convex subset of a Hilbert space $H$ , is the distance from $A$ always attained at a unique point in $A$ ?

Let $A$ be a non-empty closed convex subset of a Hilbert space $H$ , then is it true that for every $b \in H$ , $\exists$ unique $x_b \in A$ such that $||x_b-b||=d(b,A)=\inf \{||b-x||:x \in A\}$ ?
0
votes
2answers
23 views

Is this definition correct for the inverse of a function?

Is this definition correct for the inverse of a function? Let $f:X\to Y$ be a function. The inverse of $f$ is the function $g:Y\to X$ such that $g\circ f=i_X$ and $f\circ g=i_Y$. We denote the ...
0
votes
0answers
17 views

Point-set topology

I am about to begin a self-study project in point-set topology. I am a final year undergraduate. I am looking for suitable resources, I have so far come across Munkres' textbook and would like to find ...
0
votes
0answers
10 views

Which statistics test to be used?

I want to run a statistics test in SPSS that tests the potential differences in two sets of data. The first set contains the following ten values: 13.7, 34.0, -68.0, 19.8, 20.9, 23.1, 22.5, 18.4, ...
1
vote
0answers
11 views

Calculating complicated expectation

I need to calculate $\operatorname{E}( X_2 \mid X_1=x, Y=y)$, where $Y=\max\{X_2,X_3\}$ and joint density of $X_1$, $X_2$ and $X_3$ is given by: ...
0
votes
1answer
5 views

Numerical Analysis: Spectral Methods

I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not ...
0
votes
1answer
20 views

Prob. 3 (b), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How does the $K$-topology on $\mathbb{R}$ differ from the usual topology?

Let $$ K \colon= \left\{\ \frac{1}{n} \ \colon \ n \in \mathbb{N} \ \right\},$$ and let the $K$-topology on $\mathbb{R}$ be the one having as basis all open intervals $(a,b)$ and all sets of the form ...
0
votes
0answers
15 views

Is group theory a generalization of number theory

The applications of group theory are abound. Many mathematical objects are examined by associating groups to them and studying the properties of the corresponding groups. But number theory and Graph ...
0
votes
0answers
11 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
1
vote
0answers
11 views

Density of linear combination

Let $r_1, \ldots, r_n$ be a set of positive reals. Define $S = \{a_1r_1+\cdots+a_nr_n : a_i\in \mathbb{N}\}$. Define $\pi(x)= |\{a\in S:a<x\}|$. Is there an asymptotic formula for $\pi(x)$ as $x ...
0
votes
0answers
5 views

Prove that the no of solutions of the equation $f(z)=w$,counted with Multiplicities for w varying in $D_2$, is constant on $D_2$

Def:A map $f:X \to Y$ is said to be proper if $f^{-1}(K)$ is compact for every compact set $K$ in Y. Domain-Open connected Set Let $D_1$ and $D_2$ be domains in $ \mathbb C$. Suppose $f:D_1 \to D_2 ...
-2
votes
0answers
18 views

Call spread derivative

delete this tbh lads, this is not the site for finance questions apparently Using the notation $V(E)$ to mean the value of a European call option with strike $E$, what can you say about ...
1
vote
1answer
9 views

A question on metric spaces which does not have Lebesgue covering property

Let $X$ be a metric space , $\{U_{\alpha}\}$ be an open cover of $X$ which has no Lebesgue number . So for every $r>0 $ , there is an open ball of radius $r$ which is not contained in any open set ...
1
vote
0answers
34 views

why this field is either $\mathbb{R}$ or $\mathbb{C}$?

If $\mathbb{K}$ is any field and is endowed with the discrete topology, then $\mathbb{K}$ is a local field (*). further, if $\mathbb{K}$ is connected, then $\mathbb{K}$ is either $\mathbb{R}$ or ...
0
votes
1answer
15 views

How many rows & columns do 1,028 equal spaces create…

I have a board that is 17.5" wide and 67" long. I need to divide this board into 1,028 equal spaces. How many rows and how many columns will this equate to?
1
vote
2answers
25 views

How to compute $\lim_{x \to 1} \frac{sin^a{\pi x}}{(1-x)^a} , a \in \Re$

I'm trying to compute this limit: $$ \lim_{x \to 1} \frac{sin^a{\pi x}}{(1-x)^a} , a \in \Re $$ It should be quite easy, however, I just can't see it now... if it would be without $a$, it would be ...
0
votes
2answers
28 views

What result multiplying 2 3D-vectors?

I understood that multiplying two vectors by cross multiplication(!) results in a third vector which is orthogonal to the two first. What does multiplying 2 3D-vectors give us as a result? What ...
-1
votes
0answers
20 views

A problem of Taylor series

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
0
votes
1answer
17 views

How does this prove that the ring homomorphism is surjective?

The course notes on rings have the below lemma Let $R$ be a ring and $I$ a two sided ideal. Define $\pi : R \rightarrow R/I$ by $\pi(r)=r+I$. Then $\pi$ is a surjective ring map and ker $\pi=I$. ...
0
votes
1answer
6 views

Upperbound on the following logarithmic function with matrix

I am trying to upperbound the expression below with a function $f$ that is a function of the identity matrix as below $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
3
votes
0answers
37 views

Proving this inequality

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}. ...
1
vote
1answer
16 views

A Problem on the Prime Counting Function $\pi(x)$

Let $\pi(x)$ denotes the number of primes less than or equal to $x$. Also suppose that for some fixed $N$ we have $\pi(x+y)\ge\pi(x)+\pi(y)$. The problem is, Show that the equality in the above ...
0
votes
0answers
8 views

How to prove the inequality on relative entropy?

Here is the definition of Relative Entropy Now I am only interested in the simplest condition that the index set is finite and discrete, as the naive probability distribution vectors. Now if the ...
0
votes
0answers
5 views

how to solve this exponential optimization problem?

Let $v_{ik}=\log ({\lambda _{ik}}) - {\lambda _{ik}}{T_k}$, $s_{ik}=\log (\frac{1}{C-\sum_{k=1}^{K}{N_k}}) - {\lambda _{ik}}{T_k}$. Consider the following optimization problem: $$ \text{minimize ...
0
votes
0answers
6 views

Showing the attractor of an IFS is either connected or totally disconnected

I came across this execise in a problem set about Iterated Function System (IFS) and fractals: "Show that the attractor of an IFS of the form {$\mathbb{R}$; ax+b, cx+d} where a,b,c and d are in ...
3
votes
2answers
23 views

Limit of cos function in a sequence

In my assignment I have to calculate to following limit. I wanted to know if my solution is correct. Your help is appreciated: $$\lim_{n \to \infty}n\cos\frac{\pi n} {n+1} $$ Here's my solution: ...
1
vote
2answers
42 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
0
votes
2answers
25 views

Find all numbers divisible by 25, that begin with 6.

please, help me solve this problem: Find all numbers divisible by 25, that begin with 6. Regards.
0
votes
0answers
4 views

Doleans measure for local martingales

I came across the following question in my textbook and something in it doesn't quite make sense to me. Let $M$ be a local $L^2$ martingale. Then $X,Y \in \mathcal{L}(M,\mathcal{P})$ are ...
2
votes
1answer
21 views

Proving a matrix $A$ is of certain form

Let $A\in M_n(\mathbb{C})$, and $A=A^3$, prove that $A^2$ is of form $\begin{pmatrix} I_r & 0\\ 0 & 0 \end{pmatrix}$ where $1\leq r\leq n$. It make sense. My initial thought was to say that ...
0
votes
1answer
14 views

Calculate contour integral

Can somebody help me with this question please or give me a hint on how to get started, as I have never seen a question with gamma like this and I have no idea how to start. Thanks

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