0
votes
0answers
5 views
0
votes
0answers
2 views

The covering radius of $C=\{0000,1111\}$

Lets say I have the code $C=\{0000,1111\}$. Why is the covering radius $\rho=2$? How is the covering radius worked out?
0
votes
0answers
5 views

Punctured plane is not simply connected

Adapt the following definition of "simply connected space" (taken from Wikipedia): A space $X$ is simply connected if it's path connected and for any continuous map $f:S^1\rightarrow X$ can be ...
0
votes
0answers
9 views

Prove that $\int_E fd\mu = \lim \int_E f_n d\mu$ for all measurable set $E$

This is problem 4T in Bartle's The elements of integration and Lebesgue measure. Suppose $f_n$ are non-negative measurable function such that $(f_n)$ converges to $f$, and that $$\int fd\mu = ...
0
votes
0answers
5 views

necessary and sufficient condition for the lines of curvature

I'm reading the book "Differential Geometry of Curves and Surfaces" of Menfredo Carmo, and this part confuses me: We have the differential equation of the lines of the curvature: ...
0
votes
0answers
7 views

Finding fifth and tenth roots of unity in rectangular form.

Exercise: Find the fifth and tenth roots of unity in algebraic form. This is an early exercise in Ahlfors Complex Analysis. What I have tried so far: For the fifth roots I have tried reducing the ...
1
vote
0answers
6 views

If $(E,\mathcal E)$ is a measurable space and $(X_t)_{t≥0}$ is a $E$-valued process, then $σ((X_{t_1},…,X_{t_n}))=σ(X_{t_1},…,X_{t_n})$

Let $(E,\mathcal E)$ be a measurable space and $(X_t)_{t\ge 0}$ be a $(E,\mathcal E)$-valued stochastic process. How can we show that $$\mathcal ...
0
votes
1answer
22 views

Prove that, if both solutions of ${ x }^{ 2 }+ax+b=0$ are even integers, then a and b are both even integers.

Prove that, if both solutions of$${ x }^{ 2 }+ax+b=0$$are even integers, then a and b are both even integers. I don't even know where to start on this problem.
0
votes
0answers
14 views

Show sums of complex $\sin$ and $\cos$ series

By considering the series $\sum_{n=0}^\infty r^ne^{in\theta}$ for $0<r<1$ show that $$\sum_{n=1^\infty}r^{n}\cos(n\theta)=\frac{1-r\cos(\theta)}{1-2r\cos(\theta)+r^2} \text{ and } ...
0
votes
0answers
13 views

Let G = {e, g1, g2, …, gn} be a finite abelian group, proof that for any x ∈ G, the product x^n g1 g2 etc = g1 g2 etc.

Let $G = {g_1, g_2, ..., g_n}$ be a finite abelian group, prove that for any $x ∈ G$, the product $xg_1 \cdot xg_2 \cdot \cdot \cdot xg_n = g_1 \cdot g_2 \cdot \cdot \cdot g_n$. I can easily see why ...
0
votes
0answers
5 views

Representation of $SL(2,\mathbb{C})$ over Grassmann algebra

I've noticed that when doing the classical Dirac field, sometimes $\psi(x)$ can be treated as a complex-valued spinor field, but when dealing with canonical or path-integral quantization, it should be ...
-1
votes
0answers
18 views

Formal proof that $\lim_{x\to \infty}\left(\frac{\ln^2(x)}{\ln(\sin x)}\right)$ doesn't exist

How can I formally prove that the following limit is nonexistent because it's periodic? $$\lim_{x\to \infty}\left(\frac{\ln^2(x)}{\ln(\sin x)}\right)$$
0
votes
0answers
5 views

Binomial Distribution and Proof Relating to Factorials

I am studying probability and statistics at my university but haven't had a solid math course in awhile(mostly forget algebra dealing with factorials)thus I am stuck with the following proof. ...
1
vote
0answers
5 views

Prove the set of subsequential limits of ($x_t$) is closed

A sequence ($x_t$) in metric space ($X,d$). Let $S$ be the set of subsequential limits of ($x_t$) ($S$ could be empty), prove that $S$ is closed. I need to prove the problem with one of the ...
5
votes
2answers
13 views

How many sewings are there in a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many ...
0
votes
0answers
6 views

Every vector in a Hilbert space has a Fourier representation wrt an orthonormal sequences?

I'm reading Kreyszig's text, and there is a Theorem in section 3.5 stating: Theorem: Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$. Then 1) If $\sum_{k=1}^\infty \alpha_k e_k$ ...
-2
votes
0answers
34 views

Prove that $\frac{z + w}{zw + 1}$ is a real number. [duplicate]

Let $z$ and $w$ be complex numbers such that $|z| = |w| = 1$ and $zw \neq -1$. Prove that $$\frac{z + w}{zw + 1}$$ is a real number.
0
votes
1answer
15 views

For which $a$ does the integral $\int_B||x||^{-a}dx$ exist, where $B:= \{ x\in \mathbb{R^2} :||x|| \leq1\} $?

For which $a$ does the integral $\int_B||x||^{-a}dx$ exist. $B:= \{ x\in \mathbb{R^2} :||x|| \leq1\} $ My solution was transforming into polar coordinates, $x=\cos(\phi)*r,y=\sin(\phi)*r$ such that ...
1
vote
1answer
18 views

Linear Algebra matrices question.

Let $A,B$ be 2 square matrices of the same size. And the following holds true $AB=A+B$ How do I prove that $(I-B)$ and $(I-A)$ are invertible
0
votes
1answer
9 views

equality of two natural transformations and two morphisms

When are two natural transformations between two functors considered the same? When are two morphisms in a category considered the same? Do we have such a notion like we have in Set theory? Thanks
2
votes
2answers
11 views

Basic Probability Question about number selected at random intervals with a modulus

This is my thought process on solving this question, but the answer sheet is not available and I am unsure about my thought process. Please correct me if I am wrong. Question: A number x is selected ...
-1
votes
0answers
17 views

Integer Factorization problem - New Idea

I been thinking about slightly different approach of solving the problem, and I want you to tell me if my idea is reasonable and if it's original(If someone already thought about this, I would be ...
-1
votes
0answers
14 views

What does a subscripted norm mean in context of functional spaces?

While studying fundamentals of Finite Elements, I encounter this notation very frequently. $$\|u\|_{H}$$ I understand that it is a norm, but what does the subscript mean? Does it have to do ...
1
vote
0answers
7 views

Processes $(X_t)_{t≥0}$ and $(Y_t)_{t≥0}$ are independent iff $(X_t)_{t∈I}$ and $(Y_t)_{t∈J}$ are independent for all finite $I,J⊆[0,∞)$

Let $\mathcal I:=\left\{I\subseteq[0,\infty):I\text{ is finite}\right\}$ and $(X_t)_{t\ge 0}$ and $(Y_t)_{t\ge 0}$ be stochastic processes. Clearly, if $X$ and $Y$ are independent, i.e. ...
0
votes
1answer
20 views

The magnitude of $5 \vec u + 4 \vec v + 2 \vec w$ where $\vec u,\vec v,\vec w$ are mutually perpendicular and of unit magnitude

Of course the answer is $\sqrt{25+16+4} = \sqrt{45}$. It is easy to see it when we consider the (extended) Pythagorean Theorem, or even more easily just taking $\vec u = \hat i, \ \vec v = \hat j$ ...
0
votes
0answers
8 views

How to quantify differences/ similarities between groups of like objects?

First let me apologize if this is not an appropriate forum for this question. Researching all the available SE sites led me here. Also, my background is not in mathematics, so layman's terms are ...
0
votes
1answer
19 views

Show that the open disk $D(a,r)$ is connected.

Let $a\in\mathbb{C}$ and $r>0$. Show that the open disk $D(a,r)=\{z\in\mathbb{C}\colon \vert z-a\vert<r\}$ is connected. The disk is connected if there exists a path between any two points ...
0
votes
0answers
14 views

Show that $g$ is analytic and discuss the properties of $g$

Let $f$ be analytic in $\overline{B}(0; R)$ with $f(0)=0$, $f'(0) \neq 0$ and $f(z) \neq 0$ for $0<|z| \leq R$. Put $\rho=\min\{|f(z)|:|z|=R\}>0$. Define $g: B(0; \rho) \rightarrow \mathbb{C}$ ...
-1
votes
2answers
32 views

Given that ${f(x)=2x^2+4x+3}$, find the range of the function [on hold]

Given that ${f(x)=2x^2+4x+3}$, find the range of the function $\frac{3}{f(x)}$.
-1
votes
0answers
8 views

Combining parameters for Douglas-Beucker Simplification

NOTE : I'm not sure if this is the right forum for this question. if not, please advice. Context : I am collecting a huge amount of data using an android app that is placed on a vehicle. I collect ...
1
vote
0answers
16 views

A simple way to solve that inner product and functional question

Let $V$ be the polynomials space over $R$ of degree less than 3 with inner product $$\langle f,g\rangle= \int^{1}_{0} f(x)g(x) dx $$ if $x \in R$, compute $g_{x}$ such that $\langle f,g_{x}\rangle ...
0
votes
0answers
10 views

Branch point at infinty condition?

Is it a necessary and sufficient condition for the function $f(z)$ to have a branch point order $n$ at $\infty$ that $f(1/z)$ has a branch point order $n$ at $0$?
0
votes
0answers
5 views

Gramian Matrix Eigenvalues--Stronger Statement than Non-Negative

I'm struggling to find conditions under which this holds: $AA^T - B \succeq 0\,.$ If it helps, A is not necessarily square and $A_{ij} \in \{-1, 0,1\}$. B is diagonal and I would like to find ...
0
votes
1answer
16 views

The probability to log on a computer from a remote terminal is 0.7.

The probability to log on a computer from a remote terminal is $0.7$. Let $X$ denote the number of attempts that must be made to gain access to the computer. Find: (a) The distribution of $X$ and ...
2
votes
0answers
24 views

What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
-1
votes
0answers
35 views

Are there chain rules for integration possible?

In "A Quotient Rule Integration by Parts Formula" and in "Quotient-Rule-Integration-by-Parts", the authors integrate the product rule of differentiation and the quotient rule of differentiation and ...
0
votes
1answer
37 views

Prove the series $\sum_n \left(n \ln(1+\frac{1}{n})-\cos(\frac{1}{\sqrt{n}})\right)$ is convergent

I would like to prove the series $\sum ( n \ln(1+\frac{1}{n})-\cos(\frac{1}{\sqrt{n}}))$ is convergent. How can this be done? I have tried the ratio test but it did not work. Thanks.
0
votes
0answers
14 views

Uniform continuity preserving measure zero

Suppose $f$ is uniformly continuous on $[a,b]$ and $E\subset [a,b]$ such that $\mu(E)=0$ (lebesgue measure) then prove that $f(E)$ is of measure zero. What i have tried so far is : As $f$ is ...
0
votes
1answer
17 views

Finding the lub and glb of set.

the question is , find lub and glb of set $S = \{ m + \frac{1}{n} \mid m,n \in \mathbb{N}\}$ my approach : clearly glb of set is $1$ because if put $m = 1$ and when $n$ tends to infinity then $m + ...
0
votes
0answers
9 views

How to derive a truth value from the following formula, one where the formula is T and one where it is F

Hi Guys I am trying to derive a structure from the formula where truth value for one structure is F and for another is T. I haven't seen a formula like this before so I am slightly confused. this is ...
1
vote
2answers
18 views

Analyticity of $\tan(z)$ and radius of convergence

Define $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$ Where is this function defined and analytic? My answer: Our function is analytic wherever it has a convergent power series. Since (I am assuming) $\sin(z)$ ...
1
vote
1answer
13 views

Euler totient function and unramified extension of $\mathbb{Q}_p$. A clarification.

I'm studying the construction of unramified extensions, and many references say that it's enough to attach the $p^n-1$ primitive root of unity to $\mathbb{Q}_p$ in order to obtain the unique degree ...
1
vote
1answer
22 views

Proving divergence of a series via Taylor Expansion:$\sum\left(\sqrt{1+\frac{(-1)^n}{\sqrt{n}}}-1\right)$

I would like to prove using a taylor expansion that the serie $\sum \ \sqrt{1+\frac{(-1)^n}{\sqrt{n}}}-1$ is divergent for $n\geq 1$. What is the expansion to prove it ? Thanks
0
votes
1answer
16 views

Algorithm to find the irreducible polynomial

What algorithm can be used find an irreducible polynomial of degree $n$ over the field $GF(p)$ for prime $p$. The reason I ask is I want to make a program for finite field arithmetic, but creating a ...
0
votes
0answers
9 views

System of rational polynomial equations with complex root also has a solution of algebraic numbers

Consider a system of equations $$f_1(x_1,...,x_k)=0,...,f_n(x_1,...,x_k)=0$$ where $f_1,...,f_n$ are polynomials in $\mathbb{Q}[x_1,...,x_k]$. Suppose the system has a solution in $\mathbb{C}^k$. ...
0
votes
3answers
25 views

A simple approximation algorithm?

I'm not sure if this method works perfect, but I have found it to work in approximating things easily, that is, you need no more than simple algebra to understand this method. Suppose you are trying ...
1
vote
1answer
57 views

Finding $\lim_{x \to 0} \frac{180\sin x}{x}$

I am in ninth grade so I am an amateur in mathematics and with no training in limits. I self derived this limit to find the value of $\pi$. I imagined a circle to be composed of infinitely small ...
0
votes
1answer
22 views

Study convergence of $f_n(x)=\frac{\sin(nx)}{\sqrt{n}}$

$$f_n(x)=\frac{\sin(nx)}{\sqrt{n}}$$ Pointwise convergence: $$\lim_{n \rightarrow \infty } \ f_n (x)=0$$ It converges to the function $f(x)\equiv0$ Uniform convergence $$f'_n(x)=n ...
0
votes
0answers
6 views

Prove that all linegraphs L(G) are claw free

For any graph G, prove that the line graph L(G) is claw-free. I have a fairly good intuition for this one but it's hard to put into words. I really need help with this one! I feel that I should use ...
0
votes
0answers
5 views

Randomized Quick Sort and Partition Probability?

We know about Quick Sort and Randomized Version and Partition. I ran into a Fact when I read my notes. Let $0 < a < 0.5$ be some constant. We have an $n$-element array as input. Randomized ...

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