0
votes
0answers
4 views

What function does the power series $\sum_{n=0}^{\infty}\frac{z^n}{n^2}$ converge to in its disc of convergence?

For the power series $\sum_{n=0}^{\infty}\frac{z^n}{n^2}$, its radius of convergence is 1 which implies that this series is absolutely convergent in the the unit ball $\{z:|z|<1\}$. Since it is ...
0
votes
0answers
5 views

determining the limits in $u$-substitution

I have a fundamental question regarding $u$-substitution: Suppose that want to calculate the integral $$ \int_{a}^{b}f(x)\,dx $$ and we wish to use a $u$-substitution of the form $x=g(u)$ (rather ...
0
votes
0answers
10 views

Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then ...
0
votes
0answers
4 views

If a subset S of a vector space, V spans V then there exists a subset of S that also spans V. Prove?

Additional related question: Can span of a subset, S of a vector space, V ever be a superset of V. Answer is No! Because V will no longer be a vector space then as V will not be closed under vector ...
0
votes
0answers
4 views

Find $\int_{C}{\bf{F}}\cdot d{\bf{s}}$ through the line segment

Let $F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right].$ Let $C$ by the curve consisting of the line segments from $$(-1,0)\to (0,-2)\to (2,0)\to (3,4)\to (0,5)\to (-1,0)$$ Find ...
-3
votes
2answers
20 views

Factoring $12e^{2x} - 32e^x + 16$

Can you help me solve the quadratic equation $12e^{2x}-32e^x+16$ by factoring please?
0
votes
0answers
4 views

Ionofs Problem Solving

This was a practice question given to me but I can't seem to find the answer please help! Mostly need help with b,c,d. The Ionof (Integer on number of factors) of an integer is the integer divided by ...
0
votes
3answers
10 views

State the domain of $f^{-1}$

$$f(x)=\sqrt{2x+5}$$ $$x \geq -2.5$$ State the domain of $f^{-1}$ \begin{align} \ x & = \sqrt{2y+5} \\ \ \Rightarrow f^{-1}(x) & = \frac{x^2-5}{2} \\ \end{align} The Mark Scheme says that ...
0
votes
0answers
3 views

Determining the degree of a root of unity over a cyclotomic expansion

For $\xi_{n} = e^{2\pi*i/n}$ , determine the degree of $\xi_{7}$ over the field $\Bbb{Q}(\xi_{3})$ How would I approach this problem? I'm having trouble starting this problem and can't find any ...
0
votes
0answers
10 views

Prove that the set of matrices $\{I,A,A^2,\ldots A^m\}$ is linearly independent.

Let $A$ denote the adjacency matrix of a connected graph $G$ with $n$ vertices and $e$ edges.If $i $ and $j$ are vertices of $G$ with $d(i,j)=m$. Then prove that the set of matrices ...
1
vote
0answers
12 views

Determining parity of a number

I have this function: $$f(n) = \frac{-1^n + 1}{2}$$ For $n \in Z$ It seems be equal to 1 if $n$ is an even number and 0 otherwise: $$ \begin{array}{c|c} n & -3 & -2 & -1 & 0 & 1 ...
0
votes
1answer
6 views

Which is the correct explanation? $family ⊇ set$ or $family \Leftrightarrow $a set whose elements are sets themselves

I'm not sure about the concept of family. One book explains it as a broader concept containing set,e.g. {a, a, a}={a} another explains family as a set whose elements are sets themselves. Indexed ...
0
votes
0answers
7 views

$\int(\int\phi(a-z)dz)dz=\Phi(a-z)$

Lets assume $\phi(a-z)$ is integrable. Can I conclude that the following integral $$\int\left(\int\phi(a-z)dz\right)dz$$ Can be expressed by a function $$\Phi(a-z).$$ So in result: ...
1
vote
1answer
12 views

Dividing by something Undefined

I was thinking about trigonometry ratios, in particularly $\cot(\theta)$, which can be defined as $\cot(\theta) = \frac {1}{\tan(\theta)} = \frac {cos(\theta)}{sin(\theta)}$. Though $\tan(90)$ is not ...
0
votes
0answers
11 views

Are eigenvalues (resp. unit eigenvectors) dependent continuously on elements $a_{ij}$ of a symmetric matrix $A$?

Let $A(t)=(a_{ij}(t)),~(t\in \mathbb R)$ is a symmetric matrix such that $a_{ij}(t)=a_{ji}(t)$ is a real-valued continuous function. Let $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ is all of the ...
0
votes
0answers
7 views

Non-rational G-modules

Let me recall the definition of a rational $G$-module from M. Brions notes Introduction to actions of algebraic groups (Def. 1.6) Let $G$ be an affine group scheme over $\mathbb{C}$. A rational ...
1
vote
0answers
6 views

Why Fibonacci(prime-1) or Fibonacci(prime+1) is divisible by that prime?

Why Fibonacci(prime-1) or Fibonacci(prime+1) is divisible by that prime and Fibonacci(nonprime-1) or Fibonacci(nonprime+1) is not divisible by that nonprime? Is there any elegant proof of that?
0
votes
0answers
3 views

AR(2) process covariance stationarity - what am I doing wrong?

Say I specify an AR(2) process as $X_t = 0.5 + 0.7X_{t-1} + 0.4X_{t-2} + e_t$. I would not expect this process to be covariance stationary. Indeed, if I project this series stochastically, it grows ...
0
votes
0answers
9 views

Trace of the product of two matrices is 0. Prove one matrix is the 0 matrix.

In a paper I'm reading, we're given three $n \times n$ matrices, $B,Y,S$ where $S$ is skew hermitian. Given: $tr([B,Y]S) = 0$, the paper concludes that $[B,Y] = 0$, but gives no proof. How do I prove ...
0
votes
0answers
8 views

How do you define the restriction of a sheaf?

Just to be clear with the notations: Recall that the pullback of $\mathcal{F}\in\mathcal{O}_B\text{-Mod}$ via $f:A\rightarrow B$ (morphism of schemes) is defined as ...
0
votes
0answers
4 views

Chain map with one-sided inverse between isomorphic chain complexes quasi

Suppose that $C_\bullet$ and $D_\bullet$ are isomorphic chain complexes. Let $f:C_\bullet\rightarrow D_\bullet$ a chain map with a one-sided inverse. Is it true that f is a quasi-isomorphism?
1
vote
0answers
5 views

Method of mirror charges applied to diffusion equation

The equation $\frac{\partial f}{\partial t} = \frac{\partial^2f}{\partial x^2}$ has the fundamental solution (in one dimension) $f(x,t) = \frac{1}{2\sqrt{t}}\exp (-x^2/4t)$ if there are no boundary ...
0
votes
1answer
8 views

Natural Deduction Proof (c ∧ n) → t, h ∧ ¬s, h ∧ ¬(s ∨ c) → p |− (n ∧ ¬t) → p

I'm trying to do a question from Huth and Ryan's book 'Logic in Computer Science' and I am stuck on the following natural deduction proof: prove by natural deduction that the sequent (c ∧ n) → t, h ...
0
votes
2answers
19 views

I don't understand the algorithm for solving equations of the form $x^n \equiv 1 \mod m$

Given a congruential equation of the form $x^n \equiv 1 \mod m$, according to my course notes all I have to do is to find a primitive root $a \mod m$ and then the solutions to the equation are of the ...
0
votes
0answers
9 views

Prove: $\bar{a}^2 = \bar{0}$ in $\mathbb{Z}_n \rightarrow \bar{a}=0$

For this summer, I am teaching myself abstract algebra and I've been working on a proof for the following statement. I just need someone to confirm whether it is sound. (Note: Here, $\bar{a}$ denotes ...
0
votes
0answers
14 views

Help in this proof of the argument principle

I'm reading Conway's complex analysis book and on page 123 he made the following comment: Afterwards in order to prove the argument principle he said that we can get the following equality from ...
1
vote
1answer
7 views

Finding the Centre of an Abritary Set of Points in Two Dimensions

I am currently working on a program that needs to transform one set of coordinates by shifting them to the center of the screen. The points are offset from the middle of the screen - either to the ...
-4
votes
0answers
22 views

How do I solve these questions?

A person receives Rs 6,000 per month salary.If his monthly salary is incremented by Rs 300 every year,what amount he would receive in 30 years? a 43,10,000 b 37,26,000 c 23,10,000 d 52,92,000 Q2 ...
-4
votes
1answer
18 views

I want to know the difference between Greatest and least element in Maths [on hold]

Please diggerentiate between Maximal/Minimal element and Greatest and least element in Discrete Mathematics.. Please help me
0
votes
0answers
11 views

mean-deviation form, why orthogonal?

This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...
1
vote
0answers
13 views

is the real root of a higher order equation continuous with the parameters?

I come across one equation that $a_{0}x^{2}+x(a_{1}\sqrt{f(x)}+a_{2})+a_{3}\sqrt{f(x)}+a_{4}=0$, in which $f(x)=b_{2}x^{2}+b_{1}x+b_{0}$ and $b_{1}^{2}-4b_{2}b_{0}\leq 0$ and $b_{2}>0$. I cannot ...
0
votes
0answers
2 views

$0<h(y)\le c_1|y|+c_2$ implies that: $y'=g(x)h(y)$ with $y(x_0)=y_0$ has unique solution on $\mathbb{R}$

How would I prove the above statement? $c_1$ and $c_2$ are constants bigger than zero and $g,h$ are functions $\mathbb{R}\rightarrow\mathbb{R}$, while the condition applies for all $y\in\mathbb{R}$.
0
votes
0answers
5 views

Bandwidth from a transfer function

Currently I'm having problem wrapping my head around the following. Suppose you have a dynamical system described by the transfer function $$ G(s)=\frac{as}{(s+b)(s+c)} $$ depending on the variables ...
0
votes
0answers
11 views

Continuity of Holder functions

If a function taking values in $\mathbb{R}^n$ is $\alpha$-Holder continuous along lines parallel to the axes (uniformly on a compact set), is it continuous?
3
votes
0answers
22 views

Restriction to equivalence relation is equivalence relation

Let $\mathcal{R}$ be relation on $A$ and $A_0 \subseteq A$. The $\mathbf{restriction}$ of $\mathcal{R}$ to $A_0$ is defined to be the relation $\mathcal{R} \cap (A_0 \times A_0) $. ...
1
vote
1answer
32 views

Integers tables $6\times6$ and $7\times7$

Can fill integers table $6\times 6$ so that the sum of all the numbers in each square $3\times3$ equal $2016$, and the sum of all the numbers in each square table $5\times5$ equal $2015$? ...
0
votes
1answer
13 views

finite spectrum eigenvalue

Let $T:X \to X$ be a linear bounded operator where X is Banach space ,and $\sigma (T)$ is a finite set.Then does the spectrum consist of eigenvalues only? Any hint or counterexample is ok. thanks in ...
1
vote
0answers
18 views

Number theory proof denumerable sets

Assume |X| = n ∈ N≥2 and Y is a denumerable set. Prove $Y^{|X|}$ is denumerable. Recall, $Y^{|X|}$ is the set of all functions from X to Y Approach By definition, I have to find a bijection from the ...
1
vote
1answer
10 views

How do I rearrange an adjacency matrix of an acyclic digraph so its non-zero elements are above the diagonal?

Any graph can be represented by an adjacency matrix. The matrix for an acyclic digraph can be represented as a matrix with all its non-zero elements above the diagonal. However, if I were to take an ...
0
votes
0answers
7 views

Find change in gross profit after price change

How do you solve this question? A company has 3 products. They contribute to 30%, 30% and 40% of sales respectively. They have profit margins of 15%, 30%, and 50% respectively. If the ...
0
votes
1answer
27 views

Show $f$ is Riemann integrable on $R$.

Let $R=[a_1,b_1]\times\cdots\times[a_n,b_n]\subset \mathbb{R}^n$. Let $f:R\to\mathbb{R}$ be a continuous function. I want to show that $f$ is Riemann integrable on $R$. I know that Riemann integrable ...
3
votes
0answers
16 views

Determinant of a large block matrix

$\newcommand{\lmt}{\left[\begin{matrix}}$ $\newcommand{\rmt}{\end{matrix}\right]}$ Hi, I was reading through a proof of the number of domino tilings of a $(2n)\times(2n)$ chessboard, and somewhere ...
1
vote
1answer
8 views

Relationship between subset medians and the median

Suppose we have a set of data $A = \{a_1, a_2, \dots, a_n\}$ and $B = \{b_1, b_2, \dots, b_n \}$. So there are $2n$ elements in total. Further suppose the median of $A$ and $B$ is $a$ and $b$, ...
0
votes
0answers
15 views

Show $||v\times w||=||v\wedge w||$

Let $v,w\in\mathbb{R}^3.$ Let $v\times w\in\mathbb{R}^3$ be the standard cross product.I'm trying to show that $||v\times w||=||v\wedge w||.$ I know that $\wedge^2\mathbb{R}^3$ has a basis ...
1
vote
1answer
18 views

The Second Hearts Problem

Intro: According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is ...
0
votes
2answers
22 views

On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will ...
3
votes
4answers
28 views

$a_n = \frac{1}{n}b_n$, $\lim b_n = L>0, L\in\mathbb{R}$, prove $\sum a_n$ diverges

I have to prove that if $$a_n = \frac{1}{n}b_n$$for $n\ge 1$ and $$\lim_{n\to\infty} b_n = L>0, L\in\mathbb{R}$$ then $$\sum_{n=1}^{\infty} a_n$$ diverges. My idea was to show that it's not true ...
0
votes
0answers
13 views

Dense Subgroup Of Reals with addition

let $G$ be a non-trivial subgroup of $ (R,+) $. Prove that $G$ to be a dense or $(nZ)$. I know this result is true of course .But problem is to prove it. Please hint to prove it.
0
votes
0answers
5 views

inverse Mapping in Transformation of a random variable

I have a question concerning the the inverse mapping in the image . text extracted from Casella Statistical inference $g^{-1}(A) = \{ x \in \chi : g(x) \in A\}$ I know the idea that they want to ...
0
votes
0answers
17 views

Why is the function $f_n(x)=\sum_{k=0}^n\frac{(-1)^k}{2^k}\lfloor 2^kx\rfloor$ bounded for some $n$ but unbounded for others?

The function is unbounded for $n=0,1,2,4$ but bounded for $n=3,5,6,7,8$, at least if desmos.com is to be believed. https://www.desmos.com/calculator/lyre5002v0 It's past my bedtime so I have not ...

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