0
votes
0answers
2 views

How do I show that an element is in $\mathbb{Z}/p\mathbb{Z}$ given two of its powers?

I have that an element $x$ is in the algebraic closure of $\mathbb{Z}/p\mathbb{Z}$ with $x^m$ and $x^n$ given, and we know $(m, n)=1$. I suppose to show that $x$ is in $\mathbb{Z}/p\mathbb{Z}$ I ...
0
votes
0answers
3 views

expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
0
votes
0answers
3 views

Spencer and Shelah zero-one law for Erdos-Renyi random graph $G(n,p)$

In Erdos-Renyi random graph $G(n,p(n))$; set $p(n)= (\frac{ln n}{n})^2$. We know that already Spencer and Shelah have proved that zero-one law doesn't hold for $p(n)= \frac{ln n}{n}$. Now the question ...
0
votes
0answers
3 views

Find k if given the constant term of a binomial expression?

Consider the expansion of $x^2(3x^2+\frac{k}{x})^8$. The constant term is 16,128. Find k. This is simply an example of a type of question I cannot understand how to do. I have many questions: 1) ...
-1
votes
0answers
11 views

Can a balloon in flight change its surface topology without tearing?

It is very well known that an inflated balloon in flight will not change its surface topology without tearing; the elastic deformations in its surface will be like "smooth transformations" in its ...
0
votes
0answers
15 views

Homeomorphism between $\mathbb{R^2}$ and $S^2-N$, the sphere without its north pole

How would one approach the following problem? Write down a homeomorphism and its inverse from $\mathbb{R^2}$ to the sphere $S^2-N$ without its north pole So I need a function $f(x,y) : ...
0
votes
0answers
13 views

Using Cauchy Integral Formula to solve an integral

Question: Evaluate $$\int_\Gamma \frac{\sin(z)}{(z-\pi)^2} dz$$ Where $\Gamma$ consists of the sides of the rectangle with vertices at $(1,\pm3i)$ and $(4,\pm2i)$ My attempt: The only ...
0
votes
0answers
7 views

secretary problem partial information

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic i research: You are supposed to choose the best item presented to you in a row of n items. Any ...
0
votes
0answers
8 views

Integral of convex function applied on a function

Let $f$ be an integrable function of $\mathcal{L}(\mathbb{C},\mathbb{R})$, measure Lebesgue. I want to prove that there exists an increasing convex function $H:\mathbb{R}^+\rightarrow\mathbb{R}^+$ ...
0
votes
1answer
30 views

Why does $E[X] = \sum^\infty_{r=0}P(X > r)$?

Why does $E[X] = \sum^\infty_{r=0}P(X > r)$? I understand if it was $E[X] = \sum^\infty_{r=0}rP(X = r)$, but for this I don't understand why.
0
votes
0answers
4 views

Probability the difference of two iid random variables following Type 1 Extreme Value Distribution

X and Y follow iid Type 1 Extreme Value distribution with mean $\mu$ and variance $\sigma^2$ (the shape parameter $\xi = 0$). How to find the following probability $$\Pr(X - Y < B),$$ where $B$ ...
0
votes
1answer
19 views

Finding the limits of a trig function

I have been struggling with finding the following limit: $$ \lim_{x\to \pi} \frac{\cos x + 1}{x - \pi} $$ Use of L'Hospital's rule is not permitted. Thanks
0
votes
0answers
6 views

Homotopy of two circles contained in an open ball.

The following question is on my homework assignment and I have no idea how to even start answering it: Are any two circles contained in an open ball B(0,r) Homotopic? You can assume the circles are ...
0
votes
2answers
28 views

Find the eigenvector and eigenvalues for the following 3 x 3 Matrix?

$$ \pmatrix{5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 &-4 & -11} $$ I already got the eigenvalues that is $\lambda = 1$ and $-3$. And I managed to solve the eigenvector corresponding to ...
-1
votes
1answer
40 views

Integrate from zero to infinity 1/(xe^x)

I cannot solve the integral $$\int_{x=0}^{\infty}\frac{dx}{xe^x}.$$ I tried it by use integration by parts and gama function.
0
votes
1answer
17 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
0
votes
0answers
14 views

Seeking: for-credit online (or DC-area) grad-level Differential Equations course

I need to take Differential Equations to apply for economics Ph.D programs. In the meantime, I'm starting an M.S. program in math and statistics soon and just learned I can count my already-planned ...
2
votes
0answers
9 views

Constraints on a Chebyshev series representation of a CDF

My question is about deriving constraints for coefficients of a Chebyshev series which represents a CDF. Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know ...
0
votes
1answer
21 views

Show that if $|s_{mn}-S|<\varepsilon$ then $|\lim_{n\to\infty}s_{mn}-S|\le\varepsilon$

This is the exercise 2.8.5 of the book Understanding analysis of Abbott. To put you in context I have that The sequence of partial sums $(s_{mn})$ is absolutely convergent, and the definition is ...
0
votes
0answers
7 views

Representing Covering Spaces by PErmutations

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher. Hatcher starts by ...
-1
votes
0answers
23 views

Existence of solution of the heat equation on closed Riemannian manifolds

Let $M$ be a closed manifold and $\phi(x,t):M\times(0,T)\rightarrow\Bbb R$ be a smooth family of smooth functions. does the heat equation $$\partial_t\phi(t)=\Delta_{g(t)}\phi(t)$$ with the initial ...
0
votes
0answers
21 views

5x5 Bingo Puzzle [Logical thinking problem]

5 people participate in a custom game. They are given blank cards, in which they have to fill numbers from 1-25 in a 5x5 table. The host of the game, then calls out random numbers (between 1-25, ...
-1
votes
1answer
18 views

Show that $\cup _{n=1,2,3,…} (-1+1/n,0)=(-1,1)$.

My proof. Initially, we will show that $\cup _{n=1,2,3,...} (-1+1/n,0)\subseteq (-1,1)$. For every $n=1,2,3,...$ sicnce $-1<-1+1/n<1-1/n<1$, $(-1+1/n,1-1/n)\subseteq (-1,1)$. Now, we will ...
0
votes
0answers
19 views

When deriving the Fourier Series, how is $a_1$ calculated?

I am having difficulty understanding how the Fourier series is calculated. It starts like this; For any $f ∈ C_2π$ we would like to find coefficients $c_n(f)$ such that $$f(x) = \sum_{n=0}^{\infty} ...
0
votes
1answer
22 views

Determinant of orthogonal matrix

If $A$ is orthogonal. how do I show that $\det(A-2I)\not=0$. I tried writing $A-2I=A-2AA^T=A(I-2A^T)=A(A^TA-2A^T)=AA^T(A-2I)$ but it seems that I am just doing loops after loops.
0
votes
2answers
24 views

Why do we have the following implication if $\phi$ is injective

If $\phi: G \rightarrow H$ is a homorphism, and if $\phi$ is injective, why do we have the following: $\phi(g) = e_h \implies g=e_g$
0
votes
1answer
16 views

Definite Gaussian/exponential integral

I've been revising some quantum mechanics, and I was wondering how I would calculate a specific standard deviation. The wave function I am working with is ...
-1
votes
0answers
4 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
0
votes
0answers
6 views

incremental knapsack

Is there a way to compute the knapsack problem incrementally? Any approximation algorithm? I am trying to solve the problem in the following scenario. Let D be my data set which is not ordered and ...
1
vote
1answer
14 views

Tarski's schema T

On Wikipedia, Tarski schema T says: A sentence of the form "A and B" is true if and only if A is true and B is true A sentence of the form "A or B" is true if and only if A is true or B is true A ...
3
votes
3answers
52 views

$f \in C(\mathbb R)$ such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ ; is $f$ increasing?

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ , then is it true that $f$ is ...
0
votes
2answers
24 views

Using Mean Value Theorem to solve an inequality involving $\cos^{-1}(x)$

Question: Using the Mean Value Theorem, show that for all $-\frac{1}{2}\lt a,b \lt \frac{1}{2}$ with $a\lt b$ $$\lvert \cos^{-1}(a)-\cos^{-1}(b)\rvert \lt \frac{2\sqrt{3}}{3}\lvert ...
-1
votes
0answers
25 views

Relation between area of a regular hexagon and an equilateral triangle. [on hold]

Suppose that a regular hexagon and an equilateral triangle is have the same circumference. How much larger (in terms of percentage) is the larger of the two areas?
0
votes
0answers
10 views

Prove two matrices are similar?

Let $G_1=[I(k), \mathcal G_1]$, $G_2=[I(k), \mathcal G_2]$, $H_1=[\mathcal H_1,I(m)]$ and $H_2=[\mathcal H_2,I(m)]$, where $\mathcal H_1, \mathcal H_2$ are transpose of $\mathcal G_1,\mathcal G_2$, ...
0
votes
0answers
8 views

Two collineations

Give collineations to prove the following (in the extended projective plane): a, One cannot contruct the midpoint of two points using a straightedge only. b, One cannot construct the reflection of a ...
1
vote
0answers
14 views

Is a bundle morphism which restricts to homeomorphisms of the fibers a bundle isomorphism?

If $f$ is such a map between total spaces (assume a common base space) then it is a bijective and continuous and the inverse will be fiber preserving so that all we would need to prove is that the ...
2
votes
1answer
52 views

Pure mathematics research

What can a first year mathematics undergraduate, who wants to pursue research in pure mathematics, learn in 67 days that will help him in the future?
0
votes
0answers
24 views

Proof of L'Hopital's rule

I'm attempting to prove L'Hopital's rule. My solution so far is the following: Let $f,g:(a,b)\to\mathbb{R}$ be differentiable and continuous on $[a, b]$ with $f(a)=g(a)=0$ and $g(x)\neq0$ for ...
1
vote
1answer
18 views

Compact set contained in the interior of another compact set

Let $X$ be a locally compact Hausdorff space. Does the property "every compact set is contained in the interior of some compact set" has a special widely known name? Is it related to paracompactness?
0
votes
0answers
18 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...\omega_n$ of $K$, st ...
0
votes
1answer
26 views

How many diagonals does a decagon have?

How many diagonals does a decagon have? I have just learnt permutations, dispositions, combinations. How can I solve it with these concepts? I drew it and it was $35$ diagonals. How can I prove ...
3
votes
1answer
18 views

“Shape” of solutions of 2nd order homogeneous ODEs

Consider a second order homogeneous ODE: $$P(x)y''+Q(x)y'+R(x)y=0.$$ If $P,Q,R$ are constant functions, then we know that the general solution has the form $$y=c_1e^{r_1x}+c_2e^{r_2x},$$ ...
-4
votes
0answers
15 views

Multivariate Gaussian distribution

I done parts (a) and (b) but I am stuck on part (c). I think the joint distribiution of R^2 is a chi squared r.v but I am not sure
3
votes
1answer
16 views

Discriminant of a trinomial $x^n+ax^m+b$

I am trying to compute the discriminant of the trinomial $x^n+ax^m+b$. I have tried using resultants but cannot see how to approach it. Any hints?
0
votes
0answers
11 views

Show that if a Markov chain is irreducible and has a state $s_i$ such that $P_{ii}>0$, then it is also aperiodic.

Show that if a Markov chain is irreducible and has a state $s_i$ such that $P_{ii}>0$, then it is also aperiodic. Proof: Let $X=(X_0, X_1, X_2, \dots)$ be an irreducible Markov chain with a state ...
0
votes
0answers
6 views

Aligning matrices, normalization. Calculating coefficients.

So as a pre-task for my upcoming exam this is one of the rehearsal assignments. I can't wrap my head around this one at all, haven't seen anything like it earlier, and I can't seem to find any ...
0
votes
2answers
10 views

Show pointwise convergence

I'd like to show that $\sum_{n=0}^{\infty}e^{-(x-n)^2}$ converges pointwise I can see that for $x=0$ the sum can be written as a geometric sum which then convergences, but I don't know how to ...
0
votes
1answer
16 views

Problem on Inclusion & Exclusion Principle

Book has the following & solution to it too, pls clear my confusion: On rainy day , five gentlemen A, B, C,D, E attend a party after leaving their umbrellas in a checkroom. After the party is ...
0
votes
0answers
11 views

Quaternary numeral system: fractions

I have a question related to the expression of a real number in base 4. Consider the table here: it is clear to me how all columns of the table are obtained except the fourth one: how do they get the ...
1
vote
0answers
12 views

Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)

On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula: $\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$ where $s>1$, $\chi$ is a ...

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