1
vote
1answer
59 views

Taylor series of an implicit function

Suppose the function $s:[-\delta, \delta] \to \mathbb{R}$, $\delta > 0$, is defined implicitly by $$s(t) = 1 - c\beta t (s(t))^{\beta}$$ for some $c > 0$, $0 <\beta < 1$. Can an ...
2
votes
1answer
43 views

$Z_n \stackrel{a.s.}{\to} 0$ and $E(|Z_n|) \to 0$

Let $Z_n$ be a sequence of random variables with finite expectation. Is the following statement true? i) $Z_n \stackrel{a.s.}{\to} 0$ implies $E(|Z_n|) \to 0$ ii) $E(|Z_n|) \to 0$ implies $...
0
votes
3answers
37 views

How to solve $n$ from $c \leq 1.618^{n+1} -(-0.618)^{n+1}$

I need to solve the bound for $n$ from this inequality: $$c \leq 1.618^{n+1} -(-0.618)^{n+1},$$ where $c$ is some known constant value. How can I solve this? At first I was going to take the ...
1
vote
2answers
127 views

Multiplicity of an Eigenvalue of the Exponential of an Operator

I am trying to prove the following: Let $T:\mathbf R^n\to\mathbf R^n$ be a linear operator. Then $$\det e^T=e^{\text{Trace}(T)}$$ To do this I took the following apporach. Let $\lambda\in\...
2
votes
1answer
33 views

confudes with Dijkstra's algorithm.

I have tried to understand the question but I got really confused. So starting from node 3, the distance to other nodes are 3 to 1 = 3 3 to 2 = 1 3 to 4 = 4 3 to 5 = 2 3 to 6 = 3 3 to 7 = 2 ...
2
votes
1answer
95 views

Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}$ is martingale.

Let $\xi_1 \neq const.$ be a random variable with moment-generating function $\phi(\theta) = Ee^{(\theta \xi_1)}$. Let $S_n = \xi_1 + \dots \xi_n$. Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}...
0
votes
1answer
39 views

Prove inequality holds

Show that: $\displaystyle 2! \cdot 4! \cdot... \cdot(2n)!>[(n+1)!]^n $ for $n>1$ where $n$ is natural I tried by induction but I stuck when I have to show that: $(2n+2)!>(n+2)!(n+2)^n$
2
votes
1answer
122 views

there doesn't exist any sequence of polynomials which converge to $\sin x$ uniformly on $\mathbb{R}$

Show that there doesn't exist any sequence of polynomials which converge to $\sin x$ uniformly on $\mathbb{R}$. Suppose there is a sequence of polynomials $\{p_n\}$ which converges to $\sin x$ ...
0
votes
1answer
52 views

probability: best linear predictor $\hat{Y} = aX + b$

Let $X\sim\mathcal{U}(-1, 1)$ and $Y = X^2$. Since the best linear predictor is defined as $$ \hat{Y} = E_Y[Y] + \frac{\text{cov}(X, Y)}{\text{var}(X)}(x - E_X[X]) $$ Can I simple just write it as $$...
1
vote
0answers
37 views

Number of triples $(a, b, c)$ with $1 \leq a,b,c \leq n$ which are coprime ($gcd(a,b,c)=1$)

Number of ordered triples $(a, b, c)$ with $gcd(a, b, c) = 1$ and $1 \leq a, b, c \leq n$ can be computed using the following formula: $$ C(n) = \sum_{k=1}^n\mu(k) \left \lfloor \frac{n}{k} \right \...
2
votes
2answers
2k views

union and difference of convex set

suppose X,Y are two convex sets x1, x2 in X and y1, y2 in Y defn of X and Y being convex: tx1+(1-t)x2 in X ty1+(1-t)y2 in Y it is clear that: 1) X+Y is convex. 2) X intersection Y is convex 3) ...
1
vote
0answers
44 views

Boolean algebra-dual of an expression

Can anyone think of an expression that is equal to its dual ? I've been trying to solve this for the past 2 hours, but nothing comes to mind.
1
vote
1answer
39 views

How can i learn when to use which multiplication rule: Probability

Hey guys im studying for a math exam and was wondering if anyone has some easy techniques to remember in what kind of scenario to use these equations. These are I believe called multiplication rules. ...
0
votes
1answer
59 views

Regularity of the eigenvalues of a matrix-valued function

Suppose I have a $2 \times 2$ Hermitian matrix-valued function $m$ defined on $\mathbb{R}^{2}$ with entries $m_{jk} \in C^{\infty}(\mathbb{R}^{2})$. Denote by $m_{+}$ and $m_{-}$ the greater and ...
1
vote
1answer
37 views

Continuity of $\sup_{x\in\Omega}\varphi(x,\cdot)$

Let $\Omega\subset\mathbb{R}^n$ be open,bounded and (I don't know if this matter) of class $C^{1+\alpha}$. Let $\varphi:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $\varphi(x,\cdot)$ is ...
1
vote
1answer
46 views

Solution to a higher order ordinary differential equation

Let $q$ be a strictly positive integer and let $\beta \neq q$ be a real number. Consider a following Ordinary Differential Equation(ODE): \begin{equation} \frac{d^q r_t}{d t^q} + \frac{1}{t^\beta} r_t ...
0
votes
1answer
41 views

Help in this easy equivalence

If $C$ is a curve with genus $g$ and $k$ a field, I'm stuck in something I'm sure easy, I think I'm forgetting some basic things. Define $\Omega(D)=\{\omega\in\Omega;div(\omega)\ge D\}$ and $\delta(...
0
votes
1answer
82 views

Challenging Question: for Expected Value of a particular probability density function

I've been stuck on this for a while and it's been driving me crazy. Any help would be greatly appreciated. I am trying to find the Expected Value of the following Probability Density Functions (where ...
1
vote
0answers
137 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 &-\...
1
vote
2answers
119 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then $\mathrm{esssup}_{0≤...
0
votes
1answer
184 views

Total time spent travelling, given distance and speed functions

I have the following situtation: An object is traveling a certain (known) distance in a straight line. The object starts at rest, accelerates to its preset maximum speed then spends some time ...
1
vote
2answers
1k views

Permutation question on alphabets

Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is Well I do understand some ...
1
vote
1answer
36 views

get the angle between a and b using their coordinates

I am working on some simulation software, and I want to get the bearing of one point in the simulation from the position of another. I have a point A at position (lat, lon), I also have an entity B ...
0
votes
1answer
35 views

Enumeration of integers are in increasing order which have gaps

I want to solve the following: Calculate the number of ways of selecting five distinct integers $x_1,x_2,x_3,x_4,x_5$ where $0\leq x_1 \lt x_2 \lt x_3 \lt x_4 \lt x_5 \leq 20$ I think this may ...
2
votes
1answer
58 views

Fourier Transform of $ f(t) = e^{-kt}$

I am trying to calculate the fourier transform of the following function: $$ f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases} $$ where $k > 0$ is a real number. ...
2
votes
2answers
62 views

Interesting combinatorics

There is $n*n$ square grid. How many ways to fill it with $1$ and $0$ do we have, in case the sum in every row and every column should be even. The problem seems to be easy, but after some time and ...
1
vote
3answers
164 views

How can we use $(im(A))^\perp = \ker(A^T)$ to prove $rank(A)=rank(A^T)$?

Why is it that these two statements are essentially equivalent? $(im(A))^\perp$ represents all vectors orthogonal to the $im(A)$. Yet I'm not sure what this being equal to $\ker(A^T)$ exactly means, ...
2
votes
1answer
63 views

show inequality holds for all positive reals

Let $\displaystyle a_1, a_2, ... , a_n$ be positive real numbers and we know that $\displaystyle a_{n+1}=a_1$ Show that: $\displaystyle \sum_{i=1}^{n} \frac{a_i^3}{a_{i+1}^2} \ge \sum_{i=1}^{n}\...
0
votes
2answers
91 views

Show a bound sequence with a cluster point is indeed convergent

I have been wondering for a while now: How do I show that $(i) \space (a_{n})_{n \in \mathbb{N}}$ is convergent. $(ii)\space (a_{n})_{n \in \mathbb{N}}$ is bounded and has a cluster point. ...
2
votes
0answers
103 views

2 logarithmic integrals twins

This question also has the value of an answer to the first integral here How to evaluate $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1-x}dx$ and $\int_0^1\frac{\ln(x)\ln(1+x^2)}{1+x}dx$ $$\int_0^1\frac{\ln(x)\...
2
votes
3answers
198 views

Limit of a converging sequence

Let the following sequence:$$u_0=1, \, \forall n\in\mathbb{N},\,u_{n+1}=\sqrt{u_n^2+\dfrac{1}{2^n}}$$ I try to find its limit. Well we can prove that $\forall n\in\mathbb{N},\,u_{n+1}-u_n\le \dfrac{1}...
0
votes
1answer
30 views

A question about a theorem derived from a given set of postulates

Currently, I am reading the book 'Godel's proof' by Ernest Nagel and James Newman, with the forward by Douglas Hofstadter. In that book, on page 15, the authors give an example of an axiomatic system ...
1
vote
1answer
156 views

Diffeomorphism between the Grassmannian manifolds $\mathbf{Gr}(n,k)$ and $\mathbf{Gr}(n,n-k)$.

This seems to be a common exercise question, however I am having trouble with it. The hint is to use a map that associates the k-plane to its orthogonal complement. But I have not been able to show ...
0
votes
1answer
82 views

Prove by induction on a sequence

We have $$ n \in R $$ And an arithmetic sequence of natural numbers different than 0 that the sum of all its members = n $$ a_1,a_2,...,a_k $$ $$ a_1+a_2+...+a_k = n $$ I need to prove by ...
0
votes
1answer
72 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
2
votes
1answer
2k views

Prove sin(1/x) is discontinuous at 0 using epsilon delta definition of continuity

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ using the $\epsilon \delta$ definition of continuity. I ...
0
votes
1answer
120 views

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$?

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$? Intuitively I think it's wrong, but I failed to come up with a single counterexample. Can ...
0
votes
1answer
37 views

How to compute n(mod c) when n(mod a),n(mod b),a,b,c are given?

Given a(prime) > b (prime) > c(any number), is there any way to compute n(mod c) ? n%a,n%b,a,b,c are known.
0
votes
1answer
116 views

the argument of function 'argmax'

In mathematics, argmax stands for the argument of the maximum, that is to say, the set of points of the given argument for which the given function attains its maximum value.[From Wiki] We have the $...
0
votes
3answers
58 views

functional equation for $x^2$ $f(f(x))=x^4$

If $f(f(x))=x^4$ for all real $x$ and $f(1)=1$ find $f(0)$. It seems that $f(x)=x^2$ but can we solve without this explicit form of $f$?
1
vote
1answer
40 views

First part of the proof that $F^*d\beta=dF^*\beta$

Where has the $dy^j$ gone in the highlighted equation? I would have thought the highlighted equation should be $\displaystyle (F^*dg)(x) = \frac{\partial F^j}{\partial x^i}(x)\frac{\partial g}{\...
0
votes
2answers
351 views

Volume of pyramid intersection

Suppose that there are two square pyramids on the $xyz$-plane. Both have base coordinates of $(0,0,0)$, $(30,0,0)$, $(0,30,0)$, and $(30,30,0)$. One pyramid has its apex at $(10,10,30)$, while the ...
2
votes
1answer
46 views

How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
0
votes
2answers
84 views

Find a cubic interpolating polnomial that fits the data points using Vandermonde matrix

The data points are (-1,0), (1,2), (3,20), (5,102) I'm not sure how to set up the matrix for this question. I understand the matrix equation is $Va=y$ where $V$ is a Vandermonde matrix and I have ...
0
votes
2answers
29 views

divisibility relation $a|b^2 + 10c.$

Use divisibility relation to show that for all integer $a$, $b$, $c$, $a \ne 0$ counts if $a|b$ and $a|c$ then $a|b^2 + 10c$. Use direct proof. Ok, $a|6$ then there is integer $k$. $$a*k=6,$$ $$...
1
vote
1answer
35 views

Quadratic nonresidues mod p

The question asks to find congruence conditions on prime $p$ such that $7$ is the least quadratic nonresidue mod p. Also, find the least such prime. I solved it for $1,2,3,4,5,6$ mod $p$ and got ...
1
vote
1answer
60 views

Toeplitz equality constrained least-square optimization

What is the fastest known algorithm for least-square optimization problem with a linear equality constrain \begin{align*} &\min \|K x - y\|^2 + \mu \|x\|^2\\ \text{s. t. }& Q x = v \end{align*}...
0
votes
1answer
45 views

Component formula for pullback of one forms

Should it not be $\Big(F'(x)v \Big)^j = \frac{\partial F^j}{\partial x^k }(x)v^k$? Then also how is $F^*dy^j = \frac{\partial F^j}{\partial x^i}dx^i$ derived? I cannot what $\beta_j$ has been taken ...
3
votes
1answer
104 views

Conditional Probability in Poisson- How To Calculate The Probability That A Soccer Team Wins From Behind

I have been cracking my head trying to get a solution to this problem for awhile, but I can't seem to get it: "In a soccer match, given team A and team B are both expected to score 1.3 goals each, ...
1
vote
4answers
51 views

Arithmetic of a combinations formula

I am trying to study, and I'm not quite sure how: $$ \binom{5}{3} \cdot \binom{7}{3} = 350 $$ From my understanding the formula is $$ \binom{n}{r} = \frac{n!}{r!(n-r)!} $$ Therefore: $$ \binom{5}...

15 30 50 per page