2
votes
1answer
12 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
4
votes
1answer
26 views

Is it mathematically valid to separate variables in a differential equation?

I read this statement in a book on Calculus as a part of my mathematics course and one of the pages concerning Differential Equations contained the statement: Technically this separation of ...
1
vote
0answers
38 views
+50

How do I determine if a statistical relationship exists in a real life problem: honor roll class assignments?

My question relates to Determine if being on honor roll gives an advantage in being assigned to the math class (I've seen similar questions involving hits on a webpage) which I believe should be ...
-1
votes
1answer
24 views

On the minimal set of generators of ideals in $\mathbb{C}[x,y]$. [on hold]

I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry": i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of ...
0
votes
1answer
16 views

When is $\theta$ obtuse or acute in sin, cosine, tan when they are positive, negative or both?

My textbook gives a non intuitive answer and tells us to memorize when the ratios are positive or negative or both based on some arbitrary rule that I don't understand. I know how to do both of ...
1
vote
0answers
15 views

Markov factorization of the density of an AR(1) process

Suppose we have a causal, stationary AR$(1)$ process with i.i.d. innovations $Z_t$. Then we know that it is a Markov as future value $X_{t+1} = \phi X_t + Z_{t+1}$ given the past $X_1,\ldots X_t$ ...
1
vote
1answer
11 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
0
votes
1answer
14 views

Balanced Latin Square

For making a good Between-Object user study, this is suggested to use a Latin Square to give all the different conditions, ...
1
vote
1answer
87 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
0
votes
0answers
8 views

What is $h^{-1}(L)$?

Let $L = L((00 + 1)∗)$ and $h : \{a, b\}^* \to \{0, 1\}^*$ be defined by $h(a) = 01$ and $h(b) = 10$. What is $h^{−1}(L)$? In this context "$+$" means "$\cup$". So the language $L$ is all the ...
2
votes
3answers
43 views

Solve for $\frac{dy}{dx}$?

Q: Given $3^{x+y} = x^3 + 3y$, find $\frac{dy}{dx}$. I am convinced that, since it is not possible to algebraically solve for $y(x)$, one can't find $\frac{dy}{dx}$. Am I correct? Thanks!
0
votes
0answers
5 views

If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then $a^b$ is transcedental.

For my math study, I have to prove the following: Let's donate the set of algebraic numbers with $\mathbb{A}$. Prove: If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then ...
2
votes
1answer
102 views

Partials sums of cosh(x) and sinh(x)

Ok, i asked this question yesterday but then hit a snag again. Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = ...
0
votes
1answer
14 views

Schauder basis and Eigenbases

There are several question in this site comparing different basis functions including Schauder basis and others, but I could not connect the difference between the Schauder basis and Eigenbasis ...
3
votes
2answers
133 views

Maximum-Value Secretary Problem

Background: The classic secretary problem has the simple solution of rejecting the first 1/e applicants and then selecting anyone who was better than the best in the rejected set. However, in the ...
0
votes
0answers
32 views

Complex problem of complex numbers

Compute the value of $$1+2\alpha+3\alpha^{2}+...+n\alpha^{n-1}$$ in the form of a complex number where $\alpha$ is a non-real complex $n^{th}$ root of unity. The answer given is : ...
0
votes
2answers
18 views

Power series- Radius of convergence : HELP!

Calculate the radius of convergence of the following: $$ \sum \frac{\ln(1+n)}{1+n} (x-2)^n $$ Will you please help me figure out how to calculate: $$ \lim_{n\to \infty} \frac{\ln(2+n)}{2+n} ...
0
votes
0answers
6 views

Proof of an inequality with entropy and mutual information.

Entropy of a random variable (a) is (h) : $H(a) = h$. Mutual information of (a) and (b) is (3h/4) : $I(a;b) = 3h/4$. Mutual information of (a) and (c) is (3h/4) : $I(a;c) = 3h/4$. It is needed to ...
0
votes
0answers
8 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ when $b \in \mathbb W$ and $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
1
vote
0answers
7 views

SVD: find $\vec{v_2}$ when $\lambda{_2} = 0$ given the relation $A^t \vec{u_2} = \lambda{_2} \vec{v_2}$

An SVD of A given by: Where D is a diagonal matrix containing the singular values $s_1 and s_2$ on the diagonal. Given these related to the SVD of A: $ A = \left(\begin{array}{rrr} -3 & 1 \\ ...
1
vote
3answers
43 views

Prove that $\phi(n)=\frac{n}{2}$ iff $n=2^k$ for some integer $k\geq 1$

Prove that $\phi(n)=\frac{n}{2}$ iff $n=2^k$ for some integer $k\geq 1$ Attempt: Let $n=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_k^{\alpha_k}$. Then $\phi(n)=\frac{n}{2} \implies ...
2
votes
1answer
26 views

Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...
0
votes
2answers
27 views

linear transform of functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...
1
vote
0answers
9 views

Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[n,n]$ and $b$ is $[1,n]$ matrix. These all ...
3
votes
2answers
46 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
3
votes
1answer
33 views

How find function $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all function $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
2
votes
2answers
19 views

Steiner Triple System

A Steiner Triple System, denoted by $STS(v),$ is a pair $(S,T)$ consisting of a set $S$ with $v$ elements, and a set $T$ consisting of triples of $S$ such that every pair of elements of $S$ appear ...
0
votes
2answers
17 views

How find the length of the third of the chord for of a circle of radius r?

Three chord of a circle of radius $r$ are the sides of the triangle inscribed in the circle. Two of these chords have a length $\frac{r}{2}$ and $r\sqrt{3}$. How find the length of the third of the ...
2
votes
0answers
66 views
+200

Is it true that $\sup_{y'\in Y'}h_d(T,U^{-1}(y'))=0$, i.e. $h(T')=h(T)$? (Bowen, Topological entropy)

First I have to give the background to my question: Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the map $T\colon X\to X$ which describes the following dynamics: For $x\in X$, which is a ...
0
votes
2answers
21 views

Probability - very difficult combinatorial question - don't have the theoretical background

Combinatorics - a (very) old question (which I hope I have remembered correctly) from a Cambridge Math Tripos. "A student sits 6 examination papers, each worth 100 marks. In how many possible ways can ...
2
votes
0answers
12 views

How to show that $p(t|x,\mathbf x,\mathbf t)= \int p(t|x,\mathbf w)p(\mathbf w|\mathbf x, \mathbf t)d\mathbf w $

The following paragraph is approximately cited from Bishop's book, Pattern Recognition and Machine Learning. In carve fitting problem, we have training data $\mathbf x$ and $\mathbf t$, along ...
0
votes
0answers
9 views

Determinitic finite automaton (DFA) that accepts natural numbers divisible by 6

I'm new to Formal Systems and Automata and I'm working on some exercises to get familiar with the concepts. I want to create a DFA that accepts natural numbers divisible by 6. I know that a number ...
1
vote
0answers
12 views

Compatibility of direct product and quotient in group theory

This question came to me when I tried comparing direct product and quotients of groups with products and quotients of natural numbers. When we divide a number by another and multiply the result with ...
0
votes
1answer
24 views

Is a limit point compact subspace of a Hausdorff space necessarily closed?

I think the answer should be "no", but I can't give a counter-example.
0
votes
3answers
41 views

How to finish proof of $ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$

I'm trying to prove the identity $$ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$$ What I've done so far: From geometric series ...
0
votes
2answers
73 views

complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $ 0\le \arg(z+2)\le 45\deg $ I was able to sketch and shade the region that satisfies both ...
0
votes
0answers
17 views

Prob. 4, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to decide which cases to consider?

We need to show that the ordered square satisfies the first countability axiom. I'm not able to decide as to which separate cases to consider. By definition the ordered square is the product $I ...
0
votes
0answers
12 views

“Permutation” of squared norm and sum

In Problems and Solutions in Mathematics, 2nd edition, by Ta-Tsien, exercice 4312. Let $f$ be a periodic function on $\mathbb{R}$ with period $2 \pi$ such that $f|_{[0, 2 \pi]}$ belongs to $L^2(0, 2 ...
1
vote
1answer
12 views

Is $T_n(R) \cong T_n(R)^{op}$?

I am working on the following problem: Let $R$ be a commutative ring, and $T_n(R)$ be the ring of $n \times n$ upper triangular matrices. Is $T_n(R) \cong T_n(R)^{op}$? I have already shown ...
1
vote
1answer
15 views

Trigonometric limits as x approaches infinity. Need help.

I know how to solve limits with polynomials but i'm not sure how to tackle these limit problems involving trig and the constant, e. I can't seem to find any similar examples either. Any help? Thanks. ...
0
votes
2answers
2k views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
0
votes
1answer
48 views

How to prove a cube minus a cube is never a cube (in whole numbers) [duplicate]

How to prove $x^3-y^3\neq z^3$ where $x$, $y$, and $z$ are whole numbers (integers greater than zero)?
1
vote
1answer
10 views

Proving strong stability of semigroup

$X$ is the Hilbert space $L^{2}(0,\infty)$ and let $T(t):X\to X$ with $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$. I want to prove that the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ is strongly ...
1
vote
3answers
42 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
0
votes
1answer
19 views

Riemann integral property proof using the definition

We say that a function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if for every $\epsilon>0$, there are two step functions $g_1,g_2$ such that $g_1 \leq f \leq g_2$ and $\int_a^b ...
0
votes
0answers
10 views

centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
1
vote
0answers
12 views

Krull's height theorem in non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $ht I\le n$. When $R$ is not Noetherian, this is not true. I wonder if ...
1
vote
0answers
11 views

Hilbert's double series theorem $\sum_{n,m}\frac{a_n b_m}{m+n}$ and its generalizations.

Let $l_p$ be the space of all complex sequences $x=\{x_n\}$ equipped with the finite norm $\|x\|_p=\left(\sum_n |x_n|^p\right)^{1/p}$. Hilbert's double series theorem states that $$\sum_{n,m}\frac{a_n ...
0
votes
0answers
19 views

If an isomorphism can be expressed as a composition of morphisms, what can we say about it's components?

Suppose $f:X\to Y$ is an isomomorphism, and $f=g\circ h$ where $h:X\to Z$ and $g:Z\to Y$. Can we infer that either of these component morphisms is an isomorphism as well? And does this change ...
0
votes
0answers
18 views

Finding normalized eigenfunctions for $y'' + \lambda y = 0$

Find the normalized eignefunctions for $$y'' + \lambda y = 0$$ $$y(0)=0, y(\pi)-2y'(\pi)=0.$$ My teacher gives me this hints: Consider$$(py')'+qy+\lambda ry=0$$ where $p, p', q, r$ are ...

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