0
votes
0answers
1 view

Splitting Probability Within a Finite Time

Let $\mathcal{X}=\{x_i\}_{i=1}^{N}$ be a subset of $\mathbb{Z}$. For $j\in(1,N)$, what is the probability that the first element of $\mathcal{X}$ encountered by a simple 1D random walk is $x_j$ and ...
0
votes
1answer
4 views

Problem in finding examples of linear operators.

Find the example of two linear operators $T$ and $U$ such that $TU = O$ but $UT$ is not a zero operator. But I fail to find out proper example.Please help me in finding the example.Thank you in ...
2
votes
0answers
5 views

Complex measure

Let $(X,\mathcal{S}, \mu)$ be a measure space with $\mu$ a complex measure. Suppose that $$ \int_X f \ d\mu \in \mathbb{R} $$ for all $f\in C( X \to \mathbb{R} )$ i.e. the continous functions from $...
0
votes
0answers
26 views

What's the mistake on my answer for this inequality $ \frac{\left(x+1\right)}{\sqrt{x^2+1}}>\frac{\left(x+2\right)}{\sqrt{x^2+4}} $

Good evening to everyone! I have the following inequality: $$ \frac{\left(x+1\right)}{\sqrt{x^2+1}}>\frac{\left(x+2\right)}{\sqrt{x^2+4}} $$. I don't know what's wrong with my answer: $$ \frac{\...
0
votes
0answers
3 views

Necessary and sufficient condition for argmax/argmin

Let $x_1,\dots,x_n$ be real variables and let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a differentiable function with a unique maximum (or mininum). Is there a necessary and sufficient condition ...
0
votes
0answers
4 views

Prove that these pairs of complex numbers have real part 1/2 if they are symmetric in the complex plane.

Let matrix $A$ be defined as: $\Large A(n,k)=k^{-a_k + 1/2 + ib_k}$ if $k$ divides $n$, else $A(n,k)=0$ Let matrix $B$ be defined as: $\Large B(n,k)=\mu(n) n^{a_n+1/2 -ib_n}$ if $n$ divides $k$, ...
1
vote
0answers
4 views

Convergence of a stationary iteration method for linear systems

Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, ...
1
vote
0answers
10 views

Looking for an algorithm to generate int 0- 255 when provided with an arbitrary pair of numbers between 1 - 99 (upper limit can be you your choice)

This is intended to uniquely number the links between a mesh of an arbitrary number o nodes (for my needs anywhere between 1 and 50 is OK. The upper limit can be your choice as long as it is higher ...
3
votes
0answers
17 views

When do $n+2$ points in $\mathbb{R}^n$ lies on a same $(n-1)$-sphere?

When $n=2$, the following results are well-known: Proposition 1. Let $A,B,C,D$ be $4$ distinct points in $\mathbb{R}^2$. They are aligned or cocyclic if and only if: $$\left(\overrightarrow{CA},\...
0
votes
0answers
15 views

For every prime $p$ and a fixed integer $k$, are there infinitely many values of $n$ such that $p$, $p^n+k$, $kp^n+1$ are all primes?

This is similar to my last question, but may or may not be the case: For every prime $p$ and a fixed integer $k$, are there infinitely many values of $n$ such that $p$, $p^n+k$, and $kp^n+1$ are all ...
0
votes
0answers
15 views

Subtle difference between statemetns invloving negation in set theory.

What is the difference between the statements $x$ is not in an infinite number of sets $E_n$ and ...
0
votes
0answers
5 views

Discrete non-linear map from $n$-dim cube to an $n$-dim orthogonal simplex, preserving volume

The question is very similar to Looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex I am still not 100% sure if a small variation of such answer is sufficient, or I ...
0
votes
3answers
20 views

Finding Theta in this trigonometric problem

Actually , I'm new to trigonometry .. So i want help in this question $2 \sin \theta = 2 - \cos \theta $ My attempt -> $2 \sin \theta = 2 - \cos \theta $ $2 \sin \theta = 2 - ({(1-\sin^{2}\theta)}^...
1
vote
4answers
23 views

Factorize a third degree polynomial

It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ...
0
votes
0answers
12 views

What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to do so but couldn't.
2
votes
4answers
37 views

When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
3
votes
0answers
12 views

Bounded linear operator counterexample?

Does there exist a continuous linear operator $T: (C[0,1],||.||_2) \to (C[0,1],||. ||_2)$ such that $T$ is discontinuous if $C[0,1]$ is considered with $||.||_\infty$ instead of $||.||_2$ ? I think ...
1
vote
0answers
19 views

Real analysis and Topology book recommendetion

I hold a bachelor's degree in mathematics and I have taken undergraduate real analysis course. But I'm interested to know more about it, especially the stuff that will help me to understand more of ...
1
vote
0answers
7 views

Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...
0
votes
0answers
10 views

Relations, Ordered Pairs, Naive set theory by Halmos

I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?"
-3
votes
1answer
15 views

Question of P &C

There are 3 pots and 3 coins. All thesecoins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed such that no pot ...
-1
votes
0answers
6 views

$\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$

We consider two i.i.d. random variables $U_1$ and $U_2$ such that $\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$ and $\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any $c > 0$ the ...
0
votes
1answer
17 views

Proving homotopy equivalence of a torus with points removed

Suppose I'd like to show that a Torus with $n$ points removed is homotopy equivalent to a wedge sum of $n+1$ circles. I depict it in a usual way - as a rectangle with $n$ points removed inside. Now it ...
1
vote
2answers
23 views

Notation for a sum over a set of variables

I have a vector of variables $y=(y_1, \ldots, y_n)$ whose elements are either zero or one. I would like to express the sum over all variables belonging to a subset $S$. For example, if $n=4$ and $S=\{...
0
votes
0answers
14 views

The first step in the proof of the Pólya-Vinogradov Inequality.

The well-known Pólya-Vinogradov Inequality states: $\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p$. I would like ...
1
vote
2answers
19 views

which set is including $k$

$A=\{x^2+k \mid x \in \mathbb Z,-3 \leq x<k\}$, where $k$ is a constant. If $\{6,9\}\subseteq A$, then which set below includes $k$? $\{5x+1\mid x \in \mathbb Z\}$ $\{4x+3\...
0
votes
1answer
21 views

Real roots for exponential-polynomial equations

I am trying to find the number of real roots of an equation such as $k_1 x e^x-e^{k_2 x}-k_3x+k_4=0.$ Setting the first derivative equals to zero is analytically unsolvable, unfortunately. Do you ...
1
vote
3answers
43 views

is there a non unit real matrix satisfied $A^n=I$?

If A is a real matrix and $A^{2016}$ is a symmetric positive definite matrix , prove that $A$ also is a symmetric positive definite matrix I wonder if this property is wrong and so I came up with ...
0
votes
1answer
16 views

Tournament Of The Towns King and the 1000 wizard's

So i was doing one of the question's of TOURNAMENT OF THE TOWNS and I was not able to understand the solution given by them. The problem is: The King decided to reduce his Council consisting of ...
2
votes
1answer
59 views

What is the integral of $e^x a^x$

Can you confirm that my answer below is correct? $$\int (a^x e^x)dx $$ My attempt: $$\int a^x e^x \, dx = \int (ae)^x \, dx $$ $$\int a^x e^x \, dx = \frac{(ae)^x}{\ln(ae)} + C $$ $$\int a^x e^x \,...
2
votes
2answers
27 views

Uniform continuousness: are only the “extremes” interessant? (poles and infinity)

Help me please understand the concept of uniform continuousness. $f(x)=x$ is it, and I get that. But what is with $g(x)=x^3/(x^2+1)$ (in reals) Just arguementing unrigorously, i'd say this isn't ...
0
votes
1answer
12 views

volume of surface of revolution around y axis

Can anyone help walk me through this problem style? I have a lot of homework problems like this and I really want to understand how to do these problems. Find the volume of the solid generated by ...
0
votes
1answer
11 views

what is the conditional probability that the card following it is the ace of spades?

i have seen a couple of different answers for this question, but i still cant understand why. Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a time ...
3
votes
2answers
34 views

Finite commutative ring with unity and without nilpotent elements

Let $R$ be a commutative ring with unity such that for each $x \in R$ there exists a $n \in \mathbb{N}$, $n>1$, such that $x^n = x$. Then show that $$ R\simeq F_{1}\times F_{2}\times \cdots\times ...
0
votes
0answers
14 views

Construction of the classifying space of a group $G$.

Assume that $G$ is a given (topological) group, then we know that there is an induced $G$-bundle of the form $ G \hookrightarrow EG \rightarrow BG$. I do know some properties and I do understand into ...
0
votes
2answers
12 views

Direct vs Iterative solvers choice

Is there any other reason except “the big size of matrix” that makes me prefer the use of iterative solvers than direct ones, for (linear algebraic systems)? Thanks
2
votes
1answer
34 views

Milnor's definition of smooth manifold

In Milnor's book "Topology from a differential viewpoint" on page one he defines a smooth manifold to be a subset of $\mathbb R^n$ which is locally diffeomorphic to some open subset of $\mathbb R^k$. ...
0
votes
0answers
13 views

Which probability is greater, given minimal info

which probability is greater, given that $X$ and $Y$ are independant, positive random variables? There is also the option that it's impossible to know as we don't have enough information. I'd ...
1
vote
0answers
20 views

Help in this proof from Hoffman and Kunze's Linear Algebra book

I'm reading Hoffman and Kunze's Linear Algebra book and on page 177 they stated and proved the following theorem: It's a big proof which I didn't understand only a very little part of it: I ...
0
votes
2answers
24 views

Find third coordinate for a right triangle with 45degree angles

I have a right triangle with two 45degree angles. I know the points for the two coordinates opposite the right angle. I need to calculate the missing point. I have seen similar questions here, but ...
-4
votes
0answers
29 views

proving: Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/q [on hold]

4) Let $x, y, p, q$ be positive numbers with $ 1/p + 1/q = 1$. Prove that $$xy ≤ x^p/ p + y^q/q $$
0
votes
1answer
7 views

Interpolate a rectangular surface with given edges

I need to interpolate a surface by filling a rectangular hole. The height values of the edges are given. I would like to fill the rectangular surface patch by somehow interpolating the edge values. ...
2
votes
0answers
30 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real elipsoid, an ...
3
votes
0answers
17 views

4-D lattices and quaternions

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
0
votes
3answers
25 views

Solving $(x^2-1)\ddot y-2x\dot y +2y=1$

Solving $$(x^2-1)\ddot y-2x\dot y +2y=1$$ I've solved the homogenous equation: $$y=A(x^2+1)+Bx$$ Where A and B are constants of integration, but I can't for the life of me seem to remember how to ...
0
votes
0answers
14 views

Interaction of a functor with internal hom

An additive functor between abelian categories $F: \mathscr{C} \to \mathscr{D}$ induces a functor on categories of chain complexes $F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet$. The internal hom ...
-2
votes
1answer
14 views

Uniform Integrability - different characterisation - prove (ii)

Probability with Martingales: For the 'only if' part assuming the hint is true, then I guess we have $\forall \varepsilon_1 > 0, \exists K \ge 0$ s.t. $$E[|X|1_{|X| > K}] < \...
1
vote
1answer
21 views

Given a polynomial with integer coefficients and prime independent term, show that any root has absolute value greater than 1.

I was looking at exercises about algebraic structures, and in ring theory I stumbled upon this problem. Given $p$ a prime number and $f(x)=\pm p + a_{1}x+\cdots+x^{n} \in \mathbb{Z}[x]$ so that $\...
-1
votes
1answer
9 views

Uniform Integrability - different characterisation - prove hint

Probability with Martingales: For the 'only if' part how to prove the hint? i'm guessing it's something to do with $$E[X 1_F] \le E[X1_{\Omega}]$$ $$= E[X 1_{|X| > K}] + E[X 1_{|X| \le K}]...

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