0
votes
0answers
1 views

Properties of expectation values

I have a complicated problem in which I have to compute the expectation in order to see if my estimator is unbiased or not. After a lot of calcutions at the end I have found: $$ ...
1
vote
0answers
5 views

About the smoothness of a non-reduced scheme.

Consider a $k$-scheme $X$ with the following properties: $k$ is algebraically closed, $X$ is irreducible, non-reduced, separated and of finite type ($X$ is a non reduced variety). I want to show ...
0
votes
1answer
6 views

Divide matrix using left division

In matlab, I defined a=[1;2;3] b=[4;5;6] both a and b are not square matrix. and execute a\b will return ...
0
votes
1answer
11 views

Convergent sequence as series, maximum of sequence as limit

I'm currently studying for my math exams. I came across two exercises about sequences and series for which I have no clue. So any hints would be appreciated. First problem: $(a_n)_{n\in\mathbb{N}}$ ...
0
votes
0answers
10 views

Find $o (\frac{G}{Z (G)}) $

Let G ={${a^k, a^k.b|0\le k\lt 9} $} s..t o(a)=9 and o(b)=2 and ba=$ a^{-1}b $, if Z (G) denotes center of group G. Then find oorder of G/Z(G). I ve lot of time its not getting solved. In book answer ...
0
votes
2answers
12 views

Property of a differentiable function

Which one of the following is true: 1.If a function real valued function $f$ satisfies $|f(x)-f(y)|\leq |x-y|^{\sqrt2}$ for all $x,y\in \mathbb R$ is $f$ a constant? 2.If $f$ is ...
1
vote
3answers
19 views

Intermediate field extension

Consider de field $\mathbb F_2$ of two elements. Let $\theta$ be a zero of the irreducible polynomial $t^4+t+1$ and consider the extension $\mathbb F[\theta]$. May question is: Can we find a ...
-1
votes
1answer
15 views

Find range of $p$ such that the series converges

let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent. My try: to show the series it is ...
1
vote
1answer
17 views

Splitting an integral

Why is the following equality true? $$ \int_1^{2e} \left| \ln x - 1 \right| dx = \int_1^e(1-\ln x) dx + \int_e^{2e} (\ln x - 1) dx$$
0
votes
1answer
9 views

Finding flux across surface

Let S be surface {$(x,y,z) : x^{2} + y^{2} + 2z =2 . z \geq 0 $} Given F = $(y,xz,x^{2}+y^{2})$ n is outward normal .I have to find net flux through S . Since its closed surface so i applied Gauss ...
5
votes
3answers
27 views

Finding $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$

Find $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$. I tried using l'Hospital rule with the continuity of $e$ function. Also tried using Taylor expansion with no success. What ...
0
votes
0answers
10 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
1
vote
0answers
16 views

Couple of questions from Takeuti's Proof Theory book

I am reading Gaisi Takeuti's Proof Theory (Second Edition, Dover), and I have a couple of questions: I) Right after the first (1.1.) definition, the author says that "In any case it is essential that ...
7
votes
2answers
46 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

A group $G$ and a function $f:G\longrightarrow G$ are given and for all $x,y\in G$ satisfying $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?
0
votes
1answer
15 views

Show that $G$ contains elements $a,b$ s.t. $a^2=b^3=e$ and $ aba=b^2=b^{-1}$

Given that $|G|=6$ and is not commutative. I've showed that generators of $G$ have periods either $2$ or $3$, since $|H_a|=\text{period of generator a}$ divides $|G|$. Since $G$ is not commutative it ...
2
votes
1answer
21 views

How to show that a polynomial maps an algebraic set to an algebraic set?

Sorry if this is an ignorant question. I am studying algebraic geometry. This isn't an exercise problem. It is an assumption I can use to prove something else. I think it must be obvious, but I don't ...
0
votes
2answers
27 views

no. of all ordered tuples (x,y,z) such that x,y,z are all positive integers that satisfy the equation x + 2y + 3z = 30?

How do I find the number of all ordered tuples (x,y,z) such that x,y,z are all positive integers that satisfy the equation x + 2y + 3z = 30 ? Is there any easy and less time taking method to solve ...
0
votes
0answers
12 views

Isolate points of a metric space with some properties?

Suppose that all dense subspace of a metric space $(X,d)$ is open. Prove that the set of the isolate points of $X$ is dense in $X$. My Idea: all isolate points of $X$ are in any dense subspace, ...
0
votes
0answers
7 views

Find the image of a vector

I have an endomorphism and I have to study the image of a vector $v$. How can I do this? Can you please give an example? I know how to calculate the inverse image but I have many doubts on the ...
0
votes
1answer
9 views

Short proof of sequential Banach Alaoglu for Hilbert spaces

Do you know of a short proof of the fact that bounded sequences in Hilbert spaces admit weakly converging subsequences? If the space is separable, then the common sequential-version proof is what I ...
1
vote
1answer
15 views

On the greatest norm element of weakly compact set

Let $X$ be a Banach space and $K\subset X$ be a nonempty weakly compact set. I would like to know if there exists a point $u_0\in K$ such that $\|u_0\|\geq \|u\|$ for all $u\in K$. Thank you for all ...
2
votes
0answers
7 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
0
votes
2answers
17 views

Find x,y,z where multiplication of them equals 36 and sum equals to the square of the sum of two of them

I need to find three numbers x, y, z where: 1) the multiplication of all these numbers equals 36 2) the sum of these three equals to the square of the sum of the two. The question goes if there enough ...
0
votes
0answers
13 views

Relations commuting with logical equivalence.

I'm looking for theorems of the form, 'Relation X commutes with logical equivalence', where X is NOT uniform substitution. What's the best place to find such theorems?
0
votes
1answer
11 views

Partition of not-so-distinguishable objects into indistinguishable bins

Every textbook on combinatorics seems to deal with either totally indistinguishable objects and bins, or completely distinguishable objects and bins. What I have is something in between: objects are ...
0
votes
3answers
44 views

Use induction to prove that sum of the first $2n$ terms of the series $1^2-3^2+5^2-7^2+…$ is $-8n^2$.

Use induction to prove that sum of the first $2n$ terms of the series $1^2-3^2+5^2-7^2+…$ is $-8n^2$. I came up with the formula $\displaystyle\sum_{r=1}^{2n} (-1)^{r+1}(2r-1)^2=-8n^2$ but I got ...
0
votes
0answers
5 views

Drift of Brownian motion conditioned on Hitting Time

Suppose we have a Brownian motion started from height b>0, with constant negative drift $\lambda$. We can 'calculate' the drift in the following seemingly ridiculous way. We condition on the first ...
1
vote
0answers
15 views

If $a+bi$ is in $E_k$ then $a-bi$ is also in $E_k$?

I'm currently studying the properties of the Motzkin sets $E_k$, $k\in\mathbb{N}\cup\{0\}$ of the ring $\mathbb{Z}[i]$. The definition of $E_k$ is as follows: $E_0=\{0\}$, $E_1=$units of ...
0
votes
1answer
33 views

$f$ is differentiable at $x_0 \in \dot{I} $

Let $f\in \mathbb{R}^{I}$ and $x_0 \in \dot{I} $ (adherent point ) Show that $$\left.\begin{matrix} f \text{ is continuous at } x_0 \\ f \text{ is differentiable at all } x \in I \setminus ...
0
votes
0answers
16 views

Understanding why the private exponent $e$ is chosen the way it is in RSA

I am trying to get a better understanding of RSA. At the moment I am unable to understand the difference between the correctly chosen value of the private exponent $e$ and other possibilities ...
0
votes
0answers
15 views

Is there a relation to determine condition (positive or negative definite) of C, if C = A+B and A, B are positive and negative definite?

I have a question: Matrix A and B are positive and negative definite, respectively. Is there a relation to determine whether C is positive or negative definite, if C = A+B?
1
vote
0answers
16 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
2
votes
1answer
8 views

Given a number of vertices , a radius, and rotation calculate vertices' coordinates for regular polygons

So, I know half the answer to this but I don't know how to adjust it for rotation. I believe formula the below is correct if I did not have to take into account rotation. $r \cos(2 \pi i / n) = y$ ...
0
votes
1answer
6 views

Counting Inversion Pair using Merge Sort

An inversion is a pair of places of a sequence where the elements on these places are out of their natural order. I understood the naive approach where we take an element from 1 list and compare with ...
-4
votes
0answers
22 views

This question is based on sum of series. [on hold]

What will be result of $\sum\left ( 1+\left ( \frac{\lambda }{\xi } \right )^{-1} \right )$ please help to find it.
1
vote
1answer
33 views

Show how $\frac{\partial}{\partial x} \left[\int_0^x (x-t)g(t)\,\mathrm{d}t\right] = \int_0^x g(t)\,\mathrm{d}t$

It has something to do with the second part of the Fundamental Theorem of Calculus right? I've always had trouble with this theorem ever since I learned it several years ago :\ Would somebody please ...
1
vote
1answer
11 views

Chromatic number and vertex covering number

For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. Let $\chi(G)$ denote the chromatic number of $G$. Is there a graph $G$ with $\tau(G) < \chi(G) - 1$?
0
votes
0answers
9 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
0
votes
0answers
16 views

About the $d\mathbf{s}$ notation

Let $\mathbf{F}$ be a vector field that changes with time, that is, written in components:$$\mathbf{F}(\mathbf{x},t)=(F_1(\mathbf{x},t),F_2(\mathbf{x},t),F_3(\mathbf{x},t))$$ where ...
1
vote
1answer
9 views

Struggling with connection between Clifford Algebra (/GA) and their matrix generators

As I thought I understood things, the Gamma matricies behave as the 4 orthogonal unit vectors of the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb C)$, (also the Pauli matricies are for the 3 of ...
0
votes
1answer
29 views

Showing $\sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty$ implies $f_{n}\rightarrow f$ almost everywhere.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then ...
2
votes
1answer
39 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
1
vote
0answers
13 views

Find Formal Proof on Loci Theorems

Please help me to prove this theorems of loci Theorem 12 The locus of a point at a given distance from a given line is two lines parallel to the given line and at the given distance from it. Theorem ...
-2
votes
0answers
6 views

Dynamical systems,forward invariant

Show that the complement of a forward invariant set is backward invariant, and vice versa. Show that if f is bijective, then an invariant set A satisfies f t (A)= A for all t. Show that this is false, ...
0
votes
1answer
22 views

First-order nonlinear differential equation

How would I solve this differential equation for $y(x)$? $\frac{dy}{dx} = \frac{y-xy}{x-xy}$ $y -\ln(y) = x - \ln(x) + C$ I'm not sure what to do at this point. I looked it up on WolframAlpha and ...
2
votes
1answer
18 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field field of one variable $x$ where $$y^2 = f(x) \in K[x]$$. We want to compute the full constant field of $F$ ...
1
vote
3answers
22 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
0
votes
0answers
10 views

Skewness of a difference of random variables?

In this article( http://www.diva-portal.org/smash/get/diva2:302313/FULLTEXT01.pdf )page 28 explains how to derive the skewness of a sum of random variables; I haven't been able to derive this ...
2
votes
3answers
32 views

Combination Problem: Arranging letters of word DAUGHTER

The number of ways in which we can form a 8 letter word from the letters of the word DAUGHTER such that all vowels are never occur together is My approach: As ...
0
votes
0answers
11 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...

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