All Questions

0
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1answer
2 views

Congruencies with different moduli

How can I solve x congruent 4 (mod 5) x congruent 9 (mod 7) If my answer needs to be mod 35??
0
votes
0answers
13 views

Show that $deg(f\cdot g)=n+m$

I started learning about rings and I was asked to proof some claims. I don't understand how I may prove the last one. I have proven that if $f$ and $g$ are polynomials over some ring of polynomials, ...
-1
votes
1answer
17 views

Maximum negative value?

What is the maximum negative value known in mathematics? How can I reach that maximum negative value on StackOverFlow? Thank You.
0
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0answers
5 views

Association, Commutation and Identity Elements on Binary Operations?

Is the following closed, associative or commutative? f(a, b) = (a+b)/2, where a, b ∈ Z. I found that it is not closed but I am not sure how to find whether or not it is associative (I was confused ...
0
votes
2answers
8 views

how to proceed ? Have I correctly transform the question to an equation?

Amongst Teaching staff of ABC university the ratio of men and women is 5 to 2. Amongst the women three seventh are not married . If the number of married women teacher is 56 then the total number of ...
0
votes
2answers
14 views

Which solution is the right one??

If we want to solve the equation $sec^2(x)$ for finding the all roots(real and complex), we have two ways: 1-Direct solving for $sec^2(x)=0$ 2-Or by convert the above equation to polynomial series as ...
0
votes
0answers
7 views

Matrix challenge

We know that, for v a vector and P, O a linear operator v'=P$^{-1}$v and O'=P$^{-1}$OP Prove these relationships. I've found that these are equivalent to: v=Pv' and O=PO'P$^{-1}$ And $$ |v>=\sum ...
0
votes
0answers
7 views

Enumerate all the combinations given a constraint

Assume there are $N$ non-negative integer numbers: $a_1,\ldots,a_N$. How can one enumerate all the possible combinations of $(a_1,\ldots,a_N)$ which satisfy the following inequality: $\sum _{n=1}^N ...
0
votes
0answers
6 views

Conformality of a map

A conformal mapping is a map $f:U\to V$ with $U,V\subseteq\mathbb{C}$ such that the angles are locally preserved. This can be reformulated saying the jacobian matrix is everywhere a scalar multiple of ...
0
votes
2answers
8 views

Gradient of a vector field

We did in lectures gradient of a scalar field and I am wondering how is the grad of a vector field. I tried the following Let V=f(x,y,z)i+g(x,y,z)j+h(x,y,z)k Then by definition grad(V)=iVx + jVy +kVz ...
0
votes
1answer
19 views

$x \cdot x$ in inner product space is a quadratic form

Given an inner product space with some inner product $\cdot$ , how can I prove that $x \cdot x$ for any vector $x= (x_1,... x_n)$ is a quadratic form in $x_i$? I know how to recover an inner product ...
0
votes
2answers
13 views

Proof by induction that certain number is an integer

Proof that the number $\frac{2n^5}{5} + \frac{n^4}{2} - \frac{2n^3}{3} - \frac{7n}{30}$ is an integer $\forall n \in \mathbb{N}$
0
votes
2answers
19 views

$\sum{\frac{1}{\sqrt{4n+1}}+\frac{1}{\sqrt{4n+3}}-\frac{a}{\sqrt{2n}}}$ converges?

Can we find a constant $a$ such that $\sum{\frac{1}{\sqrt{4n+1}}+\frac{1}{\sqrt{4n+3}}-\frac{a}{\sqrt{2n}}}$ converges? Try: I am trying to compare the n th term with $\frac{c}{\sqrt{n}}$ where c is ...
0
votes
0answers
5 views

Find the distribution function for random vector and what if it has discontinuities

Let $X,Y$ be random variable. Suppose $Z:=(X,Y)$ be random vector and $P(X=Y)=1$ and $P(a \le X \le Y) = b-a$ with $0 \le a \le b \le 1$ Question 1: what is the distribution function of $Z$? ...
0
votes
1answer
14 views

sum of two equal digit numbers vs. sum of those digits

if I take 5688+6984=12672 then sum the result 1+2+6+7+2=18 then sum that result 1+8=9. vs. this. same digits from above. 5+6+8+8+6+9+8+4=54 then sum that result 5+4=9. using this method where the ...
0
votes
0answers
4 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
0
votes
3answers
18 views

Norm of a functional

I'm facing the problem of calculating the norm of the following functional: $\displaystyle \phi : L_p([0,1]) \rightarrow \mathbb{R}, ~~ \phi (f) = \int\limits_{0}^{1} e^x f(x) dx $ I have no idea ...
0
votes
3answers
15 views

Isomorphisms in finite abelian groups 1

True of false? If G and H are two groups with the same order and both are abelian, then they are isomorphic.
2
votes
1answer
37 views

Inequality $(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$

Let $n\geq 2$, and $x_1,x_2,\ldots,x_n>0$. Show that $$(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$$ For $n=2$, this reduces to ...
0
votes
0answers
5 views

Why does discretization work to estimate Mutual Information

Suppose I want to compute the mutual information (MI) between two variables, $X$ (continuous) and $Y$ (making no assumptions about its distribution) using their (differential) entropy ($H$) : ...
0
votes
0answers
5 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
0
votes
1answer
3 views

finding clusters in a network from eigengaps

I have a usual Laplacian matrix, which describes a network. From the matrix I get the eigenvalues and from these I can compute a metric of modularity in my network based on the largest eigengap. Let's ...
0
votes
0answers
10 views

Integeration of 2nd derivative

I have a question regarding solving the integral of the partial. Here is the equation: $$R_m = \int \psi^m \frac{\partial^2}{\partial\psi^2} \left[\left\langle \epsilon_\phi | \psi \right\rangle ...
0
votes
0answers
9 views

Integrability vs. Estimate

Given a finite measure space $\mu:\Sigma\to\mathbb{R}_+$. Consider measurable functions $f:\Omega\to\mathbb{C}$ and $g:\Omega\to\mathbb{C}$. Then the equivalence holds: ...
0
votes
0answers
4 views

estimation of condition number for column equilibration

I have trouble with the following problem: Let $A$ be an invertible square matrix. Let $D$ be the diagonal matrix with entries $d_j=\dfrac{||A||_1}{\sum_i a_{i,j}}$. Show that $||D||^{-1}_\infty ...
1
vote
1answer
28 views

Why am I obtaining an imaginary part for my integration

I try to solve an integration as follows, $$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$ as you can see its pretty complicated, and I get an answer like the following using Wolfram ...
1
vote
0answers
9 views

Does this notion of the “directed area” of a closed curve in $\mathbb R^3$ have a standard name?

Given an oriented surface $\Omega$ in $\mathbb R^3$, consider the quantity $\mathbf A=\int_\Omega\hat n\,\mathrm dA$. We may call this the "directed area" of the surface because, when $\Omega$ is ...
1
vote
0answers
8 views

Outer measure is not finitely additive

I know similar questions have been asked before, but I'm looking for clarification of a proof. In Royden's book on real analysis, he proves that every set of positive measure contains a non-measurable ...
0
votes
1answer
14 views

CHECK: Let $G=\frac{(\mathbb{Z},+)/12\mathbb{Z}}{3\mathbb{Z}/12\mathbb{Z}}$. How many elements are there in $G$?

Let $G=\frac{(\mathbb{Z},+)/12\mathbb{Z}}{3\mathbb{Z}/12\mathbb{Z}}$. How many elements are there in $G$? Write them down explicitly. \end{prob} By the 2nd Isomorphism Theorem, ...
0
votes
1answer
14 views

Simple Question: division of sums

I am a little bit confused about the following simple task. Given some functions $f(x), g(x), h(x), l(x), m(x)$. We know that $\frac{f(x)}{g(x)}= m(x)$. We further know that $h(x), l(x)$ are ...
1
vote
1answer
10 views

Intersection of two lines and the minimum of the sum of the two.

We use a formula in my Operations Research class for finding the 'Economic Order Quantity', given the cost function (sum of Holding and Ordering costs) $$C = \frac{Q}{2}H+\frac{D}{Q}S$$ where $Q$ is ...
3
votes
0answers
9 views

Range of vectors that turn into eigenvectors after recursive multiplication by a matrix

Suppose $\mathbf{x}$ is a vector, and $\mathbf{A}$ is a square matrix. Which $\mathbf{x}$'s will satisfy the equation $\mathbf{A}^n\mathbf{x} = \lambda\mathbf{A}^{n-1}\mathbf{x}$, where $\lambda$ is ...
0
votes
1answer
10 views

Bound on function

Suppose a set $|I_{r}| \leq O(r^{2})$ but also $|I_{r}| \geq \rho^{r}$ where $\rho=(1+\epsilon).$ Why is it true that there must always be an $r^{*}=f(\rho)$ such that the condition is true for all ...
0
votes
2answers
24 views

Continuous function at [-1,1] not differentiable at infinite points at [-1,1].

Is anybody that can answer the following exercise?: "Give a continuous function at [-1,1] not differentiable at infinite points at [-1,1]." Thank so much!
0
votes
3answers
18 views

extending definition of a function to make it continuous

I have two functions on hand, namely $$f(x) = x^3\cos\left(\frac1x\right)$$ and $$f(x) = \frac{1-\cos(x)}{x^2}$$ I would like to know where are these functions discontinuous, and how do I extend the ...
0
votes
1answer
20 views

Subgroups of the simple groups [duplicate]

True of false? Every subgroup of a simple group is itself simple. May i also get some examples on this as well to verify my answer.
1
vote
1answer
13 views

Find relations on the real number: transitive and/or antisymmetric

$$I\ am\ searching\ for\ a\ relation\ on\ the\ real numbers\ (\mathbb R ),\ which\ sould\ be:$$ antisymmetric and transitive antisymmetric and NOT transitive NOT antisymmetric ,but ...
0
votes
3answers
23 views

Multiplicative inverse of 5 modulo 8 [on hold]

Can someone help me with this? What is the multiplicative inverse of 5 modulo 8?
1
vote
2answers
22 views

Swimming Problem - How can I do this?

I have a problem with the following diagram. The triangle is just a possible path and may not be the shortest. If the boy has to get to the destination in the fastest way possible: What ...
1
vote
0answers
12 views

Trace norm identity (in bra-ket notation)

I came across the following identity in a paper: $$ \|\hspace{0.3em}|v\rangle\langle v| - |w\rangle\langle w|\hspace{0.3em}\|_{tr}=2\sqrt{1-|\langle v|w\rangle |^2}$$ where the norm on the left is ...
0
votes
0answers
6 views

Looking for resources: Generalizations of martingales to $\mathbb R^2$

In most introductory courses, a martingale $Y$ is defined as a stochastic process $$Y: T \times \Omega \to S$$ ,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration ...
1
vote
3answers
14 views

Conceptual problem in solving quadratic equation

The sum of all real roots of the equation $$|x-2|^2 + |x-2| - 2 = 0$$ is? I tried this problem by taking two cases $x<2$ and $x>2$ and solving the corresponding equations and I got $8$ as the ...
5
votes
1answer
24 views

A problem on the sum of the reciprocals of two derivatives

If $f(x)$ is continuous in the closed interval $[a,b]$ and differentiable in the open interval $a<x<b$, and if $f(a)=a$, $f(b)=b$, prove there exist points $x_1$ and $x_2$ with ...
0
votes
0answers
21 views

problem in linear algebra [on hold]

Prove: If $A$ is invertible, then $AB^{-1}$ and $1+BA^{-1}$ are both invertible OR both not invertible
0
votes
1answer
20 views

Find the values of k for which $\sqrt{1+\frac{k}{n}}$ is irrational.

I would like to find the positive integers $k$ for which $\sqrt{1+\frac{k}{n}}$ is irrational for all $n\in\mathbb{N}$. I was led to this question when I was making up an example for my class, and I ...
0
votes
0answers
6 views

Specific Type of Dominated Convergence (Spectral Measures)

Reference See Birman and Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, chapter 5 subparagraph 4.1, page 133... Question It is introduced a specific type of convergence, ...
-2
votes
0answers
17 views

domain of a complex analysis function 3 [on hold]

what is the domain of the function $f(z) = \frac{6z+1}{z+i} ?Prove that $f(z)$ has an inverse function and determine the inverse function and its domain.
0
votes
1answer
10 views

Robinson arithmetic and its incompleteness

Wikipedia in Italian has a sketch-of-proof that Robinson arithmetic is not complete, since commutativity of addition is undecidable. The sketch of proof creates a model that adds two elements, $a$ and ...
2
votes
2answers
20 views

indefinite integral and inequality

Let $$ f(x) \leq g(x), \forall\, x.\hspace{0.5cm} (1)$$ Moreover, considering the indefinite integrals $$\int f(x)\,dx= F(x) + C_1 \hbox{ and } \int g(x)\,dx = G(x) + C_2.$$ My question: If we ...
2
votes
2answers
18 views

Why is $\vec{s}=\frac{\vec{r}}{V^\frac{1}{3}} \Leftrightarrow d\vec{s}=\frac{d\vec{r}}{V}$?

I am following a course which contains a part in statistical thermodynamics. One of the questions involves the partition function $Q_N$. I could not figure out the answer of the question myself, so I ...

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