0
votes
0answers
3 views

Measure on Product Set

Consider a finite sequence of $\sigma$-finite measure spaces $(\Omega_i, \mathcal{F}_i, \mu_i)$. Constructing the product measurable space $$ (\Omega_1 \times \cdots \times \Omega_n, \mathcal{F}_1 ...
0
votes
2answers
20 views

After switching a lamp on and off infinitely many times in one minute, is it on or off?

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
0
votes
1answer
7 views

Eligibility for geometric progressions containing few terms

Question: Does there exists a GP containing 8, 12 and 27 as three of its terms? If it exists how many such progressions are possible? What I have tried: First of all I have made three equations ...
-5
votes
0answers
17 views

Can you answer these Polynomial questions please? [on hold]

I want to ask some Q(s). 1.) Find the values of p and q if (x+3) and (x-4) are the factors of x3-px2-qx+24 2.) Factorize :- a.) x2+x/4-1/8 b.) (x-a) 3+(x-b) 3+(x-c) 3 where x=(a+b+c)/2 c.) ...
0
votes
0answers
4 views

Autocorrelated, discrete, bounded and symmetric random walk with no edge attraction

I need to move over a discrete set of linearly organized.. let's say "Japan steps" $S=\{0,\dots,c\}, c \in \mathbb{N}^*$. My current position is given by $d \in S$. On each time step, I need to draw ...
0
votes
0answers
6 views

Is the saturation function globally Lipschitz?

I have a question for you. Is the following function sat(s) = su_0/||s|| for all ||s||\geq u_0 s for all ||s||\leq u_0 where u_0>0 and ||.|| denotes the L^2 norm for instance, ...
0
votes
1answer
7 views

Manifolds: show that this map is not a coordinate patch

Let $S^1$ be the subset of $\mathbb{R}^2$ given by {$(x,y)|x^2+y^2=1$}. We all know that $S^1$ is a 1-manifold in $\mathbb{R}^2$. I'm trying to prove that the following map:$$\alpha:[0,1) \to ...
0
votes
0answers
10 views

Is $H^2(\Omega)$ compactly embedded on $H_0^1(\Omega)$?

Considering $\Omega$ bounded and $\partial \Omega$ smooth. I already know that $H^2(\Omega)$ is continuously embedded on $H_0^1(\Omega)$, thus if I take a bounded sequence in $H^2(\Omega)$ it is also ...
1
vote
2answers
23 views

Big-O vs. Best Big-O

Is there a difference between the method to find a big-O function and the method to find the best big-O function. Take for example the following function: $f(n) = 1 + 2 + 3 + ... + n$ It is easy to ...
1
vote
0answers
13 views

Probablity: Is my way of thinking correct?

Problem Consider the model such as: The computer has not infected with any virus in the initial state. Every morning, the computer has infected with an new virus with a probability of $p$ ($0 < ...
0
votes
0answers
7 views

Conway’s Functional Analysis, VIII §3 Exercise 11

This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a): For ...
1
vote
0answers
8 views

Dimension of a measure in terms of linear form on continuous functions

So this might be a bit of a weird question, but here goes. It is well-known (Riesz representation theorem) that the dual space of continuous functions on a compact $K$ identifies with the space of ...
0
votes
0answers
10 views

Divisor sum function for integral values

Let $d(n)$ be the number of divisors of a positive integer $n$. From the analytic theory it is known that the sum function of $d(n)$ up to non-integral $x$ is given by ...
0
votes
1answer
21 views

Epsilon-Delta proof of lim(x->2) x^2=4

I have seen and understand the delta-epsilon proof of the limit of x^2 for x->2, such as explained here: https://www.youtube.com/watch?v=gLpQgWWXgMM Now I am wondering, is there also another way? How ...
0
votes
1answer
16 views

Solving a quadratic equation with complex coefficient

Express $z4$=-$\sqrt{3}$+i in polar form. Hence solve the equation $Z^2$=$z4$ for $z$ a complex number. You may leave the answer in polar form. My answer: $z4$ in polar form is 2cis-30$^{\circ}$ and ...
0
votes
2answers
15 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ I get an answer of $\frac ...
1
vote
0answers
13 views

Fibre of a local homeomorphism can be covered by disjoint open sets.

Let $f\colon X\rightarrow Y$ be an open local homeomorphism and $y\in Y$. Do there exist pairwise disjoint open neighborhoods $U_x$ for $x\in f^{-1}(y)$? If not, what would be mild topological ...
0
votes
0answers
10 views

Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, ...
1
vote
1answer
23 views

An open interval as a union of closed intervals.

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I ...
0
votes
0answers
4 views

Generalized eigenvalue problem equivalence

I would like to show that there exists a Laplacian matrix $H$ for every Laplacian matrix $L$ such that $$Lx = \lambda D x \equiv Lx = \lambda Hx,$$ where $$D=\sum_{i=1}^n d_ie_ie_i^\top,$$$e_i$ is the ...
1
vote
2answers
15 views

Relationship between vectors in convex quadrilateral

We have the triangle $ABC$. $M$ is the center of the line segment $BC$. $D$ is a point in the triangle's plane so that $ABDC$ is a convex quadrilateral. $N$ is center of the line segment $AD$. ...
-1
votes
0answers
13 views

probability class 12

Three groups of children contain 3 girl and 1 boy;2 girls and 2 boys;and 1 girl and 3 boys. One child is selected at random from each group.find the chance that the three children selected comprise 1 ...
2
votes
0answers
13 views

Existence of differentiable functions on $\mathbb R$ whose derivative is constant on the complement of uncountable set but not everywhere

Let $ A $ be a countable subset of the set of real numbers and $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f'$ is constant on $\mathbb R \setminus A$ , then I know that $f'$ is ...
0
votes
0answers
5 views

Usage of Phase Portrait of a system of 2 linear first order ODEs

Let's say have a linear system $\frac{\mathrm{d}\underline{y}}{\mathrm{d}t} = A\cdot \underline{y}$, let say 2 dimensional, and I have $\lambda_1,\lambda_2$ eigenvalues of $A$ and ...
0
votes
0answers
7 views

How to drive the membership function in fuzzy clustering nean

I am learning about Fuzzy c-means (FCM) which is a method of clustering which allows one piece of data to belong to two or more clusters. This method (developed by Dunn in 1973 ) is frequently used in ...
0
votes
2answers
50 views

Integral of cos(1/x) dx

Is the following integral expression correct (neglecting the constant of integration)? $$ \int\cos\left(\frac{1}{x}\right)dx = x^2\sin\left(2x\right) $$ When I take the derivative, it returns to the ...
2
votes
2answers
35 views

Why is the integral starts from $0$?

Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$ It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem: $$\lim_{x\to 1^-} f(x) = ...
0
votes
0answers
10 views

Tensor product of representations of a product group?

Given some group $G$ that can be written as product of two other groups $$G = G_1 \times G_2 $$ and some representation of this group written in terms of representations of $G_1$ and $G_2$ $$R = ...
1
vote
0answers
20 views

Homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$

Let $\phi_1$ and $\phi_2$ be two injective ring homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$. Show that there exists a $g\in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\phi_1(x)g^{-1}$ for all ...
0
votes
1answer
18 views

Intervals of integers modulo n

Do the following related concepts appear anywhere in literature? Denoting an "interval" in the integers modulo $n$ by $[i,j] = \{i, i+1, \dotsc, j\}$. For example, in modulo 6, $[5,3] = ...
-4
votes
1answer
28 views

problem on circles

Find the radius of the circle which touches both the coordinate axes, and touches the line x/a + y/b = 1, $a>0,b>0$, and its centre lies in the first quadrant.
7
votes
0answers
66 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
1
vote
0answers
17 views

A simplified formula for area of triangle when equations of the sides are given

For i = 1, 2, and 3, let $a_ix + b_iy + c_i = 0$ be three equations of 3 (non-special cased) straight lines. From which, the co-ordinates of the vertices can be found. Using these co-ordinates, via ...
1
vote
1answer
30 views

What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...
1
vote
0answers
28 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
0
votes
1answer
35 views

Determine the digit in a consecutive sequence of numbers

All positive integers are written in order, one after another $$1234567891011121314151617...$$ Which digits appears in the 206 787th position?
-3
votes
2answers
38 views

What is n (Natural number) if the function has to have a limit not equal to zero or infinite?

$$\lim_{x \to 0} \frac{(\tan(x))^n - x^n}{x^6}$$ What is n (Natural number) if the function has to have a limit not equal to zero or infinite?
0
votes
1answer
7 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
0
votes
0answers
6 views

$K_0$ of a ring via idempotents

As it is described here, the group $K_0(R)$ of a unital ring can also be described in termes of conjucation classes of idempotents. In this text it is shown that the semigroup of isomorphism classes ...
5
votes
3answers
59 views

Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges

Let $x_n, y_n > 0$. I need an example of convergent sequences $x_n,y_n$ such that $x_n^{y_n}$ diverges. Could you help me?
-1
votes
0answers
15 views

Find MGF of random variable X

We are given rth raw moment i.e. $E(X^r)=(r+1)!* (2^r)$. We have to find MGF of random variable $X$. so plzz provide me a solution to this problem.
1
vote
0answers
18 views

Q($\sqrt[3]{2}$) - Unique Factorisation Domain?

I am considering the set of "integers" of the from $$ a+b\sqrt[3]{2} + c\sqrt[3]{4} $$ where $a,b,c$ are integers. It is easy to show this field is closed under addition and multiplication. I then ...
0
votes
1answer
43 views

Prove the limit is $e^\alpha$

prove that $\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=e^\alpha$ $$\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=\lim_{n \to \infty} \left(\left(1+{\alpha\over n}\right)^{n\cdot ...
0
votes
0answers
17 views

About definition of Sobolev spaces

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a open subset and $dx$ be a Lebesgue measure on $U$. We often define \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
-1
votes
0answers
9 views

About Imre Hermann's book 'Parallelismes'

Apparently, in his book Parallelismes, Imre Hermann discusses Hilbert, Brouwer en Russell from the viewpoint of psycho-pathology. Does anyone know whether the entire book is about ...
0
votes
0answers
21 views

When calculating joint probabilities using double integrals…

When calculating joint probabilities using double integrals, do we use $dx\ dy$ or $dy\ dx$ ? I thought it was the former, but then my book abruptly changes to using $dy\ dx$ without an explanation ...
0
votes
0answers
11 views

Total derivative of a complex function using Wirtinger derivatives

The mathworld.wolfram page on Cauchy-Riemann Equations states that given a complex function $f(z) = f(x,y) = u(x,y) + iy(x,y)$ has the derivative: $$\frac {df} {dz} = \frac {\partial f} {\partial x} ...
0
votes
1answer
14 views

How to prove infinite solution vs no solution for singular matrix problem.

In the problem Ax=B My coefficient matrix is \begin{bmatrix} α-1 & 1-α\\ α & -α \end{bmatrix} x is \begin{bmatrix} ln K\\ ln L \end{bmatrix} b ...
0
votes
0answers
16 views

Integration of convolution

I'm trying to solve the following equation $$\int\limits_{-\infty}^t \,(f\ast g)(t')dt'.$$ $f$ could be a kind of $\delta$-function: $f(t) = \delta(t)$ but should not be limited to be one. $g$ is ...
0
votes
1answer
8 views

How to find maximal dimension abelian subalgebra in finite Lie Algebra?

Is there any well known algorithm how to find maximal dimension abelian subalgebra in finite dimension Lie Algebra? If there is a built-in routine in some computer algebra system, it is the most ...

15 30 50 per page