All Questions

12
votes
0answers
150 views
+50

A characterization of Henselian rings

It is well known that if $(A, \mathfrak m)$ is a Henselian local ring with residue field $\kappa$, then base-change from $A$ to $\kappa$ determines an equivalence of categories $$F: \{\text{Finite ...
18
votes
1answer
453 views
+50

An interesting formula for $\pi$

Looking through some old notebooks I found this monster of a formula: For any integer $r>1$, we have $$\pi=(-1)^{\left\lfloor\frac{r}{2}\right\rfloor-\left\lfloor\frac{2r-1}{4}\right\rfloor}\...
8
votes
1answer
153 views
+100

Closed form expression or asymptotic expansion for (periodic) generalized harmonic numbers?

In contrast with the series $\sum_{k=1}^n k$ and $\sum_{k=1}^n1$, there does not (as far as I know) exist a pure closed form expression (or a nice asymptotic expansion other than the Euler-Maclaurin ...
6
votes
1answer
137 views
+50

Difference between Bernoulli random variables

Given are $n$ independent Bernoulli random variables with parameters $p_1,\dots,p_n$. We want to split them into two parts so as to minimize the expectation $\mathbb{E}[|X-Y|]$, where $X$ is the sum ...
6
votes
0answers
98 views
+50

Bounding the number of certain (translation) subsets in $\mathbb N \times \mathbb N$ with respect to given subsets

Let $\mathbb N = \{0,1,2,\ldots\}$ be the monoid of natural numbers with zero. Suppose $S \subseteq \mathbb N \times \mathbb N$ be some subset such that the number of sets of the form $\{ (i,j) \mid (...
2
votes
1answer
61 views
+50

Number of points inside a shape on the Cartesian plane

I was wondering if there is a formula which can help in finding number of points on a shape on the Cartesian plane, knowing that the shape isn't a rectangle nor a square. For example: consider an ...
0
votes
1answer
64 views
+50

Optimal Transport: Showing Map is Optimal (simple question: more of an analysis/measure theory question)

I'm reading these lecture notes (not for homework, for fun) http://www.maths.gla.ac.uk/~gbellamy/LMS/BourneLectures.pdf I'm trying to figure out what it means for a transport means to be optimal. I'm ...
0
votes
0answers
78 views
+50

How does this convolution algorithm work mathmatically

I stumbled across this code which describes how you can construct smooth functions with compact support. Unfortunately, I'm not familiar with the programming language used so I can only guess what ...
3
votes
0answers
57 views
+50

Semisimple linear algebraic group

Let $G$ be a linear reductive algebraic over an algebraically closed field of characteristic $0$. I know that there is a surjective map with finite Kernel $$G' \times T \to G$$ where $G'$ is ...
3
votes
3answers
670 views
+50

Proving that the focus of a parabola lies on the circumcentre of a triangle

I recently came across the fact that if a parabola touches the three sides of a triangle then the focus of such a parabola lies on the circumcircle of the above triangle. I tried to prove it but ...
-2
votes
0answers
53 views
+50

Prove periodicity is a class property

Prove that if state $i$ in a class has period $p$ then all states in that class have period $p$. The proof is given on this answer is this: One way to define the period of state $i$ is as the ...
2
votes
2answers
89 views
+50

Prime ideals of $R[x]$ that intersect $R$ in $P$

Let $R$ be a noetherian ring and $P$ a prime ideal of height $h$. Show that the prime ideals $Q\subset R[x]$ that intersect $R$ in $P$ are of the following two kinds, with height as shown: $Q=...
0
votes
0answers
51 views
+50

How to calculate quantity of Hamilton cycles

If I have $n^2$ vertices and each vertex is adjacent to $2n-2$ vertices, how may I calculate the quantity of possible Hamilton cycles? Would I need to modify the computation if I stipulated that I'm ...
4
votes
0answers
135 views
+50

Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity .

I was working on this Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$ when I have discovered the following identity : $$\Bigg|\Big(\frac{1}{2}\Big)^{\frac{x}{8}}\pm\Big(\frac{1}{4}\Big)^{\frac{x}{8}}...
2
votes
0answers
39 views
+200

Is the consequence relation of a finite set of boolean connectives finitely generated?

Let $S$ be the set of all n-ary functions on $\{0,1\}$ for all n, including the 0-ary functions. Let $F$ be a finite subset of $S$. Consider a countably infinite set of propositional constants $PROP$. ...
3
votes
0answers
64 views
+50

Integral form of the conservation law $u_t+f(u)_x=0$

Consider the conservation law given by $$u_t+f(u)_x=0$$ We know that in general weak solutions are not smooth but are bounded in $L^{\infty}$ norm (they do not belong to Sobolev spaces). However ...
7
votes
1answer
171 views
+300

How do I prove that the second derivative of a function $f:M\to\mathbb{R}$ defined on a surface $M\subset\mathbb{R}^n$ is well defined?

QUESTION. How do I prove that the second derivative of a function $f:M\to\mathbb{R}$ defined on a surface $M\subset\mathbb{R}^n$ is well defined? QUESTION. If, as says the user Amitai Yuval, ...
24
votes
3answers
566 views
+100

Find function $f(x)$ satisfying $\int_{0}^{\infty} \frac{f(x)}{1+e^{nx}}dx=0$

I am looking for a non-trivial function $f(x)\in L_2(0,\infty)$ independent of the parameter $n$ (a natural number) satisfying the following integral equation: $$\displaystyle\int_{0}^{\infty} \frac{f(...
2
votes
2answers
161 views
+150

Evaluation of ratio of two binomial expression

If $\displaystyle A = \sum_{k=0}^{24}\binom{100}{4k}.\binom{100}{4k+2}$ and $\displaystyle B = \sum_{k=1}^{25}\binom{200}{8k-6}.$ Then $\displaystyle \frac{A}{B}$ $\bf{My\; Try::}$ For evaluation of $...
5
votes
0answers
155 views
+100

Book recommendation for learning image processing as an application of Fourier analysis

I have searched around this website for some references of applications of Fourier transform in image processing, but did not find any satisfactory ones. I have a major in maths. In years I always ...
2
votes
0answers
41 views
+50

Proving a property of a solution to a set of nonlinear polynomial equations

Consider the following system of equations for $R_{i}(\lambda)$ \begin{align} R_1 &= \frac{\lambda}{4}(1 + R_3 + 2 R_2 R_1) \tag{1.1}\\ R_2 &= \lambda \left[q + \left(\frac{1}{2} - q\right)...
1
vote
0answers
38 views
+50

On proving that if $X,Y$ are CW-complexes and $f : A \to Y$ is continuous with $A \leq X$, $f(A^n) \subset Y^n$, then $X \cup_f Y$ is a CW-complex.

As the title says, I have a question regarding the following exercise, Let $A,X,Y$ be CW-complexes with $A$ a subcomplex of $X$. If we have $f : A \to Y$ such that $f(A^{(n)}) \subset Y^{(n)}$ for ...
0
votes
1answer
30 views
+50

Integral of a differential 2-form in a parametrized surface

I want to calculate the integral $\mathop{\iint}_{\phi}xdx\land dy+ydy\land dz+z dz\land dx$ considering $\left\{ \phi\left(u,v\right):=\left(u+v,uv,u^{2}-v^{2}\right):u,v\in\left[0,1\right]\right\} $ ...
2
votes
2answers
81 views
+50

How to see if a subgroup is normal from Cayley graph

Let be a Cayley diagram of group $G$. Let $H$ be the orbit of element p. Is $H$ a normal subgroup of $G$? Is there a simple way to check that because going by definition seems complicated. I tried ...
2
votes
3answers
85 views
+50

For $\langle Tx, x \rangle \geq \|x\|^2$, prove a solution exists for $Tx = y$.

Edit I've posted this a couple other times, so I now plan on deleting those, and just using this one. Here's the original problem: $H$ is a real Hilbert space. Let $T: H\longrightarrow H$ be a ...
6
votes
1answer
106 views
+100

Uniqueness of spanning tree on a grid.

When I was at the Graduate Student Combinatorics Conference earlier this month, someone introduced me to a puzzle game called Noodles!. The game starts with a collection of "pipes" on a grid (...
2
votes
0answers
61 views
+50

If $f_n:\Omega\to\Omega$ are homeomorphisms of a planar domain $\Omega$ such that $f_n\to f$, $f_n^{-1}\to g$ in $L^1$, is $f=g^{-1}$?

In Marchioro and Pulvirenti's book Mathematical Theory of Incompressible Nonviscous Fluids, the proof of global well-posedness of the 2D Euler equation in a bounded domain $\Omega\subset\mathbb R^2$ ...
1
vote
1answer
135 views
+100

Fitting points to curve $g(t) = \frac{100}{1+\alpha e^{-\beta t}}$ by thinking about projections and inner products

This is a reinterpretation of my old question Fit data to function $g(t) = \frac{100}{1+\alpha e^{-\beta t}}$ by using least squares method (projection/orthogonal families of polynomials). I need to ...
2
votes
1answer
62 views
+50

Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
4
votes
1answer
56 views
+50

Is this stochastic Picard iterator well-defined?

Preliminaries Let $x_0 \in \mathbb{R}^d$. Let $T \in (0, \infty)$. Let $$ \sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$$ and $$\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}$$ be affine ...
1
vote
1answer
96 views
+50

Find the numbers of ordered array $(a,b,c,d)$ such $a^2+b^2\equiv c^3+d^3\pmod p$

Let $p$ be prime number,and such $p\equiv 1\pmod {12}$,Find the numbers of ordered array $(a,b,c,d)$ that satisfies the following conditions: (1):$a,b,c,d\in \{0,1,2,\cdots,p-1\}$ (2):$a^2+b^2\equiv ...
6
votes
1answer
93 views
+50

Topological Algebraic Independence of power series

Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers: $a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^...
4
votes
1answer
94 views
+50

Finding the area enclosed by curve defined by $\arcsin x+\arcsin y=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})$

If $\arcsin x+\arcsin y=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})$ Then the area represented by the locus of point $(x,y)$ if it is given that $|x|,|y|\leq 1$ My Try: Put $x=\sin \alpha$ and $y=...
2
votes
1answer
56 views
+50

Space of vector measures equipped with the total variation norm is complete

Let $\Omega$ be a set $\mathcal A\subseteq2^\Omega$ with $\emptyset\in\mathcal A$, $E$ be a $\mathbb R$-Banach space and $\mu:\mathcal A\to E$ be additive. Now, for $A\subseteq\Omega$, let $$|\mu|(A):=...
4
votes
0answers
48 views
+50

Matrix decomposition into 2x2 elementary transforms

Rotations matrices can be decomposed into a product of n(n-1)/2 elementary rotations operating on only two coordinates. Similarly, can any square matrix be decomposed into a product of n(n-1)/2 ...
2
votes
0answers
35 views
+50

Conditionnal entropy : intuitive interpretation

Consider two system $X$ and $Y$ described by probabilities distribution. We define the conditionnal entropy of $X$ knowing $Y$ as : $$S_{X|Y}=\sum_y p(y) \left( - \sum_{x} p(x|y) \log(p(x|y)) \right)...
3
votes
1answer
66 views
+50

Does this type of ODE satisfies unicity? ( right hand side f(t,y) = g(t) y, with g in L1 )

I have the following ordinary differential equation on $(0,1)$ $$ \dot y(t) = g(t) y(t) \quad \text{ almost everywhere in } (0,1) $$ Where $g(t)$ only satisfies being integrable ($\int_0^1 |g(x)|dx &...
6
votes
1answer
116 views
+150

Understanding Optimal Transport in One Dimension.

I'm trying to understand these lecture notes. https://sites.ualberta.ca/~mathirl/IUSEP/IUSEP_2018/lecture_notes/Pass1.pdf I understand the formulation of the Monge Problem. However, I'm having trouble ...
1
vote
0answers
57 views
+200

Sheaf Cohomology vs Singular Cohomology in Locally contractable Space

Let $X$ be a space and we denote by $\mathbb{Z}_X$ the sheaf of local constant sections on $X$. We are going to compare the sheaf cohomology $H^i(X,\mathbb{Z}_X)$ with singular cohomology $H^i(X,\...
15
votes
6answers
523 views
+50

Very indeterminate form: $\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x \longrightarrow (\infty-\infty)^{\infty}$

Here is problem: $$\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x$$ The solution I presented in the picture below was made by a Mathematics Teacher I tried to solve this Limit ...
1
vote
1answer
143 views
+50

What is $PQ^{-1}$ if $P=\int^{\pi}_{0}\frac{\sin(994 x)}{\sin x}\sin(1332x)\,dx$ and $Q=\int^{1}_{0}\frac{x^{338}(x^{1988}-1)}{x^2-1}\,dx$?

If $\displaystyle P=\int^{\pi}_{0}\frac{\sin(994 x)}{\sin x}\cdot \sin(1332x)\,dx$ and $\displaystyle Q=\int^{1}_{0}\frac{x^{338}(x^{1988}-1)}{x^2-1}\,dx$. Then $P\cdot Q^{-1}$ is Try: put $x=e^{i\...
2
votes
0answers
35 views
+50

non-uniformly hyperbolic definition

I'm working on Uniformly hyperbolic finite-valued $SL(2,R)$ -cocycles( article from Arthur Aveila and Jairo Bochi)and at the beginning of my researches,i want to know the exactly meaning of the title ...
7
votes
1answer
90 views
+50

Why is the permutohedron simple?

I am working with the permutohedron in $\mathbb{R}^n$ which is defined as the convex hull of $n!$ vectors as follows: $$\Pi_n := conv\{(\sigma(1), \ldots, \sigma(n))\ |\ \sigma \text{ permutation of }...
3
votes
0answers
27 views
+50

Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative? More precisely, I'd like to see an example of a function $$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$ ...
6
votes
0answers
38 views
+50

Heuristic on Sobolev and BV functions

Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field. A heuristic that I've heard frequently is the following: $f$ is almost Lipschitz on a large "good" set but ...
0
votes
1answer
67 views
+50

Using finite differences to solve BVP

I have the following ODE $$ u'' = -(1 + e^u), u(0)=0, u(1)=1$$ Using a second order accurate finite difference I obtain $$ -(1+e^{u_i}) \approx \frac{ u_{i+1} - 2 u_i + u_{i-1} }{h^2} $$ and $...
3
votes
1answer
94 views
+250

Using collocation method to solve a nonlinear boundary value ODE

I have the following ODE $$ u'' = -(1 + e^u), \quad u(0)=0,\quad u(1)=1$$ I want the divide the interval $[0,1]$ into $n-1$ equal subintervals each with length $h=1/(n-1)$ and we take approximate ...
4
votes
1answer
62 views
+50

Show convergence of an algorithm within $m$ steps

I am trying to show that the following algorithm outputs the solution to the problem $Ax=b$. Assumptions $A$ is symmetric positive definite of size $n \times n$ with $m$ distinct eigenvalues. The ...
0
votes
1answer
43 views
+50

Three statements regarding a normal and a tangential plane

I am new to calculus, and was given the following question to answer. I have worked out an answer, but am not 100% sure about the details. Any feedback would be great! Many thanks in advance. Given ...
6
votes
1answer
255 views
+50

Calculate $\int_{0}^{2\pi} \frac{\cos((2n+1)t)}{\cos(t)}dt$

Calculate the following integral for $n \in \mathbb{Z}$ with the residue theorem $$\int_{0}^{2\pi} \frac{\cos((2n+1)t)}{\cos(t)}dt$$ So far I have tried two approaches. Firsty, for $n\geq 0$: $$\...

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