4
votes
0answers
83 views
+50

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
4
votes
2answers
132 views
+50

Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, ...
1
vote
0answers
166 views
+50

$\Delta$-complex structure of the cone and the suspension

I am going around in circles trying to answer the following question: Let $Y$ be a $\Delta$-complex. Describe a $\Delta$-complex structure of its cone $CY=(Y\times[0,1])/(Y\times\{0\})$ its ...
0
votes
0answers
32 views
+50

Holomorphic semigroup can be extended to a strongly continous semigroup on $L^{p}$?

I have a question about $C_{0}$-semigroup theory. Let $(H,(\, , \,))$ be a real Hilber space and $(H_{\mathbb{C}}, (\, , \,))$ the its complexification. Any linear operator $(L,D(L))$ on $H$ can be ...
0
votes
0answers
40 views
+100

Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
3
votes
2answers
87 views
+50

How does a short exact sequence say something about a group?

I have a follow-up question to my question here: How are simple groups the building blocks? In that question I asked about what it means when we say that the simple (finite) groups are the building ...
12
votes
2answers
403 views
+100

Dividing books between two couples

Two couples of boys and girls, $(b_1,g_1)$ and $(b_2,g_2)$, are dividing a pile of books. Every book will go to one of the couples, and they'll read it together. Each person has a (nonnegative) value ...
1
vote
1answer
38 views
+50

Find limits of integration for the interior region of sphere with center $(a,0,0)$ and radius $a$ using spherical coordinates

I am asked to find limits of integration for the interior region of sphere with center $(a,0,0)$ and radius $a$ using spherical coordinates. How can one do that? I know that one may use $$ x = r ...
20
votes
6answers
463 views
+50

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$ This inequality has been verified to be correct according to Mathematica. $\frac12$ is not the best bound. I try to do AM-GM ...
1
vote
2answers
83 views
+100

Lipschitz-type estimate… True or false?

I have two parameters $\alpha,\varepsilon>0$ and the following difference: ...
1
vote
0answers
43 views
+50

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
5
votes
2answers
136 views
+50

Kernel of a Vandermonde like matrix

I am wondering how to show that the following matrix has trivial kernel: $$\begin{bmatrix} 1&1&1&1&1&1 \\ s_1&s_2&s_3&s_4&s_5&s_6 \\ ...
3
votes
0answers
53 views
+50

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
-2
votes
1answer
101 views
+100

About two polynomials $f,g$ such that $f=\pm g$

Let $R$ be an infinite commutative ring with unit and with characteristic zero. Assume that $f,g\in R[x_1,...,x_n] $ are nonzero and such that $f(x_1,...,x_n)=s(x_1,...,x_n) g(x_1,...,x_n)$, where ...
3
votes
1answer
131 views
+50

How can we define $\partial x_{i_r}^p(X_p^r)$?

Suppose $M$ is a manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ and $\mathcal ...
3
votes
0answers
53 views
+50

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...
1
vote
1answer
62 views
+50

closed path, winding number, Jordan contour

If $ D$ is a domain in $\Bbb C$, $z_0\in \Bbb C\setminus D$, and $\gamma$ is a closed, piecewise smooth path in $ D$ for which the winding number $n(\gamma, z_0)\ne0$, show that there is a Jordan ...
14
votes
0answers
953 views
+50

A constrained topological sort?

Suppose that one has a directed, acyclic graph G, and each vertex $v$ contains a (positive) value $a_v$. Additionally, let $r$ be a constant. For my purposes, $r>1$, but this might not matter. ...
5
votes
1answer
80 views
+50

Eigenvalue of block matrix of order $2n$

How to find eigenvalues of following block matrix? $$P=\begin{bmatrix} A & B \\ B & A \end{bmatrix}$$ Where, $A=\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & \cdots & ...
3
votes
2answers
191 views
+50

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
1
vote
1answer
63 views
+100

Existence of operator

I want to show that for $ s> \frac{1}{2} $ there is a bounded linear operator $ T: H^s(\mathbb{R}^n) \to H^{s-\frac{1}{2}}(\mathbb{R}^{n-1})$ following the below steps: Consider that $ u \in ...
2
votes
0answers
67 views
+50

Inequality with analytic functions on the unit ball

Let $g(z) = \sum_{n\geqslant 0} a_nz^n$ be an analytic function where $a_n$ only take values in $\{0,1\}$ (not sure if it is a necessary condition, it is just the case I'm considering). Let ...
0
votes
0answers
25 views
+50

What is the number of interior faces adjacent to an interior vertex in a triangulation in $\mathbb{R}^3$?

Let $\Omega$ be a polygonal domain in $\mathbb{R}^3$. Assume $\Omega$ is partitioned into tetrahedra using the most common admissible triangulation, that is, roughly speaking, two adjacent tetrahedra ...
1
vote
1answer
38 views
+50

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
2
votes
0answers
37 views
+50

Two questions about Li-Yau-Hamilton estimate

Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I can't see $Q\ge 0$ when $t=0$. Besides, how to ...
3
votes
0answers
76 views
+50

Intuitively what is the second directional derivative?

I'm thinking that the second directional derivative, if both dd's are evaluated in the same direction, will just give you the concavity (the second scalar derivative) in that direction. Is that ...
3
votes
3answers
211 views
+50

Definition of $\frac{dy}{dx}$ where we cannot write $y=f(x)$

Informally, I can say that "for a small unit change in $x$, $\frac{dy}{dx}$ is the corresponding small change in $y$". This is however a bit vague and imprecise, which is why we have a formal ...
6
votes
1answer
69 views
+50

How to evaluate this limit about Bernoulli number?

First,we define $\displaystyle I_{1}\left ( x \right )=\frac{\sin x}{x}$, then $\displaystyle \lim_{x\rightarrow 0^+}I_{1}\left ( x \right )=1$, also we have \begin{align*} I_2\left ( x \right ...
0
votes
0answers
55 views
+100

Are there bounded nonconvergent sequences satisfying this recurrence relation?

In this question, we ask about convergent sequences $(p_n)$ satisfying $p_{i} = 2p_{i+6} - {1 \over 2}p_{i+1} - {1 \over 4}p_{i+3} - {1 \over 8}p_{i+7} - {1 \over 16}p_{i+15} - {1 \over 32}p_{i+31} - ...
2
votes
2answers
103 views
+200

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
2
votes
1answer
46 views
+50

Representation of the Fréchet derivative of $〈f,e_n〉$, where $f:H→H$, $H$ is a Hilbert space and $(e_n)_{n∈ℕ}$ is an orthonormal basis of $H$

Let $H$ be a $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $f:H\to H$ be Fréchet differentiable and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$ ...
5
votes
0answers
142 views
+100

A tough coordination game

Consider the following game in normal form: $$\begin{bmatrix} & x_B=0& x_B=1 \\ x_A=0& 1-\theta_A,1-\theta_B & 0,0 \\ x_A=1 & 0,0 & 1+ \theta_A,1+\theta_B \end{bmatrix}$$ ...
4
votes
2answers
74 views
+50

Roots of Sum of Two Polynomials (with Known Roots)

I am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are ...
2
votes
1answer
73 views
+50

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
4
votes
2answers
220 views
+50

A proviso in l'Hospital's rule

L'Hospital's Rule states that $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ can be applied when: (1) $f$, $g$ are differentiable; (2) $g'(x) \neq 0$ for $x$ near $a$ (except ...
7
votes
1answer
243 views
+100

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
-1
votes
0answers
154 views
+50

How is this fact implicated?

Let $K$ be a field and $\overline{K}$ its algebraic closure, then we define the $n$-dimensional affine space as $$\mathbb{A}^{n}=\{(x_1, \ldots, x_{n})\mid x_1, \ldots, x_{n} \in \overline{K}\}$$ So ...
1
vote
2answers
174 views
+100

Show that the sequence is exact

We have that $R$ is a commutative ring. Suppose that $0\rightarrow A\rightarrow B\overset{f}{\rightarrow} C\rightarrow 0$ and $0\rightarrow C\overset{g}\rightarrow D\rightarrow E\rightarrow 0$ are ...
3
votes
0answers
59 views
+50

Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e ...
0
votes
1answer
33 views
+50

Initial Value Problem for $(\cos x -x\sin x +y^2)dx + 2xy\,dy =0$, $y(\pi )=1$

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following question. If it's any indication of difficulty, the exercise is only Part 1-B of the sheet, ...
2
votes
0answers
85 views
+50

Multiplication of unitary matrices to make symmetric off-diagonal elements zero

Context Starting with a unitary matrix $U$ of size $m \times m$, I have read of a way to obtain a diagonal matrix by sequentially multiplying $U$ from the right by unitary matrices $V$ of a certain ...
0
votes
2answers
62 views
+50

intersection of boundaries have Lebesgue measure 0

Suppose $A$ and $B$ are two sets in $R^n$ such that $\overline{A}\cap B \cup \overline{B}\cap A$ is empty then $\partial A \cap \partial B$ has $n$-dimensional Lebesgue measure $0$ (where $\partial ...
2
votes
2answers
34 views
+50

Proof that a discrete space (with more than 1 element) is not connected

I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its ...
-2
votes
0answers
12 views
+50

Does the initial prior affecct the asymptotic performance of Kalman filter, why?

More specifically, can the observed data can gradually correct the incorrect initial prior? Thank you very much!
3
votes
1answer
171 views
+50

Definition of Infinite Nesting?

I have read about nested radicals like $$\sqrt{a+\sqrt{a+\cdots}},$$ and they define the expression as the limit of sequence defined by $a_1=\sqrt a$ and $a_n=\sqrt{a+a_{n-1}}$. Why instead isn't it ...
4
votes
3answers
68 views
+50

General Principles of Solving Radical Equations

What are the general ways to solve radical equations similar to questions like $\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$ ...
2
votes
1answer
64 views
+50

Analytic continuation of power series on the unit whose terms tends to 0

This problem is from complex analysis. Set $$f(z)=\sum_{n=0}^{\infty}a_nz^n$$ with convergence radius of 1, and $$\lim_{n \to \infty}a_n=0$$ Prove that if $z_0 \in \partial B(0,1)$ is not a singular ...
4
votes
0answers
29 views
+50

can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ...
0
votes
0answers
29 views
+150

Recurrence relation for Fourier-Legendre series.

So for the function $f(x) = \exp(-x)$ I have the formula for the coefficients of $$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$ which is(by using Rodrigues formula) $$a_n = \frac{2n+1}{2} ...
3
votes
1answer
38 views
+50

Jensen-like averaging inequality on integers

Let $\mathbb{Z}^*=\mathbb{Z}^+\cup\{0\}$. Let $f:\mathbb{Z}^*\rightarrow\mathbb{R}$ be a nondecreasing function such that $f(a+b)\leq f(a)+f(b)$ for all $a,b\in\mathbb{Z}^*$. Is it true that for all ...

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