# All Questions

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### 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 ...
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### 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, ...
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### $\Delta$-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 ...
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### 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 ...
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### $x,y,z \geqslant 0$, $x+y^2+z^3=1$, 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 ...
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### Lipschitz-type estimate… True or false?

I have two parameters $\alpha,\varepsilon>0$ and the following difference: ...
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### 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 ...
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I am wondering how to show that the following matrix has trivial kernel: $$\begin{bmatrix} 1&1&1&1&1&1 \\ s_1&s_2&s_3&s_4&s_5&s_6 \\ ... 0answers 53 views +50 ### Is right this application of Hadamard three-lines theorem for  \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}? Let the complex variable s=\sigma+it, then from the following identity valid for \sigma=\Re s>1$$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$where \zeta(s) is the Riemann Zeta function, ... 1answer 101 views +100 ### About two polynomials f,g such that f=\pm g Let R be an infinite commutative ring with unit and with characteristic zero. Assume that f,g\in R[x_1,...,x_n]  are nonzero and such that f(x_1,...,x_n)=s(x_1,...,x_n) g(x_1,...,x_n), where ... 1answer 131 views +50 ### How can we define \partial x_{i_r}^p(X_p^r)? Suppose M is a manifold and X_r^s:M\to T_r^s(M) is a section. Let P be an open set and X_1,\ldots,X_s\in\mathcal T(P) and X^1,\ldots,X^r\in\mathcal T^*(P) where \mathcal T(P) and \mathcal ... 0answers 53 views +50 ### Monte Carlo with non uniform weighting So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution p_{z}(k) over the whole \mathbb{Z}^+. We are interested in approximating p_v(v) over ... 1answer 62 views +50 ### closed path, winding number, Jordan contour If  D is a domain in \Bbb C, z_0\in \Bbb C\setminus D, and \gamma is a closed, piecewise smooth path in  D for which the winding number n(\gamma, z_0)\ne0, show that there is a Jordan ... 0answers 953 views +50 ### A constrained topological sort? Suppose that one has a directed, acyclic graph G, and each vertex v contains a (positive) value a_v. Additionally, let r be a constant. For my purposes, r>1, but this might not matter. ... 1answer 80 views +50 ### Eigenvalue of block matrix of order 2n How to find eigenvalues of following block matrix?$$P=\begin{bmatrix} A & B \\ B & A \end{bmatrix}$$Where, A=\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & \cdots & ... 2answers 191 views +50 ### Is there any solution to find a condition for f(x)=a+bx^n+cx^2-dx>0 to always hold true? Okay, I am interested to know the criteria for a function to always hold$$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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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} - ... 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. ... 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 andf_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$... 0answers 142 views +100 ### A tough coordination game Consider the following game in normal form:$$\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}$$... 2answers 74 views +50 ### Roots of Sum of Two Polynomials (with Known Roots) I am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are ... 1answer 73 views +50 ### On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions Let \phi : C \to D be an injective ring homomorphism such that \phi(1)=1, where 1 denotes the constant function 1 and C,D are the rings of continuous and differentiable functions on ... 2answers 220 views +50 ### A proviso in l'Hospital's rule L'Hospital's Rule states that$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$can be applied when: (1) f, g are differentiable; (2) g'(x) \neq 0 for x near a (except ... 1answer 243 views +100 ### Surface area from indicator function I know that the volume and the surface area of a sphere of radius R are related by a derivative:$$V(R)=\frac{4}{3}\pi R^3A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$I am asking if an ... 0answers 154 views +50 ### How is this fact implicated? Let K be a field and \overline{K} its algebraic closure, then we define the n-dimensional affine space as$$\mathbb{A}^{n}=\{(x_1, \ldots, x_{n})\mid x_1, \ldots, x_{n} \in \overline{K}\}So ... 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 ... 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 ... 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, ... 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 ... 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 ... 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 ... 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! 1answer 171 views +50 ### Definition of Infinite Nesting? I have read about nested radicals like\sqrt{a+\sqrt{a+\cdots}},$$and they define the expression as the limit of sequence defined by a_1=\sqrt a and a_n=\sqrt{a+a_{n-1}}. Why instead isn't it ... 3answers 68 views +50 ### General Principles of Solving Radical Equations What are the general ways to solve radical equations similar to questions like \sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x ... 1answer 64 views +50 ### Analytic continuation of power series on the unit whose terms tends to 0 This problem is from complex analysis. Set$$f(z)=\sum_{n=0}^{\infty}a_nz^n$$with convergence radius of 1, and$$\lim_{n \to \infty}a_n=0$$Prove that if z_0 \in \partial B(0,1) is not a singular ... 0answers 29 views +50 ### can a convex polygon have only one boundary point at locally maximum distance from its centroid? It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ... 0answers 29 views +150 ### Recurrence relation for Fourier-Legendre series. So for the function f(x) = \exp(-x) I have the formula for the coefficients of$$f(x) = \sum_{n=0}^{\infty}a_n P_n(x)$$which is(by using Rodrigues formula)$$a_n = \frac{2n+1}{2} ...
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 ...