All Questions
6,014 questions
0
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0
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99
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Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
8
votes
1
answer
376
views
Is this inequality in two variables true?
It it true that for all $p\in(0,1/3]$ and all real $t$ we have
$$4
\ln(1-p +p\cosh t)
\ln\frac{1+\sqrt{1-2p}}{1-\sqrt{1-2p}}
\le t^2 (1+c p) \sqrt{1-2p} ,$$
where $c:=2\sqrt{3}\, \ln(2+\sqrt{3})-3$?
...
- 128k
0
votes
1
answer
139
views
A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$
Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, ...
- 1,295
3
votes
1
answer
401
views
What does the Jacobian of a vector field at an equilibrium tell you about local behavior of integral curves when the Jacobian is not a stable?
I have a soft question regarding the Jacobian of vector fields and isolated equilibria, and what they imply about local behavior of nearby integral curves near.
Let $V:U \subset_{open} \mathbb{R}^n \...
- 518
1
vote
2
answers
163
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
- 515
2
votes
1
answer
281
views
Global control of locally approximating polynomial in Stone-Weierstrass?
Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\...
- 463
16
votes
3
answers
1k
views
A kernel 'more analytic' than $\exp(-x^2)$
I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits_{k=0}^{\infty} |c_k| \varepsilon^k (2k)! < \...
- 1,295
1
vote
3
answers
159
views
Estimating the integral $\int_{\epsilon}^1 \Bigl\lvert \int_0^x \frac{f(y)}{\lvert x-y\rvert^{1/2}} dy\Bigr\rvert^2 dx$ for $L^2$ function $f(y)$?
I guess the chances are slim but still curious about the integral in the title.
Let $f : [0, \infty) \to \mathbb{R}$ be a locally "square-integrable" function on $[0,\infty)$.
Then, for any $...
- 3,477
0
votes
0
answers
101
views
Does the tensor product of mollifiers work for $L^{p,q}$ spaces?
Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively.
For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \...
- 3,477
0
votes
0
answers
94
views
Oscillation of a polynomial
Recently I came across a statement in a paper that I am unable to verify. Namely, it roughly says that the oscillation of a polynomial on a cube can be controlled by the oscillation of the polynomial ...
- 1,045
2
votes
1
answer
219
views
Is the Lipschitz constant of $f$ equal to $\|\nabla f\|_{L^\infty}$?
Let $f : \mathbb R^d \to \mathbb R$ be Lipschitz and $[f] := \sup_{x,y \in \mathbb R^d; x\neq y} \frac{|f(x) - f(y)|}{|x-y|}$ its Lipschitz constant. By Rademacher theorem, $f$ is differentiable a.e., ...
- 825
2
votes
0
answers
134
views
Extensions of Euler–Maclaurin formula
There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function $f(x)$ to be continuous, but there are several ways to extend the formula to ...
- 67
0
votes
0
answers
138
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
3
votes
0
answers
124
views
Leibniz rule bound for the inverse of the Laplacian?
Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\...
- 3,477
1
vote
1
answer
185
views
Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$?
Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1_\text{loc}$ function.
Then, I wonder if the following series
\begin{equation}
\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert
\, dt
...
- 3,477
0
votes
1
answer
92
views
If $f(x,t)=\sum_{n \in \mathbb{Z}} a_n(t) e^{in x}$ is $C^\infty$ in $x$ and all $a_n(t)$ continuous, $x$ derivatives of $f$ are continuous in $t$?
This question seem a bit elementary, but I find it more subtle than its looks. So, I post the question here.
Let $f(x,t) : [0,2\pi] \times [0,1] \to \mathbb{C}$ be a function such that $f(0,t)=f(2\pi,...
- 3,477
1
vote
1
answer
197
views
Does convolution with heat kernel converge to pointwise evaluation?
Let $G(t, x) := \frac{1}{\sqrt{4 \pi t}} \exp\left( -\frac{x^2}{4 t }\right)$ for all $(t, x) \in (0, T) \times \mathbb{R}$ be the fundamental solution to the heat equation $\partial_tu = \partial_{...
1
vote
1
answer
145
views
Convolution with the Jacobi Theta-function on "both the space and time variables" - still jointly smooth?
Let $\Theta(x,t)$ be the Jacobi-Theta function:
\begin{equation}
\Theta(x,t):=1+\sum_{n=1}^\infty e^{-\pi n^2 t} \cos(2\pi n x)
\end{equation}
Usually, the heat equation with the periodic boundary ...
- 3,477
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
1
vote
0
answers
167
views
Sobolev embedding into the space of bounded, uniformly continuous function?
Let $\Omega$ be a sufficiently smooth open domain of $\mathbb R^d$. Is any embedding of the Sobolev spaces $W^{s,p}(\Omega)$, $s>0$, into the target space $C^0(\overline{\Omega})$ (the space of ...
- 3,388
3
votes
1
answer
108
views
$L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality
For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}...
- 3,477
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
3
votes
0
answers
52
views
Closely related definitions with and without approximation built-in
Let us say that a (real) function class $A$ has 'approximation built-in' in case for every $f:\mathbb{R}\rightarrow\mathbb{R}$ in $A$ and any $x\in \mathbb{R}$, we can approximate $f(x)$ using only $f(...
- 4,359
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
4
votes
1
answer
254
views
$\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$
Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$
Compute $f(1)$ and $f(2)$.
2
votes
0
answers
103
views
Schwartz kernel theorem for restricted operators
Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
- 1,171
3
votes
1
answer
219
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
5
votes
1
answer
508
views
Generalized Wigner 3-j symbol and Legendre functions
Let $P_{n}(x)$ the $n-th$ Legendre polynomial. It is well-knonw that $$\int_{-1}^1 P_n(x) P_m(x) P_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{...
- 219
2
votes
1
answer
127
views
Partition of unity of simplex
Let $$\chi_S(x,y)=\begin{cases}1&\text{ if }0< x<y< 1\\0&\text{ else }\end{cases}$$
be the indicator function of the simplex $S=\{(x,y)\in (0,1)^2:x<y\}$. I am interested in an ...
- 1,904
1
vote
2
answers
180
views
An inequality for a real function
Let $$f(z)=(1+z)^{3/4}-\left(\frac{3}{8}+\frac{\sqrt{3}}{4}\right)^{1/4}-\frac{\left(3 z+\sqrt{6} \sqrt{-1+z^2}\right)^{3/4}}{\left(2 \left(2+\sqrt{3}\right)\right)^{3/4}}.$$ Is there a simple proof ...
- 37
3
votes
1
answer
379
views
Convergence of a power series
Consider the numbers $$a_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B_k$ are the Bernoulli ...
- 463
10
votes
1
answer
1k
views
Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent?
So, I ask whether from the ZFC axioms one can prove X that every uncountable set has strictly more than continuum many subsets, or whether X is independent of the ZFC axioms. Note that (within ZFC) ...
- 3,584
5
votes
0
answers
141
views
Maximum of a function
Let $p,q\in\Bbb N$ with $p\not=q$. Put $$M=\sup_{x\in[0,1]} \left|\cos(2 p\pi x)-\cos(2 q\pi x)\right|.$$
What is the value of $M$.
Thanks
- 521
2
votes
1
answer
179
views
Definition of integral over level sets in coarea formula
This is probably a simple question, maybe more suited for MSE. In the coarea formula, you have
$$\int_{{\mathbb{R}}^n} g (x) |\nabla f(x)|\, dx= \int_\mathbb{R} \left(\int_{\{f=t\}} g d \mathcal{H}^{n-...
- 697
0
votes
2
answers
140
views
Two-Sided Bounds on Binomial Sum
I came across this partial sum which I cannot find reasonable bounds on; I feel this must be known in the literature, but I do not know where to look. Here is the problem:
Let $s\in (0,1)$ and ...
- 364
0
votes
1
answer
175
views
Asymptotic of ratio between l1 / l2 norm of a structured vector
As suggested in this discussion, I would like to inquire about the following question:
Consider a matrix B of size $n\times n$ defined as:
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\...
- 405
0
votes
1
answer
127
views
asymptotic of ratio between two summations (l1 / l2 norm)
Let $B$ as a $n\times n$ matrix where
$$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\theta_j), 1\leq i<j\leq n$$ and other entries equals to $0$, and $$\theta=[\theta_1,\cdots,\theta_n]\...
- 405
1
vote
1
answer
233
views
Continuity of a rational function
This is a simple question. Given a real valued rational function
$$
f (x) = \frac{p(x)}{q(x)}\quad x\in\mathbf R^N,
$$
this is called regular on a point if the denominator $q$ does not vanish there. ...
- 29
0
votes
2
answers
197
views
Convergence of the infima of convex functions on $\mathbb{R}^m$
Any thoughts on proving the following statement, which is a generalization of the result in convergence of the infima of convex functions from domain $\mathbb{R}$ to $\mathbb{R}^m$ and also Theorem 1 ...
1
vote
0
answers
89
views
Pre-images of the critical point of $3.83 x(1-x)$
This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
- 333
6
votes
3
answers
536
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.2k
1
vote
1
answer
76
views
Upper bounds for the spatial differential of the inverse of a flux
It is well known that given a regular velocity field $b: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ (say, continuous in time and uniformly Lipshitz in space), the flux $X$ associated to $b$ is a ...
- 697
5
votes
1
answer
366
views
Quantitative Lebesgue density theorem
Let $A \subset [0, 1]$ be a measurable set, and $\mathbf 1_A$ its indicator function, viewed as a function on $\mathbb R$. Define for each $\delta > 0$, the function $f_{A, \varepsilon}: \mathbb R \...
- 6,223
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
3
votes
1
answer
166
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,689
3
votes
0
answers
75
views
Separate holomorphicity implies holomorphicity on analytic varieties
Suppose that $M$ and $N$ are two complex analytic varities and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
- 513
2
votes
1
answer
117
views
When is a $p$-th order stationary point of a polynomial actually a local minimum?
Definition: For integer $p\geq 1$, we say $x\in \mathbb{R}^d$ is a $p$-th order stationary point of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ if there exists a $C>0$ and an $\epsilon>0$ ...
- 637
4
votes
1
answer
288
views
A lower bound for the $L^1$ norm of real trigonometric polynomials
This question is somewhat similar to Minimizing the L1 norm of odd-term trigonometric polynomial. The context of the question is based on the paper Hardy's Inequality and the $L^1$ norm of Exponential ...
- 270
2
votes
1
answer
186
views
Local equality of functions implies global equality?
The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
- 1,117
0
votes
0
answers
102
views
Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \Delta u=0, $$
the quantity $|\nabla u|^2$ is subharmonic, i.e.
$$\Delta (|\nabla u|^2) \geq 0. $$
Reason:
$$\Delta (|\nabla u|^2)= 2 \nabla (...
- 371