All Questions
5,857 questions
2
votes
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Uniform $L_1$ convergence implies uniform convergence pointwise a.e.
Let $\Omega$ be a measure space (which can be assumed to be an interval with Lebesgue measure).
It is well known that for a sequence $(f_n)$ in $L^1(\Omega)$ which converges to zero (in $L^1(\Omega)$,...
- 2,280
0
votes
1
answer
558
views
Is the limsup or liminf of n-wise independent events independent?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
2
votes
0
answers
114
views
Is there an explicit version of Morse Lemma used in stationary phase method?
In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
- 3,625
4
votes
2
answers
655
views
Differentiability: Partially Defined Functions
These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds.
(Cf. discussion on p. 45.)
Definition
Let $E$ and $F$ be two Banach spaces together with a plain subset $A\subseteq ...
- 598
6
votes
1
answer
409
views
Can the potential of a complete Kahler metric be bounded?
Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...
- 285
0
votes
1
answer
3k
views
Is the sum sin(n) bounded? [closed]
I wonder whether the sequence $s_{n} = \sum_{k = 1}^{n} \sin k$ is bounded.
The answer seems no, but I have no idea how to prove this from the irrationality of $\pi$.
- 1
1
vote
0
answers
448
views
Largest possible variance for log-concave distributions on a bounded interval
Let $f$ be the density of a log-concave probability distribution on the interval $[0,1]$ (with respect to Lebesgue measure). To be concrete, suppose that $f(x) = \exp( - \varphi(x))$, for some convex ...
- 251
-8
votes
2
answers
1k
views
why do we need algorithms, and why is non-convex optimization difficult? [closed]
A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...
- 1
3
votes
1
answer
222
views
Asymptotic for binomial sums
Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use ...
- 33
7
votes
1
answer
532
views
how wiggly is a generic level set?
Typical level sets of smooth real-valued functions are manifolds, so they cannot be fractals. If we coarse grain a bit though, sometimes we get space-filling behavior, eg. every point could be within ...
- 73
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 43
6
votes
2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 1,649
2
votes
0
answers
2k
views
Orthogonal complements of intersections of closed subspaces
Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$.
$\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...
- 285
0
votes
1
answer
51
views
Strict positive type function on hypersurface also of positive type in neighborhood?
Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
- 13
0
votes
1
answer
59
views
Improved maximum principle estimates (deleting first mode)
Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write
$$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$
where $ r=|x|$ and $ \theta = \frac{...
- 1,385
7
votes
1
answer
942
views
Kakeya and Nikodym maximal functions
I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more specifically,...
- 213
3
votes
1
answer
446
views
floating point representation via the perspective of TTE/computable analysis
Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ...
- 459
10
votes
2
answers
1k
views
Does Rolle's Theorem imply Dedekind completeness?
I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...
- 19.7k
2
votes
2
answers
373
views
Question on the number of equilibria
Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or ...
user31317
2
votes
1
answer
102
views
Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 115
3
votes
0
answers
104
views
Rank relation to maximum subpermanent and subdeterminant?
Given a $\pm1$ matrix $M$ of rank $r$ let the largest subdeterminant be $d$ and let the largest subpermanent be $p$.
Are there relations/bounds that connect $r$, $d$ and $p$?
Are there geometric and ...
- 13.9k
3
votes
1
answer
151
views
The weakest condition guarantees some Separation-type of convex sets in Banach spaces
Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and ...
- 369
3
votes
1
answer
681
views
measure zero in R but not in R^2
I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?
- 33
11
votes
0
answers
1k
views
How the idea of adjugate matrix has been designed? [closed]
I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...
- 219
4
votes
2
answers
3k
views
Chain rule for fractional laplacian
Does anyone know a formula of chain rule for fractional laplacian?
say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...
- 41
3
votes
0
answers
155
views
Does one need Second Order Logic to do Calculus?
Second order Logic (SL) is required to define the Reals (otherwise they were at most countable). Based on this, SL is involved in the definition of the limit operator, as the 'core' of all Calculus.
...
- 39
7
votes
2
answers
2k
views
Baire Category Theorem Application
In Antoine Henrot Michel Pierre -
Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
- 2,222
4
votes
0
answers
125
views
Properties of solution to Schrödinger equation
Given a Schrödinger equation with, let's say continuous, periodic potential
$$-y''(x)+V(x)y(x)=\lambda y(x)$$
where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
- 501
9
votes
1
answer
224
views
Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
- 19.6k
2
votes
2
answers
4k
views
a limit of the laplace transform and its derivative
If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
5
votes
3
answers
1k
views
Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.
Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
- 698
3
votes
0
answers
232
views
When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
- 34.3k
0
votes
1
answer
843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
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7
votes
1
answer
233
views
Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
- 541
1
vote
1
answer
211
views
Representation of Hilbert transform by a singular integral
Hilbert transform defines as follow:
$$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$
$$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$
Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...
- 371
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 11
0
votes
1
answer
151
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A Bi-Lipschitzian application
We say that $\Omega$ is a star-shaped domain (with respect to the origin) of $\mathbb R ^n$ if :
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x \right \|})\}\; \...
- 291
2
votes
2
answers
6k
views
Derivative indicator function
I am wondering what is the derivative of the following function with respect to $x(t)$ in sense of distributions.
$$
I\left(\int_0^t x(\tau)d\tau \leq c\right)
$$
where $I$ is the indicator function ...
- 175
0
votes
1
answer
110
views
Number theory for operator bound
Let $\gamma_i$ be such that for even $i$ $\gamma_i=1$ and for odd $i$ $\gamma_i$ shall have absolute value $1$ and the product of all of the odd ones is also on the complex unit circle but not 1 or -1....
- 501
5
votes
1
answer
481
views
A continuous path between two Sobolev functions
Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...
- 679
2
votes
1
answer
383
views
Is this a log-concave function?
Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
- 79
1
vote
0
answers
49
views
On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
2
votes
0
answers
269
views
Implicit Function Theorem, parametrized - how can we get uniform domains? (from math.se)
(This question is a duplicate from here)
Consider a family of continously differentiable functions $F_r\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (where $r\in[0,1]$). For every parameter $r$, we ...
- 141
2
votes
1
answer
250
views
Density in the Space of absolutely convergent Fourier series
It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : https://math.stackexchange.com/questions/67334/...
- 125
3
votes
1
answer
193
views
Inequality of a concave function
Let $G:\mathbb R\to\mathbb R$ be a concave function, define $G_{\epsilon}: \mathbb R\to\mathbb R$ by
$$G_{\epsilon}(x)~~:=~~\max_{y\in [x-\epsilon, x+\epsilon]}G(y).$$
My question is the following: ...
- 1,835
5
votes
5
answers
2k
views
Cardinality of Equivalence Classes of Cauchy Sequences
What's the cardinality of a single equivalence class of Cauchy sequences in ℚ?
To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
- 153
0
votes
0
answers
337
views
Pfaffian minors of skew symmetric matrix under perturbation
Suppose $A$ be a skew-symmetric matrix whose entries are positive numbers. A perturbation of $A$, $A'$, is obtained by adding another skew-symmetric matrix whose entries are positive integers.
My ...
- 493
-1
votes
1
answer
1k
views
A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
- 698
2
votes
0
answers
147
views
Interchanging limit and infinite product in Euler product for Dedekind function s=1
For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation
$$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
- 21
1
vote
0
answers
100
views
Higher order derivative of negative power of cosine function
This is a question I encountered in my own research on Generalized
Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the
basis measure for this family is
$$L\left( \...
- 984