All Questions
5,909 questions
2
votes
2
answers
111
views
points separation and dimensions [closed]
Suppose $\mathcal{A}$ is a sub-algebra of $C([0,1],\mathbb{R})$.
If $\mathcal{A}$ separates points in $[0,1]$, does it follow $\dim\mathcal{A}=\infty$?
- 577
0
votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
2
votes
2
answers
693
views
Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities
This question is an expansion of another question that I asked over at Math Stack Exchange.
In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [...
CommunityBot
- 1
3
votes
1
answer
168
views
Uniform bound for an oscillatory sum
I am wondering if there is a uniform bound $C$ (independent of $\lambda>10$):
$$\sum_{k=-\infty}^{-1}\Big|\int_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\le C.$$
Remark: (1) An easy upper ...
- 24.2k
1
vote
0
answers
206
views
About filters on real numbers
While trying to solve an open problem which I formulated earlier (currently "conjecture 1857" in this file), I met the following problem:
Let $\Delta_{\geq}+\alpha$ be the filter on $\mathbb{R}$ ...
- 765
11
votes
1
answer
767
views
Generalized limits on $\ell^\infty(\mathbb{N})$
Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a_n)_{n\in\mathbb{N}}$. The $\lim$ operator is a partial linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With ...
- 4,713
3
votes
1
answer
2k
views
Whether $\varphi(E)$ is a Jordan measurable set?
Definition: A set $S \subset \mathbb {R^{n}}$ is Jordan measurable if it is bounded in $\mathbb {R^{n}}$ and its boundary is a set of Lebesgue measure zero.
The following conclusion has been ...
CommunityBot
- 1
0
votes
0
answers
140
views
Lipschitz extensions preserving the convex hull of the range
We assume that $X$ is a metric space and that $A \subseteq X$ is a subset. Let $f : A \rightarrow \mathbb R$ be a Lipschitz-continuous function with Lipschitz constant $L$.
By the Kirszbraun theorem, ...
- 5,327
5
votes
2
answers
840
views
Decompostition of a Lipschitz domain
We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \...
- 4,713
1
vote
0
answers
116
views
Eigenvalues of elliptic operator analytic with respect to a parameter
I am interested when one can say the eigenvalues of an elliptic operator
are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but ...
- 1,385
8
votes
1
answer
629
views
Bi-Lipschitz version of Kirszbraun's extension theorem
Kirszbraun's theorem for $\mathbb{R}^2$ states the following:
Given any set $S\subset \mathbb{R}^2$ and any Lipschitz function $f:S\rightarrow \mathbb{R}^2$ with Lipschitz constant $k$, $0< k<...
- 45k
13
votes
3
answers
820
views
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The ...
CommunityBot
- 1
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 63.9k
1
vote
0
answers
186
views
Using continuity + commutativity to define "limit"
Let $S$ be the set of real sequences, and $S_0\subset S$ be the set of convergent real sequences. For each $f:\mathbf R\to\mathbf R$, we define $\bar{f}:S\to S$ by $$\bar{f}(\{x_n\})=\{f(x_n)\}.$$
It ...
- 1,630
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
3
votes
0
answers
77
views
Elliptic operator applied to the distance function
Let $\Omega$ and open subset of $\mathbb{R}^n$. Let us consider the following operator:
$$
\Delta_A (u)\, \, \colon= \text{div}(A \nabla u ), \qquad u \in C^{\infty}(\Omega)
$$
where $A(x)$ is a ...
- 823
1
vote
0
answers
45
views
Measurability of functions continuous on the right [closed]
Let $f\colon (0,1)\to \mathbb{R}$ be a function continuous on the right, i.e.
for any $a\in(0,1)$ one has $\lim_{x\to a+0}f(x)=f(a)$.
Is it true that $f$ is measurable?
I apologize if this ...
- 21.8k
2
votes
1
answer
283
views
Why should these two functions approximate each other so well?
I am a physicist and during some of my research I realized that $\log_2 \sec(\pi x)$ is really well approximated by $\frac{6x^2}{1-x}$ for small but positive $x$. Is there any reason this should be so?...
- 19.6k
1
vote
0
answers
60
views
Optimizing sum of approximate and exact functions
This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers.
I have two real values functions, where one ($g(w;x):\...
- 19.6k
0
votes
1
answer
116
views
Given reals $p_1,p_2$ and a function $f_1$ with minimal period $p_1$.Existence a function $f_2$ with minimal period $p_2$,$f_1-f_2$ periodic?
Given any two positive real numbers $p_1,p_2$ and a function $f_1 : \mathbb{R} \to \mathbb{R}$ such that $f_1$ has a minimal positive period of $p_1$ .
Then is it true that whatever be the choice of $...
- 850
3
votes
1
answer
315
views
Product of sequences "slowly" converging to $0$
The starting point of this question is that $\sum_{n=1}^\infty \frac{1}{n^{\alpha}} < \infty$ if and only if $\alpha > 1$.
Let $(a_n)_{n\in\mathbb{N}}$ be a non-negative sequence. We say that $(...
- 109k
3
votes
1
answer
941
views
What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
CommunityBot
- 1
2
votes
0
answers
122
views
Which functions $f: \mathbb{R} \to \mathbb{R}$ is injective over some subinterval of $(x,y)$ whenever $x<y$ and $f(x) \ne f(y)$?
Under what conditions on a function $f: \mathbb{R} \to \mathbb{R}$ can we say that given any real numbers $x,y$ with $x<y$ if $f(x) \ne f(y)$ then there is a sub-interval $S_{(x,y)}$ of $(x,y)$ ...
- 850
1
vote
0
answers
105
views
Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
- 19.6k
4
votes
2
answers
261
views
Is Laplace's method applicable to this integral?
Consider the following integral
$$
pv\int_0^{\infty}e^{N(-2Ax+A\log x)}\frac{e^{-B\log x}}{1-2x}dx
$$
where $A,B>0$ and we take the Cauchy principal value at $x=1/2$. I am interested in obtaining ...
- 109k
2
votes
1
answer
186
views
Equivalence of definitions of the space of test functions
Let $\Omega\subseteq\mathbb{R}^n$ be some nonempty open, and use the notation $U\Subset V$ to imply that $U$ is a compact subset of $V$. Then, for all $K\Subset\Omega$, we can define the space $$\...
- 23k
1
vote
1
answer
102
views
Expansions in terms of a variable in the integration limit in a point where the integration limit is not differentiable
Consider the following integral
$$
\int_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx
$$
where $a,b>0$. The integral can be solved exactly but I am not interested in that. I want to ...
- 188k
3
votes
1
answer
228
views
The sheaf of generalized functions on compact subsets
For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual ...
- 16.5k
5
votes
1
answer
219
views
Uniqueness from orthogonality relation?
This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...
- 53.3k
13
votes
2
answers
539
views
$f$ real-rooted forbid truncated $\frac1f$ to be so?
Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as
$$\frac1{f(x)}=\sum_{k=0}^{\infty}a_kx^k,$$
and ...
- 43.2k
1
vote
2
answers
111
views
A two-parameter inequality on product of linear terms
I would like to ask about a certain inequality that I need and which came out of some work in here.
Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof?
$$6\prod_{j=1}^k(...
- 43.2k
0
votes
2
answers
319
views
Fixed point theorem that does not require the hemi-continuity of the set valued map?
All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...
CommunityBot
- 1
3
votes
1
answer
239
views
Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying following
Suppose $U \subsetneq \mathbb{R}^d$ is open. How do I see that the distance function$$u(x) = \min_{y \in \mathbb{R}^d \setminus U} |x - y|$$is the unique nonnegative continuous function on $\mathbb{R}^...
- 5,169
3
votes
1
answer
146
views
Radial Kernel with Bounded Support and Norm of Gradient Bounded by a Dimension-free Constant
I was wondering if it is possible to construct a compactly supported radial kernel function in $\mathbb{R}^d$ such that the norm of the gradient is bounded by some dimension-free constant. That is, ...
CommunityBot
- 1
9
votes
2
answers
537
views
Comparing the growth of $f\circ g$ and $g\circ f$
I asked this Question on Math.StackExchange without success. Then I learned, that this might be the better place to ask. So, sorry for crossposting. I would agree on deleting my old question.
Let $\...
- 23.2k
3
votes
1
answer
205
views
Understand the properties of this function
We define a function
$f(t):=\sum_{n=0}^{\infty}e^{-nt}= \frac{1}{1-e^{-t}}= \frac{e^{\frac{t}{2}}}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}=\frac{2e^{\frac{t}{2}}}{\sinh\left(\frac{t}{2} \right)}$
observe ...
- 7,193
2
votes
1
answer
287
views
Regularity of the reparametrization map between curves [closed]
I am looking for a reference for the following kind of results.
Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm.
Let $B$ be a Borel subset of ...
- 53.3k
3
votes
0
answers
63
views
Is the collection of Schur convex functions sequentially compact?
We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
- 185
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
3
votes
0
answers
148
views
When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
- 39
7
votes
1
answer
450
views
Convergence of Lagrange interpolation polynomials to entire functions
Consider an entire function $\ f:\mathbb C\rightarrow\mathbb C.\ $ Let $\ (a_n\in\mathbb C:n=0\ 1\ \ldots)\ $ be an infinite sequence, where $\ a_k\ne a_n\ $ whenever $\ k\ne n.\ $ Let $\ L_n\ $ be ...
- 91.8k
2
votes
1
answer
242
views
Conditions for a monotonic integral average
I am looking for conditions that ensure that an integral average of a function from $\mathbb R^n$ to $\mathbb R$ is a monotonic function of the averaging set.
To be more specific, let me start with ...
- 36
2
votes
2
answers
261
views
Prove a family of series having integer coefficients
I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$
I've convincing date to believe the following is true,...
- 34.3k
1
vote
0
answers
105
views
Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
- 623
2
votes
0
answers
192
views
Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
- 623
2
votes
0
answers
519
views
When will the upper regularization of a bounded function not defined?
Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...
- 623
1
vote
0
answers
41
views
Necessary additive and multiplicative properties to characterize a mildly growing function
Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...
- 13.9k
1
vote
1
answer
245
views
Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?
Let $X$ be a metric space.
In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
2
votes
1
answer
266
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 4,713
1
vote
0
answers
252
views
Contour integration and application of residue theorem [closed]
I found the following contour integration done in an article, but I do not fully comprehend what has actually been calculated here? Contour integration
The argument of the function $s$ is supposed to ...
- 11