All Questions
5,628 questions
2
votes
2
answers
152
views
Name of a generalized version of semi-continuity
I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.
We will say a ...
- 799
1
vote
1
answer
126
views
Evaluation of the multiple integral [closed]
Would you give me any suggestions or comments on evaluating the following $n$-dimensional
integral? $$ \int_{[0,t]^n} h(x) dx $$
where
$ x=(x_1 ,x_2 , \cdots, x_n ), h(x)= \prod_{k=1}^n min( \bar{...
- 245
1
vote
1
answer
1k
views
A question about "nice" functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ''...
- 45
6
votes
0
answers
206
views
Degree of Chebyshev polynomial necessary
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
- 13.9k
5
votes
1
answer
225
views
Extending Jordan loops
I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.
Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\...
- 32.5k
-1
votes
1
answer
237
views
Theorem with an example [closed]
i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
- 277
4
votes
0
answers
451
views
Why does it seem that $rca=rba$? [closed]
The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-...
- 459
1
vote
1
answer
273
views
Does this variable have an upper bound?
Let $x$ be a positive scalar variable whose time derivative satisfies
$$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$
where $|\cdot|$ denotes the ...
- 61
2
votes
1
answer
2k
views
Modified Lebesgue differentiation theorem
Let $\Omega\subset \mathbb{R}^n$ an open set and $u:\Omega\to \mathbb{R}$ be a (locally) $L^1$-function. Then it is well known that the Lebesgue differentiation theorem holds: For almost every $x\in \...
- 2,270
1
vote
1
answer
281
views
On the Hölder regularity of an integral function
Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
- 301
2
votes
3
answers
913
views
A definite integral
Hello,
I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...
- 1,649
2
votes
0
answers
814
views
Quantifying the “flatness” of functions which are the Fourier transforms of positive functions
Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
- 21
0
votes
1
answer
125
views
Discussion for the sign of a specific sum
Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to ...
- 1,257
1
vote
0
answers
260
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
4
votes
0
answers
188
views
Evaluate a multiple integral
I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed.
$$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) dr_1\...
- 41
0
votes
2
answers
106
views
Expected summation of dropped intervals?
For each $n\in\mathbb{N}$, let $I_n$ be an interval of length $1/2^{n}$. We drop each $I_n$ onto the interval $[0,1]$ uniformly at random (so that there is "wraparound" if need be). What is the ...
3
votes
3
answers
281
views
Which real scalings of the natural numbers approximately accommodate the unbounded powers of a noninteger?
That is, what are the possible values of a real number $\lambda$ for which there exists a nonintegral real $\alpha >1$ such that, given any $\varepsilon >0,$ all but finitely many powers of $\...
- 2,437
0
votes
1
answer
182
views
Surjectivity of "nice maps" from local properties
What tools are available from real algebraic geometry, analysis and
topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$
from local properties and maybe function values?
...
- 1,256
1
vote
0
answers
45
views
Measurability of functions continuous on the right [closed]
Let $f\colon (0,1)\to \mathbb{R}$ be a function continuous on the right, i.e.
for any $a\in(0,1)$ one has $\lim_{x\to a+0}f(x)=f(a)$.
Is it true that $f$ is measurable?
I apologize if this ...
- 21.8k
5
votes
0
answers
271
views
Is every $C^1$-domain which is homeomorphic to the unit ball in $\mathbb{R}^d$ Lipschitz equivalent to the unit ball?
Suppose we have a domain $\Omega\subset \mathbb{R}^n$ which is homeomorrphic to the unit ball $B(0,1)\subset \mathbb{R}^n$ and such that $\partial \Omega$ is of class $C^1$ (technically, this means ...
- 729
2
votes
0
answers
81
views
Convolution of decaying polynomials [closed]
I conjecture that if the functions $f$, $g$ defined on $\mathbb{R}^n$ satisfying
$$|f(x)| ≤ A(1+|x|)^{−M}, \quad |g(x)| ≤ B(1+|x|)^{−N}$$for some
$M$, $N > n$, then$$|(f * g)(x)| ≤ ABC(1+|x|)^{−L},$...
- 355
5
votes
1
answer
185
views
Existence of an equivariant Morse function
Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse ...
- 53
2
votes
1
answer
289
views
Can a simple curve intersect every subspace of dim 2 and avoid the origin?
Is there, e.g. in $\mathbb R^4$ a simple curve that does not contain the origin and intersects every subspace of dimension 2?
Sorry if the question is too easy, but I just cannot figure it out.
In ...
- 19k
1
vote
0
answers
120
views
Interpolation functional for BV spaces?
Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
- 121
1
vote
1
answer
237
views
Interpolation and embeddings for parabolic function spaces
I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
- 632
1
vote
1
answer
1k
views
Calculating the Lebesgue decomposition of a measure [closed]
How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
- 13
1
vote
0
answers
109
views
Pointwise convergence of a sequence of approximate limits of BV functions
So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
0
votes
1
answer
308
views
Limits of functions with converging zeros
What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros?
More precisely, suppose that $f_i: R^n \to R^m$ is a ...
- 1,628
2
votes
0
answers
224
views
Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental [closed]
I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts.
At page $6$, the ...
- 4,795
3
votes
1
answer
105
views
How to show monotonocity and the limit? [closed]
Let me reformulate my recent question.
Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density:
$$\phi(x) = C\left\{ \begin{array}{lcc}
\sqrt{...
- 31
2
votes
0
answers
890
views
Obtaining a pointwise bound on the convolution of two singular measures
I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids.
We are in Section 7, near equation (34) (pag.16 of the arxiv).
Notations and ...
- 692
1
vote
1
answer
137
views
Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ [closed]
Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant. My question is: Find sufficient and necessary ...
- 1,197
5
votes
1
answer
1k
views
Notions related to De Rham Cohomology
In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures.
Closed forms/exact forms
real parts of analytic functions/harmonic ...
- 51
2
votes
0
answers
181
views
Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?
Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable).
Also, let $f:D_1\cup D_2=D\...
- 121
2
votes
0
answers
343
views
continuity with respect to weak-${\ast}$ topology
Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
- 1,835
6
votes
1
answer
218
views
Approximating an iteratively defined function
Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows:
$$f_0(x) =1+2x$$
$$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } f_{n-1}\left(\frac{x}{t}...
- 13k
1
vote
0
answers
93
views
Multimodal property of polynomial logistic distribution
Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$
Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
- 783
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):\...
- 189
4
votes
1
answer
261
views
Weak continuity of Lebesgue decomposition
Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function $f:X\to\...
- 1,205
3
votes
1
answer
1k
views
ordered fields with the bounded value property
Say that an ordered field $F$ satisfies the bounded value property if, for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]_F := ${$x \in F: a \leq x \leq b$} to $F$, there ...
- 19.7k
1
vote
1
answer
192
views
Characterization of a subset of $[0,1]$
Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property:
For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...
- 1,835
0
votes
1
answer
259
views
How to perturb a function to separate points
Consider two smooth functions $f,g\in C^\infty(\Omega)$ with $\partial \Omega$ smooth and $\Omega\subset \mathbb{R}^3$. Assume that $f=g$ on $\partial \Omega$.
For any given $\varepsilon>0$, how ...
- 35
2
votes
1
answer
158
views
Positive kernel property
Let $k:[0,1]^2\rightarrow (0,+\infty)$ be a continuous function and let
$f,g:[0,1]\rightarrow (0,+\infty)$ be measurable functions. We assume that
$$\forall x\in [0,1],\quad
f(x)=\int_0^1 k(x,y) g(y) ...
- 16.2k
2
votes
0
answers
113
views
Continuous inclusions Sobolev theorem, question [closed]
How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...
- 369
0
votes
1
answer
195
views
Existence of bounded $n-$th derivative of the solution of differential equation
This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)...
- 133
1
vote
0
answers
189
views
Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$
Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,
Let $f$ be a periodic ...
- 698
3
votes
1
answer
975
views
Generalized Cesàro means of a bounded sequence
While studying the convergence of a certain iterative algorithm, I have come across the following generalization of the Cesàro mean: given a sequence $\{a_k\}$ and an integer $m\geq 0$, define
$c_k^{(...
2
votes
0
answers
251
views
Volume of bounded regions in hyperplane arrangements
I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...
- 357
1
vote
0
answers
84
views
extension for a complex operator
Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
- 135
0
votes
2
answers
179
views
Analyticity of Logarithmic Integrals
Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
- 1,583