All Questions
5,857 questions
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
5
votes
2
answers
719
views
Darboux function on $[0,1]$ with interesting property
I have proved a few years ago the following proposition:
There exists $f: [0,1] \to [0,1]$ with Darboux property such that there exist $A,B \subset[0,1]$ with $A\cap B=\emptyset,\ A \cup B=[0,1]$ ...
- 2,222
1
vote
2
answers
576
views
Dini condition and integrability condition
Assume that $A$ is an arbitrary positive integrable function on $[0,1]$. Whether exists a convex function $f_A(x)=x g(x)$ of $(0,+\infty)$ into itself (depending on $A$) such that $\lim_{x\to +\...
- 303
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
4
votes
1
answer
529
views
Find a continuous function with a prescribed continuity set
It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.
In the book "Understanding Analysis" by Abbott is stated in page 128 ...
- 193
1
vote
1
answer
715
views
Importance of Denjoy-Carleman classes as a class.
Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:
Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\...
- 807
3
votes
0
answers
166
views
Monotone version of one-dimensional Whitney extension theorem
Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...
- 29.1k
3
votes
1
answer
403
views
Is there a probability density function providing the least expected value?
Fix constant reals $A>1$ and $D>0$. Let $f:\mathbb{R}\to[0,\infty)$ be a probability density function on $\mathbb{R}$, i.e. $\int_{-\infty}^\infty f(x)\, dx=1$, that is continuous almost ...
- 31
2
votes
1
answer
310
views
Boundedness of an Oscillating Integral
Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded as $\lambda\...
- 173
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
- 16.6k
3
votes
2
answers
188
views
Seeking a class of functions for which sums approximate integrals well
Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
- 19.7k
3
votes
0
answers
91
views
Nonlinear smooth bijection from $\mathbb Q$ to itself [duplicate]
Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that
$\phi$ is nonlinear: different from $ax+b$,
$\phi$ is smooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?
...
- 303
9
votes
2
answers
954
views
Extremal properties of the determinant for matrices with entries in a fixed subset of $[-1,1]^{n^2}$?
Given a multiset $S\subset [-1,1]^{n^2}$, we set
$$m(S)=\min\vert \det(M)\vert$$
where the minimum is over all matrices with entries forming the multiset $S$
and
$$a(n)=\max m(S)$$
where the maximum ...
- 17.9k
2
votes
2
answers
711
views
Power function inequality
Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...
- 373
5
votes
0
answers
596
views
Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
- 1,421
3
votes
0
answers
290
views
Does there exist a supersmooth non-polynomial function?
Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
0
votes
2
answers
190
views
Is real analytic function good enough (see problem)? [closed]
Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then ...
- 67
2
votes
2
answers
1k
views
Characterization of Weakly measurable functions
I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong ...
- 23
1
vote
2
answers
939
views
Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
- 1,471
0
votes
1
answer
156
views
Prove a function, defined by integration of a harmonic function, is log-convex [closed]
Let $u$ be a harmonic function and we define
$$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$
The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function ...
- 679
7
votes
1
answer
2k
views
Hanner's inequalities: the intuition behind them
Hanner's inequalities in the theory of $L^p$ spaces (see http://en.wikipedia.org/wiki/Hanner's_inequalities) look hard to come-up with at the first glance. Their proof (say, the one in Lieb & Loss ...
- 5,019
1
vote
0
answers
58
views
A question on Integral inequality
Let $0 < \epsilon < 1$. Consider $\{a_n\}_{n \geq 1} \in l_2$ and $L(t) = 1+\epsilon t$. Let $x$ be fixed such that $0 < x < L(t)$. Does there exist $\tau \geq 0$ such that the following ...
- 11
3
votes
1
answer
99
views
Is the variation of two BV functions the same in the set in which they coincide?
Given two real $BV$ functions $u$ and $v$ in an open interval $(a,b)$ consider the set
$A=\{x: \text{both } u \text{ and } v \text{ are continuous at } x \text{ and } u(x)=v(x)\}$
is it true that $|...
- 33
1
vote
1
answer
547
views
When does the finite union of convex sets have a hole in it?
Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) \...
- 693
-2
votes
1
answer
212
views
A calculus question [closed]
Fix $q>1$. Define the function
$$
f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$
The problem is whether the following is true,
$$
\lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
- 1,649
3
votes
0
answers
131
views
What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?
For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...
- 111
0
votes
1
answer
55
views
On 1-iso maps and subsets of the unit circle
Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
- 97
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
1
vote
3
answers
293
views
Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?
The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
- 41
7
votes
2
answers
2k
views
What's the correct notion of determinant of a bilinear pairing?
By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
- 54.6k
4
votes
2
answers
371
views
Heights of several interesting posets
Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$).
Define several sets of total functions, in each ...
- 6,819
2
votes
1
answer
238
views
Asymptotic behaviour of eigenvalues
If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
5
votes
0
answers
310
views
Reference for Hodge decomposition
Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
- 562
4
votes
1
answer
410
views
Using a quadratic kernel instead of a linear kernel in the Laplace transform
Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
- 19.7k