All Questions
5,909 questions
5
votes
1
answer
599
views
An inequality related to Lagrange's identity and $L_p$ norm
Let $a_1, a_2, \cdots , a_n$, $b_1, b_2, \cdots, b_n$ be real numbers, $p \in [1, +\infty)$, prove that
$$\sum_{1\leq i < j \leq n} |a_ib_j - a_jb_i|^p \leq c_p \sum_{i=1}^n |a_i|^p \sum_{i=1}^n |...
- 1,991
1
vote
0
answers
116
views
Class of Borel mapping of multivalued map
At the students scientific conference I seen paper where were this propositions:
Let $X$, $Y$ $-$ compact metric spaces, $2^X$ $-$ the set of all closed subsets of $X$.
Proposition 1. Let $f:X→Y$ ...
- 32.5k
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
3
votes
1
answer
125
views
Propagation error for ODEs
I am looking for a generic estimate to the following problem coming from biology:
I am solving the ODE
$$y'(t)=Ay(t)+zf(t), y(0)=0.$$
where $f$ is an external force determined by us and $z$ a ...
- 395
6
votes
2
answers
2k
views
non-maximal prime ideal in the ring of continuous functions
Let $A=C(0,1)$ be the ring of continuous real valued functions on the open interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\...
CommunityBot
- 1
7
votes
2
answers
913
views
Optimal Talmudic Zigzag
I have a finite sequence of positive real numbers $p_1,\dots, p_n$ and I am looking for a monotonically ascending sequence of indices $z_1,\dots, z_k$ that starts with $z_1 = 1$ and ends with $z_k = n$...
2
votes
0
answers
73
views
Closed set containing infinite arithmetic progressions of ANY gap
Let $A\subseteq [0,\infty)$ be a set containing infinite arithmetic progressions of ANY gap, that is, for any $d>0$, there is $t>0$ such that $t+kd\in A$ for all $k\in \mathbb N$.
Molter and ...
- 751
1
vote
3
answers
125
views
Weighted sum of binomials with $r$-th power of lower index
Given $r\in(0,1),$ what is the best upper (asymptotic) bound for the following expression
$$S(n,r):=\sum_{k=0}^n{n\choose k}k^r?$$
Holder's inequality gives $S(n,r)\le 2^n(\frac{n}{2})^r$ but I guess ...
- 1,273
-2
votes
1
answer
99
views
A question on the zeros involving the equation containing exponential factor [closed]
I recently encounter a puzzle that: how to show that for any constant $c_1,c_2,c_3,c_4 \in \mathbb{R}$ the equation
$$c_1 e^t+c_2e^{-t}+c_3 e^{\alpha t}+c_4 e^{-\alpha t}=0$$
has at most only one ...
- 188k
0
votes
1
answer
115
views
Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
- 1,913
1
vote
0
answers
50
views
Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
9
votes
1
answer
299
views
Can all contours of a function on a disk be made arbitrarily small?
Denote $D=\{x^2+y^2\le1\}\subset\mathbb R^2$ a disk.
Let $f:D\to\mathbb R$ be a continuous function on it. I am interested in restrictions of simple Morse functions on $\mathbb R^2$, but I suspect ...
- 504
2
votes
1
answer
311
views
Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
1
vote
1
answer
165
views
Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?
Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know ...
- 128k
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
1
vote
1
answer
338
views
Bochner measurable; continuous operator
It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $...
- 66
2
votes
1
answer
224
views
Strongly continuous semigroup: continuous or continuous componentwise?
Let $T(t)_{t \ge 0}$ be a strongly continuous semigroup on a Hilbert space $H.$
Then, one can consider the function
$f(t_1,t_2):= T(t_1)S T(t_2)x$ where $x$ is a fixed element of the Hilbert space ...
- 5,005
0
votes
0
answers
112
views
On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
CommunityBot
- 1
1
vote
0
answers
170
views
non-analytic functions with arbitrary large derivatives [closed]
This may be a trivial question but I can't see it immediately.
Suppose $\{a_k\}$ is an increasing sequence of positive reals. Does there exist a smooth function $f \in C^{\infty}([0,1])$ such that $\...
- 4,115
6
votes
1
answer
353
views
Harmonic maps are light
Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map
and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?
I hope that the answer is yes.
But actually I ...
- 45k
2
votes
0
answers
76
views
Bounding algebraic numbers away from the Gaussian integers
Let $\alpha$ be an algebraic number with degree $\leq d$ and (absolute multiplicative) height $\leq H$. Then we can say a couple of things about such $\alpha$:
(1) We know the set of all $\alpha$ ...
- 31
0
votes
1
answer
139
views
Change of variables for double integral [closed]
Thank you for your time.
My basic question is whether the following change of variables allowed
$$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$
I fail to ...
- 128k
4
votes
1
answer
877
views
Arranging squares without overlap
What is the smallest positive real $r\in\mathbb{R}$ with the following property?
Every finite collection of squares such that the sum of their areas equals $1$ can be arranged without overlap ...
- 13.6k
1
vote
1
answer
163
views
Argument for differentiability of a certain quotient of smooth functions
I have what is in essence a basic analysis question.
To make working out a certain example a bit easier I found that I need to find existence of a function $f\in C^\infty(\mathbb{R})$ with the ...
- 128k
1
vote
2
answers
435
views
Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]
I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$
is it sufficient to prove the norm of ...
- 159
2
votes
1
answer
244
views
Are there many "cusps" in a rectfiable star-shaped set?
Let me first recall the definition of density with respect to a measurable set $E$ as follows:
A point $x \in \mathbb{R}^n$ is a point of density $\alpha$ for $E$ if
$$\lim_{r \rightarrow 0} \frac{...
- 28k
7
votes
3
answers
603
views
Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$
During research involving the Born–Jordan quantization I came across the expression
$$
\frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|_{x=1}\tag1
$$
for $k\in\mathbb N_0$. It is not too ...
- 34.3k
5
votes
1
answer
2k
views
Commuting with self-adjoint operator
Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...
- 42.8k
3
votes
0
answers
172
views
Nekrasov Partition function and the leading term of Prepotential
I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...
- 425
10
votes
2
answers
836
views
Functions that are approximately differentiable a.e
The classical definition of an approximately differentiable function is as follows:
Definition.
Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
- 3,146
5
votes
1
answer
477
views
An inequality involving a sum of power terms
I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows.
Consider a positive ...
- 128k
1
vote
1
answer
1k
views
Approximation of a continuous function by a smooth one on an open set
I'm interested in the following kind of theorems :
Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
- 26.3k
0
votes
0
answers
60
views
Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
3
votes
1
answer
166
views
Proof of $L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$
It is known that $L^1(\mathbb{R}) \ast f$ is dense in $L^1(\mathbb{R})$ for some $f\in L^1(\mathbb{R})$.
So for such $f$ the closure of $L^1(\mathbb{R}) \ast f$ in the $L^1$ norm is $L^1(\mathbb{R})$....
- 41.1k
2
votes
2
answers
857
views
Hölder functions dense in space of bounded continuous functions (for non-compact manifolds)
Let $M$ be a non-compact manifold and denote by $C_b(M)$ the space of bounded continuous functions on $M$. Is it true that the space of Hölder functions is dense in $C_b(M)$ (in the $C^0$ norm: $||f||=...
- 60.6k
25
votes
2
answers
2k
views
$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$
Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a ...
- 91.8k
2
votes
1
answer
250
views
Absolute continuity of infinite product of probability measures
Let $(A_i,\mathcal{B}_i,\mu_i)$ for $i=1,2,\ldots$ be a sequence of probability spaces. Let $\nu_i$ be another sequence of probability measures on the same underlying measurable spaces. Assume that $\...
- 17k
7
votes
1
answer
337
views
Flows in Hilbert spaces
Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in ...
- 25.3k
1
vote
0
answers
101
views
Non standard Lipschitz extension
Consider a ball B and let $f(x) \in L^1(B)$ such that $\int_B f(x) dx = 0$. Furtheremore, there exists a closed set $E \subset B$ such that $f|_E$ is Lipschitz. The standard Lipschitz extension ...
- 455
0
votes
1
answer
269
views
Limit of eigenvalues of a matrix perturbation sequence
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
- 135
5
votes
0
answers
195
views
What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
- 13.8k
3
votes
1
answer
209
views
A particular measure of noncompactness?
I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...
- 128k
2
votes
1
answer
1k
views
From bounded variation to 1-Lipschitz function
Let $f\colon[0,1]\to \mathbb{R^2}$ be continuous such that $f(0)=f(1)$.
If want to find a 1-Lipschitz function $g : [0,a]\to f([0,1])$ such that $g(0)=g(a)$ and $g$ is surjective ($a>0$).
I had ...
- 28k
2
votes
0
answers
125
views
Constant periodic Sobolev embedding
Dear mathoverflowers,
I would like to have a reference regarding the optimal constant in the Sobolev embedding
$$
\|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s},
$$
($H^s$ denotes the standard L^2 ...
- 28k
8
votes
0
answers
334
views
Criterion for smooth functions [duplicate]
Let $f:\mathbb{R}→\mathbb{R}$ a real-valued function and $m,n∈\mathbb{N}^∗$ coprime, i.e. greatest common divisor of m and n is 1, and define $f^m:=f\cdot f\cdot\ldots\cdot f.$
Show that
$$f^m,f^n\in ...
- 81
1
vote
1
answer
642
views
Interchange of integration order (of a not absolutely convergent integral with sinus)
Can we interchange the integral order of this integral to start integration on $x$ ? (Taking $g$ and $f$ two functions of rapid decrease which are $o(x^2)$ near zero)
$$A=\int_{0}^\infty \int_0^{\...
- 128k
1
vote
0
answers
138
views
A Gagliardo--Nirenberg inequality in $H^2(\mathbb R^4)$
Does the following inequality hold in $H^2(\mathbb R^4)$
$$
\sup_{u \in H^2(\mathbb R^4), u\not\equiv 0} \frac{\|u\|_4^4}{\|\Delta u\|_2^2 \|u\|_2^2} > \frac1{16 \pi^2}?
$$
- 17.2k
1
vote
2
answers
333
views
What gives a "Parseval like" equation mixing cosine and sine Fourier transforms ?
Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:
$$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$
$$\mathcal{F}^s [f (x)]=\...
- 1,199
1
vote
2
answers
630
views
Are rationals everywhere equally dense? [closed]
I would like to know is there any notion of density over the rationals with which we could determine are rationals everywhere equally dense on the real line, because, for example, I am not sure would ...
- 30.1k
5
votes
1
answer
877
views
Density of intersection with shifted sets
Given a subset $S$ of the positive integers $\mathbf{N}$, let $\mathrm{d}^\star(S)$ be its upper asymptotic density, that is,
$$
\mathrm{d}^\star(S)=\limsup_{n\to \infty}\frac{|S \cap [1,n]|}{n}.
$$
...
- 4,713