All Questions
5,909 questions
8
votes
3
answers
747
views
How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
- 2,912
4
votes
1
answer
350
views
Complex Structure on Manifold of Maps
Suppose $M$ is a compact smooth manifold and $V$ is a compact complex manifold. I want to show that the spaces $C^{k,\alpha}(M,V)$ and $W^{k,p}(M,V)$ (the latter for $kp>\dim M$) are complex ...
CommunityBot
- 1
4
votes
1
answer
151
views
Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm
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 $...
- 4,034
1
vote
1
answer
165
views
Morrey condition (integral condition) and (local) Holder condition
Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$)
$$\limsup_{r \to 0} r^{-\alpha \beta}\frac{...
CommunityBot
- 1
4
votes
1
answer
225
views
Bounded growth of functions vs bounded growth of functions on countable sets
I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $...
- 42.8k
141
votes
17
answers
38k
views
Why is differentiating mechanics and integration art?
It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/...
3
votes
2
answers
373
views
Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$
This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question
Let tangents be drawn to the curve $y=\sin x$ from ...
- 34.3k
5
votes
1
answer
379
views
Hilbert representation of a bilinear form
Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...
- 7,893
2
votes
0
answers
263
views
Are $C^1$ immersions dense in $C^1$?
Let $M$ be a closed compact manifold.
Is the space of all $C^1$ immersions from $M$ to $\mathbb{R}^m$ ($m> \dim M$) dense in $C^1(M; \mathbb{R}^m)$ (in the $C^1$ topology)?
- 2,558
17
votes
1
answer
794
views
Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?
Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
- 42k
1
vote
1
answer
153
views
Optimal estimate in trace norm
Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...
- 128k
5
votes
2
answers
875
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
- 11.2k
0
votes
1
answer
269
views
Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]
In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-...
- 2,558
1
vote
0
answers
49
views
On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
-2
votes
1
answer
209
views
Strong estimates for the zeta function on natural numbers
Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
be the Riemann zeta function (here we just consider real $s$).
We do have a description given by
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
- 131
2
votes
1
answer
95
views
Quotient with positive second derivative in the limit?
I am studying the quotient of
$$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$
and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$
for some $\...
- 188k
4
votes
0
answers
101
views
Injectivity of product functions on natural number sequences
Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...
3
votes
1
answer
507
views
Chain rules for Dini Derivative
Could someone provides some references for the chain rule concerning Dini derivatives. For example, let $f(\cdot) \in \mathcal{C}^1\left( \mathbb{R} ; \mathbb{R}\right)$, and $g(\cdot) \in \mathcal{C}\...
- 128k
4
votes
0
answers
128
views
Anderson Localization and Homogenization theory
I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here.
The question is mostly related to homogenization theory in mathematical physics.
$\textbf{...
- 2,821
3
votes
1
answer
569
views
Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$
I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?
This seems to be a well-known result, but I can ...
- 114k
-2
votes
1
answer
165
views
Relationship between "Radial" Fourier transform and Fourier transform, especially at infinity
Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support.
What is the relationship between
$$
\widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
- 17.2k
4
votes
1
answer
114
views
a compact set with nonempty convex sections
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x \in X$ and every coordinate $i=1,2,\ldots,d$ denote by $x_{-i} := (x_j)_{j \neq i}$.
Given a set $Y \subseteq X$ ...
- 627
6
votes
2
answers
509
views
Summing Bernoulli numbers
Consider the Bernoulli numbers denoted by $B_n$, which are rational numbers.
It is known that the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$ are not integers once $n>1$.
I am curious about the ...
- 85.6k
0
votes
1
answer
103
views
How to prove that the following function is monotonically decreasing?
$f(x) = \frac{\int_{\alpha x}^{x} e^{-t} t^{b+1}\ dt}{x \int_{\alpha x}^{x} e^{-t} t^b\ dt}:\ ]\ 0,+\infty\ [\ \to \mathbb{R}$
where $\ 0<\alpha<1\ $ and $\ b>0$
- 39.1k
2
votes
1
answer
324
views
Direct proof a property of hyperstonean spaces
First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
- 11.4k
7
votes
2
answers
298
views
Bound on sum of coefficients of polynomials w.r.t a weighted integral
Fix $k\in\mathbb{N}$ and assume $f(x)$ is a real polynomial of degree $n$ such that we have the normalization
$$\int_{-1}^1f(x)^2\,(1-x)^kdx=1.$$
I am interested in the optimal size of the sum of the ...
- 43.2k
2
votes
2
answers
152
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 128k
3
votes
2
answers
150
views
Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$
I am working in data science and I have to deal with the following problem for which I would like to find a simplification:
We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,...
- 489
7
votes
1
answer
348
views
Descartes' rule of signs for infinite series
Consider the function given by
$$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$
where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
- 54.2k
0
votes
0
answers
171
views
What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?
I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...
- 723
10
votes
1
answer
3k
views
Can the supremum of continuous functions be discontinuous at every point of an interval?
Pether Luthy gave an example of a sequence of continuous real valued functions whose supremum was discontinuous on a set of positive measure. But does it exist a sequence of continuous real valued ...
- 4,713
5
votes
1
answer
571
views
Schrödinger operator with Coulomb potential
The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
- 24.2k
4
votes
1
answer
168
views
Method of characteristics beyond the Lipschitz setting
I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
- 1,277
1
vote
0
answers
922
views
A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
1
vote
1
answer
142
views
About a Dirichlet series [closed]
I would like to know if the following assertion is true:
Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
- 4,713
3
votes
1
answer
115
views
Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?
Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$.
Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-...
- 6,741
9
votes
0
answers
180
views
Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
- 43.2k
54
votes
3
answers
4k
views
Does every real function have this weak continuity property?
In my research I came across the following question :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
- 303
1
vote
1
answer
112
views
Orthogonal complement vector space
Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...
- 29.8k
2
votes
1
answer
393
views
Positive Solutions of second-order ODE
Consider second-order ordinary differential equations of the form
$u''(t)=a(t)u(t)-2$
I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence ...
- 54.2k
5
votes
2
answers
648
views
Dominated convergence 2.1?
After this question : Dominated convergence 2.0?
I want to know, what about the case when $h\in L^1([0,1])$.
The completed question :
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging ...
- 29.8k
3
votes
1
answer
449
views
Riemann-Stieltjes integral as a limit of Riemann integrals
Let us suppose that $f, g:(A, B)\to \mathbb{R}$ are both continuous on $(A, B)$ and for $[a, b]\subset (A, B)$, suppose that $g$ is of bounded variation on $[a, b]$ (we may add, if necessary, that ...
- 128k
0
votes
0
answers
75
views
Dense Egoroff theorem
Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...
- 5,405
1
vote
0
answers
242
views
Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]
[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
- 35.5k
6
votes
2
answers
389
views
asymptotic for li(x)-Ri(x)
Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$
where
$$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
- 105k
7
votes
1
answer
552
views
Dominated convergence 2.0?
During my research, I came across the following question.
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that:
$\forall n\in\mathbb N, f_n''<h$, ...
- 41.1k
2
votes
1
answer
141
views
small perturbation of BV function
consider an interesting real analysis question:
define average operator on $[0,1]$:
$A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $
( may clarify ...
- 555
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
- 993
5
votes
0
answers
696
views
Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)
Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...
- 151
0
votes
0
answers
124
views
Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]
Recall a very famous theorem due to Hartogs for complex analytic functions of several variables.
Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...
- 671