All Questions
5,908 questions
5
votes
1
answer
641
views
Characterizing invertible nonnegative matrices with bounded sums
Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
CommunityBot
- 1
1
vote
0
answers
94
views
Estimating convolutions of powers
I would like an asymptotic estimate of
$$
\sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}}
$$
that does not involve any infinite summation. In order to lighten the notation, I ...
- 562
2
votes
1
answer
382
views
Continuous real function on germs
Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
- 42.8k
1
vote
1
answer
183
views
Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?
I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{...
CommunityBot
- 1
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 ...
- 3,312
3
votes
0
answers
69
views
Almost everywhere in a rectangle [duplicate]
I would like to ask a question about the product (Lebesgue) measure on rectangle. I tried to solve the problem but I couldn't.
Let $S$ be a subset of a region, say $R$ which is enclosed by a ...
2
votes
1
answer
275
views
The class of uniformly accelerated curves and surfaces
Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...
2
votes
1
answer
346
views
Alike looking matrices imply convergence of eigenvalues?
This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...
- 15.8k
3
votes
1
answer
284
views
Measure of the same set in different models of ZF
Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.
Is $\mu^\mathcal{M}(A^\mathcal{...
- 32.5k
18
votes
2
answers
1k
views
An Entropy Inequality (generalized)
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability measure ...
CommunityBot
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2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
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18
votes
3
answers
1k
views
Example of a space for which $V \cong Hom(V,V)$
Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology.
Is there a non-trivial ...
- 201
2
votes
2
answers
219
views
Boundedness of ratio of linear functions
Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \...
- 1,991
1
vote
1
answer
176
views
Linear map with two "incompatible" representations
Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ $$...
- 42.8k
18
votes
2
answers
2k
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Generalization of Darboux's Theorem
Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let $U\...
- 2,933
9
votes
2
answers
2k
views
Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
CommunityBot
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1
vote
0
answers
153
views
On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$
Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property
$$
\left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert
...
- 1,697
1
vote
0
answers
115
views
Inequality for an integral [closed]
How to prove that the function $$f(r)=(1 - r^2) \int_0^{2\pi}|Re[\frac{e^{i a} (2 - e^{-i s} r)}{(-e^{i s} + r)^2}]|ds$$ for real $a$ and $r\in[0,1]$ attains its maximum for $r=0$ with $f(0)=8$.
- 11
4
votes
0
answers
451
views
Why does it seem that $rca=rba$? [closed]
The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-...
- 459
10
votes
1
answer
772
views
Nondifferentiability set of an arbitrary real function
A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...
CommunityBot
- 1
11
votes
2
answers
802
views
Functions that Calculate their $L_p$ Norm
are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\...
- 148k
49
votes
2
answers
3k
views
Is a function with nowhere vanishing derivatives analytic?
My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to a ...
CommunityBot
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2
votes
1
answer
416
views
What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]
I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$?
Only I've calculated the following identity:
$$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln \...
- 655
1
vote
0
answers
248
views
Uniform bound for an alternating series of functions
I have mainly two questions, the first one being motivated by the second one.
1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
CommunityBot
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2
votes
0
answers
112
views
Asymptotic analysis involving a circular multiple integral
Let $t_1,\ldots,t_m>0$, and $m\ge 4$ be an even integer. Consider the function:
$$
f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m} |x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots |x_{m-1}-...
CommunityBot
- 1
-1
votes
1
answer
1k
views
derivatives and uniformly convergence [closed]
Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has
$$f(x)=\sum_{n\...
- 24.2k
2
votes
1
answer
151
views
A question on existence of solutions of a linear ODE system
I am working on a problem of harmonic functions on surfaces, and in one step I got the following system of ODEs with prescribed asymptotes. I was wondering what methods could give us the existence or ...
12
votes
2
answers
1k
views
Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 1,748
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 43.7k
4
votes
1
answer
185
views
Reference: Hardy space regularity of the Jacobian determinant
I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes.
Theorem:
For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, $\...
- 188k
21
votes
0
answers
658
views
A multiple integral
Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos {(s_{2n-1}...
CommunityBot
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2
votes
2
answers
1k
views
Relationship between largest eigenvalue of a positive matrix $A$ and $A∘A^T$
I'm wondering whether there is certain relationship between the largest eigenvalue of a positive matrix(every element is positive, not neccesarily positive definite) $A$, $\rho(A)$ and that of $A∘A^T$,...
- 28.6k
1
vote
2
answers
259
views
Markov-Bernstein like inequalities for monotone polynomials
Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are ...
- 311
12
votes
1
answer
2k
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Comparing Krein-Rutman theorem and Perron–Frobenius theorem
Krein–Rutman theorem is a generalization of Perron–Frobenius theorem, I know that things could be more subtle in infinite dimension, yet there's an important result in Perron–Frobenius that's missing ...
- 273
3
votes
1
answer
1k
views
For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1
This is related to a question asked on mathstackexchange https://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...
CommunityBot
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6
votes
1
answer
216
views
Estimates of Hausdorff dimension (and its derivatives)
For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
- 2,093
3
votes
0
answers
193
views
Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
- 133
0
votes
0
answers
206
views
About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
1
vote
0
answers
94
views
Determining the exact form of a projection in a Hilbert space
Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue square-...
- 108
4
votes
0
answers
896
views
A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
CommunityBot
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5
votes
0
answers
1k
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Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
5
votes
3
answers
1k
views
Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.
Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
CommunityBot
- 1
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 $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
- 516
3
votes
1
answer
354
views
What does this ODE have to do with the associated Legendre polynomials?
I am currently struggeling with the following differential equation:
$$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$
where $a \in \mathbb{R}$ constant, $\phi \...
- 54.2k
4
votes
1
answer
581
views
A question on null sequences
Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero?
In case the answer is affirmative (...
CommunityBot
- 1
3
votes
1
answer
693
views
Equivalence of negative Sobolev norm of derivative to $L^2$-norm
Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the $L^2(S)...
CommunityBot
- 1
2
votes
2
answers
133
views
formula for repeated finite differences
I am looking for a proof of a well-known fact, whose proof must be very easy, though I've been struggling to find it. Let $\Delta$ be the map from real-valued functions of a real variable, given by $(\...
- 301
7
votes
2
answers
5k
views
Relationship between the derivative of a matrix and its eigenvalues
Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing?
...
- 6,210
1
vote
2
answers
923
views
Spectrum of Mathieu equation
I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
- 9,853
0
votes
1
answer
165
views
Uniform boundedness in $L^1[0,1]$ implies finite $\limsup$ almost everywhere for a subsequence? [closed]
Given a sequence of functions $f_k \in L^1([0,1])$ such that $||f_k||_{L^1(0,1)}\leq C$.
Is there a subsequence $\{k_l\,|\,l\in \mathbb N\}\subseteq \mathbb{N}$ such that for $\mathcal{L}^1$-almost ...
- 24.2k