All Questions
5,857 questions
2
votes
3
answers
2k
views
LU decomposition
Consider a $N \times N$ symmetric real matrix $A$: $A_{ij} = (\sum_{k=1}^N n_{ik}) \delta_{ij} - n_{ij}$, where $n_{ij}$ is a real symmetric matrix whose elements are equal to $1$ or $0$. $A$ has one ...
- 151
3
votes
2
answers
487
views
Integrating over the Intersection of Convex Regions
Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?
The ...
- 13.2k
3
votes
0
answers
148
views
When a PDE add a Laplacian term
I went to a talk today and the speaker mentioned when you add a Laplacian term to a PDE, the Laplacian will dominate (in what sense?), which I don't quite understand. I know this question is a bit ...
- 39
11
votes
1
answer
1k
views
Proof of the "Neo-classical Inequality", a fractional extension of the binomial theorem
I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$:
$$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\...
- 4,952
5
votes
0
answers
432
views
Points of continuity of a lower semicontinuous function have non empty interior
Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous.
I know that the set of discontinuities of such a function is contained in a meager set, and ...
user17697
3
votes
1
answer
640
views
Relationship between LlogL and Hardy spaces
I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in L^1\,s.t.\,\...
- 843
19
votes
0
answers
775
views
A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
- 433
8
votes
3
answers
786
views
truth vs. provability for ordered fields
In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...
- 19.7k
10
votes
1
answer
1k
views
Real analytic function, injective, non surjective and preserving the rationals ?
I'd like to prove the non-existence of a real analytic function, injective, non-surjective
that sends rationals to rationals.
Is it a classical result ? If not, any hints on how to prove it ?
Thanks ...
- 103
2
votes
1
answer
301
views
Simplicity of eigenvalues
Consider the Sturm-Liouville operator$$Au = -(pu')' + qu \text{ on }I = (0, 1),$$where $p \in C([0, 1])$, $p \ge \alpha > 0$ on $I$, and $q \in C([0, 1])$. No further assumptions are made; in ...
- 369
1
vote
1
answer
115
views
Well-definedness for a singular integral
Let $T_\alpha$ be a singular integral operator defined by
$$
T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds
$$
for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$.
...
- 201
0
votes
3
answers
320
views
Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]
Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
user81900
2
votes
0
answers
195
views
A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 557
5
votes
1
answer
215
views
Smoothening a measure, II
There is an almost invisible, but significant difference between the question below and that recently answered by Boris Bukh.
Given a probability measure $\mu$ supported on a finite set $S\subset{\...
- 23k
3
votes
1
answer
657
views
Banach space of discontinuous functions(Killing continuous functions)
Edit: According to the comment of Prof. Majer, I revise the question:
For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$
$$\...
- 366
4
votes
0
answers
139
views
The class of all iterated antiderivatives of rational functions
Consider the following property of a function $f$:
There exists a non-negative integer $n$ such that the $n$'th derivative of $f$ is a rational function.
Question 1: Is there a name in the ...
- 5,637
4
votes
1
answer
1k
views
Can't figure out "standard application" of the Garsia-Rodemich-Rumsey Lemma
I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
- 617
2
votes
3
answers
755
views
On matrices in linear forms with vanishing determinant
This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...
- 4,902
6
votes
2
answers
812
views
A dual theory to the theory of currents?
The k-currents are defined as dual space to the spaces of all smooth k-forms.
(These monsters are used to work with the minimal k-surfaces.)
Assume I want to look at the generalized k-forms;
they can ...
- 1,785
3
votes
0
answers
105
views
Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?
Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
- 6,051
2
votes
1
answer
113
views
estimation of a vector-function
Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that
1) $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and
2) for some real $c_1>0$ and all $t>0$ one has $\|x(t)\|\le c_1\...
- 21
7
votes
0
answers
628
views
Proving Richardson's theorem for constants
(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
user5810
0
votes
0
answers
58
views
$N-$Green function in $\mathbb R^N$
Let $N \geq 3$. Does there exist solution of the following equation
$$-\Delta_N G + G^{N-1} = \delta_0,$$
where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
- 379
2
votes
1
answer
96
views
Isometry between punctured sphere and punctured triangle?
Setup:
Let $C_n$ be a closed $n$-simplex in $\mathbb{R}^n$ and let $r \in (0,R)$ where $R$ is the distance any one of the vertices $\{v_1,\cdots , v_{n+1}\}$ of $C_n$ to the centroid $\frac{v_1+ \...
- 5,405
3
votes
1
answer
6k
views
Classical Derivative, Weak Derivative and Integration by Parts
Hello,
While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.
QUESTION
Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $...
- 895
22
votes
3
answers
3k
views
Splitting the determinant polynomial into linear factors - a Dedekind problem
Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...
- 33.8k
3
votes
1
answer
321
views
Is this parametric inequality true?
Puzzled by this still open question, I tried comparing the arithmetic mean $A(x,y)=(x+y)/2$ with a mean intermediate between a geometric-type mean $G(X)=(x^a y^{1-a}+x^{1-a} y^a)/2\;$ for $0\le a \le ...
- 5,103
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
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
2
votes
0
answers
229
views
Analytic varieties for the primes and the twin primes
I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...
- 25.4k
4
votes
0
answers
493
views
Hodge duality and the determinant of the product of two matrices
I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are ...
- 13.6k
2
votes
1
answer
104
views
Limits of a quasiperiodic function with two pseudoperiods
Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
...
- 378
0
votes
2
answers
721
views
Is there a probability density function satisfying the following conditions?
I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here.
...
- 5,470
0
votes
1
answer
169
views
Unimodality of a certain parametric integral
Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$.
Is it true that the map
$$
F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx
$$
has exactly one ...
- 191
1
vote
1
answer
401
views
linear recurrence inequality of positive terms
This is a follow up on my previous linear recurrence inequality question.
I have some matrices which satisfy a linear recurrence formula of the form
$$
A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad n\...
- 145
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
10
votes
1
answer
837
views
Factor a sum of products of cofactors
Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...
- 2,902
0
votes
2
answers
348
views
Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians?
This question originates an engineering application.
There is a certain process that is presumed to be a sequence of diffusions and is usually modelled as a sum of Gaussians:
$$\Sigma_n w_ne^{-\...
- 2,205
1
vote
1
answer
307
views
Ordinal of injectivity for a smooth regular curve with a finite arc-length
Let $\gamma: [a,b]\to\mathbb{R}^d$ defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)) $$ be a smooth (i.e., $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve ...
- 43
3
votes
1
answer
1k
views
For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1
This is related to a question asked on mathstackexchange https://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo. This question is ...
- 516
2
votes
2
answers
233
views
Is the domain of symmetric derivative borel set?
Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$.
Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim_{r\to 0} \frac{\lambda (B(x,...
- 337
1
vote
1
answer
121
views
A property of a nonlinear ODE under periodic boundary conditions
Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $$u_1(0) = u_1(1),u_1'(0)= u_1'(1)$$ and $$u_2(0) = u_2(1),u_2'(0)=u_2'(1)$$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \...
- 698
3
votes
1
answer
459
views
Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?
I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.
- 520
8
votes
2
answers
2k
views
Expression for the sum of square roots of zeros of a polynomial
Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$.
General question. Does there exist a simple expression for the ...
- 34.3k
0
votes
1
answer
482
views
Complement of a finite union of convex sets
Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...
- 2,933
1
vote
1
answer
238
views
Does the bounded extension of the Fourier multiplier operator agrees with its original explicit definition?
We consider the Fourier multiplier operator $T_0$ defined by the explicit expression
$$(T_0f)(x)=\int_{\mathbb{R}^n}{e^{ix\cdot \xi}m(\xi)\hat{f}(\xi)d\xi}, \ f\in S(\mathbb{R}^n),$$ where $S(\mathbb{...
- 171
1
vote
1
answer
251
views
Is there a way to solve this integral equation?
I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem.
For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
- 291
5
votes
1
answer
921
views
About generalized Minkowski inequality
For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + f^{-1}\left(\sum\...
- 51
3
votes
1
answer
146
views
Radial Kernel with Bounded Support and Norm of Gradient Bounded by a Dimension-free Constant
I was wondering if it is possible to construct a compactly supported radial kernel function in $\mathbb{R}^d$ such that the norm of the gradient is bounded by some dimension-free constant. That is, ...
- 1,127
-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\...
- 3,998