All Questions
5,850 questions
3
votes
1
answer
362
views
Cartesian product of test function spaces
Mini introduction
Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \...
-3
votes
2
answers
260
views
On \ell_3 norm in R^2
Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$,
in which $\left\Vert \left(x,y\right)\right\Vert _{3}:=\sqrt[3]{\left|x\...
- 65
1
vote
0
answers
133
views
Extension of a function
Hello,
Given a $\mathcal{C}^\infty$ function $\varphi$ defined on a portion of a surface $\Sigma^-$ and let $\Sigma$ be a closed surface or union of surfaces bounding a compact volume $\Omega \...
user16974
1
vote
1
answer
275
views
Shift operator that generates separable orbit
Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator.
How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T_{n_k}(f)$...
- 696
3
votes
1
answer
121
views
Second difference
Is there an elementary example of a function f, such that $|f(x+t)+f(x-t)-2f(x)|/|t|^a\le C$, where $a>1$, such that $f$ is not $C^1$?
- 95
4
votes
0
answers
213
views
The ring generated by measures
Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
- 8,659
1
vote
1
answer
224
views
Can symmetrizing a contraction increase the speed of convergence?
Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
- 199
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
4
votes
1
answer
222
views
a closed-form for mean/integral, but weighting positive differences between values and "mean" differently from negative differences?
Given a curve $f(x)$ (for $x \in [0,1]$), and a line $y=a$, let $U$ be the total area below $f$ and above $a$, and let $L$ be the total area above $f$ and below $a$. If $L=U$, this means that $a =\...
- 41
1
vote
0
answers
57
views
Looking for CDFs that I can integrate a particular transformation of
I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate
$$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
- 11
4
votes
1
answer
346
views
approximately linear functions -- more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
- 123
0
votes
0
answers
155
views
Convexity of a Certain Set of Covariance Matrices
Hello,
My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
- 155
2
votes
0
answers
470
views
Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?
If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
- 10.1k
5
votes
0
answers
369
views
Independent Events Inducing Probability Measures
Let $\mathcal{F}$ be a sigma algebra over $\Omega$ and $M$ the set of all probability measures on $\mathcal{F}$. Let $\mathcal{C}$ be some collection of pairs $(A,B)$ with $ \ A,B\in\mathcal{F}$. Now ...
- 4,952
0
votes
0
answers
94
views
Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
- 1
0
votes
0
answers
67
views
Proof that Newton expansion over derivatives has the properties of an integral [duplicate]
Let's consider a Newton expansion over consecutive derivatives of a function:
$$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
Can it be proven that such ...
- 10.1k
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 143
3
votes
0
answers
181
views
Example showing that area is discontinuous in the 2-variation seminorm
The $p$-variation seminorm (where $p \ge 1$) of a continuous curve $\alpha: [0,1] \to \mathbb{R}^2$ is defined as the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_n = 1$ of:
$\left(\...
- 4,304
2
votes
0
answers
292
views
Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices
I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:
$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ ...
- 379
5
votes
0
answers
558
views
continuous selection of a multivalued function?
The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...
- 1,839
3
votes
0
answers
211
views
Elementary analysis: reference request
Given the continuous maps $[0,\infty) \to \mathbb R$ define the following "truncation at level $K$ operator", $T$:
$T(f)(t) = f(\min(t, S_f))$, where $S_f = \inf \{ s : f(s) \ge K \}$
So essentially ...
- 2,895
1
vote
0
answers
115
views
A question about smoothness
$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold :
$\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all ...
- 1,441
1
vote
0
answers
163
views
On explicit eigenfunctions
Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta f\left(...
- 11
3
votes
0
answers
302
views
functions on intervals with endpoints
Would most analysts say that $(2/3) x^{3/2}$ is an antiderivative of $x^{1/2}$ on $[0,\infty)$, or
just on $(0,\infty)$?
More generally, is there a standard interpretation of the assertion "$F$ is an ...
- 19.7k
3
votes
0
answers
237
views
Monotonicity of a certain parametric integral
I would like to ask for some help (hints, ideas) in solving the following problem:
Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that
if we define for any $x\in\mathbb{R}...
- 696
0
votes
0
answers
92
views
Class of integrable 0/1-functions "with no null sets."
I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable.
Let $f$ be in this class and $E$ be the set of all points where $f$ is equal to ...
- 567
2
votes
0
answers
517
views
When deRham curve is bijection?
Motivation: Suppose we have deRham curve. From wikipedia:
Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M:
$d_0:\ M \...
- 1,626
0
votes
1
answer
116
views
Root and sign of a complicated bivariate function
Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let
$$
\Phi(p,i) := \frac{1}{2^p+1}
+ \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right),
$$
where $\lg x$ is ...
2
votes
0
answers
131
views
Bounding an integral with a small parameter by log
I have been working through Erdos & Yau's `Linear Boltzmann equation as the weak coupling limit of a random Schrodinger Equation,'
(arXiv link: http://arxiv.org/abs/math-ph/9901020), and for an ...
- 21
0
votes
0
answers
165
views
minimizing the integral of a function over square sets.
Hi!
I'm interested in some problems, but to be honest i'm not sure of the field they belong to.
Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
- 1,352
1
vote
0
answers
265
views
"Lift and project" procedure for matrices
Definition. Let us call $n\times n$ matrix with non-negative entries good if sum of every row and column is equal to $1/n$.
Suppose we have a good matrix $A$. Let us consider the following strange "...
- 1,791
0
votes
0
answers
60
views
Relative homology of interlevel set
Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
H_{\...
- 181
10
votes
0
answers
439
views
Evaluating Shintani cone zeta functions
Hi everyone
I am trying the evaluate sums of the form
$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$
...
- 265
2
votes
0
answers
354
views
What is this effect in Fourier/additive synthesis called?
Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...
- 165
2
votes
0
answers
114
views
Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.
Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.)
...
- 88
0
votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
3
votes
1
answer
263
views
Asymptotically multiplicative functions and matrices
Hi,
Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
- 7,127
4
votes
0
answers
109
views
rank of a C^1 map
I saw this three star problem in Hirsch ..
If we have open sets $U \subset R^3$ ,$V \subset R^2$ and $f:U \to V$ is $C^1$ and onto...Prove there is at least one point in $U$ where $f$ has full rank
...
- 153
0
votes
0
answers
92
views
Lower bound for double sums with power law decay terms.
This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion.
The motivation to ask here if the inequality below ...
- 2,044
4
votes
0
answers
162
views
Symmetric functions and regularity (II)
My previous question (where $n=2$) was a bit too naive. I think that this one, which is the one being of genuine interest to me, is more involved.
Let $f=\mathbb R^n\rightarrow\mathbb R$ be a ...
- 52.3k
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
3
votes
0
answers
107
views
Linear relations with small coefficients
NOTE: Slightly more general question follows my specific one at the top
For $1 \leq i,j \leq n$, let $e_{ij}$ be the vector in $\mathbb{R}^{n^{2}}$ with a 1 in entry $(n-1)i + j$, and 0's everywhere ...
- 83
0
votes
0
answers
44
views
Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
0
votes
0
answers
36
views
Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
- 113
0
votes
0
answers
31
views
What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
0
votes
0
answers
49
views
ODE satisfied by a special function
Posted on MSE
Context
I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica . ...
- 393
0
votes
0
answers
66
views
Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
0
votes
0
answers
57
views
Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
- 75
0
votes
0
answers
22
views
An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
0
votes
0
answers
122
views
Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
