All Questions
5,857 questions
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)?
- 43
1
vote
1
answer
170
views
Root problem involving error function
I ran into this problem in my research:
Let $y_0$ be the root of
$$-(y+a)e^{y^2}\mathit{erfc}(y)+\frac{b}{\sqrt{\pi}}=0$$
on interval $[-a,\infty)$, while $a>0$ and $0<b<1$.
How can I ...
- 389
4
votes
1
answer
343
views
Eigenvalues and eigenvectors of the q-Bernstein operator
The Bernstein operator maps $f\in C[0,1]$ to its Bernstein
polynomial $B_n f.$ The eigenvalues and eigenfunctions of the
Bernstein operator on $C[0,1]$ have been described in [1]. Similar description ...
- 783
2
votes
2
answers
172
views
An analogue of the equidistribution theorem?
Suppose that $(n_k)_{k\in \mathbb{N}}$ is a given increasing sequence of positive integers.
Does there exist an (irrational) number $a$ such that
$\{an_k\}:=(a n_k)\text{mod }1 \rightarrow 1/2$ as $...
- 21
8
votes
2
answers
3k
views
Finite measure on the power set
Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...
- 1,199
10
votes
1
answer
539
views
Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?
In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$.
Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are
...
- 7,034
3
votes
1
answer
169
views
On the values of an entire function
Let $0<q<1$ and consider the entire function $f(z)=\displaystyle \sum_{k=0}^\infty q^{k^2}z^k$. For $a>1,$ denote $m_j=f(a^j),\; j=0,1,2,\dots.$
Question: Does there exist an entire function ...
- 783
2
votes
0
answers
74
views
Can we approximate this matrix field with an invertible matrix field?
Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...
- 6,741
3
votes
1
answer
941
views
What is the mathematical characterization of sufficient statistics of a given $\sigma$-dominated probability model?
Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\...
- 8,071
4
votes
0
answers
289
views
A uniform Riemann sum approximation of the integral of the Fejer kernels
Let $F_N(t)$ denote the Fejer kernel
$$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$
Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
- 136
-1
votes
1
answer
83
views
On probabilistic extension for Bernstein polynomials
Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
- 101
1
vote
0
answers
97
views
Dependency of the Wasserstein metric on its parameters
Let the population on some region $\Omega\subset\mathbb R^d$ be modeled by a density function $\rho:\Omega\to (0,+\infty)$. Provided $n\ge 1$ food trucks labeled by their capacity $p_1,\ldots, p_n\in (...
user128095
2
votes
2
answers
152
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 479
0
votes
0
answers
51
views
A set of zero harmonic measure 2
Let 𝑉 be a bounded open set in $\mathbb{R}^{m}$, $m\geq 2$, and 𝑊 the interior of the closure of 𝑉. Let 𝐸 be a subset of ∂𝑉∩𝑊 (∂ means boundary) such that:
1) $E$ has positive ($m-$dimensional) ...
- 411
4
votes
1
answer
259
views
The integrable condition for distance function
Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$. We denote
$d\left(x\right)$ the distance from $x$ to the boundary of $\Omega$,
that is
$$
d\left(x\right):=\inf\left\{ \left\Vert x-y\right\...
- 41
7
votes
0
answers
266
views
When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?
Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
13
votes
1
answer
1k
views
Is it necessary to use AC to solve this problem ?
Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, $A\...
user30230
0
votes
0
answers
117
views
Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
1
vote
0
answers
88
views
Approximating $3SAT$ by polynomials
Given $3SAT$ formula $\phi$ in $n$ variables and a $\mu\in(0,1/2)$ what is the smallest degree of polynomial $f\in\mathbb Z[x_1,\dots,x_n]$ such that for an $(a_1,\dots,a_n)\in\{0,1\}^n$ if $\phi(a_1,\...
- 1,826
0
votes
2
answers
137
views
Level sets and integral of functions of two variables
Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...
5
votes
0
answers
975
views
How did $\limsup$ and $\liminf$ come about? [closed]
I am aware of at least three equivalent definitions of $\limsup$ and $\liminf$. I shall only write the definitions for $\limsup$. Here $(a_n)_{n \in \mathbf{N}}$ refers to a sequence of real numbers.
...
- 151
6
votes
3
answers
11k
views
Sums of uncountably many real numbers [closed]
Suppose $S$ is an uncountable set, and $f$ is a function from $S$ to the positive real numbers. Define the sum of $f$ over $S$ to be the supremum of $\sum_{x \in N} f(x)$ as $N$ ranges over all ...
- 15.4k
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
3
votes
1
answer
461
views
Bounding the "spikiness" of a probability distribution
Are there any well-known conditions that guarantee that a probability distribution isn't too "spiky"?
I ask this question because I am interested in the families of probability distributions $f(x)$ ...
- 4,049
5
votes
1
answer
305
views
Expectation of max of Gaussian multiplied by a functional of Gaussian
Let $X \in \mathbb{R}^{d}$ follows the standard Gaussian distribution $N(0, I_d)$. Let $Y = \max_{j\in[d] } X_j$. It is not hard to see that
\begin{align}
\mathbb{E}\left [ Y \cdot X\right] = \sum_{j=...
- 1,127
5
votes
1
answer
240
views
Are Pointwise conditions studied?, do they make sense?, do they have any applications?
In weakly formulated PDE (or even ODE), we seem to be interested in solutions that satisfy or take desired values at some boundary points of a domain we are interested in. For example, Dirichlet ...
- 698
2
votes
0
answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
3
votes
1
answer
411
views
Continuation of a smooth function, whose every derivative is strictly monotonic
Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
3
votes
2
answers
2k
views
Is there an example where the error of Gauss-Laguerre quadrature does not vanish?
The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum
$$\sum_{i=1}^n f(x_i) w_i$$
where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
- 1,839
1
vote
0
answers
86
views
Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553
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
0
votes
0
answers
115
views
If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
11
votes
4
answers
5k
views
The metric space associated to a measure space
Let $(X, \mathcal{A}, \mu)$ be a measure space such that $\mu(X) < \infty$. We say that two measurable sets $A$ and $B$ are equivalent if $\mu (A \Delta B) = 0$. The equation $$ d(A,B) = \mu (A \...
- 3,830
0
votes
1
answer
386
views
Functions satisfying Neumann boundary condition
I have a question about functions satisfying a condition.
Let $D \subset \mathbb{R}^d$ be a Lipschitz domain. That is, for each $x \in \partial D$, there exists an open neighborhood $U$ of $x$ in $\...
- 721
9
votes
2
answers
537
views
Comparing the growth of $f\circ g$ and $g\circ f$
I asked this Question on Math.StackExchange without success. Then I learned, that this might be the better place to ask. So, sorry for crossposting. I would agree on deleting my old question.
Let $\...
- 13.6k
2
votes
2
answers
226
views
Orthogonal system of functions ordered by norm of second derivative
Problem setting:
Let $\Omega = [-1,1] \subset \mathbb{R}$ be an interval and consider the space of infinitely differentiable functions, that is $C^{\infty}$.
We successively define the sequence $f_k ...
- 390
1
vote
1
answer
165
views
Morrey condition (integral condition) and (local) Holder condition
Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$)
$$\limsup_{r \to 0} r^{-\alpha \beta}\frac{...
- 839
2
votes
3
answers
5k
views
Smooth approximation of the hinge loss function
I came across a paper but the smooth approximation for the hinge loss function is wrong. Can someone guide me to the proper smooth approximation (using polynomials) of the function $$h(x)=\max(0,1-x)$$...
- 111
-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 ...
- 353
1
vote
0
answers
39
views
Analytic function whose derivatives and primitives are independent from a given set of countable cardinality
Let $L=(l_j)_{j\in\mathbb{N}}$ be a set of countably many independent real analytic functions on $[0,2\pi]$. Here and in the following, independent means that a function cannot be written as finite ...
- 405
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
1
vote
0
answers
268
views
Determinant and restriction of scalar
Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...
- 833
2
votes
0
answers
164
views
Generalization of regularly varying functions
A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$,
$$
\lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a)
$$
for some function $g(a)&...
- 3,223
1
vote
1
answer
155
views
Proving a sum to be sublinear in growth
Suppose that $\alpha \in (0,1)$. The goal is to prove that the following sum is of $o(T)$ (or, if possible, give a more accurate growth rate, e.g. $O(T^{1-\alpha})$ or something like that):
$$ \sum_{t=...
- 41
2
votes
2
answers
111
views
points separation and dimensions [closed]
Suppose $\mathcal{A}$ is a sub-algebra of $C([0,1],\mathbb{R})$.
If $\mathcal{A}$ separates points in $[0,1]$, does it follow $\dim\mathcal{A}=\infty$?
- 309
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 ...
- 536
2
votes
1
answer
63
views
Can a sequence of degree one maps converges to a constant map in $W^{1,2}$ norm?
Can a sequence of degree one maps on, say, the unit circle, converges to a constant map in $W^{1,2}$ norm?
If the answer is yes, would you please provide an explicit example?
- 1,630
2
votes
1
answer
291
views
Approximation of an injective continuous curve by injective piecewise linear curves
Let $f:[a,b]\longrightarrow\mathbb{R}^2$ be an injective continuous function. For any $d>0$, does there exist a piecewise linear curve: $g:[a,b]\longrightarrow\mathbb{R}^2$ such that $g$ is also ...
- 1,997
0
votes
0
answers
77
views
Energy-minimizing set of discrete points in a bounded domain
Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain.
Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize
$$
\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}
$$
over all ...
- 401
2
votes
0
answers
144
views
Lebesgue density theorem for "doubling uniformly covering collections of subsets"
I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...
- 2,974