All Questions
18,178 questions
-1
votes
1
answer
168
views
A question in functional analysis about selfadjoint operator [closed]
In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$.
I ...
- 1
-1
votes
1
answer
148
views
Weighted sum of zero-mean random variables
Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$.
Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
- 367
-1
votes
1
answer
304
views
The category Prob of finite measure spaces does not admit all products [closed]
I am currently working in a category called Prob which has objects which are finite measure spaces and morphisms which are measure preserving maps between the spaces. A map $f:X\to Y$ is measure ...
- 91
-1
votes
1
answer
173
views
The distribution of the sum of a non-zero vector with random signs
Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find ...
- 121
-1
votes
1
answer
246
views
Density of normal elements in a C*- algebra [closed]
Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
- 149
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 131
-1
votes
2
answers
129
views
Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]
Under which condition can it form a Hilbert space? Or what space can it form?
You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
- 11
-1
votes
1
answer
215
views
Dense linear span implies closed convex hull has non-empty interior
Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
- 5,405
-1
votes
1
answer
129
views
Is it possible to regress an arbitrary function from a span?
Suppose we have three sets $A, B,C$ and a span $S := A \leftarrow C \rightarrow B$. There is a special case when the data of the span $S$ exactly specifies a function $f: A \rightarrow B$. In ...
- 1,313
-1
votes
1
answer
122
views
Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
-1
votes
1
answer
135
views
Prove that Cartesian composition $c_0 \times c_0$ is not isometric isomorphic [closed]
Prove that Cartesian composition $c_0 \times c_0$ with rate $ \Vert (x_1 ,x_2)\Vert = \Vert x_1 \Vert_{c_0} + \Vert x_2 \Vert_{c_0} $ is not isometric isomorphic to space $c_0$.
- 21
-1
votes
1
answer
346
views
Riesz representation theorem for Hilbert-to-Hilbert mappings [closed]
Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz ...
- 313
-1
votes
2
answers
614
views
Bounded difference functions and sub-Gaussian random variables
We have the following standard theorem : Let $X$ be some set and $g : X^n \rightarrow \mathbb{R}$ be a measurable function such that it satisfies the ``bounded difference property" i.e $\exists$ $\{...
- 2,246
-1
votes
1
answer
197
views
Surely recurrent random walks and the law of the iterated logarithm [closed]
Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with
$$
P(X_i=1)=P(X_i=-1)=1/2,
$$
and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As ...
- 965
-1
votes
1
answer
558
views
For i.i.d X and Y , if X + Y and X - Y are independent, show X is normally distributed [closed]
The question goes as follows:
If $X$ and $Y$ are independent and identically distributed, their density function $f(x)$ is strictly positive and second-order continuously differentiable. If $X+Y$ and $...
- 9
-1
votes
1
answer
237
views
Theorem with an example [closed]
i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
- 277
-1
votes
1
answer
280
views
Showing there is a unique spectral measure
All the books I have seen have proved that, for a normal bounded operator $T$, there is a unique spectral measure $E$ such that $\int_{\sigma(T)}^{}\lambda\,dE=T$ by first proving in it for a general ...
- 127
-1
votes
2
answers
462
views
How to deal with this Chicken-And-Egg problem ?
Let's imagine designing an odds pattern for a game, in which players bet for win or lose.
Suppose the probablity of winning is $p$, thus the probablity of losing is $1-p$.
Now imagine $n_1$ people ...
- 99
-1
votes
1
answer
1k
views
Approximating expectation [closed]
if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points:
E[f]=(1/N)(summation of f(x) over these N points).
...
-1
votes
1
answer
247
views
Concentration results for non-standard Gaussian random vectors.
Given a $c$-Lipschitz function $f(X):\mathbb{R}^d \rightarrow \mathbb{R}$, and given that $X \in \mathbb{R}^d$ is a Gausssian random vector centered at $\mathbb{w} \in \mathbb{R}^d$ (not at zero) ...
- 11
-1
votes
1
answer
571
views
Formal definition of 'useful' ?
Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...
- 11.8k
-1
votes
1
answer
80
views
Seating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
- 48.9k
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
-1
votes
1
answer
504
views
Minimum of exponential distribution
Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution:
$...
- 9
-1
votes
1
answer
128
views
Large deviation of sum of Gaussian variables
Let $X_i$ is $N(b_i,b_i^2)$, where $0<b_i\leq \alpha$. The $X_i$ are independent, but not identical (i.e. $b_i$ are not all equal). We concern the upper bound of the tail probability $P(|\sum_{i=1}^...
- 405
-1
votes
1
answer
98
views
Distribution of root with Poisson leaves
We have a pool of items, termed as item A, generated following a Poisson distribution. We use a pair of items A to produce an item B with success rate $r\in(0,1)$. My question is: is B Poisson ...
- 367
-1
votes
2
answers
158
views
Cumulants of a sequence of variables with zero mean and variance
Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
- 13
-1
votes
1
answer
77
views
Applications and motivations of resolvent for elliptic operator
Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
\...
- 1,219
-1
votes
1
answer
83
views
"Large" compact sets in separable normed space
Let $(X, \lVert \cdot \rVert)$ be a separable normed space. Can we always guarantee that there is a nonempty compact set $K \subseteq B_X$, where $B_X$ is a closed unit ball in $X$ such that:
$$\...
- 820
-1
votes
1
answer
116
views
Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse
Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $
\delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\}
$ are homeomorphic. What I'm ...
- 5,405
-1
votes
1
answer
120
views
Definition of a $\psi$-Banach space [closed]
Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
- 291
-1
votes
1
answer
90
views
A periodically independent stochastic process
Does there exist a non constant almost surely continuous stochastic process $X$ on $[0, \infty)$ with $X_t$ independent of $X_{t+1}$ for all $t \geq 0$?
- 6,215
-1
votes
1
answer
176
views
How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]
How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ?
Is the pressure limited or can it be any amount?
- 11
-1
votes
1
answer
2k
views
how to prove that the real part and the modulus of a characteristic function is still a characterisitc function? [closed]
this is a problem from Durret's probability textbook.
Show that if $\varphi$ is a ch.f., then $Re\varphi$ and $|\varphi|^2$ are also ch.f.
I am wondering how to prove this. Actually I'm not even sure ...
- 11
-1
votes
2
answers
1k
views
Does the variance of a strictly monotonically increasing function of a random variable have anything to do with the variance of the random variable? [closed]
Assume that there is a continuous random variable x, and its variance is var(x). Furthermore, there is a strictly monotonically increasing function f. Can anybody prove that the larger the var(x), the ...
- 1
-1
votes
1
answer
70
views
Is this kind of interpolation correct?
Let $f=\sum f_j$ be a finite sum. Assume that
$$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$
$$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$
Then can we conclude that for $2<p<\infty$
$$\|f\|_p\le C^{1-\...
- 1
-1
votes
1
answer
429
views
What is the expected area of the triangle? [closed]
We create the unit equilateral triangle and put one vertex on each side the of the equilateral triangle and then connect them. What is the expected value of the triangle formed by the connection of ...
-1
votes
1
answer
328
views
About the critical points of quasi-convex functions
What do we know about the structure of critical points of quasi-convex functions?
I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...
- 2,246
-1
votes
1
answer
95
views
Proving maximal entropy [closed]
It is quite easy to prove that
$$H(S) \leq \log_2(|A|),$$
where $A$ is the number of events, using the Jensen inequality
$$H(S) = E_S[\log_2(\frac{1}{P_S(s)})]\leq \log_2(E_S[(\frac{1}{P_S(s)})]) =...
-1
votes
1
answer
186
views
Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities? [closed]
This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X_i$ with different occurrence probabilities, $\mathrm{Pr}[X_i=1]=p_i>p_{\min}>0$ ...
- 307
-1
votes
2
answers
438
views
Are the coefficients of a linear combination of random vectors as random?
Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
- 271
-1
votes
1
answer
213
views
Regarding a new divergence function of two probability distributions
Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any $0 \leq \alpha \leq 1$, and any constant $\beta$ within the support of $X$ and $Y$ such ...
- 482
-1
votes
1
answer
346
views
An infinite set in a compact space
Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
-1
votes
1
answer
360
views
Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]
Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
- 7
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
-1
votes
1
answer
104
views
Question about measure lemma?
"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ $|...
- 277
-1
votes
1
answer
287
views
Property of relative entropy [closed]
For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$
...
- 35
-1
votes
1
answer
516
views
Equivalence of two definitions of Sobolev spaces
Good morning,
I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space
$$
D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) \...
- 9
-1
votes
1
answer
128
views
Proving convergence of an integral-differential equation [closed]
I have a second order nonlinear ordinary differential equation which I transformed into an integral-differential equation by multiplying the ODE by $y'$ and integrating.
My question is where can I ...
- 1,594
-1
votes
1
answer
187
views
Limit of a function in a weighted Sobolev space
I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense
$$\lim_{|x-y|\to 0} f(x)$$
? ($y$ is a fixed point)
If i have $f$ in $H^2$ I can say that
$$\lim_{|x-y|\to 0} f(x)=...
- 25