All Questions
18,179 questions
1
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309
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Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
2
votes
0
answers
82
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Description of the norm of certain interpolation space
Dear all,
I suspect that there should be some detailed description of the norm (or of the unit ball) of the following complex interpolation space (for any $0< \theta < 1$): $$\Big(B(\ell_1^n, \...
- 769
7
votes
0
answers
161
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Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
2
votes
0
answers
105
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Modelling a GI/G/1 queue with exceptional first vacation and multiple vacations
I am currently working on a stochastic modelling problem in networks. I have a G/G/1 queue where
the interarrival and inter-service times are iid
the arrival and service time distributions are ...
- 519
1
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0
answers
114
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Mappings preserving convex compactness
Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...
- 11
0
votes
0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411
2
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0
answers
153
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Reference request for a result on subsets unlikely to be hit by random walks in a group
Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some ...
- 21
2
votes
1
answer
250
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Expectation of RVs with Poisson-type decay
I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:
$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$
where $...
- 23
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
2
votes
1
answer
254
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Brownian Bridge under observational error
Suppose that $Z_t$ follows a simple discrete random walk $Z_t=Z_{t-1}+e_t$ , where $e_t$ are a bunch of uncorrelated normal variables with arbitrary variance sigma^2, and that there are observations ...
- 457
1
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0
answers
93
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Potentials of class D
A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...
- 667
2
votes
0
answers
90
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Limiting distribution of the cardinal of a Markovian set
Let $S_1=\lbrace u_1 \rbrace$ where $u_1$ is a random uniform drawing on $[0,1]$. To build $S_{n+1}$ draw $u_{n+1}$ uniformly on $[0,1]$ (independently from previous draws) and draw $v_{n+1}$ ...
3
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0
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223
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Extension of positive operators and Bauer-Namioka
When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
- 31
6
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0
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189
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average Riemannian distance between Identiity and a random point in SO(n) or SU(n)
I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...
- 4,466
4
votes
1
answer
232
views
Negative Association of Component Size in Random Hypergraph
I have a $d$-uniform hypergraph on $n$ vertices with $k$ hyperedges, where $d << k$ and $n = 4k d^2$ or so.
The hyperedges are placed independently uniformly at random. I would like to have a ...
- 241
2
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0
answers
200
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Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21
4
votes
0
answers
117
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symmetric difference of temperate zone and inscribed disk
For random domino tilings of the Aztec diamond of order $n$ or random lozenge tilings of the regular hexagon of order $n$, what's the typical order of magnitude of the area of the symmetric difference ...
- 19.7k
0
votes
1
answer
130
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Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 800
4
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1
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222
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Does positive density imply existence of the density for some part of a decomposition?
Suppose a $\mathcal{H}^{1}$ measurable set $A\subset \mathbb{R}^{n}$ has positive Hausdorff density $\Theta^{1}(\mathcal{H}^{1},A,x)=c>0$ in a point $x\in A$.
If we have a decomposition $A=B\cup ...
1
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0
answers
135
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Optimizing for a unique outcome of a probabilistic marriage problem
Let's say I have some number of individuals who are single, $(b_1, ..., b_N) \in B$, and for every possible pairing of two individuals, $b_i$ and $b_j$, I happen to know the exact probability that the ...
- 11
3
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229
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For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?
Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J \sum_{(...
- 2,252
1
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0
answers
52
views
Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\...
- 71
1
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0
answers
61
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Multi-completely monotone functions
Consider a $C^{\infty}$ nonnegative function $f(x,y,z)$, $x,y,z>0$ and let $\lambda f(\lambda x, \lambda y,\lambda z) \equiv f(z,y,z)$ for any $\lambda > 0$ (positive homogenity). Define
$$
g_{...
- 1,329
3
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0
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383
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Neglect of Compact Quantum Metric Spaces [closed]
Does anyone have an opinion on Rieffel's theory of compact quantum metric spaces? To me it seems to be a very interesting new area of mathematics. It shows how to generalise complicated geometric ...
- 1,462
8
votes
0
answers
348
views
A formula for moments of the limit distribution of singular values in the proof of the circular law
One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix
$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - zI)^\...
0
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0
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80
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relationship between different function classes
I was wondering if there is a survey of relationship between several different well-studied function classes ?
ps - The question may be vague but I am looking for something along the lines of - http:/...
- 1
0
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0
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163
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"Reverse" stochastic dominance
Let $\mu$ and $\mu'$ be probability measures on $\lbrace0,1\rbrace^\Lambda,\:\: \Lambda:= {\lbrace 0,1,\ldots,n\rbrace}$. Assume that
$\mu(X_i=1|X = \zeta \text{ on } \Lambda \setminus \lbrace i\...
- 1,491
6
votes
0
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98
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Do the translates of integrable function approximate its radial part?
For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (...
- 415
6
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0
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104
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geometric construction of uniform measure on plane partitions in a box
If one picks $a+b$ points uniformly at random (and independently) in $[0,1]$, coloring $a$ of them Red and $b$ of them Blue, then reading the points from left to right one gets a uniform random ...
- 19.7k
1
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0
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103
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Bounds on CDF for the median of samples from an exchangeable distribution
Suppose $x_1,\dotsc, x_n$ are $n = 2k-1$ samples from an EXCHANGEABLE sequence, where the common marginal distribution is assumed UNIFORM on $[0,1]$. Let $x_{(1;n)} \le \dotsc \le x_{(n;n)}$ be the ...
1
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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
0
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0
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104
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Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
1
vote
1
answer
188
views
Integral determines function behaviour
Let us define:
$f(t) = t^{-1} \int_{\mathbf{R}^{3}} Exp[-\frac{x^2}{2t}] h(x) dx,$
for a real function h. What can I say about this function if I know that
$f(t) \rightarrow 1$.
I think that the ...
- 1,491
1
vote
1
answer
336
views
A probabilistic inequality [closed]
Suppose $x_1,x_2,...,x_6$ are non-negative Independent and identically-distributed random variables, is it true that $P(x_1+x_2+x_3+x_4+x_5+x_6 \lt 3\delta) \lt 2P(x_1 \lt \delta)$ for any $\delta \...
- 35
1
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1
answer
210
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Is the metric obtained by altering the metric of a Hilbert space on a finite-dimensional subspace equivalent to the original one? [closed]
Suppose a Hilbert space W can be written as the direct sum (not necessarily orthogonal) of the closed subspaces H and V, where H is assumed to be of finite dimension. Define a new inner product via
||...
- 2,935
3
votes
0
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107
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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
2
votes
1
answer
178
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Maximal inequality over two indices
In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like:
P[$\max_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$]
in the background of ...
1
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0
answers
195
views
Lower semicontinuity of Bregman distances/divergences
For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$:
\begin{...
- 12.7k
1
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0
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225
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What is the limiting distribution of local minima of n mod i, for i up to $\sqrt{n}$, as $n \rightarrow \infty$?
The sequence n mod i
Consider the sequence n mod i for i=1...$\sqrt{n}$. If we draw the sequence as an xy-plot, we get a dense triangle (since n mod i < i). More precisely, the limiting density of ...
- 1,906
5
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0
answers
211
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Exponential tails for a functional of a subcritical branching process.
Let $(m_i, i \in \mathbb{N})$ be positive weights with $\sum_{i \in \mathbb{N}} m_i^2 < 0.1$.
Consider a subcritical branching process in discrete time and continuous space,
started from some ...
- 4,922
2
votes
0
answers
140
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WLD Banach spaces
Does anyone know of an example of a weakly Lindeloff determined (WLD) Banach space which does not contain c_0 and is not weak Asplund? I believe the example of a WLD, non-weak Asplund space by Argyros ...
- 21
1
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0
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111
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Conditional prob of value given input is drawn from subset = conditioning over subset?
Hey guys, I have a pretty basic question that I want to be sure of. I'm taking a probability over an input selected uniformly at random from binary strings of length $l(n)$. I would like to compare ...
- 113
1
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0
answers
118
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A maximum likelihood -based ranking system
Given a collection of sample rankings, what is the best way to compile them into an aggregate ranking? (Don't worry, I'm working towards a well-defined question.) There are at least two obvious ...
- 3,612
0
votes
0
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73
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A constrained prolongement
Let $\Omega$ be a domain of $R^n$, let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$
I am wondering about the existence of a function $\tilde{\theta} \in W^{2,\infty}...
- 25
2
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0
answers
156
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Holomorphic stability of inverse limit of pre-$C^*$-algebras
Let A be a C*-algebra and let At be a set of dense *-subalgebras of A, stable under holomorphic functional calculus on A, which are also Banach algebras complete with respect to the norms ||$\cdot$||t....
- 613
2
votes
1
answer
168
views
Local supporting points of Lipschitz functions
Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a
Lipschitz function. Say that a point x in X is a local supporting point
of f if there exist x^* in X^* and an open neighborhood U ...
- 31
1
vote
0
answers
129
views
A M/M/$\infty$ queue of depositors with compound interest
Hello, I'm trying to model a bank's liabilities using a queue. Suppose a bank begins with a cash reserve of $M$. Depositors are a M/M/$\infty$ queue; they arrive with rate $\lambda$ and deposit 1 ...
- 309
-2
votes
2
answers
245
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Evaluate a fair game [closed]
I'm not a mathematician, so my question may be not so clear, sorry.
Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 &...
- 1
1
vote
0
answers
134
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Renewal function - duality
Let us consider a random walk $(S_n)_n$. One denotes the instants of records of $-S_n$ by $0=T_0 < T_1 < T_2 \cdots$. Then for all $k$ one sets: $H_k=-S_{T_k}$. Finally, one define $\tau$ as the ...
- 551
0
votes
1
answer
207
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Correlation of Statistical Tests
Suppose I have a sequence $\{x_i\}_{i=1}^\infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are ...
- 1,588