1,720,973 research outputs found
Theory of the (m,σ)-general functions over infinite-dimensional Banach spaces
Full text linkIn this paper, we introduce some functions, called (m, σ)-general, that generalize the (m, σ)-standard functions and are defined in the infinite-dimensional Banach space E_I of the bounded real sequences {xn}n∈I, for some subset I of N*. Moreover, we recall the main results about the differentiation theory over E_I, and we expose some properties of the (m, σ)-general functions. Finally, we study the linear (m, σ)-general functions, by introducing a theory that generalizes the standard theory of the m × m matrices
Convergence in total variation of an affine random recursion in [0,p)^k to a uniform random vector
No full textWe study the rate of convergence of the Markov chain (mod ), where is an integer matrix with nonzero eigenvalues, is real and positive, and is a sequence of independent and identically distributed real random vectors. With some hypotheses on the law of , the sequence converges to a random vector uniformly distributed in . The rate of
convergence is geometric and depends on , , , and the distribution of . Moreover, if has an eigenvalue that is a root of , then steps are necessary to have
sampling from a nearly uniform law
Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications
No full textIn this paper we study, for some subsets I of N^{∗}, the Banach space E of bounded real sequences {x_{n}}_{n∈I}. For any integer k, we introduce a measure over (E,B(E)) that generalizes the k-dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables formula for the integration
Infinite-dimensional Gaussian change of variables’ formula
Full text linkIn this paper, we study the Banach space ∞ of the bounded real sequences, and a
measure N(a, ) over (R∞
, B∞
) analogous to the finite-dimensional Gaussian law.
The main result of our paper is a change of variables’ formula for the integration,
with respect to N(a, ), of the measurable real functions on (E∞, B∞
(E∞)), where
E∞ is the separable Banach space of the convergent real sequences. This change of
variables is given by some (m, σ) functions, defined over a subset of E∞, with values
on E∞, with properties that generalize the analogous ones of the finite-dimensional
diffeomorphisms
Asymptotic behavior of an affine random recursion in (Z_p)^k defined by a matrix with an eigenvalue of size 1
No full textIn this paper we study the rate of convergence of the Markov chain XnC1 D AXn C Bn.mod p/, where A is an integer invertible matrix, and fBngn is a sequence of independent and identically distributed integer vectors. If A has an eigenvalue of size 1, then n D O.p2/ steps are necessary and sufficient to have Xn sampling from a nearly uniform distribution
Differentiation Theory over Infinite-Dimensional Banach Spaces
Full text linkIn this paper we study, for any positive integer and for any subset\ \ of \QTR{bf}{N}^{\ast }, the Banach space of the bounded real sequences , and a measure over \left( \QTR{bf}{R}^{I},\QTR{cal}{B}^{(I)}\right) that generalizes the -dimensional Lebesgue one. Moreover, we expose a differentiation theory for the functions defined over this space. The main result of our paper is a change of variables' formula for the integration of the measurable real functions on \left( \QTR{bf}{R}^{I},\QTR{cal}{B}^{(I)}\right) . This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms
Change of variables' formula for the integration of the measurable real functions over infinite-dimensional Banach spaces
Full text linkIn this paper we study, for any subset of and for
any strictly positive integer , the Banach space of the bounded
real sequences , and a measure over
that generalizes the
-dimensional Lebesgue one. Moreover, we recall the main results about the
differentiation theory over . The main result of our paper is a change
of variables' formula for the integration of the measurable real functions on
. This change of variables
is defined by some functions over an open subset of , with values on
, called -general, with properties that
generalize the analogous ones of the finite-dimensional diffeomorphisms
Generating uniform random vectors in (Z_p)^k: the general case
No full textWe consider the rate of convergence of the Markov chain Xn+1 = AXn + Bn (modp), where A is an integer matrix with nonzero eigenvalues, and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Qk invariant under A. If |λi | =1 for all eigenvalues λi of A, then n = O((ln p)2) steps are sufficient and n = O(ln p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λi that are roots of positive integer numbers, |λ1| = 1 and |λi | > 1 for all i = 1, then O(p2) steps are necessary and sufficient
Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems
No full textIn this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large body system of hard spheres,QTR{it}{ i.e.,} formed by particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto QTR{it}{et al}., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small--size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of $arepsilon .
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