367,153 research outputs found
Random-time processes governed by differential equations of fractional distributed order
We analyze here different types of fractional differential equations, under
the assumption that their fractional order is random\ with
probability density We start by considering the fractional extension
of the recursive equation governing the homogeneous Poisson process
\ We prove that, for a particular (discrete) choice of , it
leads to a process with random time, defined as The distribution of the
random time argument can be
expressed, for any fixed , in terms of convolutions of stable-laws. The new
process is itself a renewal and
can be shown to be a Cox process. Moreover we prove that the survival
probability of , as well as its
probability generating function, are solution to the so-called fractional
relaxation equation of distributed order (see \cite{Vib}%).
In view of the previous results it is natural to consider diffusion-type
fractional equations of distributed order. We present here an approach to their
solutions in terms of composition of the Brownian motion with the
random time . We thus provide an
alternative to the constructions presented in Mainardi and Pagnini
\cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order
case.Comment: 26 page
Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds
In this paper we introduce three Markovian couplings of Brownian motions on
smooth Riemannian manifolds without boundary which sit at the crossroad of two
concepts. The first concept is the one of shy coupling put forward in
\cite{Burdzy-Benjamini} and the second concept is the lower bound on the Ricci
curvature and the connection with couplings made in \cite{ReSt}.
The first construction is the shy coupling, the second one is a
fixed-distance coupling and the third is a coupling in which the distance
between the processes is a deterministic exponential function of time.
The result proved here is that an arbitrary Riemannian manifold satisfying
some technical conditions supports shy couplings. If in addition, the Ricci
curvature is non-negative, there exist fixed-distance couplings. Furthermore,
if the Ricci curvature is bounded below by a positive constant, then there
exists a coupling of Brownian motions for which the distance between the
processes is a decreasing exponential function of time. The constructions use
the intrinsic geometry, and relies on an extension of the notion of frames
which plays an important role for even dimensional manifolds.
In fact, we provide a wider class of couplings in which the distance function
is deterministic in Theorem \ref{t:100} and Corollary~\ref{Cor:9}.
As an application of the fixed-distance coupling we derive a maximum
principle for the gradient of harmonic functions on manifolds with non-negative
Ricci curvature.
As far as we are aware of, these constructions are new, though the existence
of shy couplings on manifolds is suggested by Kendall in \cite{Kendall}.Comment: This version is a refinement expansion and simplification of the
previous versio
And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models
One of the most important results in operator theory is And\^o's \cite{ando}
generalization of dilation theory for a single contraction to a pair of
commuting contractions acting on a Hilbert space. While there are two explicit
constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the
minimal isometric dilation of a single contraction, there was no such explicit
construction of an And\^o dilation for a commuting pair of
contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In
this paper, we give two new proofs of And\^o's dilation theorem by giving both
Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation
with function-theoretic interpretation, for the general case. The results, in
particular, give a complete description of all possible factorizations of a
given contraction into the product of two commuting contractions. Unlike
the one-variable case, two minimal And\^o dilations need not be unitarily
equivalent. However, we show that the compressions of the two And\^o dilations
constructed in this paper to the minimal dilation spaces of the contraction
, are unitarily equivalent.
In the special case when the product is pure, i.e., if strongly, an And\^o dilation was constructed recently in \cite{D-S-S},
which, as this paper will show, is a corollary to the Douglas-type
construction.
We define a notion of characteristic triple for a pair of commuting
contractions and a notion of coincidence for such triples. We prove that two
pairs of commuting contractions with their products being pure contractions are
unitarily equivalent if and only if their characteristic triples coincide. We
also characterize triples which qualify as the characteristic triple for some
pair of commuting contractions such that is a pure
contraction.Comment: 24 page
Isotopic liftings of Clifford algebras and applications in elementary particle mass matrices
Isotopic liftings of algebraic structures are investigated in the context of
Clifford algebras, where it is defined a new product involving an arbitrary,
but fixed, element of the Clifford algebra. This element acts as the unit with
respect to the introduced product, and is called isounit. We construct
isotopies in both associative and non-associative arbitrary algebras, and
examples of these constructions are exhibited using Clifford algebras, which
although associative, can generate the octonionic, non-associative, algebra.
The whole formalism is developed in a Clifford algebraic arena, giving also the
necessary pre-requisites to introduce isotopies of the exterior algebra. The
flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact,
when the generators of the isotopic Lie algebra su(6) are constructed, and the
unit of the isotopic Clifford algebra is shown to be a function of the six
quark masses. The limits constraining the parameters, that are entries of the
representation of the isounit in the isotopic group SU(6), are based on the
most recent limits imposed on quark masses.Comment: 19 page
Balanced Families of Perfect Hash Functions and Their Applications
The construction of perfect hash functions is a well-studied topic. In this
paper, this concept is generalized with the following definition. We say that a
family of functions from to is a -balanced -family
of perfect hash functions if for every , , the number
of functions that are 1-1 on is between and for some
constant . The standard definition of a family of perfect hash functions
requires that there will be at least one function that is 1-1 on , for each
of size . In the new notion of balanced families, we require the number
of 1-1 functions to be almost the same (taking to be close to 1) for
every such . Our main result is that for any constant , a
-balanced -family of perfect hash functions of size can be constructed in time .
Using the technique of color-coding we can apply our explicit constructions to
devise approximation algorithms for various counting problems in graphs. In
particular, we exhibit a deterministic polynomial time algorithm for
approximating both the number of simple paths of length and the number of
simple cycles of size for any
in a graph with vertices. The approximation is up to any fixed desirable
relative error
Permutation bases in the equivariant cohomology rings of regular semisimple Hessenberg varieties
Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet
connects the well-known Stanley-Stembridge conjecture in combinatorics to the
dot action of the symmetric group on the cohomology rings
of regular semisimple Hessenberg varieties. In particular, in
order to prove the Stanley-Stembridge conjecture, it suffices to construct (for
any Hessenberg function ) a permutation basis of whose
elements have stabilizers isomorphic to Young subgroups. In this manuscript we
give several results which contribute toward this goal. Specifically, in some
special cases, we give a new, purely combinatorial construction of classes in
the -equivariant cohomology ring which form permutation
bases for subrepresentations in . Moreover, from the
definition of our classes it follows that the stabilizers are isomorphic to
Young subgroups. Our constructions use a presentation of the -equivariant
cohomology rings due to Goresky, Kottwitz, and MacPherson.
The constructions presented in this manuscript generalize past work of
Abe-Horiguchi-Masuda, Chow, and Cho-Hong-Lee.Comment: 33 page
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