20,757 research outputs found
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
Let M denote the space of Borel probability measures on the real line. For
every nonnegative t we consider the transformation
defined for any given element in M by taking succesively the the (1+t) power
with respect to free additive convolution and then the 1/(1+t) power with
respect to Boolean convolution of the given element. We show that the family of
maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of
composition and that, quite surprisingly, every is a homomorphism
for the operation of free multiplicative convolution.
We prove that for t=1 the transformation coincides with the
canonical bijection discovered by Bercovici and
Pata in their study of the relations between infinite divisibility in free and
in Boolean probability. Here M_{inf-div} stands for the set of probability
distributions in M which are infinitely divisible with respect to free additive
convolution. As a consequence, we have that is infinitely
divisible with respect to free additive convolution for any for every in
M and every t greater than or equal to one.
On the other hand we put into evidence a relation between the transformations
and the free Brownian motion; indeed, Theorem 4 of the paper
gives an interpretation of the transformations as a way of
re-casting the free Brownian motion, where the resulting process becomes
multiplicative with respect to free multiplicative convolution, and always
reaches infinite divisibility with respect to free additive convolution by the
time t=1.Comment: 30 pages, minor changes; to appear in Indiana University Mathematics
Journa
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
State-space distribution and dynamical flow for closed and open quantum systems
We present a general formalism for studying the effects of dynamical
heterogeneity in open quantum systems. We develop this formalism in the state
space of density operators, on which ensembles of quantum states can be
conveniently represented by probability distributions. We describe how this
representation reduces ambiguity in the definition of quantum ensembles by
providing the ability to explicitly separate classical and quantum sources of
probabilistic uncertainty. We then derive explicit equations of motion for
state space distributions of both open and closed quantum systems and
demonstrate that resulting dynamics take a fluid mechanical form analogous to a
classical probability fluid on Hamiltonian phase space, thus enabling a
straightforward quantum generalization of Liouville's theorem. We illustrate
the utility of our formalism by analyzing the dynamics of an open two-level
system using the state-space formalism that are shown to be consistent with the
derived analytical results
The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold
We study phase retrieval from magnitude measurements of an unknown signal as
an algebraic estimation problem. Indeed, phase retrieval from rank-one and more
general linear measurements can be treated in an algebraic way. It is verified
that a certain number of generic rank-one or generic linear measurements are
sufficient to enable signal reconstruction for generic signals, and slightly
more generic measurements yield reconstructability for all signals. Our results
solve a few open problems stated in the recent literature. Furthermore, we show
how the algebraic estimation problem can be solved by a closed-form algebraic
estimation technique, termed ideal regression, providing non-asymptotic success
guarantees
Loop algebras, gauge invariants and a new completely integrable system
One fruitful motivating principle of much research on the family of
integrable systems known as ``Toda lattices'' has been the heuristic assumption
that the periodic Toda lattice in an affine Lie algebra is directly analogous
to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper
shows that the analogy is not perfect. A discrepancy arises because the natural
generalization of the structure theory of finite-dimensional simple Lie
algebras is not the structure theory of loop algebras but the structure theory
of affine Kac-Moody algebras. In this paper we use this natural generalization
to construct the natural analog of the nonperiodic Toda lattice. Surprisingly,
the result is not the periodic Toda lattice but a new completely integrable
system on the periodic Toda lattice phase space. This integrable system is
prescribed purely in terms of Lie-theoretic data. The commuting functions are
precisely the gauge-invariant functions one obtains by viewing elements of the
loop algebra as connections on a bundle over
Electromagnetic corrections in hadronic processes
In quantum field theory, the splitting of the Hamiltonian into a strong and
an electromagnetic part cannot be performed in a unique manner. We propose a
convention for disentangling these two effects: one matches the parameters of
two theories -- with and without electromagnetic interactions -- at a given
scale mu_1, referred to as the matching scale. This procedure enables one to
analyze the separation of strong and electromagnetic contributions in a
transparent manner. We illustrate the method -- in the framework of the loop
expansion -- in a Yukawa model, as well as in the linear sigma model, where we
also investigate the corresponding low-energy effective theory.Comment: 19 pages (LaTex), 5 figures, published version. References in the
introduction added, discussion shortened, 1 figure removed, conclusions
unchange
- …