Search results for "FOS: Mathematics"
showing 10 items of 1448 documents
Assouad dimension, Nagata dimension, and uniformly close metric tangents
2013
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed in addition is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our results to equiregular subRiemannian manifolds and show that locally their Nagata dimension equals the to…
On shortening u-cycles and u-words for permutations
2017
Abstract This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for n -permutations exist of lengths n ! + ( 1 − k ) ( n − 1 ) for k = 0 , 1 , … , ( n − 2 ) ! .
Lévy flights and Lévy-Schrödinger semigroups
2010
We analyze two different confining mechanisms for L\'{e}vy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological L\'{e}vy process with the very same invariant pdf and in the reverse.
Transition densities for stochastic Hodgkin-Huxley models
2012
We consider a stochastic Hodgkin-Huxley model driven by a periodic signal as model for the membrane potential of a pyramidal neuron. The associated five dimensional diffusion process is a time inhomogeneous highly degenerate diffusion for which the weak Hoermander condition holds only locally. Using a technique which is based on estimates of the Fourier transform, inspired by Fournier 2008, Bally 2007 and De Marco 2011, we show that the process admits locally a strictly positive continuous transition density. Moreover, we show that the presence of noise enables the stochastic system to imitate any possible deterministic spiking behavior, i.e. mixtures of regularly spiking and non-spiking ti…
Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model
2015
We formulate simple criteria for positive Harris recurrence of strongly degenerate stochastic differential equations with smooth coefficients when the drift depends on time and space and is periodic in the time argument. There is no time dependence in the diffusion coefficient. Our criteria rely on control systems and the support theorem, existence of an attainable inner point of full weak Hoermander dimension and of some Lyapunov function. Positive Harris recurrence enables us to prove limit theorems for such diffusions. As an application, we consider a stochastic Hodgkin-Huxley model for a spiking neuron including its dendritic input. The latter carries some deterministic periodic signal …
Necessary and Sufficient Conditions for an Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems
2015
The notion of (non)contextuality pertains to sets of properties measured one subset (context) at a time. We extend this notion to include so-called inconsistently connected systems, in which the measurements of a given property in different contexts may have different distributions, due to contextual biases in experimental design or physical interactions (signaling): a system of measurements has a maximally noncontextual description if they can be imposed a joint distribution on in which the measurements of any one property in different contexts are equal to each other with the maximal probability allowed by their different distributions. We derive necessary and sufficient conditions for th…
Single-input perturbative control of a quantum symmetric rotor
2022
We consider the Schr\"odinger partial differential equation of a rotating symmetric rigid molecule (symmetric rotor) driven by a z-linearly polarized electric field, as prototype of degenerate infinite-dimensional bilinear control system. By introducing an abstract perturbative criterium, we classify its simultaneous approximate controllability; based on this insight, we numerically perform an orientational selective transfer of rotational population.
Probabilistic Contextuality in EPR/Bohm-type Systems with Signaling Allowed
2014
In this chapter, we review a principled way of defining and measuring contextuality in systems with deterministic inputs and random outputs, recently proposed and developed in \citep{KujalaDzhafarovLarsson2015,DKL2015FooP}.
Aperiodicity in Quantum Wang Tilings
2023
By reformulating the Wang tiles formalism with tensors, we propose a natural generalization to the probabilistic and quantum setting. In this new framework, we introduce notions of tilings and periodicity directly extending their classical counterparts. In the one dimensional case, we recover the decidability of the generalized domino problem by linking it to the trace characterization of nilpotent matrices. In the two-dimensional case, we provide extension of weak and strong aperiodicity respectively and show the equivalence of those generalized notions, extending the well known equivalence in the classical case. We also exhibit a quantum tile set being aperiodic while its underlying class…
Random tensor theory: extending random matrix theory to random product states
2009
We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in (C^d)^{otimes k}, where k and p/d^k are fixed while d grows. When k=1, the Marcenko-Pastur law determines (up to small corrections) not only the largest eigenvalue ((1+sqrt{p/d^k})^2) but the smallest eigenvalue (min(0,1-sqrt{p/d^k})^2) and the spectral density in between. We use the method of moments to show that for k>1 the largest eigenvalue is still approximately (1+sqrt{p/d^k})^2 and the spectral density approaches that of the Marcenko-Pastur law, generalizing the random matrix theory result to the random tensor case.…