7,469 research outputs found
Strong convergence rates of probabilistic integrators for ordinary differential equations
Probabilistic integration of a continuous dynamical system is a way of
systematically introducing model error, at scales no larger than errors
introduced by standard numerical discretisation, in order to enable thorough
exploration of possible responses of the system to inputs. It is thus a
potentially useful approach in a number of applications such as forward
uncertainty quantification, inverse problems, and data assimilation. We extend
the convergence analysis of probabilistic integrators for deterministic
ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\
Comput.}, 2017), to establish mean-square convergence in the uniform norm on
discrete- or continuous-time solutions under relaxed regularity assumptions on
the driving vector fields and their induced flows. Specifically, we show that
randomised high-order integrators for globally Lipschitz flows and randomised
Euler integrators for dissipative vector fields with polynomially-bounded local
Lipschitz constants all have the same mean-square convergence rate as their
deterministic counterparts, provided that the variance of the integration noise
is not of higher order than the corresponding deterministic integrator. These
and similar results are proven for probabilistic integrators where the random
perturbations may be state-dependent, non-Gaussian, or non-centred random
variables.Comment: 25 page
Intersubband transitions in pseudomorphic InGaAs/GaAs/AlGaAs multiple step quantum wells
Intersubband transitions from the ground state to the first and second excited states in pseudomorphic AlGaAs/InGaAs/GaAs/AlGaAs multiple step quantum wells have been observed. The step well structure has a configuration of two AlGaAs barriers confining an InGaAs/GaAs step. Multiple step wells were grown on GaAs substrate with each InGaAs layer compressively strained. During the growth, a uniform growth condition was adopted so that inconvenient long growth interruptions and fast temperature ramps when switching the materials were eliminated. The sample was examined by cross‐sectional transmission electron microscopy, an x‐ray rocking curve technique, and the results show good crystal quality using this simple growth method. Theoretical calculations were performed to fit the intersubband absorption spectrum. The calculated energies are in good agreement with the observed peak positions for both the 1→2 and 1→3 transitions
Invariants of differential equations defined by vector fields
We determine the most general group of equivalence transformations for a
family of differential equations defined by an arbitrary vector field on a
manifold. We also find all invariants and differential invariants for this
group up to the second order. A result on the characterization of classes of
these equations by the invariant functions is also given.Comment: 13 page
Simultanagnosia: When a Rose Is Not Red
Information regarding object identity (‘‘what’’) and spatial location (‘‘where/how to’’) is largely segregated in visual processing. Under most circumstances, however, object identity and location are linked. We report data from a simultanagnosic patient (K.E.) with bilateral posterior parietal infarcts who was unable to ‘‘see’’ more than one object in an array despite relatively preserved object processing and normal preattentive processing. K.E. also demonstrated a finding that has not, to our knowledge, been reported: He was unable to report more than one attribute of a single object. For example, he was unable to name the color of the ink in which words were written despite naming the word correctly. Several experiments demonstrated, however, that perceptual attributes that he was unable to report influenced his performance. We suggest that binding of object identity and location is a limited-capacity operation that is essential for conscious awareness for which the posterior parietal lobe is crucial
Ordinary differential equations which linearize on differentiation
In this short note we discuss ordinary differential equations which linearize
upon one (or more) differentiations. Although the subject is fairly elementary,
equations of this type arise naturally in the context of integrable systems.Comment: 9 page
Catalysis always degrades external quantum correlations
Catalysts used in quantum resource theories need not be in isolation and
therefore are possibly correlated with external systems, which the agent does
not have access to. Do such correlations help or hinder catalysis, and does the
classicality or quantumness of such correlations matter? To answer this
question, we first focus on the existence of a non-invasively measurable
observable that yields the same outcomes for repeated measurements, since this
signifies macro-realism, a key property distinguishing classical systems from
quantum systems. We show that a system quantumly correlated with an external
system so that the joint state is necessarily perturbed by any repeatable
quantum measurement, also has the same property against general quantum
channels. Our full characterization of such systems called totally quantum
systems, solves the open problem of characterizing tomographically sensitive
systems raised in [Lie and Jeong, Phys. Rev. Lett. 130, 020802 (2023)]. An
immediate consequence is that a totally quantum system cannot catalyze any
quantum process, even when a measure of correlation with its environment is
arbitrarily low. It generalizes to a stronger result, that the mutual
information of totally quantum systems cannot be used as a catalyst either.
These results culminate in the conclusion that, out of the correlations that a
generic quantum catalyst has with its environment, only classical correlations
allow for catalysis, and therefore using a correlated catalyst is equivalent to
using an ensemble of uncorrelated catalysts.Comment: 5+7 pages, 1 figure, Comments are welcom
Uniqueness of quantum state over time function
A fundamental asymmetry exists within the conventional framework of quantum
theory between space and time, in terms of representing causal relations via
quantum channels and acausal relations via multipartite quantum states. Such a
distinction does not exist in classical probability theory. In effort to
introduce this symmetry to quantum theory, a new framework has recently been
proposed, such that dynamical description of a quantum system can be
encapsulated by a static quantum state over time. In particular, Fullwood and
Parzygnat recently proposed the state over time function based on the Jordan
product as a promising candidate for such a quantum state over time function,
by showing that it satisfies all the axioms required in the no-go result by
Horsman et al. However, it was unclear if the axioms induce a unique state over
time function. In this work, we demonstrate that the previously proposed axioms
cannot yield a unique state over time function. In response, we therefore
propose an alternative set of axioms that is operationally motivated, and
better suited to describe quantum states over any spacetime regions beyond two
points. By doing so, we establish the Fullwood-Parzygnat state over time
function as the essentially unique function satisfying all these operational
axioms.Comment: 5+4 pages, comments welcom
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