45,629 research outputs found
A Theory AB Toolbox
Randomized algorithms are a staple of the theoretical computer science literature. By careful use of randomness, algorithms can achieve properties that are simply not possible with deterministic algorithms. Today, these properties are proved on paper, by theoretical computer scientists; we investigate formally verifying these proofs.
The main challenges are two: proofs about algorithms can be quite complex, using various facts from probability theory; and proofs are highly customized - two proofs of the same property for two algorithms can be completely different. To overcome these challenges, we propose taking inspiration from paper proofs, by building common tools - abstractions, reasoning principles, perhaps even notations - into a formal verification toolbox. To give an idea of our approach, we consider three common patterns in paper proofs: the union bound, concentration bounds, and martingale arguments
Ab initio few-mode theory for quantum potential scattering problems
Few-mode models have been a cornerstone of the theoretical work in quantum
optics, with the famous single-mode Jaynes-Cummings model being only the most
prominent example. In this work, we develop ab initio few-mode theory, a
framework connecting few-mode system-bath models to ab initio theory. We first
present a method to derive exact few-mode Hamiltonians for non-interacting
quantum potential scattering problems and demonstrate how to rigorously
reconstruct the scattering matrix from such few-mode Hamiltonians. We show that
upon inclusion of a background scattering contribution, an ab initio version of
the well known input-output formalism is equivalent to standard scattering
theory. On the basis of these exact results for non-interacting systems, we
construct an effective few-mode expansion scheme for interacting theories,
which allows to extract the relevant degrees of freedom from a continuum in an
open quantum system. As a whole, our results demonstrate that few-mode as well
as input-output models can be extended to a general class of problems, and open
up the associated toolbox to be applied to various platforms and extreme
regimes. We outline differences of the ab initio results to standard model
assumptions, which may lead to qualitatively different effects in certain
regimes. The formalism is exemplified in various simple physical scenarios. In
the process we provide proof-of-concept of the method, demonstrate important
properties of the expansion scheme, and exemplify new features in extreme
regimes.Comment: 41 pages, 14 figures, substantially extended version now also
covering interacting and nonlinear problem
The FermiFab Toolbox for Fermionic Many-Particle Quantum Systems
This paper introduces the FermiFab toolbox for many-particle quantum systems.
It is mainly concerned with the representation of (symbolic) fermionic
wavefunctions and the calculation of corresponding reduced density matrices
(RDMs). The toolbox transparently handles the inherent antisymmetrization of
wavefunctions and incorporates the creation/annihilation formalism. Thus, it
aims at providing a solid base for a broad audience to use fermionic
wavefunctions with the same ease as matrices in Matlab, say. Leveraging
symbolic computation, the toolbox can greatly simply tedious pen-and-paper
calculations for concrete quantum mechanical systems, and serves as "sandbox"
for theoretical hypothesis testing. FermiFab (including full source code) is
freely available as a plugin for both Matlab and Mathematica.Comment: 17 pages, 5 figure
ABCD Neurocognitive Prediction Challenge 2019: Predicting individual residual fluid intelligence scores from cortical grey matter morphology
We predicted residual fluid intelligence scores from T1-weighted MRI data
available as part of the ABCD NP Challenge 2019, using morphological similarity
of grey-matter regions across the cortex. Individual structural covariance
networks (SCN) were abstracted into graph-theory metrics averaged over nodes
across the brain and in data-driven communities/modules. Metrics included
degree, path length, clustering coefficient, centrality, rich club coefficient,
and small-worldness. These features derived from the training set were used to
build various regression models for predicting residual fluid intelligence
scores, with performance evaluated both using cross-validation within the
training set and using the held-out validation set. Our predictions on the test
set were generated with a support vector regression model trained on the
training set. We found minimal improvement over predicting a zero residual
fluid intelligence score across the sample population, implying that structural
covariance networks calculated from T1-weighted MR imaging data provide little
information about residual fluid intelligence.Comment: 8 pages plus references, 3 figures, 2 tables. Submission to the ABCD
Neurocognitive Prediction Challenge at MICCAI 201
Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector
We consider algebraic iterative reconstruction methods with applications in
image reconstruction. In particular, we are concerned with methods based on an
unmatched projector/backprojector pair; i.e., the backprojector is not the
exact adjoint or transpose of the forward projector. Such situations are common
in large-scale computed tomography, and we consider the common situation where
the method does not converge due to the nonsymmetry of the iteration matrix. We
propose a modified algorithm that incorporates a small shift parameter, and we
give the conditions that guarantee convergence of this method to a fixed point
of a slightly perturbed problem. We also give perturbation bounds for this
fixed point. Moreover, we discuss how to use Krylov subspace methods to
efficiently estimate the leftmost eigenvalue of a certain matrix to select a
proper shift parameter. The modified algorithm is illustrated with test
problems from computed tomography
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