1,612 research outputs found
Characterization of the bubble cluster and velocity field in the focal region of a lithotripter
Information mobility in complex networks
The concept of information mobility in complex networks is introduced on the basis of a stochastic process taking place in the network. The transition matrix for this process represents the probability that the information arising at a given node is transferred to a target one. We use the fractional powers of this transition matrix to investigate the stochastic process at fractional time intervals. The mobility coefficient is then introduced on the basis of the trace of these fractional powers of the stochastic matrix. The fractional time at which a network diffuses 50% of the information contained in its nodes (1/ k50 ) is also introduced. We then show that the scale-free random networks display better spread of information than the non scale-free ones. We study 38 real-world networks and analyze their performance in spreading information from their nodes. We find that some real-world networks perform even better than the scale-free networks with the same average degree and we point out some of the structural parameters that make this possible
Deep learning: an introduction for applied mathematicians
Multilayered artificial neural networks are becoming a pervasive tool in a
host of application fields. At the heart of this deep learning revolution are
familiar concepts from applied and computational mathematics; notably, in
calculus, approximation theory, optimization and linear algebra. This article
provides a very brief introduction to the basic ideas that underlie deep
learning from an applied mathematics perspective. Our target audience includes
postgraduate and final year undergraduate students in mathematics who are keen
to learn about the area. The article may also be useful for instructors in
mathematics who wish to enliven their classes with references to the
application of deep learning techniques. We focus on three fundamental
questions: what is a deep neural network? how is a network trained? what is the
stochastic gradient method? We illustrate the ideas with a short MATLAB code
that sets up and trains a network. We also show the use of state-of-the art
software on a large scale image classification problem. We finish with
references to the current literature
Algorithms and literate programs for weighted low-rank approximation with missing data
Linear models identification from data with missing values is posed as a weighted low-rank approximation problem with weights related to the missing values equal to zero. Alternating projections and variable projections methods for solving the resulting problem are outlined and implemented in a literate programming style, using Matlab/Octave's scripting language. The methods are evaluated on synthetic data and real data from the MovieLens data sets
Noncommutative geometry and stochastic processes
The recent analysis on noncommutative geometry, showing quantization of the
volume for the Riemannian manifold entering the geometry, can support a view of
quantum mechanics as arising by a stochastic process on it. A class of
stochastic processes can be devised, arising as fractional powers of an
ordinary Wiener process, that reproduce in a proper way a stochastic process on
a noncommutative geometry. These processes are characterized by producing
complex values and so, the corresponding Fokker-Planck equation resembles the
Schroedinger equation. Indeed, by a direct numerical check, one can recover the
kernel of the Schroedinger equation starting by an ordinary Brownian motion.
This class of stochastic processes needs a Clifford algebra to exist. In four
dimensions, the full set of Dirac matrices is needed and the corresponding
stochastic process in a noncommutative geometry is easily recovered as is the
Dirac equation in the Klein-Gordon form being it the Fokker--Planck equation of
the process.Comment: 16 pages, 2 figures. Updated a reference. A version of this paper
will appear in the proceedings of GSI2017, Geometric Science of Information,
November 7th to 9th, Paris (France
Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations
A discrete stochastic Razumikhin-type theorem is established to investigate whether the Euler--Maruyama (EM) scheme can reproduce the moment exponential stability of exact solutions of stochastic functional differential equations (SFDEs). In addition, the Chebyshev inequality and the Borel-Cantelli lemma are applied to show the almost sure stability of the EM approximate solutions of SFDEs. To show our idea clearly, these results are used to discuss stability of numerical solutions of two classes of special SFDEs, including stochastic delay differential equations (SDDEs) with variable delay and stochastically perturbed equations
Stereoselective palladium-catalyzed C(sp3)–H mono-arylation of piperidines and tetrahydropyrans with a C(4) directing group
A selective Pd-catalyzed C(3)–H cis-functionalization of piperidine and tetrahydropyran carboxylic acids is achieved using a C(4) aminoquinoline amide auxiliary. High mono- and cis-selectivity is attained by using mesityl carboxylic acid as an additive. Conditions are developed with significantly lower reaction temperatures (≤50 °C) than other reported heterocycle C(sp3)–H functionalization reactions, which is facilitated by a DoE optimization. A one-pot C–H functionalization-epimerization procedure provides the trans-3,4-disubstituted isomers directly. Divergent aminoquinoline removal is accomplished with the installation of carboxylic acid, alcohol, amide and nitrile functional groups. Overall fragment compounds suitable for screening are generated in 3-4 steps from readily-available heterocyclic carboxylic acids
New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation
A propagation method for the time dependent Schr\"odinger equation was
studied leading to a general scheme of solving ode type equations. Standard
space discretization of time-dependent pde's usually results in system of ode's
of the form u_t -Gu = s where G is a operator (matrix) and u is a
time-dependent solution vector. Highly accurate methods, based on polynomial
approximation of a modified exponential evolution operator, had been developed
already for this type of problems where G is a linear, time independent matrix
and s is a constant vector. In this paper we will describe a new algorithm for
the more general case where s is a time-dependent r.h.s vector. An iterative
version of the new algorithm can be applied to the general case where G depends
on t or u. Numerical results for Schr\"odinger equation with time-dependent
potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page
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