897 research outputs found
Lightweight MDS Involution Matrices
In this article, we provide new methods to look for lightweight MDS matrices, and in particular involutory ones. By proving many new properties and equivalence classes for various MDS matrices constructions such as circulant, Hadamard, Cauchy and Hadamard-Cauchy, we exhibit new search algorithms that greatly reduce the search space and make lightweight MDS matrices of rather high dimension possible to find. We also explain why the choice of the irreducible polynomial might have a significant impact on the lightweightness, and in contrary to the classical belief, we show that the Hamming weight has no direct impact. Even though we focused our studies on involutory MDS matrices, we also obtained results for non-involutory MDS matrices. Overall, using Hadamard or Hadamard-Cauchy constructions, we provide the (involutory or non-involutory) MDS matrices with the least possible XOR gates for the classical dimensions 4x4, 8x8, 16x16 and 32x32 in GF(2^4) and GF(2^8). Compared to the best known matrices, some of our new candidates save up to 50% on the amount of XOR gates required for an hardware implementation. Finally, our work indicates that involutory MDS matrices are really interesting building blocks for designers as they can be implemented with almost the same number of XOR gates as non-involutory MDS matrices, the latter being usually non-lightweight when the inverse matrix is required
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Free Probability Theory
The workhop brought together leading experts, as well as promising young researchers, in areas related to recent developments in free probability theory. Some particular emphasis was on the relation of free probability with random matrix theory
Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem
In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005)], an exact solution was reported for the probability p_{n,k} to find exactly k real eigenvalues in the spectrum of an nxn real asymmetric matrix drawn at
random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the
key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a
GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined
Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations
We give a sufficient condition for exponential stability of a network of
lossless telegrapher's equations, coupled by linear time-varying boundary
conditions. The sufficient conditions is in terms of dissipativity of the
couplings, which is natural for instance in the context of microwave circuits.
Exponential stability is with respect to any -norm, .
This also yields a sufficient condition for exponential stability to a special
class of linear time-varying difference delay systems which is quite explicit
and tractable. One ingredient of the proof is that exponential stability
for such difference delay systems is independent of , thereby reproving in a
simpler way some results from [3].Comment: To be published in SIAM Journal on Mathematical Analysis, most
probably 202
Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders
Recently developed reduced-order modeling techniques aim to approximate
nonlinear dynamical systems on low-dimensional manifolds learned from data.
This is an effective approach for modeling dynamics in a post-transient regime
where the effects of initial conditions and other disturbances have decayed.
However, modeling transient dynamics near an underlying manifold, as needed for
real-time control and forecasting applications, is complicated by the effects
of fast dynamics and nonnormal sensitivity mechanisms. To begin to address
these issues, we introduce a parametric class of nonlinear projections
described by constrained autoencoder neural networks in which both the manifold
and the projection fibers are learned from data. Our architecture uses
invertible activation functions and biorthogonal weight matrices to ensure that
the encoder is a left inverse of the decoder. We also introduce new
dynamics-aware cost functions that promote learning of oblique projection
fibers that account for fast dynamics and nonnormality. To demonstrate these
methods and the specific challenges they address, we provide a detailed case
study of a three-state model of vortex shedding in the wake of a bluff body
immersed in a fluid, which has a two-dimensional slow manifold that can be
computed analytically. In anticipation of future applications to
high-dimensional systems, we also propose several techniques for constructing
computationally efficient reduced-order models using our proposed nonlinear
projection framework. This includes a novel sparsity-promoting penalty for the
encoder that avoids detrimental weight matrix shrinkage via computation on the
Grassmann manifold
Scaling Properties of Paths on Graphs
Let be a directed graph on finitely many vertices and edges, and assign a
positive weight to each edge on . Fix vertices and and consider the
set of paths that start at and end at , self-intersecting in any number
of places along the way. For each path, sum the weights of its edges, and then
list the path weights in increasing order. The asymptotic behaviour of this
sequence is described, in terms of the structure and type of strongly connected
components on the graph. As a special case, for a Markov chain the asymptotic
probability of paths obeys either a power law scaling or a weaker type of
scaling, depending on the structure of the transition matrix. This generalizes
previous work by Mandelbrot and others, who established asymptotic power law
scaling for special classes of Markov chains.Comment: 23 pages, 2 figure
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension
The nonlinear Helmholtz equation (NLH) models the propagation of
electromagnetic waves in Kerr media, and describes a range of important
phenomena in nonlinear optics and in other areas. In our previous work, we
developed a fourth order method for its numerical solution that involved an
iterative solver based on freezing the nonlinearity. The method enabled a
direct simulation of nonlinear self-focusing in the nonparaxial regime, and a
quantitative prediction of backscattering. However, our simulations showed that
there is a threshold value for the magnitude of the nonlinearity, above which
the iterations diverge. In this study, we numerically solve the one-dimensional
NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity
contains absolute values of the field, the NLH has to be recast as a system of
two real equations in order to apply Newton's method. Our numerical simulations
show that Newton's method converges rapidly and, in contradistinction with the
iterations based on freezing the nonlinearity, enables computations for very
high levels of nonlinearity. In addition, we introduce a novel compact
finite-volume fourth order discretization for the NLH with material
discontinuities.The one-dimensional results of the current paper create a
foundation for the analysis of multi-dimensional problems in the future.Comment: 47 pages, 8 figure
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