22 research outputs found
Disjoint difference families and their applications
Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and showed that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families
Light scattering as a Poisson process and first-passage probability
A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately either achieves first-passage, leaving the medium, or it is absorbed. The KubelkaMunk model describes a flux of such particles moving perpendicular to the surface of a plane-parallel medium with a scattering rate and an absorption rate. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length λ. The reflectance of a semiinfinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential steplength distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks n in the alternating path is a step-length distribution-free combinatoric expression involving Catalan numbers. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in a combinatoric expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path combinatorics
Random matrix techniques in quantum information theory
The purpose of this review article is to present some of the latest
developments using random techniques, and in particular, random matrix
techniques in quantum information theory. Our review is a blend of a rather
exhaustive review, combined with more detailed examples -- coming from research
projects in which the authors were involved. We focus on two main topics,
random quantum states and random quantum channels. We present results related
to entropic quantities, entanglement of typical states, entanglement
thresholds, the output set of quantum channels, and violations of the minimum
output entropy of random channels
Symmetry in Chaotic Systems and Circuits
Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled âSymmetry in Chaotic Systems and Circuitsâ, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue
Pseudorandomness of the Sticky Random Walk
We extend the pseudorandomness of random walks on expander graphs using the
sticky random walk. Building on prior works, it was recently shown that
expander random walks can fool all symmetric functions in total variation
distance (TVD) upto an error, where
is the second largest eigenvalue of the expander, is the size of
the arbitrary alphabet used to label the vertices, and , where is the fraction of vertices labeled in the graph.
Golowich and Vadhan conjecture that the dependency on the term is not tight. In this paper, we resolve the conjecture in the
affirmative for a family of expanders. We present a generalization of the
sticky random walk for which Golowich and Vadhan predict a TVD upper bound of
using a Fourier-analytic approach. For this family of
graphs, we use a combinatorial approach involving the Krawtchouk functions to
derive a strengthened TVD of . Furthermore, we present
equivalencies between the generalized sticky random walk, and, using
linear-algebraic techniques, show that the generalized sticky random walk
parameterizes an infinite family of expander graphs.Comment: 21 pages, 2 figure
Zero-Knowledge Reductions and Confidential Arithmetic
The changes in computing paradigms to shift computations to third parties have resulted in the necessity of these computations to be provable. Zero-knowledge arguments are probabilistic arguments that are used to to verify computations without secret data being leaked to the verifying party.
In this dissertation, we study zero-knowledge arguments with specific focus on reductions. Our main contributions are: Provide a thorough survey in a variety of zero-knowledge techniques and protocols. Prove various results of reductions that can be used to study interactive protocols in terms of subroutines. Additionally, we identify an issue in the analogous definition of zero-knowledge for reductions. We propose a potential solution to this issue. Design a novel matrix multiplication protocol based on reductions. Design protocols for arithmetic of fixed-point values of fixed-length