442,220 research outputs found
Spectral problem on graphs and L-functions
The scattering process on multiloop infinite p+1-valent graphs (generalized
trees) is studied. These graphs are discrete spaces being quotients of the
uniform tree over free acting discrete subgroups of the projective group
. As the homogeneous spaces, they are, in fact, identical to
p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the
finite subgraph-the reduced graph containing all loops of the generalized tree.
We study the spectral problem on these graphs, for which we introduce the
notion of spherical functions-eigenfunctions of a discrete Laplace operator
acting on the graph. We define the S-matrix and prove its unitarity. We present
a proof of the Hashimoto-Bass theorem expressing L-function of any finite
(reduced) graph via determinant of a local operator acting on this
graph and relate the S-matrix determinant to this L-function thus obtaining the
analogue of the Selberg trace formula. The discrete spectrum points are also
determined and classified by the L-function. Numerous examples of L-function
calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur
Discrete Logarithms in Generalized Jacobians
D\'ech\`ene has proposed generalized Jacobians as a source of groups for
public-key cryptosystems based on the hardness of the Discrete Logarithm
Problem (DLP). Her specific proposal gives rise to a group isomorphic to the
semidirect product of an elliptic curve and a multiplicative group of a finite
field. We explain why her proposal has no advantages over simply taking the
direct product of groups. We then argue that generalized Jacobians offer poorer
security and efficiency than standard Jacobians
Solution of the generalized periodic discrete Toda equation II; Theta function solution
We construct the theta function solution to the initial value problem for the
generalized periodic discrete Toda equation.Comment: 11 page
Hard Instances of the Constrained Discrete Logarithm Problem
The discrete logarithm problem (DLP) generalizes to the constrained DLP,
where the secret exponent belongs to a set known to the attacker. The
complexity of generic algorithms for solving the constrained DLP depends on the
choice of the set. Motivated by cryptographic applications, we study sets with
succinct representation for which the constrained DLP is hard. We draw on
earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such
as generalized Menelaus' theorem for proving lower bounds on the complexity of
the constrained DLP, and construct sets with succinct representation with
provable non-trivial lower bounds
Equilibration problem for the generalized Langevin equation
We consider the problem of equilibration of a single oscillator system with
dynamics given by the generalized Langevin equation. It is well-known that this
dynamics can be obtained if one considers a model where the single oscillator
is coupled to an infinite bath of harmonic oscillators which are initially in
equilibrium. Using this equivalence we first determine the conditions necessary
for equilibration for the case when the system potential is harmonic. We then
give an example with a particular bath where we show that, even for parameter
values where the harmonic case always equilibrates, with any finite amount of
nonlinearity the system does not equilibrate for arbitrary initial conditions.
We understand this as a consequence of the formation of nonlinear localized
excitations similar to the discrete breather modes in nonlinear lattices.Comment: 5 pages, 2 figure
An Analysis of the Rayleigh-Stokes problem for a Generalized Second-Grade Fluid
We study the Rayleigh-Stokes problem for a generalized second-grade fluid
which involves a Riemann-Liouville fractional derivative in time, and present
an analysis of the problem in the continuous, space semidiscrete and fully
discrete formulations. We establish the Sobolev regularity of the homogeneous
problem for both smooth and nonsmooth initial data , including . A space semidiscrete Galerkin scheme using continuous piecewise
linear finite elements is developed, and optimal with respect to initial data
regularity error estimates for the finite element approximations are derived.
Further, two fully discrete schemes based on the backward Euler method and
second-order backward difference method and the related convolution quadrature
are developed, and optimal error estimates are derived for the fully discrete
approximations for both smooth and nonsmooth initial data. Numerical results
for one- and two-dimensional examples with smooth and nonsmooth initial data
are presented to illustrate the efficiency of the method, and to verify the
convergence theory.Comment: 23 pp, 4 figures. The error analysis of the fully discrete scheme is
shortene
On the decomposition of Generalized Additive Independence models
The GAI (Generalized Additive Independence) model proposed by Fishburn is a
generalization of the additive utility model, which need not satisfy mutual
preferential independence. Its great generality makes however its application
and study difficult. We consider a significant subclass of GAI models, namely
the discrete 2-additive GAI models, and provide for this class a decomposition
into nonnegative monotone terms. This decomposition allows a reduction from
exponential to quadratic complexity in any optimization problem involving
discrete 2-additive models, making them usable in practice
- …