1,273 research outputs found
Numerical Methods for Quasicrystals
Quasicrystals are one kind of space-filling structures. The traditional
crystalline approximant method utilizes periodic structures to approximate
quasicrystals. The errors of this approach come from two parts: the numerical
discretization, and the approximate error of Simultaneous Diophantine
Approximation which also determines the size of the domain necessary for
accurate solution. As the approximate error decreases, the computational
complexity grows rapidly, and moreover, the approximate error always exits
unless the computational region is the full space. In this work we focus on the
development of numerical method to compute quasicrystals with high accuracy.
With the help of higher-dimensional reciprocal space, a new projection method
is developed to compute quasicrystals. The approach enables us to calculate
quasicrystals rather than crystalline approximants. Compared with the
crystalline approximant method, the projection method overcomes the
restrictions of the Simultaneous Diophantine Approximation, and can also use
periodic boundary conditions conveniently. Meanwhile, the proposed method
efficiently reduces the computational complexity through implementing in a unit
cell and using pseudospectral method. For illustrative purpose we work with the
Lifshitz-Petrich model, though our present algorithm will apply to more general
systems including quasicrystals. We find that the projection method can
maintain the rotational symmetry accurately. More significantly, the algorithm
can calculate the free energy density to high precision.Comment: 27 pages, 8 figures, 6 table
Minimizing the number of lattice points in a translated polygon
The parametric lattice-point counting problem is as follows: Given an integer
matrix , compute an explicit formula parameterized by that determines the number of integer points in the polyhedron . In the last decade, this counting problem has received
considerable attention in the literature. Several variants of Barvinok's
algorithm have been shown to solve this problem in polynomial time if the
number of columns of is fixed.
Central to our investigation is the following question: Can one also
efficiently determine a parameter such that the number of integer points in
is minimized? Here, the parameter can be chosen
from a given polyhedron .
Our main result is a proof that finding such a minimizing parameter is
-hard, even in dimension 2 and even if the parametrization reflects a
translation of a 2-dimensional convex polygon. This result is established via a
relationship of this problem to arithmetic progressions and simultaneous
Diophantine approximation.
On the positive side we show that in dimension 2 there exists a polynomial
time algorithm for each fixed that either determines a minimizing
translation or asserts that any translation contains at most times
the minimal number of lattice points
The Polyhedron-Hitting Problem
We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC
'80 and JACM '86)---determining whether a target polyhedron V may be reached
from a starting point x under repeated applications of a linear transformation
A in an ambient vector space Q^m. In the context of program verification, very
similar reachability questions were also considered and left open by Lee and
Yannakakis in (STOC '92). We present what amounts to a complete
characterisation of the decidability landscape for the Polyhedron-Hitting
Problem, expressed as a function of the dimension m of the ambient space,
together with the dimension of the polyhedral target V: more precisely, for
each pair of dimensions, we either establish decidability, or show hardness for
longstanding number-theoretic open problems
Open Diophantine Problems
We collect a number of open questions concerning Diophantine equations,
Diophantine Approximation and transcendental numbers. Revised version:
corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1
(2004) dedicated to Pierre Cartie
Convex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far reaching
generalization of the standard linear combinatorial optimization problem. We
show that it is strongly polynomial time solvable over any edge-guaranteed
family, and discuss several applications
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
Quadratic compact knapsack public-key cryptosystem
AbstractKnapsack-type cryptosystems were among the first public-key cryptographic schemes to be invented. Their NP-completeness nature and the high speed in encryption/decryption made them very attractive. However, these cryptosystems were shown to be vulnerable to the low-density subset-sum attacks or some key-recovery attacks. In this paper, additive knapsack-type public-key cryptography is reconsidered. We propose a knapsack-type public-key cryptosystem by introducing an easy quadratic compact knapsack problem. The system uses the Chinese remainder theorem to disguise the easy knapsack sequence. The encryption function of the system is nonlinear about the message vector. Under the relinearization attack model, the system enjoys a high density. We show that the knapsack cryptosystem is secure against the low-density subset-sum attacks by observing that the underlying compact knapsack problem has exponentially many solutions. It is shown that the proposed cryptosystem is also secure against some brute-force attacks and some known key-recovery attacks including the simultaneous Diophantine approximation attack and the orthogonal lattice attack
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