2,582 research outputs found
More Than 1700 Years of Word Equations
Geometry and Diophantine equations have been ever-present in mathematics.
Diophantus of Alexandria was born in the 3rd century (as far as we know), but a
systematic mathematical study of word equations began only in the 20th century.
So, the title of the present article does not seem to be justified at all.
However, a linear Diophantine equation can be viewed as a special case of a
system of word equations over a unary alphabet, and, more importantly, a word
equation can be viewed as a special case of a Diophantine equation. Hence, the
problem WordEquations: "Is a given word equation solvable?" is intimately
related to Hilbert's 10th problem on the solvability of Diophantine equations.
This became clear to the Russian school of mathematics at the latest in the mid
1960s, after which a systematic study of that relation began.
Here, we review some recent developments which led to an amazingly simple
decision procedure for WordEquations, and to the description of the set of all
solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings
of CAI 2015, Stuttgart, Germany, September 1 - 4, 201
Government-Industry Cooperative Fisheries Research in the North Pacific under the MSFCMA
The National Marine Fisheries Serviceās Alaska Fisheries Science Center (AFSC) has a long and successful history of conducting research in cooperation with the fishing industry. Many of the AFSCās annual resource assessment surveys are carried out aboard chartered commercial vessels and the skill and experience of captains and crew are integral to the success of this work. Fishing companies have been contracted to provide vessels and expertise for many different types of research, including testing and evaluation of survey and commercial fishing gear and development of improved methods for estimating commercial catch quantity and composition. AFSC scientists have also participated in a number of industry-initiated research projects including development of selective fishing gears for bycatch reduction and evaluating and improving observer catch composition sampling. In this paper, we describe the legal and regulatory provisions for these types of cooperative work and present examples to illustrate the process and identify the requirements for successful cooperative research
Generalized Fock spaces and the Stirling numbers
The Bargmann-Fock-Segal space plays an important role in mathematical
physics, and has been extended into a number of directions. In the present
paper we imbed this space into a Gelfand triple. The spaces forming the
Fr\'echet part (i.e. the space of test functions) of the triple are
characterized both in a geometric way and in terms of the adjoint of
multiplication by the complex variable, using the Stirling numbers of the
second kind. The dual of the space of test functions has a topological algebra
structure, of the kind introduced and studied by the first named author and G.
Salomon.Comment: revised versio
Determining the Solution Space of Vertex-Cover by Interactions and Backbones
To solve the combinatorial optimization problems especially the minimal
Vertex-cover problem with high efficiency, is a significant task in theoretical
computer science and many other subjects. Aiming at detecting the solution
space of Vertex-cover, a new structure named interaction between nodes is
defined and discovered for random graph, which results in the emergence of the
frustration and long-range correlation phenomenon. Based on the backbones and
interactions with a node adding process, we propose an Interaction and Backbone
Evolution Algorithm to achieve the reduced solution graph, which has a direct
correspondence to the solution space of Vertex-cover. By this algorithm, the
whole solution space can be obtained strictly when there is no leaf-removal
core on the graph and the odd cycles of unfrozen nodes bring great obstacles to
its efficiency. Besides, this algorithm possesses favorable exactness and has
good performance on random instances even with high average degrees. The
interaction with the algorithm provides a new viewpoint to solve Vertex-cover,
which will have a wide range of applications to different types of graphs,
better usage of which can lower the computational complexity for solving
Vertex-cover
Fluctuations in the Site Disordered Traveling Salesman Problem
We extend a previous statistical mechanical treatment of the traveling
salesman problem by defining a discrete "site disordered'' problem in which
fluctuations about saddle points can be computed. The results clarify the basis
of our original treatment, and illuminate but do not resolve the difficulties
of taking the zero temperature limit to obtain minimal path lengths.Comment: 17 pages, 3 eps figures, revte
Fast Searching in Packed Strings
Given strings and the (exact) string matching problem is to find all
positions of substrings in matching . The classical Knuth-Morris-Pratt
algorithm [SIAM J. Comput., 1977] solves the string matching problem in linear
time which is optimal if we can only read one character at the time. However,
most strings are stored in a computer in a packed representation with several
characters in a single word, giving us the opportunity to read multiple
characters simultaneously. In this paper we study the worst-case complexity of
string matching on strings given in packed representation. Let be
the lengths and , respectively, and let denote the size of the
alphabet. On a standard unit-cost word-RAM with logarithmic word size we
present an algorithm using time O\left(\frac{n}{\log_\sigma n} + m +
\occ\right). Here \occ is the number of occurrences of in . For this improves the bound of the Knuth-Morris-Pratt algorithm.
Furthermore, if our algorithm is optimal since any
algorithm must spend at least \Omega(\frac{(n+m)\log
\sigma}{\log n} + \occ) = \Omega(\frac{n}{\log_\sigma n} + \occ) time to
read the input and report all occurrences. The result is obtained by a novel
automaton construction based on the Knuth-Morris-Pratt algorithm combined with
a new compact representation of subautomata allowing an optimal
tabulation-based simulation.Comment: To appear in Journal of Discrete Algorithms. Special Issue on CPM
200
Exceptional collections and D-branes probing toric singularities
We demonstrate that a strongly exceptional collection on a singular toric
surface can be used to derive the gauge theory on a stack of D3-branes probing
the Calabi-Yau singularity caused by the surface shrinking to zero size. A
strongly exceptional collection, i.e., an ordered set of sheaves satisfying
special mapping properties, gives a convenient basis of D-branes. We find such
collections and analyze the gauge theories for weighted projective spaces, and
many of the Y^{p,q} and L^{p,q,r} spaces. In particular, we prove the strong
exceptionality for all p in the Y^{p,p-1} case, and similarly for the
Y^{p,p-2r} case.Comment: 49 pages, 6 figures; v2 refs added; v3 published versio
Complex networks theory for analyzing metabolic networks
One of the main tasks of post-genomic informatics is to systematically
investigate all molecules and their interactions within a living cell so as to
understand how these molecules and the interactions between them relate to the
function of the organism, while networks are appropriate abstract description
of all kinds of interactions. In the past few years, great achievement has been
made in developing theory of complex networks for revealing the organizing
principles that govern the formation and evolution of various complex
biological, technological and social networks. This paper reviews the
accomplishments in constructing genome-based metabolic networks and describes
how the theory of complex networks is applied to analyze metabolic networks.Comment: 13 pages, 2 figure
Overcoming controllability problems in distributed testing from an input output transition system
This is the Pre-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 Springer VerlagThis paper concerns the testing of a system with physically distributed interfaces, called ports, at which it interacts with its environment. We place a tester at each port and the tester at port p observes events at p only. This can lead to controllability problems, where the observations made by the tester at a port p are not sufficient for it to be able to know when to send an input. It is known that there are test objectives, such as executing a particular transition, that cannot be achieved if we restrict attention to test cases that have no controllability problems. This has led to interest in schemes where the testers at the individual ports send coordination messages to one another through an external communications network in order to overcome controllability problems. However, such approaches have largely been studied in the context of testing from a deterministic finite state machine. This paper investigates the use of coordination messages to overcome controllability problems when testing from an input output transition system and gives an algorithm for introducing sufficient messages. It also proves that the problem of minimising the number of coordination messages used is NP-hard
Quantum heuristic algorithm for traveling salesman problem
We propose a quantum heuristic algorithm to solve a traveling salesman
problem by generalizing Grover search. Sufficient conditions are derived to
greatly enhance the probability of finding the tours with extremal costs,
reaching almost to unity and they are shown characterized by statistical
properties of tour costs. In particular for a Gaussian distribution of the
tours along the cost we show that the quantum algorithm exhibits the quadratic
speedup of its classical counterpart, similarly to Grover search.Comment: Published versio
- ā¦