95 research outputs found
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
Phase transition in a stochastic prime number generator
We introduce a stochastic algorithm that acts as a prime number generator.
The dynamics of such algorithm gives rise to a continuous phase transition
which separates a phase where the algorithm is able to reduce a whole set of
integers into primes and a phase where the system reaches a frozen state with
low prime density. We present both numerical simulations and an analytical
approach in terms of an annealed approximation, by means of which the data are
collapsed. A critical slowing down phenomenon is also outlined.Comment: accepted in PRE (Rapid Comm.
Physics of the Riemann Hypothesis
Physicists become acquainted with special functions early in their studies.
Consider our perennial model, the harmonic oscillator, for which we need
Hermite functions, or the Laguerre functions in quantum mechanics. Here we
choose a particular number theoretical function, the Riemann zeta function and
examine its influence in the realm of physics and also how physics may be
suggestive for the resolution of one of mathematics' most famous unconfirmed
conjectures, the Riemann Hypothesis. Does physics hold an essential key to the
solution for this more than hundred-year-old problem? In this work we examine
numerous models from different branches of physics, from classical mechanics to
statistical physics, where this function plays an integral role. We also see
how this function is related to quantum chaos and how its pole-structure
encodes when particles can undergo Bose-Einstein condensation at low
temperature. Throughout these examinations we highlight how physics can perhaps
shed light on the Riemann Hypothesis. Naturally, our aim could not be to be
comprehensive, rather we focus on the major models and aim to give an informed
starting point for the interested Reader.Comment: 27 pages, 9 figure
Towards an Intelligent Tutor for Mathematical Proofs
Computer-supported learning is an increasingly important form of study since
it allows for independent learning and individualized instruction. In this
paper, we discuss a novel approach to developing an intelligent tutoring system
for teaching textbook-style mathematical proofs. We characterize the
particularities of the domain and discuss common ITS design models. Our
approach is motivated by phenomena found in a corpus of tutorial dialogs that
were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor
for textbook-style mathematical proofs can be built on top of an adapted
assertion-level proof assistant by reusing representations and proof search
strategies originally developed for automated and interactive theorem proving.
The resulting prototype was successfully evaluated on a corpus of tutorial
dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
On some Diophantine equations
for p >= 3 and a square-free integer p(2) - 4. In addition to these, all solutions of some different Diophantine equations such as x(2) - v(2n)xy + y(2) = -(p(2) - 4)u(n)(2), x(2) - v(n)xy + y(2) = -(p(2) - 4), x(2) - v(n)xy + y(2) = 1, x(2) - v(2n)xy + y(2) = u(n)(2), x(2) - v(2n)xy + y(2) = v(n)(2), x(2) - (p(2) - 4)u(n)xy - (p(2) - 4)y(2) = 1 are identified, by using divisibility rules of the sequences (u(n)) and (v(n))
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