133 research outputs found
Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation
A generalization of the max-plus transformation, which is known as a method
to derive cellular automata from integrable equations, is proposed for complex
numbers. Operation rules for this transformation is also studied for general
number of complex variables. As an application, the max-plus transformation is
applied to the discrete Fourier transformation. Stretched coordinates are
introduced to obtain the max-plus transformation whose imaginary part coinsides
with a phase of the discrete Fourier transformation
Bilinear Equations and B\"acklund Transformation for Generalized Ultradiscrete Soliton Solution
Ultradiscrete soliton equations and B\"acklund transformation for a
generalized soliton solution are presented. The equations include the
ultradiscrete KdV equation or the ultradiscrete Toda equation in a special
case. We also express the solution by the ultradiscrete permanent, which is
defined by ultradiscretizing the signature-free determinant, that is, the
permanent. Moreover, we discuss a relation between B\"acklund transformations
for discrete and ultradiscrete KdV equations.Comment: 11 page
Ultra-discrete Optimal Velocity Model: a Cellular-Automaton Model for Traffic Flow and Linear Instability of High-Flux Traffic
In this paper, we propose the ultra-discrete optimal velocity model, a
cellular-automaton model for traffic flow, by applying the ultra-discrete
method for the optimal velocity model. The optimal velocity model, defined by a
differential equation, is one of the most important models; in particular, it
successfully reproduces the instability of high-flux traffic. It is often
pointed out that there is a close relation between the optimal velocity model
and the mKdV equation, a soliton equation. Meanwhile, the ultra-discrete method
enables one to reduce soliton equations to cellular automata which inherit the
solitonic nature, such as an infinite number of conservation laws, and soliton
solutions. We find that the theory of soliton equations is available for
generic differential equations, and the simulation results reveal that the
model obtained reproduces both absolutely unstable and convectively unstable
flows as well as the optimal velocity model.Comment: 9 pages, 6 figure
Tropical Krichever construction for the non-periodic box and ball system
A solution for an initial value problem of the box and ball system is
constructed from a solution of the periodic box and ball system. The
construction is done through a specific limiting process based on the theory of
tropical geometry. This method gives a tropical analogue of the Krichever
construction, which is an algebro-geometric method to construct exact solutions
to integrable systems, for the non-periodic system.Comment: 13 pages, 1 figur
Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation
The Yablonskii-Vorob'ev polynomials , which are defined by a second
order bilinear differential-difference equation, provide rational solutions of
the Toda lattice. They are also polynomial tau-functions for the rational
solutions of the second Painlev\'{e} equation (). Here we define
two-variable polynomials on a lattice with spacing , by
considering rational solutions of the discrete time Toda lattice as introduced
by Suris. These polynomials are shown to have many properties that are
analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce
when . They also provide rational solutions for a particular
discretisation of , namely the so called {\it alternate discrete}
, and this connection leads to an expression in terms of the Umemura
polynomials for the third Painlev\'{e} equation (). It is shown that
B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is
a symplectic map, and the shift in time is also symplectic. Finally we present
a Lax pair for the alternate discrete , which recovers Jimbo and Miwa's
Lax pair for in the continuum limit .Comment: 23 pages, IOP style. Title changed, and connection with Umemura
polynomials adde
Solitons in the Higgs phase -- the moduli matrix approach --
We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge
theory with N_F Higgs scalar fields in the fundamental representation, which
can be extended to possess eight supercharges. We propose the moduli matrix as
a fundamental tool to exhaust all BPS solutions, and to characterize all
possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices,
which are the only elementary solitons in the Higgs phase, are found in terms
of the moduli matrix. Stable monopoles and instantons can exist in the Higgs
phase if they are attached by vortices to form composite solitons. The moduli
spaces of these composite solitons are also worked out in terms of the moduli
matrix. Webs of walls can also be formed with characteristic difference between
Abelian and non-Abelian gauge theories. We characterize the total moduli space
of these elementary as well as composite solitons. Effective Lagrangians are
constructed on walls and vortices in a compact form. We also present several
new results on interactions of various solitons, such as monopoles, vortices,
and walls. Review parts contain our works on domain walls (hep-th/0404198,
hep-th/0405194, hep-th/0412024, hep-th/0503033, hep-th/0505136), vortices
(hep-th/0511088, hep-th/0601181), domain wall webs (hep-th/0506135,
hep-th/0508241, hep-th/0509127), monopole-vortex-wall systems (hep-th/0405129,
hep-th/0501207), instanton-vortex systems (hep-th/0412048), effective
Lagrangian on walls and vortices (hep-th/0602289), classification of BPS
equations (hep-th/0506257), and Skyrmions (hep-th/0508130).Comment: 89 pages, 33 figures, invited review article to Journal of Physics A:
Mathematical and General, v3: typos corrected, references added, the
published versio
Multi-indexed Wilson and Askey-Wilson Polynomials
As the third stage of the project multi-indexed orthogonal polynomials, we
present, in the framework of 'discrete quantum mechanics' with pure imaginary
shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials.
They are obtained from the original Wilson and Askey-Wilson polynomials by
multiple application of the discrete analogue of the Darboux transformations or
the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in
a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials
reported earlier.Comment: 30 pages. Three references added. To appear in J.Phys.A. arXiv admin
note: text overlap with arXiv:1203.586
Pathophysiology and pathogenesis of circadian rhythm sleep disorders
Metabolic, physiological and behavioral processes exhibit 24-hour rhythms in most organisms, including humans. These rhythms are driven by a system of self-sustained clocks and are entrained by environmental cues such as light-dark cycles as well as food intake. In mammals, the circadian clock system is hierarchically organized such that the master clock in the suprachiasmatic nuclei of the hypothalamus integrates environmental information and synchronizes the phase of oscillators in peripheral tissues. The transcription and translation feedback loops of multiple clock genes are involved in the molecular mechanism of the circadian system. Disturbed circadian rhythms are known to be closely related to many diseases, including sleep disorders. Advanced sleep phase type, delayed sleep phase type and nonentrained type of circadian rhythm sleep disorders (CRSDs) are thought to result from disorganization of the circadian system. Evaluation of circadian phenotypes is indispensable to understanding the pathophysiology of CRSD. It is laborious and costly to assess an individual's circadian properties precisely, however, because the subject is usually required to stay in a laboratory environment free from external cues and masking effects for a minimum of several weeks. More convenient measurements of circadian rhythms are therefore needed to reduce patients' burden. In this review, we discuss the pathophysiology and pathogenesis of CRSD as well as surrogate measurements for assessing an individual's circadian phenotype
Internal Ribosomal Entry Site-Mediated Translation Is Important for Rhythmic PERIOD1 Expression
The mouse PERIOD1 (mPER1) plays an important role in the maintenance of circadian rhythm. Translation of mPer1 is directed by both a cap-dependent process and cap-independent translation mediated by an internal ribosomal entry site (IRES) in the 5′ untranslated region (UTR). Here, we compared mPer1 IRES activity with other cellular IRESs. We also found critical region in mPer1 5′UTR for heterogeneous nuclear ribonucleoprotein Q (HNRNPQ) binding. Deletion of HNRNPQ binding region markedly decreased IRES activity and disrupted rhythmicity. A mathematical model also suggests that rhythmic IRES-dependent translation is a key process in mPER1 oscillation. The IRES-mediated translation of mPer1 will help define the post-transcriptional regulation of the core clock genes
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