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

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

Emergence of Noise-Induced Oscillations in the Central Circadian Pacemaker

Computational modeling and experimentation explain how intercellular coupling and intracellular noise can generate oscillations in a mammalian neuronal network even in the absence of cell-autonomous oscillators

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

The Functional Interplay between Protein Kinase CK2 and CCA1 Transcriptional Activity Is Essential for Clock Temperature Compensation in Arabidopsis

Circadian rhythms are daily biological oscillations driven by an endogenous mechanism known as circadian clock. The protein kinase CK2 is one of the few clock components that is evolutionary conserved among different taxonomic groups. CK2 regulates the stability and nuclear localization of essential clock proteins in mammals, fungi, and insects. Two CK2 regulatory subunits, CKB3 and CKB4, have been also linked with the Arabidopsis thaliana circadian system. However, the biological relevance and the precise mechanisms of CK2 function within the plant clockwork are not known. By using ChIP and Double–ChIP experiments together with in vivo luminescence assays at different temperatures, we were able to identify a temperature-dependent function for CK2 modulating circadian period length. Our study uncovers a previously unpredicted mechanism for CK2 antagonizing the key clock regulator CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1). CK2 activity does not alter protein accumulation or subcellular localization but interferes with CCA1 binding affinity to the promoters of the oscillator genes. High temperatures enhance the CCA1 binding activity, which is precisely counterbalanced by the CK2 opposing function. Altering this balance by over-expression, mutation, or pharmacological inhibition affects the temperature compensation profile, providing a mechanism by which plants regulate circadian period at changing temperatures. Therefore, our study establishes a new model demonstrating that two opposing and temperature-dependent activities (CCA1-CK2) are essential for clock temperature compensation in Arabidopsis