Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review article, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.Comment: 73 page

Expression of ZIC family genes in meningiomas and other brain tumors

<p>Abstract</p> <p>Background</p> <p>Zic zinc finger proteins are present in the developing rodent meninges and are required for cell proliferation and differentiation of meningeal progenitors. Although human <it>ZIC </it>genes are known to be molecular markers for medulloblastomas, their expression in meningioma has not been addressed to date.</p> <p>Methods</p> <p>We examined the mRNA and protein expression of human <it>ZIC1</it>, <it>ZIC2</it>, <it>ZIC3</it>, <it>ZIC4 </it>and <it>ZIC5 </it>genes in meningiomas in comparison to other brain tumors, using RT-PCR, analysis of published microarray data, and immunostaining.</p> <p>Results</p> <p><it>ZIC1</it>, <it>ZIC2 </it>and <it>ZIC5 </it>transcript levels in meningiomas were higher than those in whole brain or normal dura mater, whereas all five <it>ZIC </it>genes were abundantly expressed in medulloblastomas. The expression level of <it>ZIC1 </it>in public microarray data was greater in meningiomas classified as World Health Organization Grade II (atypical) than those classified as Grade I (benign). Immunoscreening using anti-ZIC antibodies revealed that 23 out of 23 meningioma cases were ZIC1/2/3/5-immunopositive. By comparison, nuclear staining by the anti-ZIC4 antibody was not observed in any meningioma case, but was strongly detected in all four medulloblastomas. ZIC-positive meningiomas included meningothelial, fibrous, transitional, and psammomatous histological subtypes. In normal meninges, ZIC-like immunoreactivities were detected in vimentin-expressing arachnoid cells both in human and mouse.</p> <p>Conclusions</p> <p>ZIC1, ZIC2, and ZIC5 are novel molecular markers for meningiomas whereas <it>ZIC4 </it>expression is highly selective for medulloblastomas. The pattern of <it>ZIC </it>expression in both of these tumor types may reflect the properties of the tissues from which the tumors are derived.</p

T-systems and Y-systems in integrable systems

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these correction

Role of BMP, FGF, Calcium Signaling, and Zic Proteins in Vertebrate Neuroectodermal Differentiation

More than a decade has passed since Zic family zinc finger proteins were discovered to be transcription factors controlling neuroectodermal differentiation (neural induction) in Xenopus laevis embryos. Although BMP-signal blocking has been shown to be a major upregulator of Zic genes in neuroectodermal differentiation, recent studies have revealed that FGF signaling and intracellular calcium elevation are also involved in regulating the expression of Zic genes. Different regulatory mechanisms have been found for the Zic1 and Zic3 genes, raising the possibility that functional synergism between them partly accounts for the integration of BMP-signal blocking and FGF signaling in neuroectodermal differentiation. Furthermore, mammalian Zic1 and Zic3 have been found to be neural-cell-fate-inducing and pluripotency-maintaining factors, respectively, leading us to the intriguing question of whether the mechanism underlying amphibian neuroectodermal differentiation is applicable to mammals. Comprehensive understanding of the Zic family genes is therefore essential for the study of the neuroectodermal differentiation and stem cell biology