Group representations and multinomial combinatorics of the icosahedral symmetry

The icosahedral symmetry is one of the most intriguing symmetries, as it not only presents a challenge but it appears in many fullerenes and high energetic materials such as the dodecahedral N₂₀. We have considered the combinatorics of all irreducible representations of the icosahedral symmetry for a number of multinomial partitions for vertex, face and edge colourings in this work. We have constructed the combinatorial tables for all irreducible representations for various multinomial partitions of colourings for the vertices, edge and faces of the icosahedron. These techniques should have important applications to enumerations and spectroscopy of fullerenes and high-energy materials such as N₂₀.     

The Yangians,Bethe Ansatz and combinatorics

An axiomatic definition of a quantum monodromy matrix and the representations of its corresponding Hopf algebra are discussed. The connection between the quantum inverse transform method and the representation theory of a symmetric group is considered. A new approach to the completeness problem of Bethe vectors is also given.     

Matrix integration and combinatorics of modular groups

Integration over Gaussian matrix ensembles was used to obtain the virtual Euler characteristics of mapping class groups. We present some simplifications in the combinatorial part of the calculation.Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique     

Magic moments

The moment method of Shifman, Vainshtein and Zakharov, for calculating bound-state energies in QCD, is tested in the context of potential models. For simple power-law potentials of low degree a refined version of the method works suprisingly well. The cruder version actuallye used by Shifman, Vainshtein and Zakharov for charmonium works less well, and the composite potentials usually envisaged for charmonium are less accurately dealt with than simple power potentials. We conjecture then that the magnitude of their confinement parameter φ has been substantially underestimated by those authors.     

Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

Hamilton系统的Mei对称性、Noether对称性和Lie对称性

研究Hamilton系统的形式不变性即Mei对称性，给出其定义和确定方程.研究Hamilton系统的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量.给出一个例子说明本文结果的应用. 关键词： Hamilton系统 Mei对称性 Noether对称性 Lie对称性 守恒量     

A characterization of horizontal visibility graphs and combinatorics on words

A Horizontal Visibility Graph (HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos Luque et al. [B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: exact results for random time series, Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if it is outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics Flajolet and Noy [P. Flajolet, M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204 (1999) 203-229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique, we determine asymptotically the average number of edges of HVGs.     

Quark combinatorics and meson-production ratios in hadronic Z 0 decays

A vast body of experimental data accumulated in the past few years in hadronic Z ⁰ decays allows one to check quark-combinatorics relations for a new type of processes-namely, quark jets in the decays Z ⁰ → $q\bar q$ → hadrons. In this paper, we review quark-combinatorics rules for the yields of vector and pseudoscalar mesons, V/P, in the central and fragmentation regions of hadronic Z ⁰ decays. It is emphasized that, in the central region, a direct verification of quark-combinatorics rules is rather problematic because of a considerable background associated with the decay of highly excited resonances; however, such a verification is possible in the fragmentation region, at x _hadron～0.5–1, where the contribution of resonance decays is suppressed owing to a fast decrease of the spectra with increasing x _hadron. It is shown that, in the fragmentation region, experimental data on ρ⁰/π⁰ and p/π⁺ are in reasonable agreement with the predictions of quark combinatorics. The ratios of the heavy-meson yields, B*/B and D*/D, are also discussed: the data demonstrate good agreement with the quark combinatorial results. We analyze the structure of the suppression parameters for strange-and heavy-quark production in soft processes and estimate their orders of magnitude in multiperipheral processes; the ratio K ^±/π^± at x _hadron～0.5–0.8 and the production probabilities for $c\bar c$ and $b\bar b$ mesons are in good agreement with the estimates.     

Vacco动力学方程的Mei对称性、Lie对称性和Noether对称性

研究Vacco动力学方程的形式不变性即Mei对称性，给出其定义和确定方程，研究Vacco动力学方程的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量，给出一 个例子说明结果的应用. 关键词： Vacco动力学方程 Mei对称性 Noether对称性 Lie对称性 守恒量     

Emden方程的Mei对称性、Lie对称性和Noether对称性

研究Emden动力学方程的形式不变性即Mei对称性，给出其定义和确定方程，研究Emden方程的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量，给出一个例子说明结果的应用. 关键词： Emden动力学方程 Mei对称性 Noether对称性 Lie对称性     

Magic pipe geometry

The initial section pipe geometry has been investigated experimentally with respect to the hysteresis phenomenon of laminar-turbulent transition observed recently. The requirements for pipe geometry for gas dynamic hysteresis have been formulated.     

On the combinatorics of Feynman graphs. I

As a preliminary step towards a rigorous proof ofDyson's conjectures on the combinatorics of Feynman graphs, the following lemma is proved rigorously in an explicit and elementary way: The generating functional of the time-orderedGreen's functions in perturbation theory is an exponential, the exponent of which is a sum of contributions from connected graphs only, with no contributions from vacuum graphs. The model used to demonstrate this is a scalar hermitean fieldA(x) with anA ⁴ selfcoupling.     

奇妙的左手材料

左手材料最早由前苏联科学家Veselago V G在20世纪60年代从理论上提出来的，是指一种介电常数g和磁导率μ同时为负值的材料．它具有诸如负群速度、负折射率、理想成像、逆Doppler频移、反常Cerenkov辐射等种种奇异的物理性质．经过多年的沉寂之后，近几年在实验上取得了突破性的进展，重新引起了人们的重视，尤其是在2003年，还被Science杂志评为当年十大科技进展之一．文章介绍了左手材料的概念和基本原理，并回顾了这一领域近年来的发展．     

Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D₃ invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.     

Absolute parallelism: Spherical symmetry and singularities

A one-parameter class of compatible field equations of n-frames h ^a is considered, characterized by the fact that after the substitution h ^a = HPH ^a (H = detH ^a , p is the constant depending on the dimensionality n and on the equation parameter ), the equations can be written in a trilinear form which contains the matrix H_a and its derivatives, but not H ^a . The equations remain regular for degenerated matrices H_a, if r = rank H_a ^< 2, the points with r < n being singular (if n > 1 + 1/). It is shown that there exist spherically symmetric solutions with singularities appearing for r < n; as the singularity develops, there appears a domain bounded by singular points in which, for certain values of and n, the field h ^a becomes complex.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 47–52, September, 1991.The author thanks I. L. Bukhbinder for his interest to this paper and useful discussions during the seminar.     

Topological quasi-charges in the Einstein theory of teleparallelism and particle combinatorics

A Riemann space with absolute parallelism is considered. Geometry is given by the field of n-frames h ^a . It is shown that for a localized field configuration h one can define a topological charge — an element of the homotopy group G_m= _m(SO_m), m=n–1. The most interesting case of m=6, G₆=0, is considered in detail. Even though in this case there is no topological charge, for configurations symmetrical in dimensions, one can define a quasi-charge — an element of the group G_6–. The groups G₃, G₄, G₅ may correspond to elementary particles (quasiparticles) of the photon, electron, and neutrino types, accordingly. The combinatorics emerging from this scheme is compared with the observation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 15–18, July, 1990.     

Pion-nucleon scattering: Chiral symmetry and unitarity

The chiral quark models which have been constructed over the past five years generally preserve chiral symmetry in the tree approximation. However, unitarity is then violated. We examine the problem of going beyond the tree approximation for s-wave pion-nucleon scattering within the context of the cloudy bag model.     

Magic ferritin: A novel chemotherapeutic encapsulation bullet

The dissociation of apoferritin into subunits at pH 2 followed by its reformation at pH 7.4 in the presence of doxorubicin–HCl gives rise to a solution containing five doxorubicin–HCl molecules trapped within the apoferritin. This is the first report showing that ferritin can encapsulate an anti-cancer drug into its cavity.