The Tutte polynomial of an infinite family of outerplanar,small-world and self-similar graphs

In this paper we recursively describe the Tutte polynomial of an infinite family of outerplanar, small-world and self-similar graphs. In particular, we study the Abelian Sandpile Model on these graphs and obtain the generating function of the recurrent configurations. Further, we give some exact analytical expression for the Tutte polynomial at several special points     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Differential realizations of polynomial algebras in finite-dimensional spaces of monomials

We give a complete classification of at most second-order differential ladder operators preserving finite-dimensional spaces of monomials and closing under the Lie bracket to give a cubic polynomial of the diagonal generators.     

New integrable hierarchies from vertex operator representations of polynomial Lie algebras

We give a representation–theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld–Sokolov of A, B, C, D hierarchies and to construct their soliton solutions.     

Sum of characteristic polynomial coefficients of cycloparaphenylene graphs as topological index

ABSTRACT

Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm for that of the cycloparaphenylene (CPP) graphs in matrix product form. The algorithm gives a recursive relation in expressing the sum of the CP coefficients of a CPP in terms of that of its two immediately preceding analogues which ultimately ends up with the use of transfer matrix in an analytical form. The sum of CP coefficients, being combinatorial in nature, is found to be used as a topological index showing much similarity with Hosoya index (sum all matching polynomial coefficients), cardinality and number of Kekulé valence structures of CPP graphs compared to the Wiener index which is the distance sum of all pairs of vertices in the graph. The sum of CP coefficients has been found to model the physical properties like strain energy and diameter of CPPs that are verified by the respective excellent correlations.     

Matching polynomial coefficients and the Hosoya indices of poly(p-phenylene) graphs

A general approach to determine the matching polynomial (MP) of a graph with two parts connected by an edge is presented in matrix product that is ultimately used in deducing recursion formulas for obtaining the MP coefficients of linear and cylindrical poly(p-phenylene) (PPP) graphs. The Hosoya indices of linear and cylindrical PPPs are derived in terms of that of the two immediately preceding graphs as well as in analytical forms with the use of transfer matrices. Ambient condition density and bulk modulus of linear PPPs with 2–6 phenyl rings have been found to correlate well with the logarithm of their Hosoya indices. Excellent correlations of diameters with the logarithm of Hosoya indices and strain energies with the inverse of the logarithm of Hosoya indices for cylindrical PPPr with r (= 6–16, 18, 20) phenyl rings are obtained. The linear relation between the logarithm of Hosoya indices and diameter and the inverse relation between diameter and strain energy corroborate the fact.     

Uniform topologies and strong operator topologies on polynomial algebras and on the algebra of CCR

In the paper uniform topologies and strong operator topologies on the free polynomial algebra in n Hermitian indeterminants, on the polynomial algebra in n commuting Hermitian indeterminants and on the *-algebra generated by the CCR (finite number of degrees of freedom) are investigated. It is proved that the strongest locally convex topology on these algebras is a uniform topology and a strong operator topology. For the polynomial algebra in one variable it is shown that on each algebraical realization as an Op*-algebra by an unbounded operator, the strongest locally convex topology coincides with the uniform topology. If in addition the realization is closed, then also the strong operator topology is equal to the strongest locally convex topology.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

用空间频率参量研究激光束特性

本文提出了一种新的方法,用空间频率参量解亥姆霍兹微分方程来获得激光束特性的解析数学描述,并从空间频率参数出发定义了光束束散角和激光束腰斑最小极限值。与用衍射积分方程所得解相比较,结果一致。     

A Pascal's triangle-like approach for the determination of characteristic polynomial coefficients of reciprocal graphs

Characteristic polynomial coefficients of three classes of graph, namely L_n + n(p), C_n + n(p) and K _1,n-1 + n(p), which are known to have reciprocal pairs of eigenvalues, have been shown to be generated by simple manipulation of the Pascal's triangle.     

Formulas for the characteristic polynomial coefficients of the pendant graphs of linear chains,cycles and stars

Formulas for the characteristic polynomial (CP) coefficients of three classes of (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base vertices in one-to-one correspondence (p = 0, 1, 2, …, n), have been developed. Such pendant graphs become reciprocal graphs for linear chains and cycles if p = n. The n-vertex star graphs follow the same rule as paths and cycles, they become reciprocal on adding a pendant vertex to each of n vertices. The formulas so developed have been expressed in matrix product and in analytical forms for the three classes of graphs that require only the values of n and p for calculation of the respective CP coefficients. Such formulas have the general applicability for a large variety of molecular graphs with varying n and p and have been shown to be reduced to the corresponding formulas for reciprocal graphs that are the special cases of the graphs discussed here.     

Covering and gluing of algebras and differential algebras

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C^*-algebra isomorphic to a certain Podleś sphere, as well as the gluing of U_q^1/2(sl₂)-covariant differential calculi on the discs.     

The gauging of BV algebras

A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending on the nature of the global symmetry, the gauging involves coupling to a pure ghost system with a varying amount of ghostly supersymmetry. Coupling to an N=0$N=0$ ghost system yields an ordinary gauge theory whose observables are appropriately classified by the invariant BV cohomology. Coupling to an N=1$N=1$ ghost system leads to a topological gauge field theory whose observables are classified by the equivariant BV cohomology. Coupling to higher N$N$ ghost systems yields topological gauge field theories with higher topological symmetry. In the latter case, however, problems of a completely new kind emerge, which call for a revision of the standard BV algebraic framework.     

The quasi-multiplication on rings and algebras

On a ring or an algebra we introduce the quasi-multiplication a b = ab + a + b, taking into account the multiplication and addition simultaneously. An example shows that this quasi-multiplication has remarkable and fundamental properties.     

Bicovariant quantum algebras and quantum Lie algebras

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun toU _q g, given by elements of the pure braid group. These operators—the reflection matrixYL ⁺ SL ^– being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO _q (N).This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139     

The higher rank Virasoro algebras

Higher rank Virasoro algebras are defined and their properties studied: triangular decompositions, automorphism groups and finite dimensional subalgebras.Work supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR du Québec     

光折变光子晶格中空间二次谐波的产生

用干涉法在自散焦光折变晶体 LiNbO₃：Fe中写入光子晶格的动态过程中，发现了双光束干涉条纹一分为二，四光束干涉点阵一分为四的分裂现象. 研究证明：这是干涉条纹空间频率的倍频现象，是入射的干涉光场与写入的光子晶格之间相互作用的结果. 本实验说明利用光折变效应可以容易地实现干涉光场空间频率的倍频和空间高次谐波的产生，并可利用产生的空间谐波感应出二倍频和高倍频的光折变光子晶格. 关键词： 光折变晶体 光子晶格 空间谐波