Cyclic gradings of Lie algebras and lax pairs for <Emphasis Type="Italic">σ</Emphasis>-models

We study a class of σ-models with complex homogeneous target spaces and zero-curvature representations. We find a relation between these models and σ-models with certain m-symmetric target spaces. We also describe a model with the hypercomplex target space S ¹ × S ³ in detail.     

Vust's Theorem and higher level Schur–Weyl duality for types B, C and D

Let G be a complex linear algebraic group, $\mathfrak{g}=\mathrm{Lie}(G)$ its Lie algebra and $e\in \mathfrak{g}$ a nilpotent element. Vust's Theorem says that in case of $G=\mathrm{GL}(V)$, the algebra ${\mathrm{End}}_{G_e}(V^{?d})$, where $G_e?G$ is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group ${\mathfrak{S}}_d$ and the linear maps $\{1^{?(i?1)}?e?1^{?(d?i)}|i=1,\dots ,d\}$. In this paper, we give an analogue of Vust's Theorem for $G=\mathrm{O}(V)$ and $\mathrm{SP}(V)$ when the nilpotent elements e satisfy that $\stackrel{￣}{G?e}$ is normal. As an application, we study the higher Schur–Weyl duality in the sense of [4] for types B, C and D, which establishes a relationship between W-algebras and degenerate affine braid algebras.     

A bijection for covered maps, or a shortcut between Harer-Zagier?s and Jackson?s formulas

We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked unicellular spanning submap (which can have any genus in {0,1,…,g}). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n+1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4].Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps.We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33]) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection.     

Green’s Identities,Comparison Principle and Uniqueness of Positive Solutions for Nonlinear p-sub-Laplacian Equations on Stratified Lie Groups

Potential Analysis - We propose analogues of Green’s and Picone’s identities for the p-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised...     

Transient spectral theory, stable and unstable cones and Gershgorin?s theorem for finite-time differential equations

Dynamical behaviour on a compact (finite-time) interval is called monotone-hyperbolic or M-hyperbolic if there exists an invariant splitting consisting of solutions with monotonically decreasing and increasing norms, respectively. This finite-time hyperbolicity notion depends on the norm. For arbitrary norms we prove a spectral theorem based on M-hyperbolicity and extend Gershgorin?s circle theorem to this type of finite-time spectrum. Similarly to stable and unstable manifolds, we characterize M-hyperbolicity by means of existence of stable and unstable cones. These cones can be explicitly computed for D-hyperbolic systems with norms induced by symmetric positive definite matrices and also for row diagonally dominant systems with the sup-norm, thus providing sufficient and computable conditions for M-hyperbolicity.     

Irreducible Wakimoto-like Modules for the Lie Superalgebra <Emphasis Type="Italic">D</Emphasis>(2, 1;α)

By using the idea of Wakimoto's free field, we construct a class of representations for the Lie superalgebra D(2, 1; α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter λ satisfies (λ + m)(λ-(1+α/α)≠m)=0 for any m ∈ Z₊.     

<Emphasis Type="Italic">q</Emphasis>-Hypergeometric Proofs of Polynomial Analogues of the Triple Product Identity,Lebesgue’s Identity and Euler’s Pentagonal Number Theorem

We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.Work supported by the Australian Research Council     

Hochschild cohomology of algebras of quaternionic type: The family Q(2ℬ)<Subscript>1</Subscript>(<Emphasis Type="Italic">k,s, a,c</Emphasis>) over a field of characteristic different from 2

In a previous joint paper of the author with A.I. Generalov and S.O. Ivanov, the Hochschild cohomology algebra of quaternionic-type algebras from the family Q(2ℬ)₁ over an algebraically closed field of characteristic 2 was calculated. In this paper, the Hochschild cohomology groups of algebras from this family over an algebraically closed field of characteristic different from 2 are calculated. As a corollary, the additive structure of the Hochschild cohomology of algebras of type Q(2 $ A $ A ) over a field of characteristic not 2 is described.     

Converse and Smoothness Theorems for Erdo&#x030B;s Weights inL_(0<p?∞)

We prove converse and smoothness theorems of polynomial approximation in weightedL_pspaces with norm ‖fW‖_Lp()(0<p?∞) for Erdo&#x030B;s weights on the real line. In particular we prove characterization theorems involving realization functionals and thereby establish some interesting properties of our weighted modulus of continuity.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

Qualifying for a government’s scrappage program to stimulate consumers’ trade-in transactions? Analysis of an automobile supply chain involving a manufacturer and a retailer

We investigate an automobile supply chain where a manufacturer and a retailer serve a market with a fuel-efficient automobile under a scrappage program by the government. The program awards a subsidy to each consumer who trades in his or her used automobile with a new fuel-efficient automobile, if the manufacturer’s suggested retail price (MSRP) for the new one does not exceed a cutoff level. We derive the conditions assuring that the manufacturer has an incentive to qualify for the program, and find that when the cutoff level is low, the manufacturer may be unwilling to qualify for the program even if the subsidy is high. We also show that when the manufacturer qualifies for the program, increasing the MSRP cutoff level would raise the manufacturer’s expected profit but may decrease the expected sales. A moderate cutoff level can maximize the effectiveness of the program in stimulating the sales of fuel-efficient automobiles, whereas a sufficiently high cutoff level can result in the largest profit for the manufacturer. The retailer’s profit always increases when the manufacturer chooses to qualify for the program. Furthermore, we compute the government’s optimal MSRP cutoff level and subsidy for a given sales target, and find that as the program budget increases, the government should raise the subsidy but reduce the MSRP cutoff level to maximize sales.     

On q-analogs of weight multiplicities for the Lie superalgebras \mathfrak{gl}(n,m) and \mathfrak{spo(}2n,M)

The paper is devoted to the generalization of Lusztig’s q-analog of weight multiplicities to the Lie superalgebras $\mathfrak{gl}(n,m)$ and $\mathfrak{spo(}2n,M).$ We define such q-analogs K _λ,μ (q) for the typical modules and for the irreducible covariant tensor $\mathfrak{gl}(n,m)$ -modules of highest weight λ. For $\mathfrak{gl}(n,m),$ the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For $\mathfrak{spo(}2n,M)$ , we show that the positivity property holds when μ is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor $\mathfrak{gl}(n,m)$ -module of highest weight λ and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger.     

Generalized Logan’s Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson’s Inequality in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Superscript>3</Superscript>)

We study Jackson's inequality between the best approximation of a function f ∈ L₂(R³) by entire functions of exponential spherical type and its generalized modulus of continuity. We prove Jackson's inequality with the exact constant and the optimal argument in the modulus of continuity. In particular, Jackson's inequality with the optimal parameters is obtained for classical modulus of continuity of order r and Thue-Morse modulus of continuity of order r ∈ N. These results are based on the solution of the generalized Logan problem for entire functions of exponential type. For it we construct a new quadrature formulas for entire functions of exponential type.     

Contiguous relations,Laplace’s methods,and continued fractions for $${}_3F_2(1)$$ 3 F 2 ( 1 )

The Ramanujan Journal - Using Hilbert transforms, we establish two families of sum rules involving Bessel moments, which are integrals associated with Feynman diagrams in two-dimensional quantum...     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

On exact solutions for quantum particles with spin <Emphasis Type="Italic">S</Emphasis> = 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space–time model are examined in detail. Firstly, for scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient R_ej{R_{\epsilon j}} on the background of the de Sitter space–time model is analyzed. It is shown that the determination of R_ej{R_{\epsilon j}} requires an additional constrain on quantum numbers er/ (^h/_2p) c >> j{\epsilon \rho / \hbar c \gg j}, where ρ is a curvature radius. When taken into account of this condition, the R_ej{R_{\epsilon j}} vanishes identically. It is claimed that the calculation of the reflection coefficient R_ej{R_{\epsilon j}} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space–time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.