q-Analogue of the Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials

We associate a family of Hilbert spaces H _q ^2;(D) of analytic functions on the unit disk D=z :|z|<1 the q-continuous Gegenbauer polynomials C _n (x;q) on the interval]–1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L ²(]–1;1[;(1–x ²)^– dx onto the weighted Hilbert space H ^2;(D).     

Universal Associated Legendre Polynomials and Some Useful Definite Integrals

We first introduce the universal associated Legendre polynomials, which are occurred in studying the non-central fields such as the single ring-shaped potential and then present definite integrals I_A^±(a, τ)=∫_-1⁺¹x^a[P_l'^m'(x)]²/(1±x)^τdx, a=0, 1, 2, 3, 4, 5, 6, τ=1, 2, 3, I_B(b, σ)=∫_-1⁺¹x^b[P_l'^m'(x)]²/(1- x²)^σdx, b=0, 2, 4, 6, 8, σ=1, 2, 3, and I_C^±(c, κ)=∫_-1⁺¹x^c[P_l'^m'(x)]²/[(1-x²)^κ(1±x)]dx, c=0, 1, 2, 3, 4, 5, 6, 7, 8, κ=1, 2. The superindices “±” in I_A^±(a, τ) and I_C^±(c, κ) correspond to those of the factor (1±x) involved in weight functions. The formulas obtained in this work and also those for integer quantum numbers l' and m' are very useful and unavailable in classic handbooks.     

Evaluate More General Integrals Involving Universal Associated Legendre Polynomials via Taylor's Theorem

A few important integrals involving the product of two universal associated Legendre polynomials P_(l′)~(m′)(x),P_k~n′~′(x)and x~(2a)(1-x~2)~(-p-1),x~b(1±x)~(-p-1)and x~c(1-x~2)~(-p-1)(1±x)are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial.These integrals are more general since the quantum numbers are unequal,i.e.l~′≠k~′and m~′≠n~′.Their selection rules are also given.We also verify the correctness of those integral formulas numerically.     

On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1-x2) and Their Byproducts

The associated Legendre polynomials play an important role in the central fields, but in the case of the non-central field we have to introduce the universal associated Legendre polynomials P_l'^m'(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential. We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor (1-x²)^-p-1 as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction. The calculations are obtained systematically using some properties of the generalized hypergeometric series.     

On the Recurrence Coefficients for Generalized q-Laguerre Polynomials

In this paper we consider a semi-classical variation of the weight related to the q-Laguerre polynomials and study their recurrence coefficients. In particular, we obtain a second degree second order discrete equation which in particular cases can be reduced to either the qPV or the qPIII equation.     

On the Averages of Characteristic Polynomials From Classical Groups

We provide an elementary and self-contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to Weyl and Littlewood. The first author was supported in part by the NSF grant FRG DMS-0354662. The second author was supported in part by the NSF postdoctoral fellowship and by the NSF grant DMS-0501245.     

On Spectral Polynomials of the Heun Equation. II

The well-known Heun equation has the form

Application of Eigenkets of Bosonic Creation Operator in Deriving Some New Formulas of Associated Laguerre Polynomials

Using the resolution of unity composed of bosonic creation operator＇s eigenkets and annihilation operator＇s un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L^ρm （｜z｜^2） and its generalized generating function formula. A series of recursive relations regarding to L^ρm （｜z｜^2） are also deduced in the context of the Fock representation by algebraic method.     

Application of Eigenkets of Bosonic Creation Operator in Deriving Some New Formulas of Associated Laguerre Polynomials

Using the resolution of unity composed of bosonic creation operator's eigenkets and annihilation operator's un-normalized eigenket, which is a new quantum mechanical representation in contour integration form, we derive new contour integration expression of associated Laguerre polynomials L_m^ρ (|z|²) and its generalized generating function formula. A series of recursive relations regarding to L_m^ρ(|z|²) are also deduced in the context of the Fock representation by algebraic method.     

Integrability and Transition Coefficients Related to Jack Polynomials

Integrability plays a central role in solving many body problems in physics. The explicit construction of the Jack polynomials is an essential ingredient in solving the Calogero-Sutherland model, which is a one-dimensional integrable system. Starting from a special class of the Jack polynomials associated to the hook Young diagram, we find a systematic way in the explicit construction of the transition coefficients in the power-sum basis, which is closely related to a set of mutually commuting operators, i.e. the conserved charges.     

Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized Nizhnik-Novikov-Veselov (GNNV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the GNNV equation are obtained directly, without too much trick like Hirota's bilinear method.     

Generalized q-Hermite Polynomials

We consider two operators A and A ⁺ in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ₀ and that A and A ⁺ are ladder operators for the sequence . The sequence (ψ _n /ψ₀) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001     

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co- compact free action of the integer lattice $\mathbb{Z}^{n}The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co- compact free action of the integer lattice . It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation’s support, the width of the neighborhood dependent upon the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is known in some cases and is expected to be satisfied for periodic Schr?dinger operators.     

Bell Polynomials to the Kadomtsev-Petviashivili Equation with Self-Consistent Sources

Bell Polynomials play an important role in the characterization of bilinear equation. Bell Polynomials are extended to construct the bilinear form, bilinear Bäcklund transformation and Lax pairs for the Kadomtsev-Petviashvili equation with self-consistent sources.     

Emergent Spacetime from Modular Motives

The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L?Cfunctions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n?forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L?Cfunction can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet.     

Dirichlet Polynomials and Entropy

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R⩾0×R⩾0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.     

Mating Non-Renormalizable Quadratic Polynomials

In this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1/2-limb of , is non- renormalizable, and does not have any non-repelling periodic orbits. The first author was partially supported by the Foundation Blanceflor Boncompagni-Ludovisi, née Bildt and by the Fields Institute. The second author was partially supported by an NSERC Discovery Grant.     

Auxiliary Graph for Attribute Graph Clustering

Attribute graph clustering algorithms that include topological structural information into node characteristics for building robust representations have proven to have promising efficacy in a variety of applications. However, the presented topological structure emphasizes local links between linked nodes but fails to convey relationships between nodes that are not directly linked, limiting the potential for future clustering performance improvement. To solve this issue, we offer the Auxiliary Graph for Attribute Graph Clustering technique (AGAGC). Specifically, we construct an additional graph as a supervisor based on the node attribute. The additional graph can serve as an auxiliary supervisor that aids the present one. To generate a trustworthy auxiliary graph, we offer a noise-filtering approach. Under the supervision of both the pre-defined graph and an auxiliary graph, a more effective clustering model is trained. Additionally, the embeddings of multiple layers are merged to improve the discriminative power of representations. We offer a clustering module for a self-supervisor to make the learned representation more clustering-aware. Finally, our model is trained using a triplet loss. Experiments are done on four available benchmark datasets, and the findings demonstrate that the proposed model outperforms or is comparable to state-of-the-art graph clustering models.