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In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

This paper deals with a class of singular problems involving the fractional $$p(x,\cdot )$$ -Laplace operator of the form $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^N$$ ( $$N\ge 3$$ ), $$00$$ small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $$p(x,\cdot )$$ -Laplace operators.

In this paper, we prove the strong log-concavity and the unimodality of the various sequences of an extension of elementary symmetric function. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot. As applications, we establish the strong q-log-concavity and the unimodality of q-bi $$^{s}$$ nomial coefficients and their sequences lying on rays of the associated triangle.

Two classes of determinants with their matrix entries being polynomial differences are evaluated in closed product expressions. They extend with many free parameters the Vandermonde-like determinan...

In this work, using the weighted symmetric functions $$\sigma _{k}^{\infty }$$ σ k ∞ and the weighted Newton transformations $$T_{k}^{\infty }$$ T k ∞ introduced by Case (Alias et al. Proc Edinb Math Soc 46(02):465–488, 2003), we derive some generalized integral formulae for close hypersurfaces in weighted manifolds. We also give some examples and applications of these formulae.

In this paper, we provide a family of bounds for the rate at which the functions of many inputs can be computed in-network on general topologies. Going beyond simple symmetric functions where the output is invariant to the permutation of the operands, e.g., average, parity, we describe an algorithm that is analyzed to provide throughput bounds (both lower and upper) for the general functions. In particular, we analyze our algorithm when the function to be computed is given as a binary tree schema. Our lower bounds depend on schema parameters like the number of operands and graph parameters like the second largest eigenvalue of the transition matrix of simple random walk on network graph, the maximum and minimum degree of any node in the network. The lower bounding technique that we have used is based on the network flows and can capture general multi-commodity flow settings. Our proposed algorithm uses the well-known simple random walk on a network as its basic primitive for routing. We show that the lower bound obtained on the rate of computation is tight for the complete network topology, the hypercube and the star topology. We also present an upper bound on the expected latency of any data operand in terms of the height of schema, well-studied random walk parameter like the hitting time, and the relative distance from the critical data rate. For the computation time of symmetric functions on the random geometric graph under the gossip model, our approach achieves an order-optimal $$\widetilde{O}(n)$$ time despite enforcing a binary tree schema for function computation. In general, Big-O notation represents an upper bound and $$\widetilde{O}$$ hides $$\text {poly}\log n$$ factors.

In this present paper, a new generalized classℛp,qfrom the family of function with bounded turning was introduced by usingp,q- derivative operator. Our aim for this class is to find out the upper bound of third- and fourth-order Hankel determinant. Moreover, the upper bounds for two-fold and three-fold symmetric functions for this class are also obtained.

We establish weighted$$L^p$$Lp-Fourier extension estimates for$$O(N-k) \times O(k)$$O(N-k)×O(k)-invariant functions defined on the unit sphere$${\mathbb {S}}^{N-1}$$SN-1, allowing for exponentspbelow the Stein–Tomas critical exponent$$\frac{2(N+1)}{N-1}$$2(N+1)N-1. Moreover, in the more general setting of an arbitrary closed subgroup$$G \subset O(N)$$G⊂O(N)andG-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results forG-invariant solutions to the nonlinear Helmholtz equation$$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$-Δu-u=Q(x)|u|p-2u,u∈W2,p(RN),whereQis a nonnegative bounded andG-invariant weight function.

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Karolyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.

We give a simple proof of some explicit formulas of periodic continuants by Chebyshev polynomials of the second kind given by R´ozsa. We also mention an application of these explicit for- mulas to q-analogue of rationals and continued fractions introduced by Morier-Genoud and Ovsienko.

Just as we study energy density e(u), energy E(u) of harmonic maps, extend them in Wei (An extrinsic average variational method, American Mathematical Society, Providence, 1989) to $$\Phi $$ -energy density $$e_{\Phi }(u)$$ , $$\Phi $$ -energy $$E_{\Phi }(u)$$ of a map, and $$\Phi $$ -harmonic map $$\big ($$ from the view point of the second symmetric function $$\sigma _2$$ of a pullback (0, 2)-tensor $$\big )$$ , in this paper, we introduce the notions of the $$\Phi _S$$ -energy density $$e_{\Phi _S}(u)$$ , $$\Phi _S$$ -energy $$E_{\Phi _S}(u)$$ of a map $$u: M \rightarrow N\, ,$$ $$\Phi _S$$ -harmonic map, stable $$\Phi _S$$ -harmonic map, and unstable $$\Phi _S$$ -harmonic map, that are associated with the stress-energy tensor S as discussed in (4). We investigate $$\Phi _S$$ -harmonic maps or stress-energy stationary maps between Riemannian manifolds. Liouville type theorems for $$\Phi _S$$ -harmonic maps from complete Riemannian manifolds are established under some conditions on the Hessian of the distance function and the asymptotic behavior of the map at infinity. By an extrinsic average variational method in the calculus of variations (Han and Wei in $$\Phi $$ -harmonic maps and $$\Phi $$ -superstrongly unstable manifolds, 2019), we prove that any stable $$\Phi _S$$ -harmonic map from or to a compact $$\Phi $$ -SSU manifold (to or from any compact manifold) must be constant (cf. Theorems 6.1 and 7.1). We further prove that the homotopic class of any map from any compact manifold into a compact $$\Phi $$ -SSU manifold contains elements of arbitrarily small $$\Phi _S$$ -energy, and the homotopic class of any map from a compact $$\Phi $$ -SSU manifold into any manifold contains elements of arbitrarily small $$\Phi _S$$ -energy (cf. Theorems 8.1 and 9.1). As immediate consequences, we give a simple and direct proof of Theorems 6.1 and 7.1. These Theorems 6.1, 7.1, 8.1 and 9.1 give rise to the concept of $$\Phi _S$$ -strongly unstable $$(\Phi _S$$ -SU) manifolds, extending the notions of strongly unstable $$({\text {SU}})$$ , p-strongly unstable (p-SU), $$\Phi $$ -strongly unstable $$(\Phi $$ -SU) manifolds (cf. [17, 19, 30, 32]). We also introduce the concepts of $$\Phi _S$$ -superstrongly unstable $$(\Phi _S$$ - $${\text {SSU}})$$ manifold, $$\Phi _S$$ -unstable $$(\Phi _S$$ - $${\text {U}})$$ manifold and establish a link of $$\Phi _S$$ - $${\text {SSU}}$$ manifold to p-SSU manifold and topology. Compact $$\Phi _S$$ - $${\text {SSU}}$$ homogeneous spaces are also studied.

We analytically characterize equilateral triangles. Our characterization includes the classically well-known characterizations: for a given triangle in the Euclidean plane, if the centroid and incenter of the triangle coincide, then the triangle must be equilateral; for a given triangle in the Euclidean plane, if the centroid and circumcenter of the triangle coincide, then the triangle must be equilateral. In our characterization, we consider the convolution of a radially symmetric function and the characteristic function of a triangle. The centroid, incenter and circumcenter of a triangle are described in terms of critical points of such a convolution.

Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymlE≅∇μSymmE of representations of SL2(C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when l=m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇λSymlE is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new q-binomial identity in this setting

We introduce a family of ideals $I_{n,\lambda,s}$ in $\mathbb{Q}[x_1,\dots,x_n]$ for $\lambda$ a partition of $k\leq n$ and an integer $s \geq \ell(\lambda)$. This family contains both the Tanisaki ideals $I_\lambda$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings $R_{n,\lambda,s}$ as symmetric group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi modules, and when $\lambda=(1^k)$ and $s=k$, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for $R_{n,\lambda,s}$, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono, and realize the $S_n$-module structure of $R_{n,\lambda,s}$ in terms of an action on $(n,\lambda,s)$-ordered set partitions. We also prove formulas for the Hilbert series and graded Frobenius characteristic of $R_{n,\lambda,s}$. We then connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our work, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.

Relativistic effects are quite large for the actinide 5f and 6d open shells, and in the present study we present combinatorial enumeration of $$\upomega {{-}} \upomega $$ states of all 14 actinide (lanthanide) dimers using multinomial symmetric function (S-function) techniques. The techniques presented here enumerate all possible $$\upomega {{-}} \upomega $$ states that arise from relativistic 2-component molecular spinors composed of the 7s, 6d 5f, and possibly 7p shells of the actinide dimers Th2 through Lr2. The combinatorial enumerations techniques reveal that that for Americium dimer (Am2) there are 20,058,300 S-function terms of which only 616,227 terms satisfy the Pauli exclusion principle yielding 401,166,110 $$\upomega {{-}} \upomega $$ states (142.6 MB of output), all of which arise from primarily the 7s/6d/5f shells of Am. Explicit tables are constructed for the relativistic $$\upomega {{-}} \upomega $$ states for Th2 to Am2 with several restrictions imposed on the levels of excitations primarily to reduce the table sizes, and through the electron–hole equivalence $$\upomega {{-}} \upomega $$ states of the remaining dimers of the actinide row are enumerated. The computational and mathematical techniques also enumerate the $$\upomega {{-}} \upomega $$ states of Ce2 to Eu2 and by electron–hole equivalence Gd2 to Tm2, although spin–orbit splittings are smaller for lanthanides. A unique feature of the present combinatorial technique is that it facilitates any number of orbitals and any number of open shells including the possibility of 14 singly-occupied orbitals for actinide and lanthanide dimers, and thus providing for a technique to enumerate $$\upomega {{-}} \upomega $$ states arising from both very high spin open-shell states as well as low spin states exhaustively.

The main purpose of this paper is to show that the Poincare q-polynomials admit a representation in terms of the symmetric functions and the Patterson-Selberg (or Ruelle-type) spectral functions. We have shown that the q-series elliptic genera can be expressed in terms of q-analogs of the classical special functions, specially the equivalence between the spectral Patterson-Selberg and the Ruelle functions. The main result of this manuscript is to show that this representation can be used in theoretical physics and we analyze them in terms of the Patterson-Selberg spectral function R (s).