New Expressions for Sums of Products of the Catalan Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 330
Author(s):  
Conghui Xie ◽  
Yuan He
Keyword(s):  
Arbitrary Number ◽  
Catalan Numbers ◽  
Bell Polynomials ◽  
Sums Of Products

In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. The results presented here can be regarded as the development of some known formulas.

scholarly journals The Generating Function of the Catalan Numbers and Lower Triangular Integer Matrices

2017 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Long Qin ◽  
Yong-Hong Yao
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Function ◽  
Catalan Numbers ◽  
Bell Polynomials ◽  
Nonlinear Ordinary Differential Equations ◽  
Inversion Theorem ◽  
Lower Triangular

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

Kinematic analysis and deformation of a planar lattice with an arbitrary number of panels

2020 ◽  
pp. 15-19
Author(s):  
M.N. Kirsanov
Keyword(s):  
Exact Solution ◽  
Kinematic Analysis ◽  
Arbitrary Number ◽  
Design Parameters ◽  
Lattice Deformation ◽  
Planar Lattice ◽  
Force Loading

Formulae are obtained for calculating the deformations of a statically determinate lattice under the action of two types of loads in its plane, depending on the number of panels located along one side of the lattice. Two options for fixing the lattice are analyzed. Cases of kinematic variability of the structure are found. The distribution of forces in the rods of the lattice is shown. The dependences of the force loading of some rods on the design parameters are obtained. Keywords: truss, lattice, deformation, exact solution, deflection, induction, Maple system. [email protected]

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals Free fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Marius de Leeuw ◽  
Chiara Paletta ◽  
Anton Pribytok ◽  
Ana L. Retore ◽  
Alessandro Torrielli
Keyword(s):  
Spectral Problem ◽  
Transfer Matrix ◽  
Arbitrary Number ◽  
Free Fermion ◽  
Nearest Neighbour ◽  
Bogoliubov Transformation ◽  
Free Fermions ◽  
New Models ◽  
Lower Dimensional

Abstract In this paper we first demonstrate explicitly that the new models of integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020) 031604 [36] satisfy the so-called free fermion condition. This both implies that all these models are amenable to reformulations as free fermion theories, and establishes the universality of this condition. We explicitly recast the transfer matrix in free fermion form for arbitrary number of sites in the 6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubov transformation. We then put this observation to use in lower-dimensional instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond massless and massive AdS3, mixed-flux relativistic AdS3 and massless AdS2. We also attack the class of models akin to AdS5 with our free fermion machinery. In all cases we use the free fermion realisation to greatly simplify and reinterpret a wealth of known results, and to provide a very suggestive reformulation of the spectral problem in all these situations.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

scholarly journals Some identities for degenerate complete and incomplete r-Bell polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jongkyum Kwon ◽  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim
Keyword(s):  
Bell Polynomials

On a generalization of the Rényi–Srivastava characterization of the Poisson law

2021 ◽  
Vol 58 (1) ◽  
pp. 68-82
Author(s):  
Jean-Renaud Pycke
Keyword(s):  
Life Insurance ◽  
Collective Model ◽  
New Method ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Compound Distributions ◽  
Poisson Law ◽  
Pierre Loti

AbstractWe give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

scholarly journals Language Family Engineering with Product Lines of Multi-level Models

2021 ◽  
Author(s):  
Juan de Lara ◽  
Esther Guerra
Keyword(s):  
Software Engineering ◽  
Arbitrary Number ◽  
Formal Theory ◽  
The Other ◽  
Language Family ◽  
Top Down ◽  
Product Lines ◽  
Modelling Languages ◽  
Multi Level ◽  
Definition Of

AbstractModelling is an essential activity in software engineering. It typically involves two meta-levels: one includes meta-models that describe modelling languages, and the other contains models built by instantiating those meta-models. Multi-level modelling generalizes this approach by allowing models to span an arbitrary number of meta-levels. A scenario that profits from multi-level modelling is the definition of language families that can be specialized (e.g., for different domains) by successive refinements at subsequent meta-levels, hence promoting language reuse. This enables an open set of variability options given by all possible specializations of the language family. However, multi-level modelling lacks the ability to express closed variability regarding the availability of language primitives or the possibility to opt between alternative primitive realizations. This limits the reuse opportunities of a language family. To improve this situation, we propose a novel combination of product lines with multi-level modelling to cover both open and closed variability. Our proposal is backed by a formal theory that guarantees correctness, enables top-down and bottom-up language variability design, and is implemented atop the MetaDepth multi-level modelling tool.

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