On Dual Hyperbolic Generalized Pierre Numbers

In this study, we introduce the generalized dual Edouard numbers, a novel class of numerical sequences that extend classical recurrence relations within a broader mathematical framework. Several notable special cases particularly the dual Edouard numbers and dual Edouard-Lucas numbers are examined in depth, each exhibiting rich combinatorial and algebraic structures. We derive explicit formulations for these sequences, including Binet-type expressions, generating functions, and summation identities, which provide analytical insight into their intrinsic behavior and structural characteristics. Furthermore, we investigate their matrix representations, offering a refined algebraic approach for theoretical exploration and potential applications in related fields.

In this research, we introduce the generalized dual hyperbolic Edouard numbers, a novel class of number sequences that extends existing recurrence relations into a new mathematical framework. Several special cases of these numbers are examined in detail, including the dual hyperbolic Edouard numbers and the dual hyperbolic Edouard-Lucas numbers, each revealing intriguing combinatorial and algebraic properties. Explicit expressions for these sequences are derived, such as Binet-type formulas, generating functions, and summation identities, which offer analytical insight into their behavior and structural patterns. In addition, we explore matrix representations associated with these sequences, providing an elegant algebraic tool for further theoretical development and potential applications.

This work introduces complex hybrid numbers, an extension of hybrid numbers, which are defined as Z = z0 + z1i + z2ɛ + z3ℎ where zi are complex numbers and iℎ = −ℎi = i + ɛ. We investigate their algebraic properties, classify them into ℂ1, ℂ2, D, and ℙ-types, and analyze their matrix representations. Numerical examples illustrate key findings, highlighting both their theoretical structure and potential areas of application. Key findings are demonstrated with numerical examples, emphasizing their structural importance and their uses.

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

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We study conformal Ward-Takahashi identities for two-point functions in $d(\geq3)$-dimensional finite-temperature conformal field theory. We first show that the conformal Ward-Takahashi identities can be translated into the intertwining relations of conformal algebra $\mathfrak{so}(2,d)$. We then show that, at finite temperature, the intertwining relations can be translated into the recurrence relations for two-point functions in complex momentum space. By solving these recurrence relations, we find the momentum-space two-point functions that satisfy the Kubo-Martin-Schwinger thermal equilibrium condition.

Różne sposoby rozumienia pojęcia parametru przez uczniów gimnazjum w toku badań długoterminowych w trakcie zintegrowanej nauki matematyki i informatyki

Conformal and their natural generalization to quasi-conformal mappings of surfaces, have been extensively and successfully employed in various tasks of Computer Graphics and Imaging. Due to the intrinsic differences between surfaces and objects from higher dimensions, many important results on surfaces can not be directly generalized to produce desirable mapping of volumes. Moreover, in dimension higher than 2, there are no conformal maps apart from Möbius transformations. Therefore, most of the real-world applications generate only quasi-conformal transformations, which produce some conformal distortion. Hence it is tempting and natural to measure the quality of volume deformation by the amount of a conformal distortion it produces. In this paper we examine theoretical properties of quasi-conformal mappings in 3D. We apply those conclusion to process discrete volumetric data in the terms of conformality. We present numerical methods to measure "the degree of conformality" of a transformation between a given pair of domains, represented by volumetric meshes.

The purpose of this paper is to show the elegance and power of operator algebra methods in deriving recurrence relations for the Franck–Condon Factors. The initial element to start the operation of these recurrence relations is also derived using operator algebra techniques.

This paper examines tropical recurrent sequences associated with Somos sequences. The classical Somos sequences have applications in the theory of elliptic curves. Due to the Laurent character of the classical sequences, a pattern can be inferred between the classical sequences and their tropical counterparts. The greatest interest is the increase in the dimension of the solution space of tropical sequences depending on the length of the final sequence. For a set of tropical sequences described by tropical recurrence relations, D.Yu.Grigoriev put forward a hypothesis about the stabilization of the maximum dimensions of the components of the corresponding tropical prevarieties. This hypothesis has been proven for tropical linear recurrent sequences. As part of this work, for tropical recurrent sequences associated with the sequences Somos-4 and Somos-5, the corresponding tropical prevarieties were investigated using the Gfan package in order to test Grigoriev's hypothesis.

A paperback edition of a classic text, this book gives a unique survey of the known solutions of Einstein's field equations for vacuum, Einstein-Maxwell, pure radiation and perfect fluid sources. It introduces the foundations of differential geometry and Riemannian geometry and the methods used to characterize, find or construct solutions. The solutions are then considered, ordered by their symmetry group, their algebraic structure (Petrov type) or other invariant properties such as special subspaces or tensor fields and embedding properties. Includes all the developments in the field since the first edition and contains six completely new chapters, covering topics including generation methods and their application, colliding waves, classification of metrics by invariants and treatments of homothetic motions. This book is an important resource for graduates and researchers in relativity, theoretical physics, astrophysics and mathematics. It can also be used as an introductory text on some mathematical aspects of general relativity.

Due to their recursive definition, manipulating cluster algebras in an efficient way can be hard. Several combinatorial models have been developed in order to overcome this difficulty; here we investigate some of them in the finite type case.

Reynolds' relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds' theory exists, exploiting invariance to yield theorems, non-inhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship between the two. The transformations under which properties are invariant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations.In this paper, we investigate programming languages whose types are indexed by algebraic structures such as groups of geometric transformations. Other examples include types indexed by principals--for information flow security--and types indexed by distances--for analysis of analytic uniform continuity properties. Following Reynolds, we prove a general Abstraction Theorem that covers all these instances. Consequences of our Abstraction Theorem include free theorems expressing invariance properties of programs, type isomorphisms based on invariance properties, and non-definability results indicating when certain algebraically indexed types are uninhabited or only inhabited by trivial programs. We have fully formalised our framework and most examples in Coq.

We analyze the mathcal {N}=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we callcores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. IfBnis a graph drawn uniformly at random from a suitable classof labelled biconnected graphs, then we show that with probability 1 −o(1) asn→ ∞,Bnbelongs to exactly one of the following categories:(i)either there is a uniquegiantcore inBn, that is, there is a 0 <c=c() < 1 such that the largest core contains ~cnvertices, and every other core contains at mostnαvertices, where 0 < α = α() < 1;(ii)or all cores ofBncontainO(logn) vertices.Moreover, we find the critical condition that determines the category to whichBnbelongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 andn. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particularc= 0.765. . . and α = 2/3.