scholarly journals On k-Fibonacci hybrid numbers and their matrix representations

2021 ◽  
Vol 27 (4) ◽  
pp. 257-266
Author(s):  
Fügen Torunbalcı Aydın ◽  
Keyword(s):  
Generating Functions ◽  
Matrix Representations ◽  
Binet Formula ◽  
Algebraic Properties

In this paper, k-Fibonacci hybrid numbers are defined. Also, some algebraic properties of k-Fibonacci hybrid numbers such as Honsberger identity, Binet Formula, generating functions, d’Ocagne identity, Cassini and Catalan identities are investigated. In addition, we also give 2 × 2 and 4 × 4 representations of the k-Fibonacci hybrid numbers HF_{k,n}.

Gaussian integrals. Algebraic preliminaries

2021 ◽  
pp. 1-17
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Generating Functions ◽  
Complex Conjugate ◽  
Gaussian Measures ◽  
Expectation Values ◽  
Boson Systems ◽  
Fermion Systems ◽  
The Central Limit Theorem ◽  
Algebraic Properties ◽  
Wick’S Theorem

In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. This physics textbook thus begins with a discussion of algebraic properties of Gaussian measures, and Gaussian expectation values for a finite number of variables. The important role of Gaussian measures is not unrelated to the central limit theorem of probabilities, although the interesting physics is generally hidden in essential deviations from Gaussian distributions. A few algebraic identities about Gaussian expectation values, in particular Wick's theorem are recalled. Integrals over some type of formally complex conjugate variables, directly relevant for boson systems are defined. Fermion systems require the introduction of Grassmann or exterior algebras, and the corresponding generalization of the notions of differentiation and integration. Both for complex and Grassmann integrals, Gaussian integrals, and Gaussian expectation values are calculated, and generalized Wick's theorems proven. The concepts of generating functions and Legendre transformation are recalled.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

scholarly journals Pauli–Fibonacci quaternions

2021 ◽  
Vol 27 (3) ◽  
pp. 184-193
Author(s):  
Fügen Torunbalcı Aydın ◽  
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Matrix Representations ◽  
The Matrix ◽  
Binet’S Formula

The aim of this work is to consider the Pauli–Fibonacci quaternions and to present some properties involving this sequence, including the Binet’s formula and generating functions. Furthermore, the Honsberger identity, the generating function, d’Ocagne’s identity, Cassini’s identity, Catalan’s identity for these quaternions are given. The matrix representations for Pauli–Fibonacci quaternions are introduced.

scholarly journals Algebraic Rules for the Percentage Composition of Oligomers in Genomes

2021 ◽  
Author(s):  
Sergey Petoukhov
Keyword(s):  
Dna Sequences ◽  
Genomic Dna ◽  
Gestalt Psychology ◽  
Matrix Representations ◽  
Quantum Biology ◽  
Percentage Composition ◽  
Discrete Signals ◽  
Algebraic Properties ◽  
Prokaryotic Genomes

The article presents the author's results of studying hidden rules of structural organizations of long DNA sequences in eukaryotic and prokaryotic genomes. The results concern some rules of percentages (or probabilities) of n-plets in genomes. To reveal such rules, the author considers genomic DNA nucleotide sequences as multilayers sequences of n-plets and studies the percentage contents of n-plets in different layers. Unexpected rules of invariance of total sums of percentages in certain tetra-groupings of n-plets in different layers of genomic DNA sequences are revealed. These discovered rules are candidates for the role of universal genomic rules. A tensor family of matrix representations of interrelated DNA-alphabets of 4 nucleotides, 16 doublets, 64 triplets, and 256 tetraplets is used in the study. This matrix approach allows revealing algebraic properties of the mentioned genetic rules of probabilities, which are useful for developing algebraic and quantum biology. Some analogies of the discovered genetic phenomena with phenomena of Gestalt psychology are noted and discussed. The author connects the received results about the genomic percentages rules with a supposition of P. Jordan, who is one of the creators of quantum mechanics and quantum biology, that life's missing laws are the rules of chance and probability of the quantum world. Additional attention is paid to the algebraic features of the system of structured DNA alphabets and their relationship with the methods of algebraic holography, known in the technique of processing discrete signals. The concept of algebraic-holographic genetics is being developed for the understanding of inherited holographic properties of organisms.

scholarly journals Viewing Determinants as Nonintersecting Lattice Paths yields Classical Determinantal Identities Bijectively

10.37236/2530 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Markus Fulmek
Keyword(s):  
Generating Functions ◽  
Point Of View ◽  
Lattice Paths ◽  
Binet Formula ◽  
Plücker Relations ◽  
Nonintersecting Lattice Paths ◽  
Determinantal Identities ◽  
Determinantal Identity

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot-method and the Jacobi Trudi identity together with elementary observations.After some preparations, this point of view provides "graphical proofs'' for classical determinantal identities like the Cauchy-Binet formula, Dodgson's condensation formula, the Plücker relations, Laplace's expansion and Turnbull's identity. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which might be new.

Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials and their properties

2020 ◽  
pp. 2150076
Author(s):  
Gamaliel Cerda-Morales
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Third Order ◽  
Lucas Polynomials ◽  
Binet Formula

In this paper, the Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are obtained by using the recurrence relation satisfied by them. Then, some linear and quadratic relations are given between Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials.

scholarly journals Algebraic Properties of Edge Ideals via Combinatorial Topology

10.37236/68 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Anton Dochtermann ◽  
Alexander Engström
Keyword(s):  
Generating Functions ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Vertex Degree ◽  
Combinatorial Topology ◽  
Edge Ideals ◽  
Algebraic Properties ◽  
Ferrers Graphs ◽  
Recursive Relations

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edge ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph (adding whiskers and ears) as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs, (not necessarily pure) vertex decomposability, shellability, etc.

scholarly journals Algebraic Rules for the Percentage Composition of Oligomers in Genomes

2021 ◽  
Author(s):  
Sergey Petoukhov
Keyword(s):  
Dna Sequences ◽  
Genomic Dna ◽  
Gestalt Psychology ◽  
Matrix Approach ◽  
Matrix Representations ◽  
Quantum Biology ◽  
Percentage Composition ◽  
Algebraic Properties ◽  
Prokaryotic Genomes

The article presents the author's results of studying hidden rules of structural organizations of long DNA sequences in eukaryotic and prokaryotic genomes. The results concern some rules of percentages (or probabilities) of n-plets in genomes. To reveal such rules, the author considers genomic DNA nucleotide sequences as multilayers sequences of n-plets and studies the percentage contents of n-plets in different layers. Unexpected rules of invariance of total sums of percentages in certain tetra-groupings of n-plets in different layers of genomic DNA sequences are revealed. These discovered rules are candidates for the role of universal genomic rules. A tensor family of matrix representations of interrelated DNA-alphabets of 4 nucleotides, 16 doublets, 64 triplets, and 256 tetraplets is used in the study. This matrix approach allows revealing algebraic properties of the mentioned genetic rules of probabilities, which are useful for developing algebraic and quantum biology. Some analogies of the discovered genetic phenomena with phenomena of Gestalt psychology are noted and discussed. The author connects the received results about the genomic percentages rules with a supposition of P. Jordan, who is one of the creators of quantum mechanics and quantum biology, that life's missing laws are the rules of chance and probability of the quantum world.

Gaussian Padovan and Gaussian Perrin numbers and properties of them

2019 ◽  
Vol 12 (06) ◽  
pp. 2040014
Author(s):  
Meral Yaşar Kartal
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Binet Formula

In this paper, the Gaussian Padovan and Gaussian Perrin numbers are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian Padovan and Gaussian Perrin numbers are obtained by using the recurrence relation satisfied by them. Then two relations are given between Gaussian Padovan and Gaussian Perrin numbers.

Irreducible matrix resolution for symmetry classes of elasticity tensors

2020 ◽  
Vol 25 (10) ◽  
pp. 1873-1895
Author(s):  
Yakov Itin
Keyword(s):  
Matrix Representation ◽  
Symmetric Matrix ◽  
Elasticity Tensor ◽  
Matrix Representations ◽  
Irreducible Decomposition ◽  
Symmetry Classes ◽  
Artificial Materials ◽  
Deformation Properties ◽  
Algebraic Properties ◽  
Elasticity Tensors

In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

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