scholarly journals Tribonacci Numbers and Some Related Interesting Identities

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1195 ◽  
Author(s):  
Shujie Zhou ◽  
Li Chen
Keyword(s):  
Power Series ◽  
Computational Problem ◽  
Symmetry Properties ◽  
Elementary Methods ◽  
Tribonacci Numbers

The main purpose of this paper is, by using elementary methods and symmetry properties of the summation procedures, to study the computational problem of a certain power series related to the Tribonacci numbers, and to give some interesting identities for these numbers.

scholarly journals ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

1996 ◽  
Vol 11 (18) ◽  
pp. 3257-3295 ◽  
Author(s):  
F. TOPPAN
Keyword(s):  
Power Series ◽  
Symmetric Spaces ◽  
Unexpected Result ◽  
Hermitian Symmetric Spaces ◽  
Regular Elements ◽  
Symmetry Properties ◽  
Alternating Sums ◽  
Spectral Parameter Power Series ◽  
Special Limit ◽  
Free Parameters

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.

scholarly journals Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 303 ◽  
Author(s):  
Zhao Jianhong ◽  
Chen Zhuoyu
Keyword(s):  
Calculation Method ◽  
Computational Formula ◽  
Euler Numbers ◽  
Special Sequence ◽  
Computational Problem ◽  
Elementary Methods ◽  
Symmetric Identities ◽  
Recursive Calculation

The aim of this paper is to use elementary methods and the recursive properties of a special sequence to study the computational problem of one kind symmetric sums involving Fubini polynomials and Euler numbers, and give an interesting computational formula for it. At the same time, we also give a recursive calculation method for the general case.

scholarly journals Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Author(s):  
Lan Qi ◽  
Zhuoyu Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Fourth Order ◽  
Power Series Expansion ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Elementary Methods ◽  
Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

scholarly journals The generalized quadratic Gauss sums and its sixth power mean

2021 ◽  
Vol 6 (10) ◽  
pp. 11275-11285
Author(s):  
Xingxing Lv ◽  
◽  
Wenpeng Zhang
Keyword(s):  
Gauss Sums ◽  
Power Mean ◽  
Computational Problem ◽  
Elementary Methods

<abstract><p>In this article, we using elementary methods, the number of the solutions of some congruence equations and the properties of the Legendre's symbol to study the computational problem of the sixth power mean of a certain generalized quadratic Gauss sums, and to give an exact calculating formula for it.</p></abstract>

scholarly journals Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 788 ◽  
Author(s):  
Zhuoyu Chen ◽  
Lan Qi
Keyword(s):  
Power Series Expansion ◽  
Second Order ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Symmetry Properties ◽  
Elementary Methods ◽  
Linear Recurrence Sequence ◽  
Convolution Formula ◽  
The Relationship

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

scholarly journals Some New Identities of Second Order Linear Recurrence Sequences

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1496 ◽  
Author(s):  
Yanyan Liu ◽  
Xingxing Lv
Keyword(s):  
Power Series ◽  
Second Order ◽  
Linear Recurrence ◽  
Combinatorial Method ◽  
Order Linear ◽  
Computational Problem ◽  
Characteristic Roots ◽  
Recursive Sequence ◽  
Recurrence Sequences

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.

scholarly journals A Certain Mean Square Value Involving Dirichlet L-Functions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 948
Author(s):  
Wenpeng Zhang ◽  
Di Han
Keyword(s):  
Positive Integer ◽  
Character Sums ◽  
Mean Square ◽  
Trigonometric Sums ◽  
Integer Points ◽  
Computational Problem ◽  
Elementary Methods

The main purpose of this article is using the elementary methods, the properties of Dirichlet L-functions to study the computational problem of a certain mean square value involving Dirichlet L-functions at positive integer points, and give some exact calculating formulae. As some applications, we obtain some interesting identities and inequalities involving character sums and trigonometric sums.

scholarly journals DEFINITION OF THE ELEMENTARY FUNCTIONS ON THE PRINCIPLE OF THE MATHEMATICAL ANALYSIS

2020 ◽  
Vol 1 (191) ◽  
pp. 12-16
Author(s):  
Yurii Volkov ◽  
◽  
Natalia Vojnaloviсh ◽  
Keyword(s):  
Sixteenth Century ◽  
Power Series ◽  
Mathematical Analysis ◽  
Elementary Functions ◽  
Integral Calculus ◽  
Educational Literature ◽  
Power Operations ◽  
Elementary Methods ◽  
Definition Of ◽  
Basic Level

The elementary functions occupy the special place in school maths and at the study of mathematical analysis in universities .Usually the study of elementary functions begins with the definition of basic elementary functions. Elementary methods are used, without regard to difficulties and imperfections of similar methods. Often rely on intuition, although, it will be desirable to give definition and studies of properties of functions at more logical level. This is old problem, but it is also actual today. The mathematicians were interested in the problem of introduction the definition of the power operations and the logarithms since the sixteenth century, but a main contribution to the decision of this problem belongs to Euler, his book "Introductio in analisin infinitorum - Lausanannae, 1748" became basic level in development of mathematical analysis and by inalienable part in educational literature. Later to the questions of methodology of introduction of definitions basic the elementary functions an attention has been given a number of well-known mathematicians such as F.Kiein, N.Bourbaki, R.Kurant et al. Basic idea: of using the methods of mathematical analysis for the construction of more complete theory. But this suggestion wasn’t very successful in methodology of study of the elementary functions not only at school but also at higher educational establishments. There are many different ways of studying basic elementary functions (logarithmic, exponential, sine, cosine) with the using of differential, integral calculus and the theory of power series are shown in this article.

Semi-automated methods for 3-D reconstruction of macromolecular complexes

1995 ◽  
Vol 53 ◽  
pp. 42-43
Author(s):  
B. Carragher ◽  
M. Whittaker
Keyword(s):  
Three Dimensional ◽  
Time Saving ◽  
Macromolecular Complexes ◽  
Symmetry Properties ◽  
Dimensional Reconstruction ◽  
Experienced Operator ◽  
Projection Images ◽  
The Individual ◽  
Natural Symmetry

Techniques for three-dimensional reconstruction of macromolecular complexes from electron micrographs have been successfully used for many years. These include methods which take advantage of the natural symmetry properties of the structure (for example helical or icosahedral) as well as those that use single axis or other tilting geometries to reconstruct from a set of projection images. These techniques have traditionally relied on a very experienced operator to manually perform the often numerous and time consuming steps required to obtain the final reconstruction. While the guidance and oversight of an experienced and critical operator will always be an essential component of these techniques, recent advances in computer technology, microprocessor controlled microscopes and the availability of high quality CCD cameras have provided the means to automate many of the individual steps.During the acquisition of data automation provides benefits not only in terms of convenience and time saving but also in circumstances where manual procedures limit the quality of the final reconstruction.

Symmetry properties of a universal interaction

Physica ◽  
1952 ◽  
Vol 18 (2) ◽  
pp. 1017-1019 ◽  
Author(s):  
D PURSEY
Keyword(s):  
Symmetry Properties ◽  
Universal Interaction
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