KAUFFMAN-TYPE POLYNOMIAL INVARIANTS FOR DOUBLY PERIODIC STRUCTURES

2007 ◽  
Vol 16 (06) ◽  
pp. 779-788 ◽  
Author(s):  
SERGEI A. GRISHANOV ◽  
VADIM R. MESHKOV ◽  
ALEXANDER V. OMEL'CHENKO
Keyword(s):  
Periodic Structures ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Doubly Periodic Structures

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250128
Author(s):  
KYEONGHUI LEE ◽  
YOUNG HO IM
Keyword(s):  
Knot Theory ◽  
Recursive Method ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Virtual Links

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

A general time-dependent invariant for and integrability of the quadratic system

1994 ◽  
Vol 444 (1922) ◽  
pp. 643-650 ◽  
Keyword(s):  
Direct Method ◽  
First Integral ◽  
Quadratic System ◽  
Time Dependent ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Linear Polynomial ◽  
A Direct Method ◽  
Special Case ◽  
General Time

We derive a general time-dependent invariant (first integral) for the quadratic system (QS) that requires only one condition on the coefficients of the QS. The general invariant could yield asymptotic behaviour of phase-space trajectories. With more conditions imposed on the coefficients, the general invariant reduces to polynomial form and is equivalent to polynomial invariants found using a direct method. For the special case of a linear polynomial invariant where one of the variables is analytically invertible, the solution of the QS is reduced to a quadrature.

INDEX POLYNOMIAL INVARIANTS OF TWISTED LINKS

2013 ◽  
Vol 22 (04) ◽  
pp. 1340005 ◽  
Author(s):  
NAOKO KAMADA
Keyword(s):  
Virtual Link ◽  
Alternative Definition ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Intersection Index ◽  
Definition Of

Bourgoin defined the notion of a twisted link. In a sense, it is a non-orientable version of a virtual link. Im, Lee and Lee defined a polynomial invariant of a virtual link by using the virtual intersection index. In this paper, we give an alternative definition of index polynomial by using indices of real crossings and extend it to a twisted links.

scholarly journals A POLYNOMIAL INVARIANT OF VIRTUAL LINKS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341002 ◽  
Author(s):  
ZHIYUN CHENG ◽  
HONGZHU GAO
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
The Third ◽  
Linking Number ◽  
Virtual Links ◽  
Definition Of ◽  
Knots And Links

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

scholarly journals The properties and parameters of generic Bose – Chaudhuri – Hocquenghem codes

2020 ◽  
Vol 56 (2) ◽  
pp. 157-165
Author(s):  
A. V. Kushnerov ◽  
V. A. Lipinski ◽  
M. N. Koroliova
Keyword(s):  
Error Correction ◽  
Cyclic Codes ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Close Connection ◽  
Design Parameters ◽  
Polynomial Invariants ◽  
Accurate Evaluation

The Bose – Chaudhuri – Hocquenghem type of linear cyclic codes (BCH codes) is one of the most popular and widespread error-correcting codes. Their close connection with the theory of Galois fields gave an opportunity to create a theory of the norms of syndromes for BCH codes, namely, syndrome invariants of the G-orbits of errors, and to develop a theory of polynomial invariants of the G-orbits of errors. This theory as a whole served as the basis for the development of effective permutation polynomial-norm methods and error correction algorithms that significantly reduce the influence of the selector problem. To date, these methods represent the only approach to error correction with non-primitive BCH codes, the multiplicity of which goes beyond design boundaries. This work is dedicated to a special error-correcting code class – generic Bose – Chaudhuri – Hocquenghem codes or simply GBCH-codes. Sufficiently accurate evaluation of the quantity of such codes in each length was produced during our work. We have investigated some properties and connections between different GBCH-codes. Special attention was devoted to codes with constructive distances of 3 and 5, as those codes are usual for practical use. Their almost complete description is given in the range of lengths from 7 to 107. The paper contains a fairly clear theoretical classification of GBCH-codes. Special attention is paid to the corrective capabilities of the codes of this class, namely, to the calculation of the minimal distances of these codes with various parameters. The codes are found whose corrective capabilities significantly exceed those of the well-known GBCH-codes with the same design parameters.

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