scholarly journals Weakly minimal groups with a new predicate

2020 ◽  
Vol 20 (02) ◽  
pp. 2050011
Author(s):  
Gabriel Conant ◽  
Michael C. Laskowski
Keyword(s):  
Text Structure ◽  
Linear Recurrence ◽  
Algebraic Numbers ◽  
Expansion Formula ◽  
Elementary Substructure ◽  
Structure Formula ◽  
Linear Recurrence Relation ◽  
Induced Structure ◽  
Homogeneous Linear ◽  
The Universe

Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text].

HOMFLY POLYNOMIALS OF ITERATIVE FAMILIES OF LINKS

2002 ◽  
Vol 11 (01) ◽  
pp. 1-12 ◽  
Author(s):  
TAT-HUNG CHAN
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Explicit Formulas ◽  
Linear Recurrence Relation ◽  
Homfly Polynomials ◽  
Homogeneous Linear ◽  
Torus Links

We show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. The construction embodied in the proof is applied to derive explicit formulas for the HOMFLY polynomials of some iterative families, including the torus links {Tn,3}n≥0.

Single-Boundary Iteration for Polynomial Invariants

2003 ◽  
Vol 12 (03) ◽  
pp. 355-357
Author(s):  
TAT-HUNG CHAN
Keyword(s):  
Recurrence Relation ◽  
General Formula ◽  
Lower Order ◽  
Linear Recurrence ◽  
Entire Family ◽  
Polynomial Invariants ◽  
Linear Recurrence Relation ◽  
Homfly Polynomials ◽  
Single Boundary ◽  
Homogeneous Linear

In an earlier paper [1] we show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. This also holds for Kauffman polynomials. Here we prove the result using a different construction which in some cases yields a recurrence relation of much lower order, thereby drastically reducing the amount of computation required to obtain explicit polynomials for individual members or a general formula for the entire family.

scholarly journals Matrices determined by a linear recurrence relation among entries

2003 ◽  
Vol 373 ◽  
pp. 89-99 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Suk-Geun Hwang ◽  
Seog-Hoon Rim ◽  
Seok-Zun Song
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

Structure and Substructure Connectivity of Hypercube-Like Networks

2020 ◽  
Vol 30 (03) ◽  
pp. 2040007
Author(s):  
Cheng-Kuan Lin ◽  
Eddie Cheng ◽  
László Lipták
Keyword(s):  
Text Structure ◽  
Network Architectures ◽  
Connected Subgraph ◽  
Menger's Theorem ◽  
The Family ◽  
Minimum Number ◽  
Structure Formula ◽  
Connected Subgraphs ◽  
Twisted Cubes

The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] can be determined using Menger’s theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.

scholarly journals Linear recurrences in the degree sequences of monomial mappings

2008 ◽  
Vol 28 (5) ◽  
pp. 1369-1375 ◽  
Author(s):  
ERIC BEDFORD ◽  
KYOUNGHEE KIM
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Degree Sequences ◽  
Integer Matrix ◽  
Linear Recurrences ◽  
Monomial Map ◽  
Linear Recurrence Relation

AbstractLet A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

General anharmonic oscillators

1978 ◽  
Vol 364 (1717) ◽  
pp. 265-275 ◽  
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Recurrence Relation ◽  
Quantum Number ◽  
Anharmonic Oscillator ◽  
Linear Recurrence ◽  
Anharmonic Oscillators ◽  
Anharmonicity Constant ◽  
Linear Recurrence Relation

The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H 2 μ ( k, λ ) = -d 2 / d x 2 + kx 2 + λx 2 μ , ( k, λ ) is investi­gated in this work. Very accurate eigenvalues are obtained in all régimes of the quantum number n and the anharmonicity constant λ . The eigenvalues, as functions of λ , exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W. K. B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.

Convolution Identities for Stirling Numbers of the First Kind

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Takashi Agoh ◽  
Karl Dilcher
Keyword(s):  
Recurrence Relation ◽  
Stirling Numbers ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

AbstractWe derive several new convolution identities for the Stirling numbers of the first kind. As a consequence we obtain a new linear recurrence relation which generalizes known relations.

scholarly journals Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

2020 ◽  
Vol 16 (5) ◽  
pp. 351-360
Author(s):  
Mikhail N. Kirsanov
Keyword(s):  
Energy Method ◽  
Oscillation Frequency ◽  
Frequency Equation ◽  
Linear Recurrence ◽  
Natural Oscillations ◽  
Analytical Estimate ◽  
Mass System ◽  
Symbolic Form ◽  
Upper Estimate ◽  
Homogeneous Linear

The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohrs formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

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