NEW INVARIANTS FOR GROUPS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 285-339
Author(s):  
D. A. CRUICKSHANK ◽  
S. J. PRIDE
Keyword(s):  
Commutative Ring ◽  
Higher Dimension ◽  
Group Invariant ◽  
Graphs Of Groups ◽  
Higher Dimensional ◽  
Thompson’S Group ◽  
A Chain ◽  
Finitely Generate

The aim of this paper is to begin the development of a theory of higher-dimensional Alexander ideals for groups. The classical Alexander ideals of a finitely generate group take the form of an ascending chain of ideals in a commutative ring. For a group of type FPn, we define a group invariant in every dimension up to n, also in the form of a chain of ideals, the ideals in dimension 1 being precisely the chain of Alexander ideals. We show how these invariants can be calculated for certain groups, such as R. Thompson's group, (relatively) aspherical groups and graphs of groups, and we demonstrate that the higher-dimension invariants can distinguish groups which the classical Alexander ideals cannot.We also define Alexander-type ideals for modules as well as a number of secondary group invariants in the form of polynomials and 'ranks', which are connected with work of Swan, Hattori and Stallings, K. S. Brown and Lustig. Finally, we explain how this theory can be extended to monoids.

scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

scholarly journals Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1012
Author(s):  
Innocent Simbanefayi ◽  
Chaudry Masood Khalique
Keyword(s):  
Real World ◽  
Nonlinear Partial Differential Equations ◽  
Elliptic Functions ◽  
Lie Symmetry ◽  
Conserved Quantities ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Higher Dimensional ◽  
Petviashvili Equation ◽  
Group Invariant Solutions

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

The Structure of the Polytope of Hereditary Information

2019 ◽  
Vol 8 (2) ◽  
pp. 7-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Nucleic Acid ◽  
Hydrogen Bonds ◽  
Phosphoric Acid ◽  
High Intensity ◽  
Information Exchange ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Nitrogen Bases ◽  
Higher Dimensional ◽  
Low Dimensional

The representations of the sugar molecule and the residue of phosphoric acid in the form of polytopes of higher dimension are used. Based on these ideas and their simplified three-dimensional images, a three-dimensional image of nucleic acids is constructed. The geometry of the neighborhood of the compound of two nucleic acid helices with nitrogen bases has been investigated in detail. It is proved that this neighborhood is a cross-polytope of dimension 13 (polytope of hereditary information), in the coordinate planes of which there are complementary hydrogen bonds of nitrogenous bases. The structure of this polytope is defined, and its image is given. The total incident flows from the low-dimensional elements to the higher-dimensional elements and vice versa of the hereditary information polytope are calculated equal to each other. High values of these flows indicate a high intensity of information exchange in the polytope of hereditary information that ensures the transfer of this information.

Automatic First Arrival Picking Workflow by Global Path Tracing

Geophysics ◽  
2021 ◽  
pp. 1-46
Author(s):  
Dongliang Zhang ◽  
Tong W. Fei ◽  
Song Han ◽  
Constantine Tsingas ◽  
Yi Luo ◽  
...  
Keyword(s):  
Field Data ◽  
Amplitude Ratio ◽  
Optimal Path ◽  
Two Dimensions ◽  
Higher Dimension ◽  
Path Tracing ◽  
Higher Dimensional ◽  
First Arrival ◽  
The Stability ◽  
The One

It can be challenging to pick high quality first arrivals on noisy seismic datasets. The stability and smoothness criteria of the picked first arrival are not satisfied for datasets with shingles and interferences from unexpected and backscattered events. To improve first arrival picking, we propose an automatic first arrival picking workflow using global path tracing to find a global solution for first arrival picking with the condition of smoothness of the traced path. The proposed methodology is composed of data preconditioning, global path tracing, and final addition of traced and piloted travel times to compute the total picked travel time. We propose several ways to precondition the dataset, including the use of amplitude and amplitude ratio with and without a pilot. 2D global path tracing is comprised of two steps, namely, accumulation of energy on the potential path and backtracking of the optimal path with a strain factor for smoothness. For higher dimensional datasets, two strategies were adopted. One was to split the higher-dimension data into sub-domains of two dimensions to which 2D global path tracing was applied. The alternative method was to smooth the preconditioned dataset in directions except for the one used to trace the path before applying 2D global path tracing. Next, we discussed the importance of choosing proper parameters in both data preconditioning and constraining global path tracing. We demonstrated the robustness and stability of the proposed automatic first arrival picking via global path tracing using synthetic and field data examples.

scholarly journals On valuation subalgebras and their centres

1989 ◽  
Vol 31 (1) ◽  
pp. 115-126 ◽  
Author(s):  
C. P. L. Rhodes
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Valuation Domain ◽  
A Chain

We shall extend results of Samuel [19] and Griffin [8, 9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an R-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x ∈ U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x ∉ L, which we call P+(L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions.

scholarly journals Castelnuovo bounds for higher-dimensional varieties

2012 ◽  
Vol 148 (4) ◽  
pp. 1085-1132 ◽  
Author(s):  
F. L. Zak
Keyword(s):  
Betti Numbers ◽  
Higher Dimension ◽  
Algebraic Varieties ◽  
Sectional Genus ◽  
Sharp Bounds ◽  
Higher Dimensional ◽  
Dual Varieties ◽  
Hyperplane Sections

AbstractWe give bounds for the Betti numbers of projective algebraic varieties in terms of their classes (degrees of dual varieties of successive hyperplane sections). We also give bounds for classes in terms of ramification volumes (mixed ramification degrees), sectional genus and, eventually, in terms of dimension, codimension and degree. For varieties whose degree is large with respect to codimension, we give sharp bounds for the above invariants and classify the varieties on the boundary, thus obtaining a generalization of Castelnuovo’s theory for curves to varieties of higher dimension.

INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS

2009 ◽  
Vol 19 (01) ◽  
pp. 409-417 ◽  
Author(s):  
MD SHAFIQUL ISLAM
Keyword(s):  
Invariant Measures ◽  
Stochastic Matrix ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Higher Dimension ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Higher Dimensional ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

2-chains and square roots of Thompson’s group 

2019 ◽  
Vol 40 (9) ◽  
pp. 2515-2532
Author(s):  
THOMAS KOBERDA ◽  
YASH LODHA
Keyword(s):  
General Phenomenon ◽  
Square Roots ◽  
Thompson’S Group ◽  
A Chain ◽  
Free Subgroups ◽  
Finitely Presented

We study 2-generated subgroups $\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$ such that $\langle f^{2},g^{2}\rangle$ is isomorphic to Thompson’s group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $C^{2}$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\operatorname{Homeo}^{+}(I)$.

Quantum codes from cyclic codes over F3 + vF3

2014 ◽  
Vol 12 (06) ◽  
pp. 1450042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Commutative Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Chain Ring ◽  
Arbitrary Length ◽  
Orthogonal Codes ◽  
Finite Commutative Ring ◽  
A Chain

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

Sign in / Sign up

Export Citation Format

Share Document