scholarly journals Stronger arithmetic equivalence

2021 ◽  
Vol 2021 ◽  
Author(s):  
Andrew SUTHERLAND
Keyword(s):  
Group Theory ◽  
Sufficient Condition ◽  
Local Integral ◽  
Open Questions ◽  
Integral Equivalence ◽  
Arithmetic Equivalence

This paper considers two alternative strengthenings of the notion of arithmetic equivalence, which the author calls local integral equivalence and solvable equivalence. (The latter turns out to be strictly stronger than the former.) They have the advantage of being easier to check than Prasad’s notion, which the author calls integral equivalence. Furthermore, solvable equivalence, which the author shows does not imply integral equivalence, is nevertheless a sufficient condition to imply that the invariants considered by Prasad are equal. This opens the door to easier proofs of Prasad’s result, and lessens the reliance on Scott’s construction: the author finds a generalization of this construction that yields infinitely many examples of solvable equivalence. The paper also contains several examples to clarify the relationships between the various different notions of equivalence. Some of these examples (which are mainly found with the help of a computer) answer open questions from the group theory literature.

scholarly journals On the Invariant Factors of Class Groups in Towers of Number Fields

2018 ◽  
Vol 70 (1) ◽  
pp. 142-172 ◽  
Author(s):  
Farshid Hajir ◽  
Christian Maire
Keyword(s):  
Group Theory ◽  
Number Fields ◽  
Class Groups ◽  
Logarithmic Mean ◽  
Theoretical Perspectives ◽  
Open Questions ◽  
The Mean ◽  
Compare And Contrast ◽  
Invariant Factors ◽  
Class Field Tower

AbstractFor a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Right-Angled Artin Groups

2017 ◽  
Author(s):  
Robert W. Bell ◽  
Matt Clay
Keyword(s):  
Group Theory ◽  
Broad Spectrum ◽  
Geometric Group Theory ◽  
The Other ◽  
Artin Group ◽  
Artin Groups ◽  
Research Projects ◽  
Open Questions ◽  
Free Abelian Groups ◽  
Right Angled Artin Groups

This chapter deals with right-angled Artin groups, a broad spectrum of groups that includes free groups on one end, free abelian groups on the other end, and many other interesting groups in between. A right-angled Artin group is a group G(Γ‎) defined in terms of a graph Γ‎. Right-angled Artin groups have taken a central role in geometric group theory, mainly due to their involvement in the solution to one of the main open questions in the topology of 3-manifolds. The chapter first considers right-angled Artin groups as subgroups and how they relate to other classes of groups before exploring subgroups of right-angled Artin groups and the word problem for right-angled Artin groups. The discussion includes exercises and research projects.

Functional completeness in one variable

1968 ◽  
Vol 33 (1) ◽  
pp. 105-106 ◽  
Author(s):  
James Rosenberg
Keyword(s):  
Group Theory ◽  
Sufficient Condition ◽  
Functional Completeness ◽  
Necessary And Sufficient Condition ◽  
The Subject ◽  
Complete Sets ◽  
Necessary And Sufficient

It is extremely difficult to make general statements about functional completeness. (For the main reference on the subject see Post [2].) In this paper we restrict ourselves to the case of unary functions in a finite valued logic, and prove a result concerning minimal functionally complete sets, along with a necessary and sufficient condition for completeness. A basic familiarity with group theory would be helpful.

scholarly journals Perinormality in Polynomial and Module-Finite Ring Extensions

2017 ◽  
Author(s):  
◽  
Andrew McCrady
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Finite Ring ◽  
Sufficient Condition ◽  
Local Domain ◽  
Descent Property ◽  
Open Questions ◽  
Special Cases ◽  
Adic Completion ◽  
Canonical Integral

In this dissertation we investigate some open questions posed by Epstein and Shapiro in [9] regarding perinormal domains. More specifically, we focus on the ascent/descent property of perinormality between "canonical" integral domain extensions, in particular, R [superscript] R[X] and R [suberscript] Rb. We give special conditions under which perinormality ascends from R to the polynomial ring R[X] in the case that R is a universally catenary domain. Whereas we have a characterizing result for when perinormality descends from R[X] to R, the sufficient condition for the descent is cumbersome to check. For this reason, we turn to special cases for which perinormality descends from R[X] to R. In the case of an analytically irreducible local domain (R, m) and its m-adic completion (R, b mRb), we refer to a technique for generating examples in which perinormality fails to ascend. When Rb is perinormal, we explore hypotheses under which R must be normal, perinormal, or weakly normal.

scholarly journals State Aggregation and Population Dynamics in Linear Systems

2005 ◽  
Vol 11 (4) ◽  
pp. 473-492 ◽  
Author(s):  
Jonathan E. Rowe ◽  
Michael D. Vose ◽  
Alden H. Wright
Keyword(s):  
Population Dynamics ◽  
Group Theory ◽  
Complex Systems ◽  
Artificial Life ◽  
Sufficient Conditions ◽  
Emergent Properties ◽  
Sufficient Condition ◽  
State Aggregation ◽  
Macroscopic States ◽  
Necessary And Sufficient

We consider complex systems that are composed of many interacting elements, evolving under some dynamics. We are interested in characterizing the ways in which these elements may be grouped into higher-level, macroscopic states in a way that is compatible with those dynamics. Such groupings may then be thought of as naturally emergent properties of the system. We formalize this idea and, in the case that the dynamics are linear, prove necessary and sufficient conditions for this to happen. In cases where there is an underlying symmetry among the components of the system, group theory may be used to provide a strong sufficient condition. These observations are illustrated with some artificial life examples.

The Current Situation in Chemical Stabilization of Biological Materials

1972 ◽  
Vol 30 ◽  
pp. 132-133
Author(s):  
John H. Luft
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Osmium Tetroxide ◽  
Molecular Level ◽  
Cross Linking ◽  
Chemical Stabilization ◽  
Specimen Preparation ◽  
Sufficient Condition ◽  
Radio Telescopes

With information processing devices such as radio telescopes, microscopes or hi-fi systems, the quality of the output often is limited by distortion or noise introduced at the input stage of the device. This analogy can be extended usefully to specimen preparation for the electron microscope; fixation, which initiates the processing sequence, is the single most important step and, unfortunately, is the least well understood. Although there is an abundance of fixation mixtures recommended in the light microscopy literature, osmium tetroxide and glutaraldehyde are favored for electron microscopy. These fixatives react vigorously with proteins at the molecular level. There is clear evidence for the cross-linking of proteins both by osmium tetroxide and glutaraldehyde and cross-linking may be a necessary if not sufficient condition to define fixatives as a class.

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