scholarly journals Pseudofinite groups and VC-dimension

2020 ◽  
pp. 2150009
Author(s):  
Gabriel Conant ◽  
Anand Pillay
Keyword(s):  
Group Theory ◽  
Invariant Measure ◽  
Connected Component ◽  
Vc Dimension ◽  
Counting Measure ◽  
Text Type ◽  
Definable Sets ◽  
Structure Formula ◽  
The Right ◽  
Compact Domination

We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure [Formula: see text] expanding a group, and left invariant NIP formula [Formula: see text], we prove various aspects of “local fsg” for the right-stratified formula [Formula: see text]. This includes a [Formula: see text]-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on [Formula: see text]-formulas and generic compact domination for [Formula: see text]-definable sets.

scholarly journals On the convergence of the zeta function for certain prehomogeneous vector spaces

1995 ◽  
Vol 140 ◽  
pp. 1-31 ◽  
Author(s):  
Akihiko Yukie
Keyword(s):  
Invariant Measure ◽  
Zeta Function ◽  
Complex Variable ◽  
Vector Spaces ◽  
Invariant Polynomial ◽  
Connected Component ◽  
Relative Invariant ◽  
Prehomogeneous Vector Space ◽  
Prehomogeneous Vector Spaces ◽  
Rational Character

Let (G, V) be an irreducible prehomogeneous vector space defined over a number field k, P ∈ k[V] a relative invariant polynomial, and χ a rational character of G such that . For , let Gx be the stabilizer of x, and the connected component of 1 of Gx. We define L0 to be the set of such that does not have a non-trivial rational character. Then we define the zeta function for (G, Y) by the following integralwhere Φ is a Schwartz-Bruhat function, s is a complex variable, and dg” is an invariant measure.

An Algorithmic Solution for a Word Problem in Group Theory

1964 ◽  
Vol 16 ◽  
pp. 509-516 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Group Theory ◽  
Finite Number ◽  
Word Problem ◽  
Finite Index ◽  
Unit Element ◽  
Systematic Procedure ◽  
Algorithmic Solution ◽  
Finite Set ◽  
The Right

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

scholarly journals Volume growth in the component of fibered twists

2018 ◽  
Vol 20 (08) ◽  
pp. 1850014 ◽  
Author(s):  
Joontae Kim ◽  
Myeonggi Kwon ◽  
Junyoung Lee
Keyword(s):  
Lower Bound ◽  
Symplectic Manifold ◽  
Infinite Order ◽  
Floer Homology ◽  
Volume Growth ◽  
Spectral Sequences ◽  
Connected Component ◽  
Text Type ◽  
Infinite Dimensional ◽  
Homology Groups

For a Liouville domain [Formula: see text] whose boundary admits a periodic Reeb flow, we can consider the connected component [Formula: see text] of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, in the component [Formula: see text] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [Formula: see text] has infinite order in [Formula: see text] if there is an admissible Lagrangian [Formula: see text] in [Formula: see text] whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of [Formula: see text]-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse–Bott spectral sequences.

scholarly journals Common Transversals

1966 ◽  
Vol 15 (2) ◽  
pp. 147-154 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Group Theory ◽  
Finite Index ◽  
Simple Proof ◽  
Hecke Operators ◽  
Similar Nature ◽  
The Subject ◽  
The Right

A well-known theorem in group theory [(8), p. 11, Satz 3] asserts that, whenHis a subgroup of finite index in a groupG, there exists a system of common representatives of the right cosets and the left cosets ofHinG. Various proofs and generalisations, mainly involving combinatorial rather than grouptheoretical ideas, are known, and an excellent account of the subject is to be found in Chapter 5 of Ryser's book (6), where references to the literature are given. The purpose of the present paper is to use group-theoretical ideas to prove theorems of a similar nature. The motivation for this work comes from the theory of Hecke operators, and one of the main objects is to provide a simple proof of a result given by Petersson (4), which is needed in order to prove the normality of these operators.

scholarly journals THE STRUCTURE OF RESIDUATED LATTICES

2003 ◽  
Vol 13 (04) ◽  
pp. 437-461 ◽  
Author(s):  
KEVIN BLOUNT ◽  
CONSTANTINE TSINAKIS
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
Binary Operations ◽  
Residuated Structures ◽  
Structure Formula ◽  
Left And Right ◽  
The Right ◽  
Decomposition Theorems ◽  
First Time ◽  
Monoid Structure

A residuated lattice is an ordered algebraic structure [Formula: see text] such that <L,∧,∨> is a lattice, <L,·,e> is a monoid, and \ and / are binary operations for which the equivalences [Formula: see text] hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒC that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).

CHIRAL BOSONIZED ACTION AND THE MINIMAL WESS-ZUMINO-WITTEN ACTION FOR CHIRAL TWO-DIMENSIONAL QCD FROM PATH-INTEGRAL METHOD

1994 ◽  
Vol 09 (18) ◽  
pp. 1643-1652
Author(s):  
JIAN-GE ZHOU ◽  
YAN-GANG MIAO ◽  
YAO-YANG LIU
Keyword(s):  
Invariant Measure ◽  
Path Integral ◽  
Integral Method ◽  
Poisson Brackets ◽  
Field Independent ◽  
Integration Measure ◽  
Path Integral Method ◽  
Definition Of ◽  
The Right ◽  
Constraint Surface

The path-integral method is applied to derive the chiral bosonized action and the minimal WZW action for chiral QCD2. An interesting feature in our model is that the Poisson brackets of the new set of constraints, which is a recombination of the original ones, are field-independent on the constraint surface. The correct contribution from det {Ωi, Ωj} or δ(Ωa) is considered, and it seems to affect the properties of the integration measure. Here Ωi are the original chiral constraints, Ωa are linear combination of Ωi. However, due to the determinants of Lab(ɸ) and [Formula: see text] (the definition of Lab(ɸ) and [Formula: see text] can be seen from Eqs. (22) and (41)) being ±1, the right-invariant measure is equivalent to the left-invariant measure, which is quite unlike the case of the gauge quasigroup. As a result, we can identify the integration measure in our model with the Haar measure.

The last word on elimination of quantifiers in modules

1990 ◽  
Vol 55 (2) ◽  
pp. 670-673 ◽  
Author(s):  
Hans B. Gute ◽  
K. K. Reuter
Keyword(s):  
Finite Index ◽  
Finite Family ◽  
Left Translation ◽  
Boolean Combination ◽  
Cartesian Power ◽  
Definable Set ◽  
Combinatorial Lemma ◽  
Definable Subset ◽  
Definable Sets ◽  
The Right

In what follows, a coset is a subset of a group G of the form aH, where H is a subgroup of G; H can be recovered from the coset C: it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka, since aH = aHa−1a. A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C1 = a1H1,…,Cn = anHn, then it is the union of those Ci whose corresponding Hi has finite index in H.In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power Gn is a(finite!) Boolean combination of cosets modulo definable subgroups. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.

scholarly journals On the equivalence of invariant integrals and minimal ideals in semigroups

1969 ◽  
Vol 1 (2) ◽  
pp. 269-278
Author(s):  
N. A. Tserpes ◽  
A. G. Kartsatos
Keyword(s):  
Invariant Measure ◽  
Left Ideal ◽  
Invariant Probability Measure ◽  
Continuous Functions ◽  
Minimal Left Ideal ◽  
Locally Compact ◽  
Left Group ◽  
Invariant Integrals ◽  
Invariant Integral ◽  
The Right

Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.

Degree problems for modular machines

1980 ◽  
Vol 45 (3) ◽  
pp. 510-528 ◽  
Author(s):  
Daniel E. Cohen
Keyword(s):  
Group Theory ◽  
Turing Machine ◽  
Recursive Function ◽  
Related Result ◽  
Turing Degrees ◽  
General Sense ◽  
Halting Problem ◽  
The Right

Modular machines were introduced in [1] and [2], where they were used to give simple proofs of various unsolvability results in group theory. Here we discuss the degrees of the halting, word, and confluence problems for modular machines, both for their own sake and in the hope that the results may be useful in group theory (see [4] for an application of a related result to group theory).In the course of the analysis, I found it convenient to compare degrees of these problems for a Turing machine T and for a Turing machine T1 obtained from T by enlarging the alphabet but retaining the same quintuples (or quadruples). The results were surprising. The degree for a problem of T1 depends not just on the corresponding degree for T, but also on the degrees of the corresponding problems when T is restricted to a semi-infinite tape (both semi-infinite to the right and semi-infinite to the left). For the halting and confluence problems, the Turing degrees of the problems for these three machines can be any r.e. degrees. In particular the halting problem of T can be solvable, while that of T1 has any r.e. degree.A machine M (in the general sense) consists of a countable set of configurations (together with a numbering, which we usually take for granted), a recursive subset of configurations called the terminal configurations, and a recursive function, written C ⇒ C′, on the set of configurations. If, for some n ≥ 0, we have C = C0 ⇒ C1 ⇒ … ⇒ Cn = C′, we write C → C′. We say M halts from C if C → C′ for some terminal C′.

scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

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