Quasiunitriangular groups

1993 ◽  
Vol 58 (1) ◽  
pp. 205-218 ◽  
Author(s):  
O. V. Belegradek
Keyword(s):  
Direct Summand ◽  
Algebraic Characterization ◽  
First Order ◽  
Expanded Group ◽  
The Matrix ◽  
Image Position

For a ring with unit R, which need not be associative, denote the group of upper unitriangular 3 × 3 matrices over R by UT3(R). Let e1 = (1,0,0), e2 = (0,1,0), where (α, β, γ) denotes the matrixDenote the expanded group (UT3(R), e1, e2) by (R). A. 1. Mal′cev [M] gave an algebraic characterization of the expanded groups of the form (R) as follows. Let h1, h2 be elements of a group H; then (H, h1, h2) is isomorphic to (R), for some R, if and only if(i) H is 2-step nilpotent;(ii) CH(hi) are abelian, i = 1,2;(iii) CH(h1) ∩ CH(h2) = Z(H);(iv) [CH(h1),h2] = [h1, CH(h2)] = Z(H);(v) Z(H) is a direct summand in both CH(hi).(In [M] condition (v) is a bit stronger; the version above is presented in [B2].)A pair (h1, h2) of elements of a group H is said to be a base if (H, h1, h2) satisfies the conditions (i)–(iv). A. I. Mal′cev [M] found a uniform way of first order interpreting a ring Ring(H, h1, h2) in any group with a base (H, h1, h2); in particular, Ring((R)) ≃ R.

scholarly journals Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 728
Author(s):  
Yasunori Maekawa ◽  
Yoshihiro Ueda
Keyword(s):  
Hyperbolic System ◽  
Dissipative Structure ◽  
Dissipative Structures ◽  
Algebraic Characterization ◽  
Symmetric Hyperbolic System ◽  
Full Generality ◽  
Order Linear ◽  
First Order

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

An algebraic characterization of power set in countable standard models of ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 167-170
Author(s):  
George Metakides ◽  
J. M. Plotkin
Keyword(s):  
Standard Model ◽  
Boolean Algebra ◽  
Classical Result ◽  
Algebraic Characterization ◽  
Categorical Theory ◽  
Power Set ◽  
Standard Models ◽  
Image Position ◽  
Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

scholarly journals EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

2019 ◽  
Vol 84 (02) ◽  
pp. 452-472 ◽  
Author(s):  
JAROSLAV NEŠETŘIL ◽  
PATRICE OSSONA DE MENDEZ
Keyword(s):  
Relative Size ◽  
Convergent Sequence ◽  
List Type ◽  
Sparse Graphs ◽  
First Order ◽  
Graph Modeling ◽  
Image Position ◽  
Standard Probability ◽  
Convergent Sequences

AbstractA sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:1.If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.2.A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

scholarly journals NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

2017 ◽  
Vol 82 (1) ◽  
pp. 35-61 ◽  
Author(s):  
ALLEN GEHRET
Keyword(s):  
Order Theory ◽  
Exchange Property ◽  
Independence Property ◽  
First Order ◽  
Valued Field ◽  
First Order Theory ◽  
Order Language ◽  
Complete Survey ◽  
Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

scholarly journals AN AXIOMATIC APPROACH TO FREE AMALGAMATION

2017 ◽  
Vol 82 (2) ◽  
pp. 648-671 ◽  
Author(s):  
GABRIEL CONANT
Keyword(s):  
Axiomatic Approach ◽  
Order Property ◽  
Combinatorial Characterization ◽  
First Order ◽  
Homogeneous Structures ◽  
Image Position ◽  
Rosy Theories

AbstractWe use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic Kn-free graphs are SOP3, while the higher arity generic $K_n^r$-free r-hypergraphs are simple.

A strong version of Herbrand's theorem for introvert sentences

1998 ◽  
Vol 63 (2) ◽  
pp. 555-569 ◽  
Author(s):  
Tore Langholm
Keyword(s):  
Ground Term ◽  
New Members ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Universal Sentence ◽  
Order Language ◽  
The Matrix ◽  
Image Position ◽  
Infinite Set ◽  
Herbrand's Theorem

A version of Herbrand's theorem tells us that a universal sentence of a first-order language with at least one constant is satisfiable if and only if the conjunction of all its ground instances is. In general the set of such instances is infinite, and arbitrarily large finite subsets may have to be inspected in order to detect inconsistency. Essentially, the reason that every member of such an infinite set may potentially matter, can be traced back to sentences like(1) Loosely put, such sentences effectively sabotage any attempt to build a model from below in a finite number of steps, since new members of the Herbrand universe are constantly brought to attention. Since they cause an indefinite expansion of the relevant part of the Herbrand universe, such sentences could quite appropriately be called expanding.When such sentences are banned, stronger versions of Herbrand's theorem can be stated. Define a clause (disjunction of literals) to be non-expanding if every non-ground term occurring in a positive literal also occurs (possibly as an embedded subterm) in a negative literal of the same clause. Written as a disjunction of literals, the matrix of (1) clearly fails this criterion. Moreover, say that a sentence is non-expanding if it is a universal sentence with a quantifier-free matrix that is a conjunction of non-expanding clauses. Such sentences do in a sense never reach out beyond themselves, and the relevant part of the Herbrand universe is therefore drastically reduced.

Oscillation Criteria for Matrix Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 184-199 ◽  
Author(s):  
H. C. Howard
Keyword(s):  
Existence Theorems ◽  
Symmetric Matrix ◽  
Order System ◽  
Matrix Differential Equation ◽  
First Order ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Image Position

We shall be concerned at first with some properties of the solutions of the matrix differential equation1.1whereis an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

An Algebraic Characterization of Concurrent Systems1

1988 ◽  
Vol 11 (2) ◽  
pp. 171-193
Author(s):  
Waldemar Korczyński
Keyword(s):  
Special Kind ◽  
Concurrent Systems ◽  
Algebraic Characterization ◽  
First Order ◽  
Order Language

In the paper a characterization of concurrent systems as algebras of a special kind is given. The algebras are defined by semi-equations in the first order language.

Characterization of nanographite and carbon nanotubes by polarization dependent optical spectroscopy

2002 ◽  
Vol 737 ◽  
Author(s):  
Alexander Grüneis ◽  
Riichiro Saito ◽  
Georgii G. Samsonidze ◽  
Marcos A. Pimenta ◽  
Ado Jorio ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Matrix Element ◽  
Optical Absorption ◽  
Single Wall Carbon Nanotubes ◽  
First Order ◽  
Conduction Bands ◽  
The Matrix ◽  
First Order Perturbation ◽  
Energy Contour

ABSTRACTThe optical absorption for ? electrons as a function of the electron wavevector k is investigated by first order perturbation theory in graphite and single wall carbon nanotubes (SWNTs). The matrix element connecting two states in the valence and conduction bands is found to be significantly anisotropic in k-space and polarization dependent. In the case of graphite, the absorption shows a node around the equi-energy contour, and in the case of SWNTs we obtain selection rules that allow only transitions between certain pairs of subbands. The strength of the optical absorption is not only diameter dependent but also chirality dependent. The implications of the optical absorption matrix element on the resonant conditions are discussed.

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