Algebraic types and automorphism groups

1993 ◽  
Vol 58 (1) ◽  
pp. 232-239 ◽  
Author(s):  
Akito Tsuboi
Keyword(s):  
General Model ◽  
Galois Extension ◽  
Galois Theory ◽  
Automorphism Groups ◽  
The Other ◽  
Closed Field ◽  
Prime Characteristic ◽  
Galois Correspondence ◽  
Definable Functions ◽  
Special Case

Galois theory states that if L is a certain algebraic extension (called a Galois extension) of a field K, then there is a one-to-one correspondence (called a Galois correspondence) between subfields M, K ⊂ M ⊂ L and subgroups of the automorphism groups of L fixing the elements in K.A subfield of a field L can be considered as a substructure of L in general model theory. However, a substructure is a subset closed under functions which are interpretations of function symbols in a given language, so the notion of substructure may change if we expand the language by adding definable notions. On the other hand a definably closed substructure is a subset which is closed under all definable functions, and it does not change by such expansions. If we are interested in subfields of an algebraically closed field of characteristic 0, these two notions are the same. But in a field of prime characteristic they are not equal. Speaking roughly, a Galois extension of a field K is an extension whose subfields are relatively definably closed. Poizat [4] showed that if a structure M has elimination of imaginaries there is a kind of Galois correspondence between definably closed substructures and subgroups of bijective elementary mappings of M.In this paper, using Poizat's result we study algebraic types. As is well known, one motivation for developing the Galois theory was to show the unsolvability of equations with degree ≥ 5. We want to take this unsolvability as a special case of general phenomena. For this purpose, we introduce several notions which are stronger than mere algebraicity and study relations between these notions and groups of bijective elementary mappings. (See Theorems 3.7 and 3.9.)

Galois corings and groupoids acting partially on algebras

2020 ◽  
pp. 2140003
Author(s):  
S. Caenepeel ◽  
T. Fieremans
Keyword(s):  
Galois Extension ◽  
Galois Theory ◽  
Type Theorem ◽  
Partial Action ◽  
Morita Theory ◽  
Maschke Type Theorem ◽  
Galois Corings ◽  
Partial Galois Extension ◽  
Special Case

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

scholarly journals Real Liouvillian Extensions of Partial Differential Fields

2021 ◽  
Author(s):  
Teresa Crespo ◽  
◽  
Zbigniew Hajto ◽  
Rouzbeh Mohseni ◽  
◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Galois Theory ◽  
Uniqueness Result ◽  
Real Closed Field ◽  
Closed Field ◽  
Partial Differential ◽  
Linear Algebraic Groups ◽  
Galois Correspondence ◽  
Definition Of ◽  
Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

scholarly journals An Invitation to Model-Theoretic Galois Theory

2010 ◽  
Vol 16 (2) ◽  
pp. 261-269 ◽  
Author(s):  
Alice Medvedev ◽  
Ramin Takloo-Bighash
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Order Theory ◽  
Closed Field ◽  
Finite Sets ◽  
Algebraically Closed Field ◽  
First Order ◽  
First Order Theory ◽  
Special Case ◽  
Large Model

AbstractWe carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.

scholarly journals SOME DEFINABLE GALOIS THEORY AND EXAMPLES

2017 ◽  
Vol 23 (2) ◽  
pp. 145-159 ◽  
Author(s):  
OMAR LEÓN SÁNCHEZ ◽  
ANAND PILLAY
Keyword(s):  
Differential Equations ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Differential Galois Theory ◽  
Algebraic Differential Equations ◽  
Galois Correspondence

AbstractWe make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

scholarly journals The game of Double-Silver on intervals

2001 ◽  
Vol 26 (8) ◽  
pp. 485-496 ◽  
Author(s):  
Gerald A. Heuer
Keyword(s):  
Measure Zero ◽  
Optimal Strategies ◽  
The Other ◽  
Parameter Plane ◽  
Left Endpoint ◽  
Relative Placement ◽  
Special Case

Silverman's game on intervals was analyzed in a special case by Evans, and later more extensively by Heuer and Leopold-Wildburger, who found that optimal strategies exist (and gave them) quite generally when the intervals have no endpoints in common. They exist in about half the parameter plane when the intervals have a left endpoint or a right endpoint, but not both, in common, and (as Evans had earlier found) exist only on a set of measure zero in this plane if the intervals are identical. The game of Double-Silver, where each player has its own threshold and penalty, is examined. There are several combinations of conditions on relative placement of the intervals, the thresholds and penalties under which optimal strategies exist and are found. The indications are that in the other cases no optimal strategies exist.

Silent Victims Revisited: The Special Case of Domestic Violence

PEDIATRICS ◽  
1995 ◽  
Vol 96 (3) ◽  
pp. 511-513
Author(s):  
Barry Zuckerman ◽  
Marilyn Augustyn ◽  
Betsy McAlister Groves ◽  
Steven Parker
Keyword(s):  
Domestic Violence ◽  
Community Violence ◽  
Clinical Experience ◽  
Behavioral Problems ◽  
The Other ◽  
Media Attention ◽  
Witness Violence ◽  
Primary Focus ◽  
Special Case ◽  
At Home

In a commentary published previously, we communicated our concern regarding the plight of children who witness violence.1 Research suggests that children who witness violence suffer significant psychologic and behavioral problems that interfere with their ability to function in school, at home, and with peers. The primary focus of that commentary was children who witnessed community violence. Our ongoing clinical experience, heightened by media attention on domestic violence, including the O.J. Simpson case, leads us to revisit silent victims with a sole focus on those children who witness domestic violence. Domestic violence is a particularly devastating event for a child who, in the presence of danger, typically turns to a parent for protection and for whom there is no comfort or security if one parent is the perpetrator of violence, and the other is a terrified victim.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

scholarly journals Can you reach for the planets or grasp at the stars? – Modified noun, verb, or preposition constituents in idiom processing

2019 ◽  
Author(s):  
Eva Smolka ◽  
Carsten Eulitz
Keyword(s):  
Sentence Type ◽  
The Other ◽  
Psycholinguistic Research ◽  
Idiom Comprehension ◽  
Figurative Meaning ◽  
Idiomatic Phrase ◽  
Idiom Processing ◽  
Special Case ◽  
Idiomatic Meaning ◽  
Control Sentence

Idioms are a special case of multi-word expressions in that their meaning cannot be compositionally constructed from the meaning of the single constituents. The question of how the idiomatic meaning is assembled remains an unsettled issue in psycholinguistic research. The present study examines whether the figurative meaning of an idiom is recognized if critical idiomatic constituents, such as the noun, verb, or preposition, are modified. In three paraphrase experiments, participants saw (a) the canonical idiomatic phrase (e.g., She reached for the stars), (b) the idiomatic phrase with a modified constituent (e.g., She reached/grasped for/at the stars/planets), or (c) a matched literal control sentence (e.g., She reached for the sweets) and rated on a scale from 1 (not at all) - 7 (completely) how strongly the sentence reflected the meaning of a paraphrase of the idiom (e.g., She has always aspired to unattainable goals). Sentence type and constituent type strongly affected paraphrase ratings with highest ratings for canonical idiomatic phrases, lowest ratings for control sentences, and ratings in between for idioms with modified constituents. Further, idioms with modified verbs were rated higher in matching the figurative meaning than idioms with modified prepositions or nouns. Overall these findings indicate that the figurative meaning was assembled in spite of the modifications. We conclude that idioms are not fully ‘semantically fixed’ but allow for some flexibility in the processing of idioms. Modified constituents that activate meanings similar to those of the canonical constituents will co-activate the figurative meaning of the idiom together with the other idiomatic constituents. We discuss psycholinguistic models on idiom comprehension.

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