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1,495 results on '"Algebraic closure"'

1. On the representation of fields as sums of two proper subfields.

Author

Kȩpczyk, Marek

Subjects
*FINITE fields
Abstract

In this paper, we study which field F can be represented as sums of two proper subfields. We prove that a field F is a sum of two proper subfields if and only if the transcendence degree of F over its prime subfield is infinite. This answers a question posed in the paper [M. Kȩpczyk and R. Mazurek, On the representation of fields as finite sums of proper subfields, Results Math. 75(2) (2020) 65]. [ABSTRACT FROM AUTHOR]

Published
2024
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2. Algebraic Closures in Divisible Rigid Groups.

Author

Romanovskii, N. S.

Subjects
*DIVISIBILITY groups, *SOLVABLE groups
Abstract

We prove that in a divisible -rigid group the algebraic closure of each set generating a subgroup of solvability length coincides with the elementary closure of this set. [ABSTRACT FROM AUTHOR]

Published
2024
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3. On Algebraic and Definable Closures for Theories of Abelian Groups

Author

In.I. Pavlyuk

Subjects
algebraic closure, definable closure, degree of algebraization, abelian group, Mathematics, QA1-939
Abstract

Classifying abelian groups and their elementary theories, a series of characteristics arises that describe certain features of the objects under consideration. Among these characteristics, an important role is played by Szmielew invariants, which define the possibilities of divisibility of elements, orders of elements, dimension of subgroups, and allow describing given abelian groups up to elementary equivalence. Thus, in terms of Szmielew invariants, the syntactic properties of Abelian groups are represented, i.e. properties that depend only on their elementary theories. The work, based on Szmielew invariants, provides a description of the behavior of algebraic and definable closure operators based on two characteristics: degrees of algebraization and the difference between algebraic and definable closures. Thus, possibilities for algebraic and definable closures, adapted to theories of Abelian groups, are studied and described. A theorem on trichotomy for degrees of algebraization is proved: either this degree is minimal, if in the standard models, except for the only two-element group, there are no positively finitely many cyclic and quasi-cyclic parts, or the degree is positive and natural, if in a standard model there are no positively finitely many cyclic and quasi-cyclic parts, except a unique copy of a two-element group and some finite direct sum of finite cyclic parts, and the degree is infinite if the standard model contains unboundedly many nonisomorphic finite cyclic parts or positively finitely many of copies of quasi-finite parts. In addition, a dichotomy of the values of the difference between algebraic closures and definable closures for abelian groups defined by Szmielew invariants for cyclic parts is established. In particular, it is shown that torsion-free abelian groups are quasi-Urbanik.

Published
2024
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4. Polynomial Root Extensions

Author

Anderson, D. D., Anderson, David F., Chabert, Jean-Luc, editor, Fontana, Marco, editor, Frisch, Sophie, editor, Glaz, Sarah, editor, and Johnson, Keith, editor

Published
2023
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5. Punctually presented structures I: Closure theorems.

Author

Dorzhieva, Marina and Melnikov, Alexander

Subjects
*MODEL theory, *ALGEBRA
Abstract

We study the primitive recursive content of various closure results in algebra and model theory, including the algebraic, the real, and the differential closure theorems. In the case of ordered fields and their real closures, our result settles a question recently raised by Selivanova and Selivanov. [ABSTRACT FROM AUTHOR]

Published
2023
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6. Computability of Algebraic and Definable Closure

Author

Ackerman, Nathanael, Freer, Cameron, Patel, Rehana, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Artemov, Sergei, editor, and Nerode, Anil, editor

Published
2020
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7. The minimal cone of an algebraic Laurent series.

Author

Aroca, Fuensanta, Decaup, Julie, and Rond, Guillaume

Abstract

We study the algebraic closure of K ((x)) , the field of power series in several indeterminates over a field K . In characteristic zero we show that the elements algebraic over K ((x)) can be expressed as Puiseux series such that the convex hull of its support is essentially a polyhedral rational cone, strengthening the known results. In positive characteristic we construct algebraic closed fields containing the field of power series and we give examples showing that the results proved in characteristic zero are no longer valid in positive characteristic. [ABSTRACT FROM AUTHOR]

Published
2022
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8. On binomials and algebraic closure of some pseudofinite fields.

Author

Gismatullin, Jakub and Tarasek, Katarzyna

Subjects
POLYNOMIALS
Abstract

We give a criterion when a polynomial x n − g is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic. [ABSTRACT FROM AUTHOR]

Published
2023
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10. New vectorial versions of Takahashi's nonconvex minimization problem.

Author

Khazayel, B. and Farajzadeh, A.

Abstract

In this article, some new vectorial versions of Takahashi's nonconvex minimization theorem, which involve algebraic notions instead of topological notions, are established. A nonlinear separation theorem, which extends the result derived by Gerth and Weidner (JAMA 67:297–320, 1990) to general linear spaces (not necessarily endowed with a topology), is proved. Some examples, in order to illustrate and compare the results of this article with the corresponding known results from the literature, are provided. [ABSTRACT FROM AUTHOR]

Published
2021
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11. On the Representation of Fields as Finite Sums of Proper Subfields.

Author

Kȩpczyk, Marek and Mazurek, Ryszard

Abstract

We study which fields F can be represented as finite sums of proper subfields. We prove that for any n ≥ 2 every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field F = K (x 1 , … , x n) can be represented as an unshortenable sum of n + 1 subfields. We also show that no subfield of the algebraic closure of a finite field is a finite sum of proper subfields, and no finite extension of the field Q of rationals can be decomposed into a sum of two proper subfields. [ABSTRACT FROM AUTHOR]

Published
2020
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12. Trends in Temporal Reasoning: Constraints, Graphs and Posets

Author

Daykin, Jacqueline W., Miller, Mirka, Ryan, Joe, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Kotsireas, Ilias S., editor, Rump, Siegfried M., editor, and Yap, Chee K., editor

Published
2016
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14. Algebraic Properties of Qualitative Spatio-temporal Calculi

Author

Dylla, Frank, Mossakowski, Till, Schneider, Thomas, Wolter, Diedrich, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Tenbrink, Thora, editor, Stell, John, editor, Galton, Antony, editor, and Wood, Zena, editor

Published
2013
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15. A refinement of Christol’s theorem for algebraic power series

Author

Jason P. Bell and Seda Albayrak

Subjects
Combinatorics, Power series, Base (group theory), Ring (mathematics), Finite field, General Mathematics, Field (mathematics), Rational function, Algebraic number, Algebraic closure, Mathematics
Abstract

A famous result of Christol gives that a power series $$F(t)=\sum _{n\ge 0} f(n)t^n$$ with coefficients in a finite field $$\mathbb {F}_q$$ of characteristic p is algebraic over the field of rational functions in t if and only if there is a finite-state automaton accepting the base-p digits of n as input and giving f(n) as output for every $$n\ge 0$$ . An extension of Christol’s theorem, giving a complete description of the algebraic closure of $$\mathbb {F}_q(t)$$ , was later given by Kedlaya. When one looks at the support of an algebraic power series, that is the set of n for which $$f(n)\ne 0$$ , a well-known dichotomy for sets generated by finite-state automata shows that the support set is either sparse—with the number of $$n\le x$$ for which $$f(n)\ne 0$$ bounded by a polynomial in $$\log (x)$$ —or it is reasonably large in the sense that the number of $$n\le x$$ with $$f(n)\ne 0$$ grows faster than $$x^{\alpha }$$ for some positive $$\alpha $$ . The collection of algebraic power series with sparse supports forms a ring and we give a purely algebraic characterization of this ring in terms of Artin–Schreier extensions and we extend this to the context of Kedlaya’s work on generalized power series.

Published
2021

16. Conjugation of semisimple subgroups over real number fields of bounded degree

Author

Jinbo Ren, Mikhail Borovoi, and Christopher Daw

Subjects
Linear algebraic group, Mathematics::Group Theory, Pure mathematics, Finite field, Galois cohomology, Applied Mathematics, General Mathematics, Bounded function, Field (mathematics), Algebraic closure, Real number, Conjugate, Mathematics
Abstract

Let G G be a linear algebraic group over a field k k of characteristic 0. We show that any two connected semisimple k k -subgroups of G G that are conjugate over an algebraic closure of k k are actually conjugate over a finite field extension of k k of degree bounded independently of the subgroups. Moreover, if k k is a real number field, we show that any two connected semisimple k k -subgroups of G G that are conjugate over the field of real numbers R {\mathbb {R}} are actually conjugate over a finite real extension of k k of degree bounded independently of the subgroups.

Published
2021

17. Adapting Rabin’s Theorem for Differential Fields

Author

Miller, Russell, Ovchinnikov, Alexey, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Löwe, Benedikt, editor, Normann, Dag, editor, Soskov, Ivan, editor, and Soskova, Alexandra, editor

Published
2011
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18. The Impact of Qualification on the Application of Qualitative Spatial and Temporal Reasoning Calculi

Author

Schultz, Carl, Amor, Robert, Guesgen, Hans W., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Goebel, Randy, editor, Siekmann, Jörg, editor, Wahlster, Wolfgang, editor, and Li, Jiuyong, editor

Published
2011
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22. The Multivariate Resultant Is NP-hard in Any Characteristic

Author

Grenet, Bruno, Koiran, Pascal, Portier, Natacha, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Hliněný, Petr, editor, and Kučera, Antonín, editor

Published
2010
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26. Basic Concepts of Difference Algebra

Author

Fialowski, Alice, editor, Friedlander, Eric, editor, Greenlees, John, editor, Hiss, Gerhard, editor, Moerdijk, Ieke, editor, Reiten, Idun, editor, Schweigert, Christoph, editor, Teicher, Mina, editor, and Levin, Alexander

Published
2008
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27. Compatibility, Replicability, and Monadicity

Author

Fialowski, Alice, editor, Friedlander, Eric, editor, Greenlees, John, editor, Hiss, Gerhard, editor, Moerdijk, Ieke, editor, Reiten, Idun, editor, Schweigert, Christoph, editor, Teicher, Mina, editor, and Levin, Alexander

Published
2008
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28. Preliminaries

Author

Fialowski, Alice, editor, Friedlander, Eric, editor, Greenlees, John, editor, Hiss, Gerhard, editor, Moerdijk, Ieke, editor, Reiten, Idun, editor, Schweigert, Christoph, editor, Teicher, Mina, editor, and Levin, Alexander

Published
2008
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29. Topological Properties of Concept Spaces

Author

de Brecht, Matthew, Yamamoto, Akihiro, Carbonell, Jaime G., editor, Siekmann, Jörg, editor, Freund, Yoav, editor, Györfi, László, editor, Turán, György, editor, and Zeugmann, Thomas, editor

Published
2008
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32. New representations of algebraic domains and algebraic L-domains via closure systems

Author

Lankun Guo, Qingguo Li, and Mingyuan Wu

Subjects
Pure mathematics, Algebra and Number Theory, Morphism, Closure (topology), Algebraic number, Algebra over a field, Categorical variable, Algebraic closure, Mathematics
Abstract

Closure systems (spaces) play an important role in characterizing certain ordered structures. In this paper, FinSet-bounded algebraic closure spaces are introduced, and then used to provide a new approach to constructing algebraic domains. Then, a special family of algebraic closure spaces, algebraic L-closure spaces, are used to represent algebraic L-domains. Next, algebraic approximate mappings are defined and serve as the appropriate morphisms between algebraic closure spaces, respectively, algebraic L-closure spaces. On the categorical level, we show that algebraic closure spaces (respectively, algebraic L-closure spaces,) each equipped with algebraic approximate mappings as morphisms, are equivalent to algebraic domains (respectively, algebraic L-domains) with Scott continuous functions as morphisms.

Published
2021

33. Existence of Equivariant Models of Spherical Varieties and Other G-varieties

Author

Mikhail Borovoi and Giuliano Gagliardi

Subjects
Pure mathematics, Mathematics::Operator Algebras, General Mathematics, 010102 general mathematics, Field (mathematics), 01 natural sciences, Algebraic closure, Mathematics::K-Theory and Homology, 0103 physical sciences, Homogeneous space, Embedding, Equivariant map, 010307 mathematical physics, 0101 mathematics, Variety (universal algebra), Mathematics
Abstract

Let $k_0$ be a field of characteristic $0$ with algebraic closure $k$. Let $G$ be a connected reductive $k$-group, and let $Y$ be a spherical variety over $k$ (a spherical homogeneous space or a spherical embedding). Let $G_0$ be a $k_0$-model ($k_0$-form) of $G$. We give necessary and sufficient conditions for the existence of a $G_0$-equivariant $k_0$-model of $Y$.

Published
2021

40. Field Theory

Author

Axler, S., editor, Gehring, F. W., editor, Ribet, K. A., editor, and Lang, Serge

Published
2005
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41. Applications

Author

Axler, S., editor, Gehring, F. W., editor, Ribet, K. A., editor, and Chambert-Loir, Antoine

Published
2005
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42. Roots

Author

Axler, S., editor, Gehring, F. W., editor, Ribet, K. A., editor, and Chambert-Loir, Antoine

Published
2005
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43. Weak Composition for Qualitative Spatial and Temporal Reasoning

Author

Renz, Jochen, Ligozat, Gérard, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, and van Beek, Peter, editor

Published
2005
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44. The closure issue related to liquid–cell mass transfer and substrate uptake dynamics in biological systems

Author

Jérôme Morchain, Vincent Quedeville, Rodney O. Fox, Philippe Villedieu, Toulouse Biotechnology Institute (TBI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Fédération de Recherche Fluides, Energie, Réacteurs, Matériaux et Transferts (FERMAT), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées, Iowa State University (ISU), ONERA / DMPE, Université de Toulouse [Toulouse], ONERA-PRES Université de Toulouse, ANR-11-IDEX-0002,UNITI,Université Fédérale de Toulouse(2011), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), and Université de Toulouse (UT)

Subjects
0106 biological sciences, [SDV.BIO]Life Sciences [q-bio]/Biotechnology, Bioengineering, Models, Biological, 01 natural sciences, Applied Microbiology and Biotechnology, Algebraic closure, Quantitative Biology::Cell Behavior, bioreactor, 03 medical and health sciences, Bioreactors, Chain (algebraic topology), 010608 biotechnology, Mass transfer, mass transfer, Computer Simulation, 030304 developmental biology, 0303 health sciences, Chemistry, Dynamics (mechanics), modeling, regulation, Substrate (marine biology), Closure (computer programming), Coupling (computer programming), uptake, Ordinary differential equation, Biological system, Biotechnology
Abstract

International audience; An original dynamic model for substrate uptake under transient conditions is established and used to simulate a variety of biological responses to external perturbations. The actual uptake and growth rates, treated as cell properties, are part of the model variables as well as the substrate concentration at the cell-liquid interface. Several regulatory loops inspired by the structure of the glycolytic chain are considered to establish a set of ordinary differential equations. The uptake rate evolves so as to reach an equilibrium between the cell demand and the environmental supply. This model does not contain any of the usual algebraic closure laws relating to the instantaneous uptake, growth rates, and the substrate concentration, nor does it enforce the continuity of mass fluxes at the liquid-cell interface. However, these relationships are found in the steady-state solution. Previously unexplained experimental observations are well reproduced by this model. Also, the model structure is suitable for further coupling with flux-based metabolic models and fluid-flow equations.

Published
2021

45. The lattice of algebraic closure operators on an infinite subgroup lattice

Author

Arturo Magidin and Martha L. H. Kilpack

Subjects
Pure mathematics, Algebra and Number Theory, Group (mathematics), High Energy Physics::Lattice, 010102 general mathematics, 010103 numerical & computational mathematics, Lattice of subgroups, 01 natural sciences, Algebraic closure, Mathematics::Group Theory, Lattice (module), 0101 mathematics, Algebraic number, Mathematics
Abstract

We say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)≅L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of algebraic cl...

Published
2021

50. Model Completeness

Author

Marcja, Annalisa, Toffalori, Carlo, Wójcicki, Ryszard, editor, Mundici, Daniele, editor, Orłowska, Ewa, editor, Priest, Graham, editor, Segerberg, Krister, editor, Urquhart, Alasdair, editor, Wansing, Heinrich, editor, Marcja, Annalisa, and Toffalori, Carlo

Published
2003
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