243 results on '"Algebraic closure"'
Search Results
2. Algebraic Closures in Divisible Rigid Groups.
- Author
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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
- Full Text
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3. On Algebraic and Definable Closures for Theories of Abelian Groups
- Author
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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.
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- 2024
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4. Punctually presented structures I: Closure theorems.
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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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5. The minimal cone of an algebraic Laurent series.
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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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6. On binomials and algebraic closure of some pseudofinite fields.
- Author
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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
- Full Text
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7. Rings and coefficients
- Author
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Emerton, Matthew, author and Gee, Toby, author
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- 2022
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8. New vectorial versions of Takahashi's nonconvex minimization problem.
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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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9. On the Representation of Fields as Finite Sums of Proper Subfields.
- Author
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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]
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- 2020
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10. Zermelo 1914 : On integral transcendental numbers
- Author
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Felgner, Ulrich, Zermelo, Ernst, Ebbinghaus, Heinz-Dieter, editor, Fraser, Craig G., editor, and Kanamori, Akihiro, editor
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- 2010
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11. Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes
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Sha Min, Mérai László, Fabrizio Barroero, Capuano Laura, Ostafe Alina, Barroero, Fabrizio, Capuano, Laura, Mérai, László, Ostafe, Alina, and Sha, Min
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Mathematics - Number Theory ,General Mathematics ,Multiplicative function ,Order (ring theory) ,Codimension ,Unlikely intersections ,Algebraic closure ,Combinatorics ,Elliptic curve ,Mathematics - Algebraic Geometry ,multiplicative dependence ,Finite field ,Unlikely intersections, multiplicative dependence, elliptic curves ,elliptic curves ,FOS: Mathematics ,Number Theory (math.NT) ,Variety (universal algebra) ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
For positive integers $K$ and $L$, we introduce and study the notion of $K$-multiplicative dependence over the algebraic closure $\overline{\mathbb{F}}_p$ of a finite prime field $\mathbb{F}_p$, as well as $L$-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions $\varphi_1,\ldots,\varphi_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $\mathbb{Z}$, for any sufficiently large prime $p$, for all but finitely many $\alpha\in\overline{\mathbb{F}}_p$, at most one of the following two can happen: $\varphi_1(\alpha),\ldots,\varphi_m(\alpha)$ are $K$-multiplicatively dependent or the points $(\varrho_1(\alpha),\cdot), \ldots,(\varrho_n(\alpha),\cdot)$ are $L$-linearly dependent on the reduction of $E$ modulo $p$. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety $\mathbb{G}_{\mathrm{m}}^m \times E^n$ with the algebraic subgroups of codimension at least $2$. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases., Comment: 32 pages. To appear in International Mathematics Research Notices
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- 2022
12. The Connes–Consani plane connection.
- Author
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Thas, Koen
- Subjects
- *
GEOMETRIC analysis , *ALGEBRAIC field theory , *MULTIPLY transitive groups , *INFINITY (Mathematics) , *GROUP theory - Abstract
Inspired by a recent paper of Alain Connes and Caterina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces (“Singer actions”), we consider several fundamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields F the plane P 2 ( F ) = PG ( 2 , F ) (and more generally the space P n ( F ) = PG ( n , F ) ) admits a Singer group, and show, e.g., that for any prime p and any positive integer n > 1 , PG ( n , F p ‾ ) cannot admit Singer groups ( F p ‾ an algebraic closure of F p ). One of the main results in characteristic 0, which is a corollary of a criterion which applies to many other fields, is that PG ( m , R ) with m ≠ 0 a positive even integer, cannot admit Singer groups. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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13. On the separability of elements and sets in hypergraphs of models of a theory
- Author
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S.V. Sudoplatov
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SEPARABILITY OF ELEMENTS ,HYPERGRAPH OF MODELS ,ALGEBRAIC CLOSURE ,Analysis ,QA299.6-433 ,Analytic mechanics ,QA801-939 ,Probabilities. Mathematical statistics ,QA273-280 - Abstract
We consider topological properties of hypergraphs of models of a theory. The separability of elements in these hypergraphs is characterized in terms of algebraic closures. Similarly we specify the separability of sets by the hypergraphs. The separability of finite sets is characterized for special hypergraphs defined by limit models as well as by countable models which are neither almost prime nor limit.
- Published
- 2016
14. Skolemizing algebraically closed universal classes of algebras.
- Author
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Quackenbush, Robert
- Subjects
- *
ALGEBRA , *ABELIAN groups , *BOOLEAN algebra , *SEMILATTICES , *DISTRIBUTIVE lattices , *VECTOR spaces - Abstract
Algebraically closed abelian groups are precisely the divisible abelian groups; the underlying groups of rational vector spaces are precisely the divisible abelian groups. That is, rational vector spaces are the skolemization of algebraically closed abelian groups, obtained by adding scalar multiplication by each rational to the set of operations. Algebraically closed bounded distributive lattices are precisely the complemented distributive lattices; the underlying lattices of boolean algebras are precisely the complemented distributive lattices. That is, boolean algebras are the skolemization of algebraically closed bounded distributive lattices, obtained by adding complementation to the set of operations. We explore this idea for arbitrary universal classes of algebras and focus particularly on the case of meet semilattices. The algebraically closed meet semilattices are precisely the distributive meet semilattices; their skolemization seems not to have been discussed previously in the literature. We discuss it here but reach no firm solution. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
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15. Solving determinantal systems using homotopy techniques
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Éric Schost, Jonathan D. Hauenstein, Mohab Safey El Din, Thi Xuan Vu, Department of Applied and Computational Mathematics and Statistics [Notre Dame], University of Notre Dame [Indiana] (UND), Polynomial Systems (PolSys), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Cheriton School of Computer Science [Waterloo] (CS), and University of Waterloo [Waterloo]
- Subjects
FOS: Computer and information sciences ,Computer Science - Symbolic Computation ,[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] ,Sequence ,Polynomial ,Algebra and Number Theory ,Homotopy ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,010103 numerical & computational mathematics ,Symbolic Computation (cs.SC) ,01 natural sciences ,Algebraic closure ,Polynomial matrix ,Combinatorics ,Computational Mathematics ,0101 mathematics ,Maxima ,Mathematics - Abstract
International audience; Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g_1,\dots,g_s)$ in $\K[X_1,\dots,X_n]$, a polynomial matrix $\F=[f_{i,j}] \in \K[X_1,\dots,X_n]^{p \times q}$, with $p \leq q$,and the algebraic set $V_p(F, G)$ of points in $\KKbar$ at which all polynomials in $\G$ and all $p$-minors of $\F$vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry.We provide bounds on the number of isolated points in $V_p(F, G)$ depending on the maxima of the degrees in rows (resp. columns) of $\F$. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining $V_p(F, G)$. In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.
- Published
- 2021
16. Achievable rate-region for $3-$User Classical-Quantum Interference Channel using Structured Codes
- Author
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Touheed Anwar Atif, Arun Padakandla, and S. Sandeep Pradhan
- Subjects
FOS: Computer and information sciences ,Quantum Physics ,Current (mathematics) ,Computer science ,Information Theory (cs.IT) ,Computer Science - Information Theory ,FOS: Physical sciences ,Topology ,Interference (wave propagation) ,Algebraic closure ,Scheme (mathematics) ,Encoding (memory) ,Coset ,Quantum Physics (quant-ph) ,Coding (social sciences) ,Communication channel - Abstract
We consider the problem of characterizing an inner bound to the capacity region of a $3-$user classical-quantum interference channel ($3-$CQIC). The best known coding scheme for communicating over CQICs is based on unstructured random codes and employs the techniques of message splitting and superposition coding. For classical $3-$user interference channels (ICs), it has been proven that coding techniques based on coset codes - codes possessing algebraic closure properties - strictly outperform all coding techniques based on unstructured codes. In this work, we develop analogous techniques based on coset codes for $3$to$1-$CQICs - a subclass of $3-$user CQICs. We analyze its performance and derive a new inner bound to the capacity region of $3$to$1-$CQICs that subsume the current known largest and strictly enlarges the same for identified examples., 18 pages. arXiv admin note: text overlap with arXiv:2103.02082
- Published
- 2021
17. On computable aspects of algebraic and definable closure
- Author
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Rehana Patel, Cameron E. Freer, and Nathanael L. Ackerman
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Discrete mathematics ,FOS: Computer and information sciences ,Computer Science - Logic in Computer Science ,Rank (linear algebra) ,Logic ,Computability ,Closure (topology) ,Structure (category theory) ,0102 computer and information sciences ,Mathematics - Logic ,01 natural sciences ,Algebraic closure ,Theoretical Computer Science ,Logic in Computer Science (cs.LO) ,Quantifier (logic) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Arts and Humanities (miscellaneous) ,010201 computation theory & mathematics ,Hardware and Architecture ,FOS: Mathematics ,Algebraic number ,Logic (math.LO) ,Software ,Mathematics - Abstract
We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $\Sigma^0_{n+2}$ sets. We further show that these bounds are tight., Comment: 20 pages
- Published
- 2021
18. STRONGLY MINIMAL STEINER SYSTEMS I: EXISTENCE
- Author
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John T. Baldwin and Gianluca Paolini
- Subjects
03C45, 51E10 ,Logic ,0102 computer and information sciences ,strongly minimal ,01 natural sciences ,Algebraic closure ,Combinatorics ,Integer ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Conjecture ,Linear space ,010102 general mathematics ,Mathematics - Logic ,Philosophy ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Steiner system ,010201 computation theory & mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Hrushovski construction ,Logic (math.LO) ,Trichotomy (mathematics) ,Counterexample - Abstract
A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for $k \geq 2$ ) is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary $\tau $ with a single ternary relation R. We prove that for every integer k there exist $2^{\aleph _0}$ -many integer valued functions $\mu $ such that each $\mu $ determines a distinct strongly minimal Steiner k-system $\mathcal {G}_\mu $ , whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.
- Published
- 2021
19. Homotopy techniques for solving sparse column support determinantal polynomial systems
- Author
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Thi Xuan Vu, George Labahn, Mohab Safey El Din, Éric Schost, Symbolic Computation Group (SCG), University of Waterloo [Waterloo]-David R. Cheriton School of Computer Science, Polynomial Systems (PolSys), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE33-0011,SESAME,Singularités Et Stabilité des AsservisseMEnts référencés capteurs(2018), ANR-19-CE48-0015,ECARP,Algorithmes efficaces et exacts pour la planification de trajectoire en robotique(2019), ANR-19-CE40-0018,DeRerumNatura,Décider l'irrationalité et la transcendance(2019), European Project: 813211,H2020,POEMA(2019), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), and H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019)
- Subjects
Computer Science - Symbolic Computation ,Statistics and Probability ,FOS: Computer and information sciences ,[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] ,Numerical Analysis ,Polynomial ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,010103 numerical & computational mathematics ,Symbolic Computation (cs.SC) ,01 natural sciences ,Algebraic closure ,Polynomial matrix ,Combinatorics ,Symmetric group ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F} = [f_{i,j}] \in \mathbf{K}[x_1, \ldots, x_n]^{p \times q}$, with $p \leq q$, we are interested in determining the isolated points of $V_p(\mathbf{F},\mathbf{g})$, the algebraic set of points in $\overline{\mathbf{K}}$ at which all polynomials in $\mathbf{g}$ and all $p$-minors of $\mathbf{F}$ vanish, under the assumption $n = q - p + s + 1$. Such polynomial systems arise in a variety of applications including for example polynomial optimization and computational geometry. We design a randomized sparse homotopy algorithm for computing the isolated points in $V_p(\mathbf{F},\mathbf{g})$ which takes advantage of the determinantal structure of the system defining $V_p(\mathbf{F}, \mathbf{g})$. Its complexity is polynomial in the maximum number of isolated solutions to such systems sharing the same sparsity pattern and in some combinatorial quantities attached to the structure of such systems. It is the first algorithm which takes advantage both on the determinantal structure and sparsity of input polynomials. We also derive complexity bounds for the particular but important case where $\mathbf{g}$ and the columns of $\mathbf{F}$ satisfy weighted degree constraints. Such systems arise naturally in the computation of critical points of maps restricted to algebraic sets when both are invariant by the action of the symmetric group.
- Published
- 2021
20. Computing in degree $$2^k$$ -extensions of finite fields of odd characteristic.
- Author
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Doliskani, Javad and Schost, Éric
- Subjects
FINITE fields ,SYMBOLISM in communication ,DATA encryption ,CRYPTOGRAPHY ,COMPUTER security software - Abstract
We show how to perform basic operations (arithmetic, square roots, computing isomorphisms) over finite fields of the form $$\mathbb F _{q^{2^k}}$$ in essentially linear time. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
21. Exploring Novel Cyclic Extensions of Hamilton's Dual-Quaternion Algebra.
- Author
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AMOROSO, RICHARD L., ROWLANDS, PETER, and KAUFFMAN, LOUIS H.
- Subjects
QUATERNIONS ,DIRAC equation ,NILPOTENT groups ,IDEMPOTENTS ,MATHEMATICAL physics - Published
- 2013
22. An explicit algebraic closure for passive scalar-flux: Applications in channel flows at a wide range of reynolds numbers
- Author
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Evangelos Akylas, Fotos Stylianou, Constantine Michailides, Constantinos F. Panagiotou, and Elias Gravanis
- Subjects
RANS ,Scalar (mathematics) ,Prandtl number ,Ocean Engineering ,Reynolds stress ,01 natural sciences ,Algebraic closure ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,lcsh:Oceanography ,symbols.namesake ,lcsh:VM1-989 ,0103 physical sciences ,lcsh:GC1-1581 ,0101 mathematics ,Water Science and Technology ,Civil and Structural Engineering ,Mathematics ,Turbulence ,Algebraic model ,Mathematical analysis ,lcsh:Naval architecture. Shipbuilding. Marine engineering ,Reynolds number ,channel flows ,high Reynolds numbers ,Channel flows ,Biological Sciences ,Hagen–Poiseuille equation ,High Reynolds numbers ,010101 applied mathematics ,scalar-flux ,algebraic model ,symbols ,Computer Science::Programming Languages ,Reynolds-averaged Navier–Stokes equations ,Natural Sciences ,Scalar-flux - Abstract
In this paper, we propose an algebraic model for turbulent scalar-flux vector that stems from tensor representation theory. The resulting closure contains direct dependence on mean velocity gradients and quadratic products of the Reynolds stress tensor. Model coefficients are determined from Direct Numerical Simulations (DNS) data of homogeneous shear flows subjected to arbitrary mean scalar gradient orientations, while a correction function was applied at one model coefficient based on a turbulent channel flow case. Model performance is evaluated in Poiseuille and Couette flows at several Reynolds numbers for Pr=0.7, along with a case at a higher Prandtl number (Pr=7.0) that typically occurs in water&ndash, boundary interaction applications. Overall, the proposed model provides promising results for wide near-wall interaction applications. To put the performance of the proposed model into context, we compare with Younis algebraic model, which is known to provide reasonable predictions for several engineering flows.
- Published
- 2020
23. Tractable Fragments of Temporal Sequences of Topological Information
- Author
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Quentin Cohen-Solal, Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision (LAMSADE), Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
050101 languages & linguistics ,Pure mathematics ,Relation (database) ,Topological information ,05 social sciences ,Context (language use) ,02 engineering and technology ,16. Peace & justice ,Algebraic closure ,Satisfiability ,Universal relation ,law.invention ,law ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0501 psychology and cognitive sciences ,Cartesian coordinate system ,[INFO]Computer Science [cs] ,Focus (optics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this paper, we focus on qualitative temporal sequences of topological information. We firstly consider the context of topological temporal sequences of length greater than 3 describing the evolution of regions at consecutive time points. We show that there is no Cartesian subclass containing all the basic relations and the universal relation for which the algebraic closure decides satisfiability. However, we identify some tractable subclasses, by giving up the relations containing the non-tangential proper part relation and not containing the tangential proper part relation.
- Published
- 2020
24. On the Skolem problem and some related questions for parametric families of linear recurrence sequences
- Author
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Igor E. Shparlinski and Alina Ostafe
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Sequence ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Rational function ,Algebraic number field ,01 natural sciences ,Algebraic closure ,Combinatorics ,Bounded function ,0103 physical sciences ,Skolem problem ,FOS: Mathematics ,010307 mathematical physics ,Number Theory (math.NT) ,0101 mathematics ,Parametric family ,Mathematics - Abstract
We show that in a parametric family of linear recurrence sequences $a_1(\alpha ) f_1(\alpha )^n + \cdots + a_k(\alpha ) f_k(\alpha )^n$ with the coefficients $a_i$ and characteristic roots $f_i$ , $i=1, \ldots ,k$ , given by rational functions over some number field, for all but a set of elements $\alpha $ of bounded height in the algebraic closure of ${\mathbb Q}$ , the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.
- Published
- 2020
25. On the complexity of finding tensor ranks
- Author
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Mohsen Aliabadi and Shmuel Friedland
- Subjects
Rank (linear algebra) ,Field (mathematics) ,Algebraic closure ,Combinatorics ,Integer ,Tensor (intrinsic definition) ,Linear algebra ,FOS: Mathematics ,General Earth and Planetary Sciences ,Computational Science and Engineering ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,General Environmental Science ,Mathematics - Abstract
The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $$\mathbb {F}$$ is at most k over the algebraic closure of $$\mathbb {F}$$ , where k is a given positive integer. We estimate the arithmetic complexity of our algorithm.
- Published
- 2020
26. A new proof of the ultrametric hermite-lindermann theorem
- Author
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Alain Escassut, Escassut, Alain, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Power series ,Hermite polynomials ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Algebraic closure ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,0103 physical sciences ,Integral element ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Ultrametric space ,Mathematics ,Analytic function ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience; We propose a new proof of the Hermite-Lindeman Theorem in an ultrametric field by using classical properties of analytic functions. The proof remains valid in zero residue characteristic. Definitions and notations: The Archimedean absolute value of C is denoted by |. | ∞. Let IK be an algebraically closed complete ultrametric field of characteristic 0 and residue characteristic p. We denote by |. | the ultrametric absolute value of IK and we denote by Ω an algebraic closure of Q in IK. Given a ∈ IK and r > 0, we denote by d(a, r −) the disk {x ∈ IK |x−a| < r}, we denote by A(d(a, r −)) the IK-Banach algebra of power series f (x) = ∞ n=0 a n (x − a) n converging in d(a, r −) and for all s ∈]0, r[, we put |f |(s) = sup n∈IN |a n |r n. Given a ∈ Ω, we call denominator of a any strictly positive integer n such that na is integral over Z and we denote by den(a) the smallest denominator of a. Next, considering the conjugates a 2 , ..., a n of a over Q and putting a 1 = a, we put |a| = max{|a 1 | ∞ , ..., |a n | ∞ }. Next, we put s(a) = Log(max(|a|, den(a)).
- Published
- 2020
27. The Golomb topology on a Dedekind domain and the group of units of its quotients
- Author
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Dario Spirito
- Subjects
Golomb space ,Mathematics::Number Theory ,Prime ideal ,Dedekind domain ,Commutative Algebra (math.AC) ,Topology ,01 natural sciences ,Algebraic closure ,Computer Science::Discrete Mathematics ,FOS: Mathematics ,Homeomorphism problem ,Dedekind cut ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - General Topology ,Mathematics ,Mathematics - Number Theory ,010102 general mathematics ,General Topology (math.GN) ,Dedekind domains ,Mathematics - Commutative Algebra ,Homeomorphism ,010101 applied mathematics ,Golomb coding ,Torsion (algebra) ,Geometry and Topology ,Partially ordered set - Abstract
We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the P-adic topology on R ∖ P . Under certain hypothesis, we show that we can associate to a prime ideal P of R a partially ordered set, constructed from some subgroups of the group of units of R / P n , which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of Z are the identity and the multiplication by −1. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of Q and different from Z is not homeomorphic to the Golomb space of Z .
- Published
- 2020
28. On Fuzzy Deductive Systems of Hilbert Algebras
- Author
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Gezahagne Mulat Addis and Derso Abeje Engidaw
- Subjects
0209 industrial biotechnology ,Class (set theory) ,Article Subject ,Mathematics::General Mathematics ,General Mathematics ,Fuzzy set ,02 engineering and technology ,Fuzzy logic ,Algebraic closure ,Algebra ,020901 industrial engineering & automation ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Complete lattice ,Distributive property ,Truth value ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0202 electrical engineering, electronic engineering, information engineering ,QA1-939 ,020201 artificial intelligence & image processing ,ComputingMethodologies_GENERAL ,Variety (universal algebra) ,Mathematics - Abstract
In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.
- Published
- 2020
- Full Text
- View/download PDF
29. GENERALIZED CONVEXITY AND CLOSURE CONDITIONS.
- Author
-
CZÉDLI, GÁBOR and ROMANOWSKA, ANNA B.
- Subjects
- *
CONVEX domains , *CLOSURE of functions , *SET theory , *GENERALIZATION , *REAL numbers , *ALGEBRAIC field theory - Abstract
Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
30. On n-dependent groups and fields II
- Author
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Nadja Hempel and Artem Chernikov
- Subjects
Statistics and Probability ,Pure mathematics ,Absoluteness ,Structure (category theory) ,Field (mathematics) ,0102 computer and information sciences ,Group Theory (math.GR) ,Bilinear form ,Commutative Algebra (math.AC) ,01 natural sciences ,Algebraic closure ,Theoretical Computer Science ,03C45 ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,math.GR ,0101 mathematics ,03C60 ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,Conjecture ,010102 general mathematics ,Mathematics - Logic ,Mathematics - Commutative Algebra ,math.AC ,Computational Mathematics ,math.LO ,Geometric group theory ,010201 computation theory & mathematics ,03C45, 03C60 ,Geometry and Topology ,Isomorphism ,Logic (math.LO) ,Mathematics - Group Theory ,Analysis - Abstract
We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing that every infinite $n$-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah's Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in $n$-dependent groups, generalizing Shelah's absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$-dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$-dependent fields with additional structure, showing that Granger's examples of non-degenerate bilinear forms over dependent fields are $2$-dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that $n$-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$-dependence and use it to deduce $2$-dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory $T$ by a generic predicate is dependent if and only if it is $n$-dependent for some $n$, if and only if the algebraic closure in $T$ is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem., v.3: 52 pages; the presentation was thoroughly revised; the order of the sections was changed; many proofs were expanded with additional details and clarifications; minor corrections throughout the article; accepted to Forum of Mathematics, Sigma
- Published
- 2019
31. Conical Embeddings of Steiner systems
- Author
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Scott A. Vanstone and Dieter Jungnickel
- Subjects
Combinatorics ,Steiner system ,Conic section ,General Mathematics ,Point set ,Order (group theory) ,Conical surface ,Algebra over a field ,Algebraic closure ,Mathematics - Abstract
Any classicalS(3,2 a +1;2 ab +1) is embedded intoPG(2,2 ab ) as point set one may use any conic, the blocks being determined by subplanes of order 2 a . Consequently, every classicalS(3,2 a +1;2 ab +1) is naturally embedded intoPG(2,K) whereK is the algebraic closure ofGF(2).
- Published
- 2019
32. An algebraic closure for barycentric algebras and convex sets.
- Author
-
Czédli, Gábor and Romanowska, A.
- Subjects
- *
ALGEBRA , *CONVEX sets , *BINARY number system , *IDEMPOTENTS , *LINEAR algebra - Abstract
Let A be an algebra (of an arbitrary finitary type), and let γ be a binary term. A pair (a, b) of elements of A will be called a γ- eligible pair if for each x in the subalgebra generated by {a, b} such that x is distinct from a there exists an element y in A such that b = xyγ. We say that A is a γ- closed algebra if for each γ-eligible pair (a, b) there is an element c with b = acγ. We call A a closed algebra if it is γ-closed for all binary terms γ that do not induce a projection. Let T be a unital subring of the field of real numbers. Equipped with all the binary operations $${(x, y) \mapsto (1- p)x+py}$$ for $${p \in T}$$ and 0 < p < 1, T becomes a mode, that is, an idempotent algebra in which any two term functions commute. In fact, the mode T is a (generalized) barycentric algebra. Let $${\mathcal{Q}(T)}$$ denote the quasivariety generated by this mode. Our main theorem asserts that each mode of $${\mathcal{Q}(T)}$$ extends to a minimal closed cancellative mode, which is unique in a reasonable sense. In fact, we prove a slightly stronger statement. As corollaries, we obtain a purely algebraic description of the usual topological closure of convex sets, and we exemplify how to use the main theorem to show that certain open convex sets are not isomorphic. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
33. Extending Nathanson Heights to Arbitrary Finite Fields.
- Author
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Huicochea, Mario
- Subjects
- *
PROJECTIVE spaces , *FINITE fields , *ALGEBRAIC fields , *ALGEBRA , *GEOMETRIC congruences , *MATHEMATICS - Abstract
In this paper, we extend the definition of the Nathanson height from points in projective spaces over to points in projective spaces over arbitrary finite fields. If , then the Nathanson height is where with the field norm and the element of congruent to modulo p. We investigate the basic properties of this extended height, provide some bounds, study its image on the projective line and propose some questions for further research. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
34. RANKS OF RELATIVE-UNIT-GROUPS RELATED TO redset(f).
- Author
-
OGLE, JACOB and Abhyankar, S.
- Subjects
- *
RANKING (Statistics) , *GROUP theory , *SET theory , *POLYNOMIALS , *ALGEBRAIC fields , *PROOF theory , *FACTORIZATION - Abstract
Given an irreducible polynomial f in k[X1,..., Xn] (where k is a field) such that k is algebraically closed in the quotient field of A ≔ k[X1,...,Xn]/f k[X1,...,Xn], we show that k(f) is algebraically closed in k(X1,...,Xn). Further, if n ≥ 2 and char k = 0, then we show that the number of k-translates of f that are reducible in k[X1,..., Xn] is bounded above by the rank of U(A)/U(k). Finally, we prove a similar bound for the number of reducible composites of the form Γ(f) with Γ ∈ k[T] monic irreducible. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
35. Sums of two square-zero matrices over an arbitrary field
- Author
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Botha, J.D.
- Subjects
- *
ALGEBRAIC fields , *MATRICES (Mathematics) , *MATHEMATICAL singularities , *NILPOTENT groups , *MATHEMATICAL constants , *CHARACTERISTIC functions - Abstract
Abstract: The problem to express an matrix A as the sum of two square-zero matrices was first investigated by Wang and Wu for matrices over the complex field. This paper investigates the problem over an arbitrary field F. It is shown that, if char, then is the sum of two square-zero matrices if and only if A is similar to a matrix of the form , where N is nilpotent, X is nonsingular, and each is a companion matrix associated with an even-power poly nomial with nonzero constant term. If F is of characteristic two, the term falls away. If F is of characteristic zero and algebraically closed, the term falls away and the result of Wang and Wu is obtained. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
36. Theory of equivalence systems for describing algebraic closures of a generalized estimation model. II.
- Author
-
D'yakonov, A.
- Abstract
Characteristic matrices and metrics of equivalence systems are studied that help give an efficient description of conjunctions of equivalence systems. Using these results, families of correct polynomials in the algebraic approach to classification are described. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
37. The Markov–Zariski topology of an abelian group
- Author
-
Dikranjan, Dikran and Shakhmatov, Dmitri
- Subjects
- *
ALGEBRAIC topology , *ABELIAN groups , *MARKOV processes , *MATHEMATICAL forms , *SET theory , *HAUSDORFF compactifications , *NOETHERIAN rings , *SEPARABLE algebras - Abstract
Abstract: According to Markov (1946) , a subset of an abelian group G of the form , for some integer n and some element , is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally bounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian topology on G called the Zariski, or verbal, topology of G; see Bryant (1977) . We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Fréchet–Urysohn. For a countable family of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology on G such that the -closure of each member of coincides with its -closure. As an application, we provide a characterization of the subsets of G that are -dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov (1946) . [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
38. A Priori Direct Numerical Simulation Assessment of Algebraic Models of Variances and Dissipation Rates in the Context of Reynolds-Averaged Navier-Stokes Simulations for Low Damkohler Number Partially Premixed Combustion.
- Author
-
Malkeson, Sean P. and Chakraborty, Nilanjan
- Subjects
STATISTICS ,TURBULENCE ,NUMERICAL analysis ,SIMULATION methods & models ,FLAME - Abstract
Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensity are simulated for global equivalence ratio 〈φ〉 = 0.7 and 1.0 using three-dimensional simplified chemistry based Direct Numerical Simulations (DNS). For the simulations of partially premixed flames a bimodal distribution of equivalence ratio variation about the prescribed value of 〈φ〉 is introduced in the fresh reactants. The simulation parameters are chosen in such a manner that the combustion situation in all the cases represents the thin reaction zones regime with global Damkohler number smaller than unity. The DNS data has been used to analyze the statistics of the variances [image omitted], covariances [image omitted] (where Y, ξ and c are the fuel mass fraction, mixture fraction, and reaction progress variable, respectively, and tilde and double prime represent the Favre mean and Favre fluctuation of the relevant quantities, respectively), scalar dissipation rates (i.e., [image omitted] and [image omitted]) of active scalar variances, and the cross-scalar dissipation rates ([image omitted] and [image omitted]) of the covariances of active scalar and mixture fraction ξ fluctuations in the context of Reynolds-Averaged Navier-Stokes (RANS) simulations. The performances of different algebraic models for the variances, covariances, scalar dissipation rate of active scalars, and cross-scalar dissipation rates have been assessed with respect to the corresponding values obtained from the DNS database. It has been found that root mean square turbulence velocity fluctuation u' and global equivalence ratio 〈φ〉 have significant effects on the statistics of [image omitted]. The authors found that the maximum values of [image omitted] increase with increasing u' and 〈φ〉 values. Moreover, the modeling parameters of the algebraic models for the quantities [image omitted], [image omitted] and [image omitted] show significant u' and 〈φ〉 dependence. Based on the a priori DNS assessment, the algebraic models for [image omitted], which give rise to satisfactory agreement with the corresponding quantities obtained from DNS without any change in model parameters in response to the changes in u' and 〈φ〉, have been identified. It has been found that none of the existing algebraic models for [image omitted] and [image omitted] are capable of predicting corresponding quantities extracted from the DNS data for the complete set of u' and 〈φ〉 values analyzed in the present study. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
39. Computing with algebraically closed fields
- Author
-
Steel, Allan K.
- Subjects
- *
ALGEBRAIC fields , *NUMERICAL calculations , *FACTORIZATION , *SYSTEMS theory , *FIELD extensions (Mathematics) , *MATHEMATICAL optimization , *POLYNOMIALS - Abstract
Abstract: A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
40. Theory of equivalence systems for describing the algebraic closures of a generalized estimation model.
- Author
-
D’yakonov, A.
- Abstract
The recognition problem is considered in which the initial information is given by the values of similarity functions on pairs of objects. A generalization of the estimation algorithm model for this problem is proposed. A theory for the description and analysis of algebraic closures of the generalized and classical models is developed. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
41. Estimates for the orders of zeros of polynomials in some analytic functions.
- Author
-
Dolgalev, A. P.
- Subjects
- *
POLYNOMIALS , *ANALYTIC functions , *DIFFERENTIAL equations , *ALGEBRAIC independence , *ALGEBRAIC functions , *ALGEBRA , *MATHEMATICAL analysis - Abstract
In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
42. Mixing and linear equations over groups in positive characteristic.
- Author
-
Masser, D. W.
- Abstract
We prove a result on linear equations over multiplicative groups in positive characteristic. This is applied to settle a conjecture about higher order mixing properties of algebraicZ
d -actions. [ABSTRACT FROM AUTHOR]- Published
- 2004
- Full Text
- View/download PDF
43. Algebraic closure of a rational function.
- Author
-
Ollagnier, Jean
- Abstract
We give a simple algorithm to decide if a non-constant rational fraction R=P/Q in the field $$\mathbb{K}(x) = \mathbb{K}(x_1 ,...,x_n )$$ in n≥2 variables over a field K of characteristic 0 can be written as a non-trivial composition R=U(R
1 ), where R1 is another n-variable rational fraction whereas U is a one-variable rational fraction which is not a homography. More precisely, this algorithm produces a generator of the algebraic closure of a rational fraction in the field K x. Although our algorithm is simple (it uses only elementary linear algebra), its proof relies on a structure theorem: the algebraic closure of a rational fraction is a purely transcendental extension of K of transcendence degree 1. Despite this theorem is a generalization of a result of Poincaré about the rational first integrals of polynomial planar vector fields, we found it useful to give a complete proof of it: our proof is as algebraic as possible and thus very different from Poincaré's original work. [ABSTRACT FROM AUTHOR]- Published
- 2004
- Full Text
- View/download PDF
44. Computability-theoretic and proof-theoretic aspects of partial and linear orderings.
- Author
-
Downey, Rodney G., Hirschfeldt, Denis R., Lempp, Steffen, and Solomon, Reed
- Abstract
Szpilrajn’s Theorem states that any partial orderP=〈S,
- Published
- 2003
- Full Text
- View/download PDF
45. Equivalence of differential equations of order one
- Author
-
Jakob Top, Khuong An Nguyen, L. X. Chau Ngo, M. van der Put, and Algebra
- Subjects
Discrete mathematics ,Algebra and Number Theory ,34M15, 34M35, 34M55 ,Algebraic closure ,Algebraic element ,Computational Mathematics ,Mathematics - Algebraic Geometry ,Algebraic surface ,FOS: Mathematics ,Algebraic curve ,Algebraically closed field ,Differential algebraic geometry ,Algebraic Geometry (math.AG) ,Equivalence (measure theory) ,Differential algebraic equation ,Mathematics - Abstract
The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having the Painleve property. The order one equation determines an algebraic curve. If this curve has genus zero or one, then it is difficult to verify strict equivalence. However, for higher genus strict equivalence can be tested by an algorithm sketched in the text. For autonomous equations, testing strict equivalence and the existence of algebraic solutions are shown to be algorithmic., 19 pages
- Published
- 2015
46. On algebraic automorphisms and their rational invariants
- Author
-
Philippe Bonnet
- Subjects
Linear algebraic group ,Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Function field of an algebraic variety ,Field (mathematics) ,Transcendence degree ,14R10, 14R20 ,Automorphism ,Algebraic closure ,Mathematics - Algebraic Geometry ,Algebraic group ,FOS: Mathematics ,Geometry and Topology ,Algebraically closed field ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F) be the transcendence degree of k(X)^F over k. In this paper, we study the class of automorphisms F of X for which n(F)= dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form F=A_g, where A is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(F)=1., 13 pages
- Published
- 2018
47. Hypersurface model-fields of definition for smooth hypersurfaces and their twists
- Author
-
Francesc Bars and Eslam Badr
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Hypersurface models ,Mathematics::Complex Variables ,Dimension (graph theory) ,Fields of definition ,Algebraic closure ,Mathematics - Algebraic Geometry ,Hypersurface ,Diagonal matrix ,FOS: Mathematics ,Perfect field ,Automorphism groups ,Twists ,Number Theory (math.NT) ,Mathematics::Differential Geometry ,Variety (universal algebra) ,Twist ,Algebraic Geometry (math.AG) ,Projective variety ,Mathematics - Abstract
Given a smooth projective variety of dimension n − 1 ≥ 1 defined over a perfect field k that admits a non-singular hypersurface model in Pnk− over k−, a fixed algebraic closure of k, it does not necessarily have a non-singular hypersurface model defined over the base field k. We first show an example of such phenomenon: a variety defined over k admitting non-singular hypersurface models but none defined over k. We also determine under which conditions a non-singular hypersurface model over k may exist. Now, even assuming that such a smooth hypersurface model exists, we wonder about the existence of non-singular hypersurface models over k for its twists. We introduce a criterion to characterize twists possessing such models and we also show an example of a twist not admitting any non-singular hypersurface model over k, i.e. for any n ≥ 2, there is a smooth projective variety of dimension n − 1 over k which is a twist of a smooth hypersurface variety over k, but itself does not admit any non-singular hypersurface model over k. Finally, we obtain a theoretical result to describe all the twists of smooth hypersurfaces with cyclic automorphism group having a model defined over k whose automorphism group is generated by a diagonal matrix. The particular case n = 2 for smooth plane curves was studied by the authors jointly with E. Lorenzo García in [Math. Comp. 88 (2019)], and we deal here with the problem in higher dimensions.
- Published
- 2018
48. Algorithms for orbit closure separation for invariants and semi-invariants of matrices
- Author
-
Visu Makam and Harm Derksen
- Subjects
FOS: Computer and information sciences ,matrix semi-invariants ,68W30 ,Field (mathematics) ,Computational Complexity (cs.CC) ,Commutative Algebra (math.AC) ,01 natural sciences ,Algebraic closure ,Group action ,null cone ,matrix invariants ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,14L24 ,13A50, 14L24, 68W20 ,Time complexity ,Mathematics ,Algebra and Number Theory ,Degree (graph theory) ,orbit closure intersection ,010102 general mathematics ,13A50 ,Mathematics - Rings and Algebras ,Mathematics - Commutative Algebra ,Computer Science - Computational Complexity ,Finite field ,Rings and Algebras (math.RA) ,Invariants of tensors ,separating invariants ,010307 mathematical physics ,Orbit (control theory) ,Algorithm - Abstract
We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for the degree of separating invariants in these cases., Better bounds for separating invariants and improved exposition in some sections
- Published
- 2018
49. Pr\'ufer intersection of valuation domains of a field of rational functions
- Author
-
Giulio Peruginelli
- Subjects
Ring (mathematics) ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,residually transcendental extension ,010102 general mathematics ,pseudo-convergent sequence ,Field (mathematics) ,Rational function ,Mathematics - Rings and Algebras ,pseudo-limit ,Mathematics - Commutative Algebra ,Primary 13F05, Secondary 13F20, 13A18 ,Prufer domain, pseudo-convergent sequence, pseudo-limit, residually transcendental extension, integer-valued polynomial ,01 natural sciences ,Algebraic closure ,Prufer domain ,010101 applied mathematics ,Combinatorics ,Prüfer domain ,Domain (ring theory) ,integer-valued polynomial ,0101 mathematics ,Quotient ,Mathematics ,Valuation (algebra) - Abstract
Let $V$ be a rank one valuation domain with quotient field $K$. We characterize the subsets $S$ of $V$ for which the ring of integer-valued polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ is a Pr\"ufer domain. The characterization is obtained by means of the notion of pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that ${\rm Int}(S,V)$ is Pr\"ufer if and only if no element of the algebraic closure $\overline{K}$ of $K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with respect to some extension of $V$ to $\overline{K}$. This result expands a recent result by Loper and Werner., Comment: to appear in J. Algebra. All comments are welcome. Keywords: Pr\"ufer domain, pseudo-convergent sequence, pseudo-limit, residually transcendental extension, integer-valued polynomial
- Published
- 2017
50. Computing in degree 2 k -extensions of finite fields of odd characteristic
- Author
-
Doliskani, Javad and Schost, Éric
- Published
- 2015
- Full Text
- View/download PDF
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