1,735 results on '"Dedekind cut"'
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52. Relationship of Essentially Small Quasi-Dedekind Modules with Scalar and Multiplication Modules
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Darya Jabar AbdulKareem, Mukdad Qaess Hussain, and Inas Salman Obaid
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Physics ,Pure mathematics ,General Computer Science ,Mathematics::Number Theory ,Scalar (mathematics) ,Dedekind cut ,General Chemistry ,General Biochemistry, Genetics and Molecular Biology - Abstract
Let be a ring with 1 and D is a left module over . In this paper, we study the relationship between essentially small quasi-Dedekind modules with scalar and multiplication modules. We show that if D is a scalar small quasi-prime -module, thus D is an essentially small quasi-Dedekind -module. We also show that if D is a faithful multiplication -module, then D is an essentially small prime -module iff is an essentially small quasi-Dedekind ring.
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- 2020
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53. Another regular Menon-type identity in residually finite Dedekind domains
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Ch. Ji and Y. Wang
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Pure mathematics ,Mathematics::General Mathematics ,Mathematics::Number Theory ,General Mathematics ,media_common.quotation_subject ,Mathematics::History and Overview ,010102 general mathematics ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Regular extension ,Identity (philosophy) ,Dedekind cut ,0101 mathematics ,media_common ,Mathematics - Abstract
We give a regular extension of the Menon-type identity to residually finite Dedekind domains.
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- 2020
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54. Linkage of Ideals in Integral Domains
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Abdeslam Mimouni and Salah-Eddine Kabbaj
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Noetherian ,Pure mathematics ,Property (philosophy) ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,0211 other engineering and technologies ,A domain ,021107 urban & regional planning ,02 engineering and technology ,Linkage (mechanical) ,01 natural sciences ,law.invention ,If and only if ,law ,Domain (ring theory) ,Dedekind cut ,0101 mathematics ,Valuation (algebra) ,Mathematics - Abstract
In this paper, we investigate the linkage of ideals, in Noetherian and non-Noetherian settings, with the aim to establish new characterizations of classical notions of domains through linkage theory. Two main results assert that a Noetherian domain is Dedekind if and only if it has the primary linkage property; and a domain is almost Dedekind (resp., Prufer) if and only if it has the linkage (resp., finite linkage) property. Also, we prove that a finite-dimensional valuation domain is a DVR (i.e., Noetherian) if and only if it has the primary linkage property.
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- 2020
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55. On secondary and representable modules over almost Dedekind domains
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Esmaeil Rostami, N. Pakyari, and Reza Nekooei
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Noetherian ring ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Direct sum ,Mathematics::Rings and Algebras ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Injective module ,Injective function ,Condensed Matter::Superconductivity ,Condensed Matter::Statistical Mechanics ,Dedekind cut ,0101 mathematics ,Mathematics::Representation Theory ,Indecomposable module ,Commutative property ,Mathematics - Abstract
Matlis showed that an injective module over a commutative Noetherian ring R can be completely decomposed as a direct sum of indecomposable injective submodules. In this paper, we prove the Matlis’ ...
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- 2020
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56. The Largest Group Contained in the Order Completion of a Totally Ordered Group
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Adrialy Muci and Elena Olivos
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body regions ,Combinatorics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Order (group theory) ,Dedekind cut ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Mathematics - Abstract
For a totally ordered group G we determine the largest group contained in its Dedekind completion G#. It was the result of studying the family of convex subgroups of G and some well-known properties of ordered groups.
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- 2020
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57. Nonperiodic Locally Soluble Groups with Non-Dedekind Locally Nilpotent Norm of Decomposable Subgroups
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T. D. Lukashova and F. M. Lyman
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010101 applied mathematics ,Mathematics::Group Theory ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Locally nilpotent ,Dedekind cut ,Norm (social) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We study the relationships between the properties of nonperiodic groups and the norms of their decomposable subgroups. The influence of restrictions imposed on the norms of decomposable subgroups and on the properties of the group is analyzed under the condition that this norm is non-Dedekind and locally nilpotent. We also describe the structure of nonperiodic locally soluble groups for which the norm of decomposable subgroups possesses the indicated properties.
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- 2020
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58. Analysis of the RSA-cryptosystem in abstract number rings
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Nikita V. Kondratyonok
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Statistics and Probability ,Discrete mathematics ,Polynomial ,Algebra and Number Theory ,dedekind ring ,Basis (linear algebra) ,business.industry ,lcsh:Mathematics ,ideal ,abstract number ring ,Cryptography ,lcsh:QA1-939 ,rsa-cryptosystem ,Computational Theory and Mathematics ,Factorization ,factorization ,Exponent ,Discrete Mathematics and Combinatorics ,Cryptosystem ,Dedekind cut ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,business ,Computer Science::Cryptography and Security ,Mathematics ,Quantum computer - Abstract
Quantum computers can be a real threat to some modern cryptosystems (such as the RSA-cryptosystem). The analogue of the RSA-cryptosystem in abstract number rings is not affected by this threat, as there are currently no factorization algorithms using quantum computing for ideals. In this paper considered an analogue of RSA-cryptosystem in abstract number rings. Proved the analogues of theorems related to its cryptographic strength. In particular, an analogue of Wiener’s theorem on the small secret exponent is proved. The analogue of the re-encryption method is studied. On its basis the necessary restrictions on the parameters of the cryptosystem are obtained. It is also shown that in numerical Dedekind rings the factorization problem is polynomial equivalent to factorization in integers.
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- 2020
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59. Rad-Discrete Modules
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Ergül Türkmen, Hasan Hüseyin Ökten, and Burcu Nişancı Türkmen
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Pure mathematics ,Noetherian ring ,Physics::Instrumentation and Detectors ,Direct sum ,Generalization ,Physics::Optics ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,010201 computation theory & mathematics ,If and only if ,0202 electrical engineering, electronic engineering, information engineering ,Physics::Accelerator Physics ,Pharmacology (medical) ,Dedekind cut ,Commutative property ,Astrophysics::Galaxy Astrophysics ,Mathematics - Abstract
We introduce Rad-discrete and quasi-Rad-discrete modules as a proper generalization of (quasi) discrete modules, and provide various properties of these modules. We prove that a direct summand of a (quasi) Rad-discrete module is (quasi) Rad-discrete. We show that every projective R-module is (quasi) Rad-discrete if and only if R is left perfect. We also prove that, over a commutative Noetherian ring R, every quasi-Rad-discrete R-module is the direct sum of local R-modules if and only if R is Artinian. Finally, we investigate self-projective Rad-discrete modules and $$\pi $$ -projective quasi-Rad-discrete modules over Dedekind domains.
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- 2020
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60. A characterization of large Dedekind domains
- Author
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Greg Oman
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Mathematics::Commutative Algebra ,Mathematics::General Mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Mathematics::General Topology ,Dedekind domain ,Characterization (mathematics) ,01 natural sciences ,Combinatorics ,Mathematics::Logic ,Cardinality ,0103 physical sciences ,Domain (ring theory) ,Dedekind cut ,010307 mathematical physics ,Ideal (ring theory) ,0101 mathematics ,Commutative property ,Mathematics - Abstract
Let D be a commutative domain with identity, and let $${\mathcal {L}}(D)$$ be the lattice of nonzero ideals of D. Say that D is ideal upper finite provided $${\mathcal {L}}(D)$$ is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D. Now let $$\kappa >2^{\aleph _0}$$ be a cardinal. We show that a domain D of cardinality $$\kappa $$ is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if $$\kappa $$ is a cardinal such that $$\aleph _0\le \kappa \le 2^{\aleph _0}$$ , then there is an ideal upper finite domain of cardinality $$\kappa $$ which is not Dedekind.
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- 2020
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61. ${P}$-adic approximation of Dedekind sumsin function fields
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Yoshinori Hamahata
- Subjects
Pure mathematics ,General Mathematics ,Dedekind cut ,Function (mathematics) ,Mathematics - Published
- 2021
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62. Formalization of the Equivalence among Completeness Theorems of Real Number in Coq
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Wensheng Yu and Yaoshun Fu
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formalization ,analysis ,General Mathematics ,0102 computer and information sciences ,real number theory ,01 natural sciences ,Formal proof ,Compactness theorem ,Computer Science (miscellaneous) ,Coq ,Dedekind cut ,Gödel's completeness theorem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,Real number ,Fundamental theorem ,lcsh:Mathematics ,010102 general mathematics ,Monotone convergence theorem ,lcsh:QA1-939 ,Algebra ,Automated theorem proving ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,010201 computation theory & mathematics ,completeness theorems - Abstract
The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau&rsquo, s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.
- Published
- 2021
63. Lattices with normal elements
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Jelena Jovanović, Andreja Tepavčević, and Branimir Šešelja
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Normal subgroup ,Mathematics::Group Theory ,Nilpotent ,Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,High Energy Physics::Lattice ,Lattice (order) ,Dedekind cut ,Abelian group ,Algebraic number ,Hamiltonian (control theory) ,Mathematics - Abstract
By several postulates we introduce a new class of algebraic lattices, in which a main role is played by so called normal elements. A model of these lattices are weak-congruence lattices of groups, so that normal elements correspond to normal subgroups of subgroups. We prove that in this framework many basic structural properties of groups turn out to be lattice-theoretic. Consequently, we give necessary and sufficient conditions under which a group is Hamiltonian, Dedekind, abelian, solvable, supersolvable, metabelian, finite nilpotent. These conditions are given as lattice-theoretic properties of a lattice with normal elements. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
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- 2021
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64. On orthogonally additive operators in C-complete vector lattices
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Marat Pliev and Nazife Erkurşun-Özcan
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Pure mathematics ,Algebra and Number Theory ,Operator (computer programming) ,Compact space ,Functional analysis ,Function space ,Lattice (order) ,Dedekind cut ,Operator theory ,Space (mathematics) ,Analysis ,Mathematics - Abstract
In this article, we investigate orthogonally additive (nonlinear) operators on C-complete vector lattices which strongly includes all Dedekind complete vector lattices. In the first part of the paper, we present basic examples of orthogonally additive operators on function spaces. Then we show that an orthogonally additive map defined on a lateral band of a C-complete vector lattice and taking values in a Dedekind complete vector lattice could be extended to the whole space and an extended orthogonally additive operator preserves continuity, narrowness, compactness and disjointness. In the second part of the article, we consider lateral projection operators onto lateral bands. One of our main results asserts that for a C-complete vector lattice E there is a lateral projection onto every lateral band of E. Applying the technique of lateral projections we prove that for a orthogonally additive narrow operator $$T:E\rightarrow F$$ from a C-complete vector lattice E to an order continuous Banach lattice F all elements of the order interval [0, T] are narrow operators as well. Finally we show that $$T+S$$ is a narrow operator provided that the operator T is horizontally-to-norm continuous and C-compact and the operator S is narrow.
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- 2021
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65. Dedekind's Logicism
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Erich H. Reck
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Pure mathematics ,Philosophy ,Logicism ,Dedekind cut - Published
- 2021
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66. GAPS BETWEEN ZEROS OF DEDEKIND ZETA-FUNCTIONS OF QUADRATIC NUMBER FIELDS. II.
- Author
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BUI, H. M., HEAP, WINSTON P., and TURNAGE-BUTTERBAUGH, CAROLINE L.
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ZERO (The number) ,CARDINAL numbers ,DEDEKIND sums ,NUMBER theory ,DEDEKIND cut - Abstract
Let K be a quadratic number field and ζK(s) be the associated Dedekind zeta-function. We show that there are infinitely many gaps between consecutive zeros of ζK(s) on the critical line which are >2.866 times the average spacing. [ABSTRACT FROM AUTHOR]
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- 2016
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67. A NEW SUMS AND ITS RECIPROCITY THEOREM.
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GUOHUI CHEN and DI HAN
- Subjects
- *
RECIPROCITY theorems , *DEDEKIND sums , *NUMBER theory , *DEDEKIND cut , *PARTITIONS (Mathematics) , *REAL numbers - Abstract
The main purpose of this paper is introduced a new sums analogous to Dedekind sums, then using the analytic method and the properties of Dirichlet L-functions to study the arithmetical properties of this sums, and give an interesting reciprocity theorem for it. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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68. The injective hull of ultra-quasi-metric versus q-hyperconvex hull of quasi-metric space.
- Author
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Otafudu, Olivier Olela
- Subjects
- *
QUASI-metric spaces , *INJECTIVE functions , *ORDERED sets , *CONVEXITY spaces , *DEDEKIND cut - Abstract
For any partially ordered set equipped with its natural T 0 -quasi-metric ( T 0 -ultra-quasi-metric), we study the connection between the ultra-quasi-metrically injective hull and the q -hyperconvex hull. We also observe that for a partially ordered set, its Dedekind–MacNeille completion coincides exactly with the ultra-quasi-metric injective hull of its natural T 0 -ultra-quasi metric. [ABSTRACT FROM AUTHOR]
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- 2016
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69. Dedekind $$\eta $$-function identities of level 6 and an approach towards colored partitions
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B. R. Srivatsa Kumar and Shruthi
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Combinatorics ,Colored ,General Mathematics ,Dedekind cut ,Function (mathematics) ,Mathematics - Abstract
Somos conjectured thousands of Dedekind $$\eta $$ -function identities of various levels, around 6200 in number. He did so using computational evidence but has not sought to provide any proof for these identities. In this paper, we prove level 6 of Somos’s Dedekind $$\eta $$ -function identities containing five terms in two methods. Further, as an application of these identities, we deduce colored partitions for the same.
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- 2021
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70. Dedekind harmonic numbers
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Haydar Göral and Çağatay Altuntaş
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Pure mathematics ,General Mathematics ,Dedekind cut ,Harmonic number ,Mathematics - Published
- 2021
- Full Text
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71. Domain Extension and Ideal Elements in Mathematics
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Anna Bellomo, ILLC (FGw), and Logic and Language (ILLC, FNWI/FGw)
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Algebra ,Philosophy ,Ideal (set theory) ,General Mathematics ,Closure (topology) ,Dedekind cut ,Extension (predicate logic) ,Mathematics ,Domain (software engineering) - Abstract
Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements.
- Published
- 2021
72. Infinity and the Self: Royce on Dedekind
- Author
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Sébastien Gandon, Laboratoire Philosophies et Rationalités (PHIER), Université Clermont Auvergne (UCA), Maison des Sciences de l’Homme de Clermont-Ferrand (MSH Clermont), and Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)
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Infinite set ,Pure mathematics ,Dedekind ,media_common.quotation_subject ,Self ,[SHS.PHIL]Humanities and Social Sciences/Philosophy ,Infinity ,Set (abstract data type) ,Royce ,infinity ,self ,History and Philosophy of Science ,Dedekind cut ,media_common ,Mathematics - Abstract
International audience; In Die Zahlen (1888), Dedekind defines an infinite set as a set that is isomorphic with one of its proper parts. In The World and the Individual (1900), the American philosopher Josiah Royce relates Dedekind’s notion to Fichte’s and Hegel’s concept of Self defined as an entity that reflects itself into itself. The first aim of this article is to explain Royce’s analysis and to put it in its proper context, that of a critique of Bradley’s mystical idealism. The second aim is to urge a shift in focus in Dedekind’s scholarship: instead of addressing the question of the relationship between mathematics and philosophy in Dedekind’s work through the supposed intentions of its author, it is more fruitful to analyze the reception that philosophers have made of his texts.
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- 2021
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73. On the arithmetic of stable domains
- Author
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Alfred Geroldinger, Aqsa Bashir, and Andreas Reinhart
- Subjects
Property (philosophy) ,Endomorphism ,13A05 ,Algebraic structure ,stable domains ,Catenary degrees ,010103 numerical & computational mathematics ,Commutative ring ,Mori domains ,Commutative Algebra (math.AC) ,01 natural sciences ,Stability (probability) ,factorizations ,FOS: Mathematics ,Dedekind cut ,Ideal (ring theory) ,0101 mathematics ,Arithmetic ,Mathematics ,Ring (mathematics) ,Algebra and Number Theory ,13A15 ,Mathematics::Commutative Algebra ,13H10 ,010102 general mathematics ,13A05, 13A15, 13F05, 13H10 ,Mathematics - Commutative Algebra ,sets of lengths ,13F05 ,Research Article - Abstract
A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the $2$-generator property. In the present paper we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains., Comment: This paper has been accepted for publication in the Communications in Algebra
- Published
- 2021
74. Dedekind's criterion for the monogenicity of a number field versus Uchida's and Lüneburg's
- Author
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Carlos R. Videla and Xavier Vidaux
- Subjects
Pure mathematics ,General Mathematics ,Dedekind cut ,Algebraic number field ,Mathematics - Published
- 2021
- Full Text
- View/download PDF
75. Tracing internal categoricity
- Author
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Jouko Väänänen and Department of Mathematics and Statistics
- Subjects
Gödel's incompleteness theorems ,0603 philosophy, ethics and religion ,Peano axioms ,Metatheory ,Computer Science::Logic in Computer Science ,Mathematics::Category Theory ,050602 political science & public administration ,111 Mathematics ,Dedekind cut ,Infinite regress ,Axiom ,03C85 ,computer.programming_language ,Mathematics ,second order ,Interpretation (logic) ,05 social sciences ,06 humanities and the arts ,Mathematics - Logic ,16. Peace & justice ,0506 political science ,Philosophy ,Mathematics::Logic ,060302 philosophy ,Gödel ,Mathematical economics ,computer ,categoricity ,SET - Abstract
Informally speaking, the categoricity of an axiom system means that its non-logical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the twentieth century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive their non-categoricity from Skolem's Paradox and Godel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on meta-theory, where the Skolem-Godel phenomenon re-emerges. Using second-order meta-theory to avoid non-categoricity of the meta-theory would only seem to lead to an infinite regress. In this article we maintain that internal categoricity breaks with this traditional picture. It applies to both first- and second-order axiomatizations, although in the first-order case we have so far only examples. It does not depend on the meta-theory in a way that would lead to an infinite regress. It also covers the classical categoricity results of early researchers. In the first-order case it is weaker than categoricity itself, and in the second-order case stronger. We provide arguments to suggest that internal categoricity is the "right" concept of categoricity.
- Published
- 2021
76. On the Riesz dual of L1(μ)
- Author
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A.C.M. van Rooij
- Subjects
Surjective function ,Pure mathematics ,General Mathematics ,Order (group theory) ,Homomorphism ,Dedekind cut ,Isomorphism ,Measure (mathematics) ,Injective function ,Mathematics - Abstract
In this article, ( X , A , μ ) is a measure apace. A classical result establishes a Riesz isomorphism between L 1 ( μ ) ∼ and L ∞ ( μ ) in case the measure μ is σ -finite. In general, there still is a natural Riesz homomorphism Φ : L ∞ ( μ ) → L 1 ( μ ) ∼ , but it may not be injective or surjective. We prove that always the range of Φ is an order dense Riesz subspace of L 1 ( μ ) ∼ . If μ is semi-finite, then L 1 ( μ ) ∼ is a Dedekind completion of L ∞ ( μ ) .
- Published
- 2020
- Full Text
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77. Inequalities for the arithmetical functions of Euler and Dedekind
- Author
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Horst Alzer and Man Kam Kwong
- Subjects
Combinatorics ,Mathematics::Functional Analysis ,symbols.namesake ,Euler's formula ,symbols ,Arithmetic function ,Dedekind cut ,Euler's totient function ,Mathematics - Abstract
For positive integers n, Euler’s phi function and Dedekind’s psi function are given by $$\varphi (n) = n\prod\limits_{_{p\;\text{prime}}^{p|n}} {\left( {1 - \frac{1}{p}} \right)} \;\;\text{and}\;\;\psi (n) = n\prod\limits_{_{p\;\text{prime}}^{p|n}} {\left( {1 + \frac{1}{p}} \right)} ,$$ respectively. We prove that for all n ⩾ 2 we have $${\left( {1 - \frac{1}{n}} \right)^{n - 1}}{\left( {1 + \frac{1}{n}} \right)^{n + 1}} \leqslant {\left( {\frac{{\varphi (n)}}{n}} \right)^{\varphi (n)}}{\left( {\frac{{\psi (n)}}{n}} \right)^{\psi (n)}}$$ and $${\left( {\frac{{\varphi (n)}}{n}} \right)^{\psi (n)}}{\left( {\frac{{\psi (n)}}{n}} \right)^{\varphi (n)}} \leqslant {\left( {1 - \frac{1}{n}} \right)^{n + 1}}{\left( {1 + \frac{1}{n}} \right)^{n - 1}}$$ . The sign of equality holds if and only if n is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).
- Published
- 2020
- Full Text
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78. A Sequence of Models of Generalized Second-order Dedekind Theory of Real Numbers with Increasing Powers
- Author
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V. K. Zakharov and T.V. Rodionov
- Subjects
Pure mathematics ,Sequence ,Order (group theory) ,Dedekind cut ,General Medicine ,Ultraproduct ,Mathematics ,Real number ,Non-standard analysis - Abstract
The paper is devoted to construction of some closed inductive sequence of models of the generalized second-order Dedekind theory of real numbers with exponentially increasing powers. These models are not isomorphic whereas all models of the standard second-order Dedekind theory are. The main idea in passing to generalized models is to consider instead of superstructures with the single common set-theoretical equality and the single common set-theoretical belonging superstructures with several generalized equalities and several generalized belongings for rst and second orders. The basic tools for the presented construction are the infraproduct of collection of mathematical systems different from the factorized Los ultraproduct and the corresponding generalized infrafiltration theorem. As its auxiliary corollary we obtain the generalized compactness theorem for the generalized second-order language.
- Published
- 2020
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79. The Idea of Continuity as Mathematical-Philosophical Invariant
- Author
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Eldar Amirov
- Subjects
Pure mathematics ,Dedekind cut ,General Medicine ,Invariant (mathematics) ,Real number ,Mathematics - Published
- 2019
- Full Text
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80. Dedekind $$\sigma $$-complete $$\ell $$-groups and Riesz spaces as varieties
- Author
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Marco Abbadini
- Subjects
Mathematics::Number Theory ,General Mathematics ,Sigma ,Order (ring theory) ,Mathematics - Logic ,Operator theory ,Riesz space ,Potential theory ,Theoretical Computer Science ,Combinatorics ,FOS: Mathematics ,Dedekind cut ,Variety (universal algebra) ,Logic (math.LO) ,06D20 (Primary) 03C05, 08A65 (Secondary) ,Unit (ring theory) ,Analysis ,Mathematics - Abstract
We prove that the category of Dedekind $\sigma$-complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that $\mathbb{R}$, regarded as a Dedekind $\sigma$-complete Riesz space, generates this category as a quasi-variety, and therefore as a variety. Analogous results are established for the categories of (i) Dedekind $\sigma$-complete Riesz spaces with a weak order unit, (ii) Dedekind $\sigma$-complete lattice-ordered groups, and (iii) Dedekind $\sigma$-complete lattice-ordered groups with a weak order unit., Comment: 15 pages
- Published
- 2019
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81. On 2-absorbing multiplication modules over pullback rings
- Author
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Farkhondeh Farzalipour
- Subjects
Pure mathematics ,Pullback ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Multiplication ,Dedekind cut ,010103 numerical & computational mathematics ,0101 mathematics ,Indecomposable module ,01 natural sciences ,Mathematics - Abstract
In this article, we classify all those indecomposable 2-absorbing multiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [10] to a more general 2-absorbing multiplication modules case.
- Published
- 2019
- Full Text
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82. NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF
- Author
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Wilfried Sieg and Patrick Walsh
- Subjects
Structure (mathematical logic) ,Natural deduction ,Logic ,Computer science ,010102 general mathematics ,Proof assistant ,Inference ,06 humanities and the arts ,0603 philosophy, ethics and religion ,Mathematical proof ,01 natural sciences ,Philosophy ,Mathematics (miscellaneous) ,Proof theory ,060302 philosophy ,Calculus ,Dedekind cut ,Set theory ,0101 mathematics - Abstract
Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas.To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/.We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation.Hilbert 1918
- Published
- 2019
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83. Construction of all polynomial relations among Dedekind eta functions of level N
- Author
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Ralf Hemmecke and Silviu Radu
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Polynomial ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Computational Mathematics ,Gröbner basis ,Integer ,Generating set of a group ,Dedekind cut ,Ideal (ring theory) ,0101 mathematics ,Mathematics - Abstract
We describe an algorithm that, given a positive integer N, computes a Grobner basis of the ideal of polynomial relations among Dedekind η-functions of level N, i.e., among the elements of { η ( δ 1 τ ) , … , η ( δ n τ ) } where 1 = δ 1 δ 2 … δ n = N are the positive divisors of N. More precisely, we find a finite generating set (which is also a Grobner basis) of the ideal ker ϕ where ϕ : Q [ E 1 , … , E n ] → Q [ η ( δ 1 τ ) , … , η ( δ n τ ) ] , E k ↦ η ( δ k τ ) , k = 1 , … , n .
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- 2019
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84. A hermitian analog of a quadratic form theorem of Springer
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Stefan Gille
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Surjective function ,Pure mathematics ,Exact sequence ,Number theory ,Mathematics::K-Theory and Homology ,Quadratic form ,General Mathematics ,Mathematics::Rings and Algebras ,Dedekind cut ,Algebraic geometry ,Hermitian matrix ,Discrete valuation ring ,Mathematics - Abstract
We show that the second residue map for hermitian Witt groups of an Azumaya algebra A with involution $$\tau $$ of first- or second kind over a semilocal Dedekind domain R is surjective. This proves a generalization to hermitian Witt groups of an exact sequence for Witt groups of quadratic forms due to Springer. If R is a complete discrete valuation ring and $$\tau $$ is of the first kind we show that our short exact sequence of hermitian Witt groups is split. As a corollary we prove a purity theorem for hermitian Witt groups of Azumaya algebras with involutions over a regular semilocal domain of dimension two.
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- 2019
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85. Infinite locally finite groups with the locally nilpotent non-Dedekind norm of decomposable subgroups
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T. Lukashova
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Mathematics::Group Theory ,Pure mathematics ,Algebra and Number Theory ,Norm (group) ,010102 general mathematics ,Locally nilpotent ,Dedekind cut ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The author studies infinite locally finite groups with the locally nilpotent non-Dedekind norm of decomposable subgroups. It is found out that such groups are either p-groups or Frobenius groups, w...
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- 2019
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86. Generalization of the theorems of Barndorff-Nielsen and Balakrishnan–Stepanov to Riesz spaces
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Bruce A. Watson, Bertin Zinsou, and Nyasha Mushambi
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General Mathematics ,46B40, 60F15, 60F25 ,Mathematics::Classical Analysis and ODEs ,0211 other engineering and technologies ,02 engineering and technology ,Riesz space ,Conditional expectation ,01 natural sciences ,Potential theory ,Theoretical Computer Science ,Combinatorics ,Computer Science::Logic in Computer Science ,FOS: Mathematics ,Dedekind cut ,0101 mathematics ,Mathematics ,Mathematics::Functional Analysis ,Sequence ,021103 operations research ,Probability (math.PR) ,010102 general mathematics ,Order (ring theory) ,Mathematics::Spectral Theory ,Operator theory ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Unit (ring theory) ,Mathematics - Probability ,Analysis - Abstract
In a Dedekind complete Riesz space, E, we show that if $$(P_n)$$ is a sequence of band projections in E then $$\begin{aligned} \limsup \limits _{n\rightarrow \infty } P_n - \liminf \limits _{n\rightarrow \infty } P_n = \limsup \limits _{n\rightarrow \infty } P_n(I-P_{n+1}). \end{aligned}$$This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan–Stepanov generalizations of the first Borel–Cantelli theorem.
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- 2019
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87. Sets of arithmetical invariants in transfer Krull monoids
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Alfred Geroldinger and Qinghai Zhong
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Monoid ,Pure mathematics ,Rational number ,20M13, 13A05, 13F05, 16H10, 16U30 ,Group Theory (math.GR) ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics::Category Theory ,0103 physical sciences ,Catenary ,FOS: Mathematics ,Mathematics - Combinatorics ,Arithmetic function ,Dedekind cut ,0101 mathematics ,Mathematics::Representation Theory ,Finite set ,Commutative property ,Mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,Multiplicative function ,Mathematics - Commutative Algebra ,Combinatorics (math.CO) ,010307 mathematical physics ,Mathematics - Group Theory - Abstract
Transfer Krull monoids are a recently introduced class of monoids and include the multiplicative monoids of all commutative Krull domains as well as of wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.
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- 2019
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88. On elasticities of locally finitely generated monoids
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Qinghai Zhong
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Monoid ,Rational number ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Infimum and supremum ,13A05, 13F05, 20M13 ,0103 physical sciences ,FOS: Mathematics ,Dedekind cut ,010307 mathematical physics ,0101 mathematics ,Elasticity (economics) ,Algebraic number ,U-1 ,Commutative property ,Mathematics - Abstract
Let H be a commutative and cancellative monoid. The elasticity ρ ( a ) of a non-unit a ∈ H is the supremum of m / n over all m , n for which there are factorizations of the form a = u 1 ⋅ … ⋅ u m = v 1 ⋅ … ⋅ v n , where all u i and v j are irreducibles. The elasticity ρ ( H ) of H is the supremum over all ρ ( a ) . We establish a characterization, valid for finitely generated monoids, when every rational number q with 1 q ρ ( H ) can be realized as the elasticity of some element a ∈ H . Furthermore, we derive results of a similar flavor for locally finitely generated monoids (they include all Krull domains and orders in Dedekind domains satisfying certain algebraic finiteness conditions) and for weakly Krull domains.
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- 2019
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89. Dedekind’s Criterion and Integral Bases
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Lhoussain El Fadil
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Pure mathematics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,Dedekind cut ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤ L be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤ L : R[α]], for every prime p of R. Some computational examples are given for R = ℤ.
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- 2019
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90. The Mathematics of Continuous Multiplicities: The Role of Riemann in Deleuze's Reading of Bergson
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Nathan Widder
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Literature and Literary Theory ,05 social sciences ,Multiplicity (mathematics) ,06 humanities and the arts ,0603 philosophy, ethics and religion ,050105 experimental psychology ,Epistemology ,Philosophy ,Riemann hypothesis ,symbols.namesake ,Philosophy of mathematics ,060302 philosophy ,symbols ,0501 psychology and cognitive sciences ,Dedekind cut ,Mathematics - Abstract
A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut.
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- 2019
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91. An explicit Chebotarev density theorem under GRH
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Giuseppe Molteni and Loïc André Henri Grenie
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Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Density theorem ,01 natural sciences ,Chebotarev theorem ,Generalized Riemann Hypothesis ,Prime (order theory) ,Settore MAT/02 - Algebra ,Riemann hypothesis ,symbols.namesake ,symbols ,Dedekind cut ,0101 mathematics ,Symbol (formal) ,Mathematics - Abstract
We prove an explicit version of the Chebotarev theorem for the density of prime ideals with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions. In appendix we also give some explicit formulas counting non-trivial zeros of Hecke's L-functions, in that case without assuming the truth of the Riemann hypothesis.
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- 2019
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92. Strong atoms in Krull monoids
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Gerhard Angermüller
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Pure mathematics ,Algebra and Number Theory ,Condensed Matter::Strongly Correlated Electrons ,Dedekind cut ,Extraction methods ,Algebra over a field ,Condensed Matter::Disordered Systems and Neural Networks ,Mathematics - Abstract
It is shown that strong atoms are rather abundant in Krull monoids. For the proof extraction methods are used. An application of this result yields a positive answer to a question of D. D. Anderson, D. F. Anderson and J. Park about Dedekind domains.
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- 2019
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93. An analogue of the Wielandt subgroup in infinite groups
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Maria Ferrara, Marco Trombetti, Martyn R. Dixon, Dixon, M. R., Ferrara, M., and Trombetti, M.
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f-subnormal subgroup ,f-Wielandt subgroup ,Dedekind group ,Group (mathematics) ,Applied Mathematics ,010102 general mathematics ,Generalized f-Wielandt subgroup ,01 natural sciences ,Omega ,Combinatorics ,Chain (algebraic topology) ,0103 physical sciences ,Dedekind cut ,010307 mathematical physics ,0101 mathematics ,Connection (algebraic framework) ,Abelian group ,Subsoluble group ,Mathematics - Abstract
In this paper we define analogues of the Wielandt subgroup of a group. We say that a subgroup H of a group G is f-subnormal in G if there is a finite chain of subgroups $$ H=H_0\le H_1\le \cdots \le H_n=G $$ such that either $$|H_{i+1}: H_i|$$ is finite or $$H_i$$ is normal in $$H_{i+1}$$, for $$0\le i\le n-1$$. We study in a group G the connection between the subgroups $$\overline{\omega }(G)$$ and $$\overline{\omega }_i(G)$$ which are, respectively, the sets of elements of G normalizing all f-subnormal subgroups of G and those normalizing all infinite f-subnormal subgroups of G. In particular we show that $$\overline{\omega }_i(G)/\overline{\omega }(G)$$ is always Dedekind and often abelian.
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- 2019
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94. Some characterizations of Riesz spaces in the sense of strongly order bounded operators
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Akbar Bahramnezhad, M. B. Moghimi, Kazem Haghnejad Azar, Seyed AliReza Jalili, Razi Alavizadeh, and Abbas Najati
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Pure mathematics ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Riesz space ,Operator theory ,01 natural sciences ,Potential theory ,Theoretical Computer Science ,symbols.namesake ,Operator (computer programming) ,Fourier analysis ,Norm (mathematics) ,Bounded function ,symbols ,Dedekind cut ,0101 mathematics ,Analysis ,Mathematics - Abstract
We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in $$E^{\sim \sim }$$ and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if $$T_A$$ is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.
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- 2019
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95. On the density function for the value-distribution of automorphic L-functions
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Kohji Matsumoto and Yumiko Umegaki
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Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Distribution (number theory) ,Mathematics::Number Theory ,010102 general mathematics ,Measure (physics) ,Probability density function ,Automorphic L-function ,010103 numerical & computational mathematics ,Expression (computer science) ,11F66, 11M41 ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,Value-distribution ,FOS: Mathematics ,symbols ,Congruence (manifolds) ,Density function ,Dedekind cut ,Number Theory (math.NT) ,Limit (mathematics) ,0101 mathematics ,Mathematics - Abstract
The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certaindensity function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly constructed density function, for the case of automorphic L-functions attached to primitive forms with respect to congruence subgroups Gamma_0(N)., Comment: 21pages
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- 2019
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96. Dedekind on Axiomatics and His Logicism
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Jun-yong Park
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Pure mathematics ,Philosophy ,Logicism ,Dedekind cut - Published
- 2019
- Full Text
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97. The pseudo-fundamental group scheme
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Arijit Dey and Marco Antei
- Subjects
Pure mathematics ,Fundamental group ,Algebra and Number Theory ,010102 general mathematics ,Torsors ,01 natural sciences ,Fundamental group scheme ,Section (category theory) ,Scheme (mathematics) ,0103 physical sciences ,Torsor ,Order (group theory) ,Dedekind cut ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Mathematics - Abstract
Let X be any scheme defined over a Dedekind scheme S with a given section x ∈ X (S). We prove the existence of a pro-finite S-group scheme א (X,x) and a universal א (X,x)-torsor dominating all the pro-finite pointed torsors over X. Though א (X,x) may not be unique in general it still can provide useful information in order to better understand X. In a similar way we prove the existence of a pro-algebraic S-group scheme א alg(X,x) and a א alg(X,x)-torsor dominating all the pro-algebraic and affine pointed torsors over X.The case where X→S has no sections is also considered. Universidad de Costa Rica/[820-B7-185]/UCR/Costa Rica Universidad de Costa Rica/[820-B7-185]/UCR/Costa Rica UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemática
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- 2019
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98. Nano topology induced by Lattices
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M. Lellis Thivagar and V. Sutha Devi
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Physics ,Algebraic structure ,General Mathematics ,Lattice (order) ,Open set ,Empty set ,Dedekind cut ,Partially ordered set ,Topology ,Upper and lower bounds ,Infimum and supremum - Abstract
Lattice is a partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. Dedekind worked on lattice theory in the 19th century. Nano topology explored by Lellis Thivagar et.al. can be described as a collection of nano approximations, a non-empty finite universe and empty set for which equivalence classes are buliding blocks. This is named as Nano topology, because of its size and what ever may be the size of universe it has atmost five elements in it. The elements of Nano topology are called the Nano open sets. This paper is to study the nano topology within the context of lattices. In lattice, there is a special class of joincongruence relation which is defined with respect to an ideal. We have defined the nano approximations of a set with respect to an ideal of a lattice. Also some properties of the approximations of a set in a lattice with respect to ideals are studied. On the other hand, the lower and upper approximations have also been studied within the context various algebraic structures.
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- 2019
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99. On sums of narrow and compact operators
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Mikhail Popov, O. Fotiy, A. I. Gumenchuk, and I. Krasikova
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Mathematics::Functional Analysis ,Pure mathematics ,021103 operations research ,Function space ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,02 engineering and technology ,Riesz space ,Operator theory ,Compact operator ,01 natural sciences ,Potential theory ,Theoretical Computer Science ,symbols.namesake ,Fourier analysis ,symbols ,Dedekind cut ,0101 mathematics ,Analysis ,Mathematics - Abstract
We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Kothe function spaces and Riesz spaces.
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- 2019
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100. On the density theorem related to the space of non-split tri-Hermitian forms II
- Author
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Akihiko Yukie
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Pure mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Space (mathematics) ,01 natural sciences ,Hermitian matrix ,Quadratic equation ,Number theory ,0103 physical sciences ,Dedekind cut ,Quadratic field ,010307 mathematical physics ,Cubic field ,0101 mathematics ,Mathematics - Abstract
Let $${\widetilde{k}}$$ be a fixed cubic field, F a quadratic field and $$L=\widetilde{k}\cdot F$$. In this paper and its companion paper, we determine the density of more or less the ratio of the residues of the Dedekind zeta functions of L, F where F runs through quadratic fields.
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- 2019
- Full Text
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