M-NUQAM Modulations with Constant Symbol Mean Energy

2021 ◽  
pp. 137-148
Author(s):  
Fernando Lara ◽  
Miguel Castillo ◽  
Gonzalo Olmedo
Keyword(s):  
Constant Symbol

Multiplication complexe et équivalence élémentaire dans le langage des corps (Complex multiplication and elementary equivalence in the language of fields)

2002 ◽  
Vol 67 (2) ◽  
pp. 635-648
Author(s):  
Xavier Vidaux
Keyword(s):  
Function Fields ◽  
Complex Multiplication ◽  
Closed Field ◽  
Constant Symbol ◽  
Endomorphism Rings ◽  
Modular Invariant ◽  
Elementary Equivalence ◽  
Algebraically Closed Field

AbstractLet K and K′ be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0. non k-isomorphic, and let C and C′ be two curves with respectively K and K′ as function fields. We prove that if the endomorphism rings of the curves are not isomorphic then K and K′ are not elementarily equivalent in the language of fields expanded with a constant symbol (the modular invariant). This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras.

Complete types and the natural numbers

1973 ◽  
Vol 38 (3) ◽  
pp. 413-415
Author(s):  
Julia F. Knight
Keyword(s):  
Complete Theory ◽  
Elementary Extension ◽  
Constant Symbol ◽  
Natural Numbers ◽  
Complete Type

In this paper it is shown that, for any complete type Σ omitted in the structure , or in any expansion of having only countably many relations and operations, there is a proper elementary extension of (or of ) which omits Σ. This result (which was announced in [2]) is used to answer a question of Malitz on complete -sentences. The result holds also for countable families of types.A type is a countable set of formulas with just the variable υ free. A structure is said to omit a type Σ if no element of satisfies all of the formulas of Σ. For example, omits the type Σω = {υ ≠ n: n ∈ ω}, since n fails to satisfy υ ≠ n. (Here n is the constant symbol standing for n.)A type Σ is said to be complete with respect to a theory T if the set of sentences T ∪ Σ(e) generates a complete theory, where Σ(e) is the result of replacing υ by the new constant e in all of the formulas of Σ. The type Σω is clearly not complete with respect to Th(). (For any structure Th(), Th() is the set of all sentences true in .)

scholarly journals The Transrational Numbers as an Abstract Data Type

2020 ◽  
Author(s):  
Jan Aldert Bergstra ◽  
John V. Tucker
Keyword(s):  
Rational Numbers ◽  
Data Type ◽  
Abstract Data Type ◽  
Constant Symbol ◽  
Total Function ◽  
Initial Algebra ◽  
Abstract Data ◽  
Initial Algebra Semantics

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.

Some undecidability results in strong algebraic languages

1984 ◽  
Vol 49 (3) ◽  
pp. 951-954
Author(s):  
Cornelia Kalfa
Keyword(s):  
Decision Problem ◽  
Decision Problems ◽  
Turing Machines ◽  
Constant Symbol ◽  
Finite Sets ◽  
Operation Symbol ◽  
Halting Problem ◽  
First Order ◽  
Order Language ◽  
Image Position

The recursively unsolvable halting problem for Turing machines is reduced to the problem of the existence or not of an algorithm for deciding whether a field is finite. The latter problem is further reduced to the decision problem of each of propertiesfor recursive sets Σ of equations of strong algebraic languages with infinitely many operation symbols.Decision problems concerning properties of sets of equations were first raised by Tarski [9] and subsequently examined by Perkins [6], McKenzie [4], McNulty [5] and Pigozzi [7]. Perkins is the only one who studied recursive sets; the others investigated finite sets. Since the undecidability of properties Pi for recursive sets of equations does not imply any answer to the corresponding decision problems for finite sets, the latter problems remain open.The work presented here is part of my Ph.D. thesis [2]. I thank Wilfrid Hodges, who supervised it.An algebraic language is a first-order language with equality but without relation symbols. It is here denoted by , where Qi is an operation symbol and cj, is a constant symbol.

Finite extensions and the number of countable models

1989 ◽  
Vol 54 (1) ◽  
pp. 264-270 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
General Question ◽  
Constant Symbol ◽  
First Order Theory ◽  
Consistent Extension ◽  
Branching Trees ◽  
Complete Theories ◽  
Ehrenfeucht Theory ◽  
Countable Models

An Ehrenfeucht theory is a complete first order theory with exactly n countable models up to isomorphism, 1 < n < ω. Numerous results have emerged regarding these theories ([1]–[15]). A general question in model theory is whether or not the number of countable models of a complete theory can be different than the number of countable models of a complete consistent extension of the theory by finitely many constant symbols. Examples are known of Ehrenfeucht theories that have complete extensions by finitely many constant symbols such that the extensions fail to be Ehrenfeucht ([4], [8], [13]). These examples are easily modified to allow finite increases in the number of countable models.This paper contains examples in the other direction—complete theories that have consistent extensions by finitely many constant symbols such that the extensions have fewer countable models. This answers affirmatively a question raised by, among others, Peretyat'kin [8]. The first example will be an Ehrenfeucht theory with exactly four countable models with an extension by a constant symbol that has only three countable models. The second example will be a complete theory that is not Ehrenfeucht, but which has an extension by a constant symbol that is Ehrenfeucht. The notational conventions for this paper are standard.Peretyat'kin introduced the theory of a dense binary branching tree with a meet operator [7]. Dense ω-branching trees have also proven useful [5], [11]. Both of the Theories that will be constructed make use of dense ω-branching trees.

The completeness theorem for infinitary logic

1972 ◽  
Vol 37 (1) ◽  
pp. 31-34 ◽  
Author(s):  
Richard Mansfield
Keyword(s):  
Boolean Algebra ◽  
Completeness Theorem ◽  
Complete Boolean Algebra ◽  
Infinitary Logic ◽  
Constant Symbol ◽  
Truth Values ◽  
Truth Function ◽  
Universal Quantification ◽  
Image Position ◽  
Extended Notion

It will be proven that a set of sentences of infinitary logic is satisfiable iff it is proof theoretically consistent. Since this theorem is known to be false, it must be quickly added that an extended notion of model is being used; truth values may be taken from an arbitrary complete Boolean algebra. We shall give a Henkin style proof of this result which generalizes easily to Boolean valued sets of sentences.For each infinite candinal number κ the language Lκ is built up from a set of relation symbols together with a constant symbol cα and a variable υα for each α in κ. It contains atomic formulas and is closed under the following rules:(1) If Γ is a set of formulas of power < κ ∧ Γ is a set of formulas.(2) If φ is a formula, ¬ φ is also.(3) If φ is a formula and A Is a subset of κ of power < κ then Aφ is a formula.∧Γ is meant to be the conjunction of all the formulas in Γ, while Aφ is the universal quantification of all the variables υα for α in A. We let C denote the set of constant symbols in Lκ, the parameter κ must be discovered from the context.A model is identified with its truth function. Thus a model is a function mapping the sentences of Lκ into a complete Boolean algebra which satisfies the following conditions:

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