On closure under direct product

1958 ◽  
Vol 23 (2) ◽  
pp. 149-154 ◽  
Author(s):  
C. C. Chang ◽  
Anne C. Morel
Keyword(s):  
Normal Form ◽  
Direct Product ◽  
Disjunctive Normal Form ◽  
Negative Answer ◽  
Sufficient Condition ◽  
Natural Question ◽  
Existential Quantifier ◽  
Relational Systems ◽  
Image Position

In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question.We shall discuss relational systems of the formwhere A and R are non-empty sets; each element of R is an ordered triple 〈a, b, c〉, with a, b, c ∈ A.1 If the triple 〈a, b, c〉 belongs to the relation R, we write R(a, b, c); if 〈a, b, c〉 ∉ R, we write (a, b, c). If x0, x1 and x2 are variables, then R(x0, x1, x2) and x0 = x1 are predicates. The expressions (x0, x1, x2) and x0 ≠ x1 will be referred to as negations of predicates.We speak of α1, …, αn as terms of the disjunction α1 ∨ … ∨ αn and as factors of the conjunction α1 ∧ … ∧ αn. A sentence (open, closed or neither) of the formwhere each Qi (if there be any) is either the universal or the existential quantifier and each αi, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form.

On sentences which are true of direct unions of algebras

1951 ◽  
Vol 16 (1) ◽  
pp. 14-21 ◽  
Author(s):  
Alfred Horn
Keyword(s):  
Normal Form ◽  
Wide Class ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Factor Algebra ◽  
The Matrix ◽  
Distinct Individual ◽  
Direct Union ◽  
Individual Variables ◽  
Image Position

It is well known that certain sentences corresponding to similar algebras are invariant under direct union; that is, are true of the direct union when true of each factor algebra. An axiomatizable class of similar algebras, such as the class of groups, is closed under direct union when each of its axioms is invariant. In this paper we shall determine a wide class of invariant sentences. We shall also be concerned with determining sentences which are true of a direct union provided they are true of some factor algebra. In the case where all the factor algebras are the same, a further result is obtained. In §2 it will be shown that these criteria are the only ones of their kind. Lemma 7 below may be of some independent interest.We adopt the terminology and notation of McKinsey with the exception that the sign · will be used for conjunction. Expressions of the form ∼∊, where ∊ is an equation, will be called inequalities. In accordance with the analogy between conjunction and disjunction with product and sum respectively, we shall call α1, …, αn the terms of the disjunctionand the factors of the conjunctionEvery closed sentence is equivalent to a sentence in prenez normal form,where x1, …, xm distinct individual variables, Q1, …, Qm are quantifiers, and the matrix S is an open sentence in which each of the variables x1, …, xm actually occurs. The sentence S may be written in either disjunctive normal form:where αi,j is either an equation or an inequality, or in conjunctive normal form:.

Theory of models with generalized atomic formulas

1960 ◽  
Vol 25 (1) ◽  
pp. 1-26 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Natural Number ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Elementary Class ◽  
Relational Systems ◽  
Necessary And Sufficient

IntroductionWe shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

scholarly journals Existence of nonoscillatory solutions of first order nonlinear neutral equations

1990 ◽  
Vol 32 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Lu Wudu
Keyword(s):  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Neutral Equation ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Neutral Equations ◽  
First Order ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES

2004 ◽  
Vol 14 (09) ◽  
pp. 3337-3345 ◽  
Author(s):  
JIANPING PENG ◽  
DUO WANG
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Sufficient Condition ◽  
First Order ◽  
Recursive Formulas ◽  
Unique Normal Forms

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

scholarly journals Algebraic independence properties of the Fredholm series

1978 ◽  
Vol 26 (1) ◽  
pp. 31-45 ◽  
Author(s):  
J. H. Loxton ◽  
A. J. van der Poorten
Keyword(s):  
Algebraic Independence ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Independence Properties ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

scholarly journals GROUPOIDS AND RELATIVE INTERNALITY

2019 ◽  
Vol 84 (3) ◽  
pp. 987-1006
Author(s):  
LÉO JIMENEZ
Keyword(s):  
Stable Theory ◽  
Sufficient Condition ◽  
Natural Type ◽  
Image Position ◽  
Stationary Type

AbstractIn a stable theory, a stationary type $q \in S\left( A \right)$ internal to a family of partial types ${\cal P}$ over A gives rise to a type-definable group, called its binding group. This group is isomorphic to the group $Aut\left( {q/{\cal P},A} \right)$ of permutations of the set of realizations of q, induced by automorphisms of the monster model, fixing ${\cal P}\,\mathop \cup \nolimits \,A$ pointwise. In this article, we investigate families of internal types varying uniformly, what we will call relative internality. We prove that the binding groups also vary uniformly, and are the isotropy groups of a natural type-definable groupoid (and even more). We then investigate how properties of this groupoid are related to properties of the type. In particular, we obtain internality criteria for certain 2-analysable types, and a sufficient condition for a type to preserve internality.

Sign in / Sign up

Export Citation Format

Share Document