scholarly journals DEFINABLE AND INVARIANT TYPES IN ENRICHMENTS OF NIP THEORIES

2017 ◽  
Vol 82 (1) ◽  
pp. 317-324 ◽  
Author(s):  
SILVAIN RIDEAU ◽  
PIERRE SIMON
Keyword(s):  
Sufficient Condition ◽  
Image Position ◽  
Differential Fields ◽  
Model Completion ◽  
Formula Type

AbstractLet T be an NIP ${\cal L}$-theory and $\mathop T\limits^\~ $ be an enrichment. We give a sufficient condition on $\mathop T\limits^\~$ for the underlying ${\cal L}$-type of any definable (respectively invariant) type over a model of $\mathop T\limits^\~$ to be definable (respectively invariant). These results are then applied to Scanlon’s model completion of valued differential fields.

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).

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.

scholarly journals Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

On inequalities for powers of linear operators and for quadratic forms

1981 ◽  
Vol 89 (1-2) ◽  
pp. 25-50 ◽  
Author(s):  
Vũ Qúôc Phóng
Keyword(s):  
Quadratic Forms ◽  
Linear Operators ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dense Domain ◽  
The Best Possible Constant ◽  
Bilinear Functional ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLetHbe a Hilbert space in which a symmetric operatorSwith a dense domainDsis given and letShave a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov typeand a method for calculating the best possible constantsCn,m(S).Moreover, let φ be a symmetric bilinear functional with a dense domainDφsuch thatDs⊂Dφand φ(f, g) = (Sf, g) for allf∈Ds,g∈Dφ. A necessary and sufficient condition for validity of the inequalityas well as a method for calculating the best possible constantKare obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the formThe paper is concluded by establishing the best possible constants in the inequalitieswhereTis an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

Oscillation and Global Attractivity in a Periodic Delay Equation

1996 ◽  
Vol 39 (3) ◽  
pp. 275-283 ◽  
Author(s):  
J. R. Graef ◽  
C. Qian ◽  
P. W. Spikes
Keyword(s):  
Global Attractivity ◽  
Positive Periodic Solution ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Periodic Delay ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

scholarly journals Iterative criteria for bounds on the growth of positive solutions of a delay differential equation

1978 ◽  
Vol 25 (2) ◽  
pp. 195-200
Author(s):  
Raymond D. Terry
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Sufficient Condition ◽  
All Solutions ◽  
Image Position ◽  
Delay Differential

AbstractFollowing Terry (Pacific J. Math. 52 (1974), 269–282), the positive solutions of eauqtion (E): are classified according to types Bj. We denote A neccessary condition is given for a Bk-solution y(t) of (E) to satisfy y2k(t) ≥ m(t) > 0. In the case m(t) = C > 0, we obtain a sufficient condition for all solutions of (E) to be oscillatory.

scholarly journals Remarks on a Problem of Obreanu

1963 ◽  
Vol 6 (2) ◽  
pp. 267-273 ◽  
Author(s):  
P. Erdös ◽  
A. Rényi
Keyword(s):  
Infinite Sequence ◽  
Greatest Common Divisor ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Bruijn ◽  
Image Position ◽  
Necessary And Sufficient

Let a1 < a2 < … be any sequence of integers. Assume that the infinite sequence of numbers un satisfies the following condition: To every ɛ > 0 there is an no = no (ɛ) such that for all n > no and all k1Obreanu asked (Problem P. 35 Can. Math. Bull.) under what conditions on the sequence a1 < a2 < … does (1) imply that the sequence u is convergent. N. G. de Bruijn and P. Erdos proved that a necessary and sufficient condition for (1) to imply the convergence of un is that the sequence {an} be infinite and that the greatest common divisor of the a1 should be 1.

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