scholarly journals DIFFERENTIATION IN P-MINIMAL STRUCTURES AND A p-ADIC LOCAL MONOTONICITY THEOREM

2014 ◽  
Vol 79 (4) ◽  
pp. 1133-1147 ◽  
Author(s):  
TRISTAN KUIJPERS ◽  
EVA LEENKNEGT
Keyword(s):  
Wide Class ◽  
Local Version ◽  
First Order ◽  
Almost Everywhere ◽  
Local Monotonicity ◽  
Definable Functions

AbstractWe prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions f : K → K are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.

On the variance of the sample correlation between two independent lattice processes

1981 ◽  
Vol 18 (04) ◽  
pp. 943-948 ◽  
Author(s):  
Sylvia Richardson ◽  
Denis Hemon
Keyword(s):  
Wide Class ◽  
Asymptotic Expression ◽  
Asymptotic Variance ◽  
First Order ◽  
Sample Correlation ◽  
Lattice Processes ◽  
Independent Lattice

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z 2} and {Y(u), u (Z 2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z 2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

On the notion of algebraic closedness for noncommutative groups and fields

1971 ◽  
Vol 36 (3) ◽  
pp. 441-444 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Wide Class ◽  
First Order ◽  
Division Rings ◽  
Ordered Field ◽  
Ordered Fields ◽  
Skew Fields

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

scholarly journals Spaces of polynomial and nonpolynomial spline-wavelets

2019 ◽  
Vol 292 ◽  
pp. 04001
Author(s):  
Yu. K. Dem’yanovich ◽  
I. G. Burova ◽  
T. O. Evdokimovas ◽  
A. V. Lebedeva
Keyword(s):  
Interpolation Problem ◽  
Wavelet Decomposition ◽  
Hermite Interpolation ◽  
Uniform Grid ◽  
First Order ◽  
Spline Wavelets ◽  
Almost Everywhere

This paper, discusses spaces of polynomial and nonpolynomial splines suitable for solving the Hermite interpolation problem (with first-order derivatives) and for constructing a wavelet decomposition. Such splines we call Hermitian type splines of the first level. The basis of these splines is obtained from the approximation relations under the condition connected with the minimum of multiplicity of covering every point of (α, β) (almost everywhere) with the support of the basis splines. Thus these splines belong to the class of minimal splines. Here we consider the processing of flows that include a stream of values of the derivative of an approximated function which is very important for good approximation. Also we construct a splash decomposition of the Hermitian type splines on a non-uniform grid.

Anneaux de fonctions p-adiques

1995 ◽  
Vol 60 (2) ◽  
pp. 484-497 ◽  
Author(s):  
Luc Bélair
Keyword(s):  
Prime Ideal ◽  
Residue Field ◽  
Local Rings ◽  
First Order ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Definable Functions ◽  
Quotient Rings

AbstractWe study first-order properties of the quotient rings (V)/ by a prime ideal where (V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x∣y2 ∨ y∣x2).

Undecidability in diagonalizable algebras

1997 ◽  
Vol 62 (1) ◽  
pp. 79-116 ◽  
Author(s):  
V. Yu. Shavrukov
Keyword(s):  
Wide Class ◽  
Formal Theory ◽  
First Order ◽  
Unary Operator ◽  
Diagonalizable Algebra

AbstractIf a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.

On the variance of the sample correlation between two independent lattice processes

1981 ◽  
Vol 18 (4) ◽  
pp. 943-948 ◽  
Author(s):  
Sylvia Richardson ◽  
Denis Hemon
Keyword(s):  
Wide Class ◽  
Asymptotic Expression ◽  
Asymptotic Variance ◽  
First Order ◽  
Sample Correlation ◽  
Lattice Processes ◽  
Independent Lattice

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z2} and {Y(u), u (Z2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

THE CRITERIA OF WEAK GENERALIZED LOCALIZATION FOR MULTIPLE WALSH–FOURIER SERIES OF FUNCTIONS IN ORLICZ CLASSES

2004 ◽  
Vol 02 (02) ◽  
pp. 181-185 ◽  
Author(s):  
S. K. BLOSHANSKAYA
Keyword(s):  
Fourier Series ◽  
Wide Class ◽  
Geometric Characteristics ◽  
Equal Zero ◽  
Almost Everywhere ◽  
Series Of Functions

For the wide class of measurable sets [Formula: see text], [Formula: see text], N≥1, the criteria are found (in terms of structural and geometric characteristics of sets [Formula: see text] called [Formula: see text] and [Formula: see text] properties) for validity of the weak generalized localization almost everywhere (WGL) for multiple Walsh–Fourier series of functions equal zero on [Formula: see text], in the Orlicz classes Φ(L)(IN) "lying between" L1 and Lp, p>1. In particular, it is found that in the class L( log +L)2WGL holds on the set [Formula: see text] iff [Formula: see text] has the [Formula: see text] property and in any class L( log + log +L)1-ε, 0<ε<1, WGL holds on [Formula: see text] iff [Formula: see text] has the [Formula: see text] property.

Zero-one law and definability of linear order

2009 ◽  
Vol 74 (1) ◽  
pp. 105-123
Author(s):  
Hannu Niemistö
Keyword(s):  
Linear Order ◽  
Quantifier Elimination ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Almost Everywhere ◽  
Asymptotic Probability ◽  
Finite Variable Logic ◽  
Second Order Logic

§1. Introduction. A logic ℒ has a limit law, if the asymptotic probability of every query definable in ℒ converges. It has a 0–1-law if the probability converges to 0 or 1. The 0–1-law for first-order logic on relational vocabularies was independently found by Glebski et al. [6] and Fagin [5]. Later it has been shown for many other logics, for instance for fragments of second order logic [12], for finite variable logic [13] and for FO extended with the rigidity quantifier [3]. Lynch [14] has shown a limit law for first-order logic on vocabularies with unary functions.We say that two formulas or two logics are almost everywhere equivalent, if they are equivalent on a class of structures whose asymptotic probability measure is one [7]. A 0–1-law is usually proved by showing that every quantifier of the logic has almost everywhere quantifier elimination, i.e., every formula with just one quantifier in front of it is almost everywhere equivalent to a quantifier-free formula. Besides proving 0–1-law, this implies that the logic is (weakly) almost everywhere equivalent to first-order logic.The aim of this paper is to study, whether a logic with a 0–1-law can have greater expressive power than FO in the almost everywhere sense and to what extent. In particular, we are interested on the definability of linear order. Because a 0–1-law determines the almost everywhere expressive power of the sentences of the logic completely, but does not say anything about formulas explicitly, we have to assume some regularity on logics. We will therefore mostly consider extensions of first-order logic with generalized quantifiers.

Negative-existentially complete structures and definability in free extensions

1976 ◽  
Vol 41 (1) ◽  
pp. 95-108 ◽  
Author(s):  
Volker Weispfenning
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Wide Class ◽  
First Order ◽  
Formal Degree ◽  
Atomic Relation

Let R be a commutative ring with 1 and R[X1, …, Xn] the polynomial ring in n variables over R. Then for any relation f(X) = 0 in R[X] there exists a conjunction of equations φf such that f(X) = 0 holds in R[X] iff φf holds in R; φf is of course the formula saying that all the coefficients of f(X) vanish. Moreover, φf is independent of R and formed uniformly for all polynomials f up to a given formal degree. In this paper we investigate first order theories T for which a similar phenomenon holds. More precisely, we let TAH be the universal Horn part of a theory T and look at free extensions of models of T in the class of models of TAH. We ask whether an atomic relation t1(X, a) = t2(X, a) or R(t1(X, a), …, tn(X, a)) in can be equivalently expressed by a finite or infinitary formula φ(a) in , such that φ(y) depends only on ti{X, y) and not on or a1, …, am ∈ A.We will show that for a wide class of theories T “defining formulas” φ(y) in this sense exist and can be taken as infinite disjunctions of positive existential formulas.

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