scholarly journals Wijsman Orlicz Asymptotically Ideal -Statistical Equivalent Sequences

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Bipan Hazarika
Keyword(s):  
Asymptotic Equivalence ◽  
Equivalent Sequence ◽  
Definition Of ◽  
Natural Combination ◽  
Positive Integers

An ideal is a family of subsets of positive integers which is closed under taking finite unions and subsets of its elements. In this paper, we introduce a new definition of asymptotically ideal -statistical equivalent sequence in Wijsman sense and present some definitions which are the natural combination of the definition of asymptotic equivalence, statistical equivalent, -statistical equivalent sequences in Wijsman sense. Finally, we introduce the notion of Cesaro Orlicz asymptotically -equivalent sequences in Wijsman sense and establish their relationship with other classes.

scholarly journals ON λ-ASYMPTOTICALLY WIJSMAN GENERALIZED STATISTICAL CONVERGENCE OF SEQUENCES OF SETS

2013 ◽  
Vol 56 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Bipan Hazarika ◽  
Ayhan Esi
Keyword(s):  
Statistical Convergence ◽  
Asymptotic Equivalence ◽  
Wijsman Convergence ◽  
Convergence Of Sequences ◽  
Definition Of ◽  
Natural Combination

ABSTRACT The concept of Wijsman statistical convergence was defined by [Nuray, F.-Rhoades, B. E.: Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 1-9]. In this paper we present three definitions which are a natural combination of the definition of asymptotic equivalence, statistical convergence, generalized statistical convergence and Wijsman convergence. In addition, we also present asymptotically equivalent sequences of sets in sense of Wijsman and study some properties of this concept.

scholarly journals On the(p,q)th Relative Order Oriented Growth Properties of Entire Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Manuel Sánchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal
Keyword(s):  
Quantitative Assessment ◽  
Entire Functions ◽  
Relative Order ◽  
Order Of Growth ◽  
Alternative Definition ◽  
Oriented Growth ◽  
Growth Properties ◽  
Self Similar ◽  
Definition Of ◽  
Positive Integers

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integerspandq, we wish to introduce an alternative definition of relative(p,q)th order which improves the earlier definition of relative(p,q)th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative(p,q)th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.

scholarly journals Period and toroidal knot mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750031 ◽  
Author(s):  
Seungsang Oh ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Hwa Jeong Lee ◽  
Mi Jeong Yeon
Keyword(s):  
Quantum System ◽  
Prime Number ◽  
Growth Rates ◽  
Exact Enumeration ◽  
Common Features ◽  
System A ◽  
Number Formula ◽  
And Mathematics ◽  
Definition Of ◽  
Positive Integers

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (1) ◽  
pp. 40-41 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Present Note ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Symbolic Logic ◽  
Recursive Definition ◽  
System L ◽  
Preceding Paragraph ◽  
Individual Variables ◽  
Definition Of ◽  
Positive Integers

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

Completeness in the theory of types

1950 ◽  
Vol 15 (2) ◽  
pp. 81-91 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Formal System ◽  
Axiom System ◽  
Second Order ◽  
True Proposition ◽  
Arbitrary Domain ◽  
Individual Variables ◽  
Definition Of ◽  
The Individual ◽  
Positive Integers

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

Computability and λ-definability

1937 ◽  
Vol 2 (4) ◽  
pp. 153-163 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Computable Function ◽  
General Recursive Function ◽  
Technical Details ◽  
Intuitive Idea ◽  
Short Space ◽  
Definition Of ◽  
Definable Functions ◽  
Positive Integers ◽  
Computable Functions ◽  
Definable Function

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ-K-definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ-K-definable function is computable; that every λ-definable function is λ-K-definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ-K-definability.A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.

scholarly journals On -Asymptotically Statistical Equivalence of Sequences of Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ömer Kışı ◽  
Fatıh Nuray
Keyword(s):  
Statistical Convergence ◽  
Asymptotic Equivalence ◽  
Statistical Equivalence ◽  
Asymptotically Equivalence ◽  
Natural Combination

This paper presents the notion of -asymptotically statistical equivalence, which is a natural combination of asymptotic -equivalence, and -statistical equivalence for sequences of sets. We find its relations to -asymptotically statistical convergence, strong -asymptotically equivalence, and strong Cesaro -asymptotically equivalence for sequences of sets.

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

Effective procedures in field theory

1956 ◽  
Vol 248 (950) ◽  
pp. 407-432 ◽  
Keyword(s):  
Finite Number ◽  
General Procedure ◽  
Finite Extension ◽  
Splitting Algorithm ◽  
Fundamental Field ◽  
Finite Procedure ◽  
Definition Of ◽  
Extension Ring ◽  
Positive Integers

Van der Waerden (1930 a , pp. 128- 131) has discussed the problem of carrying out certain field theoretical procedures effectively, i.e. in a finite number of steps. He defined an ‘explicitly given’ field as one whose elements are uniquely represented by distinguishable symbols with which one can perform the operations of addition, multiplication, subtraction and division in a finite number of steps. He pointed out that if a field K is explicitly given then any finite extension K' of K can be explicitly given, and that if there is a splitting algorithm for K , i.e. an effective procedure for splitting polynomials with coefficients in K into their irreducible factors in K [x], then(1) there is a splitting algorithm for K' . He observed in (1930 b ), however, that there was no general splitting algorithm applicable to all explicitly given fields K , or at least that such an algorithm would lead to a general procedure for deciding problems of the type ‘Does there exist an n such that E(n) ?’, where E is an arbitrarily given property of positive integers such that there is an algorithm for deciding for any n whether E(n) holds. In this paper we review these results in the light of the precise definition of algorithm (finite procedure) given by Church (1936), Kleene (1936) and Turing (1937) and discuss the existence of a number of field theoretical algorithms in explicit fields, and the effective construction of field extensions. We sharpen van der Waerden’s result on the non-existence of a general splitting algorithm by constructing (§7) a particular explicitly given field which has no splitting algorithm. We show (§7) that the result on the existence of a splitting algorithm for a finite extension field does not hold for inseparable extensions, i.e. we construct a particular explicitly given field K and an explicitly given inseparable algebraic extension K ( x ) such that K has a splitting algorithm but K ( x ) has not. (2) We note also (in §6) that there exist two isomorphic explicitly given fields, one of which possesses a splitting algorithm but the other of which does not. Thus the sort of properties of fields we are interested in depend not only on the abstract field but also on the particular representation chosen. It is necessary therefore to state rather carefully our definitions of explicit ring, extension ring, splitting algorithm, etc., and to introduce the concept of explicit isomorphism (3) and homomorphism. This occupies §§ 1,2 and 3. On the basis of these definitions we then discuss the existence of some fundamental field theoretical algorithms in explicit fields and their extension fields. This leads also to a classification of the types of extension fields which can be effectively constructed.

scholarly journals Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

2021 ◽  
Vol 2090 (1) ◽  
pp. 012037
Author(s):  
Kaushik Ghosh
Keyword(s):  
Line Segment ◽  
Statistical Physics ◽  
Rational Numbers ◽  
Partition Functions ◽  
Actual Process ◽  
Real Numbers ◽  
Hausdorff Topology ◽  
Straight Line ◽  
Definition Of ◽  
Positive Integers

Abstract In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

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