Number Arithmetic Research Articles

Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation

The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and p -adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock–Margulis (strong extremality of nondegenerate submanifolds of ℝ^n ) are generalized to the S -arithmetic setting.

Evolutionary computation to search Mandelbrot sets for aesthetic images

We describe an evolutionary algorithm for searching Mandelbrot sets for aesthetic images. Artistic intent is enforced through the design of fitness functions that drive the evolutionary search. Our fitness function employs a mask that specifies a desired iterative behaviour at sample points within the image. The mask used to define a specific instance of the fitness function can be derived from an existing image of the Mandelbrot set. Because of this the artist need not be skilled in programming or familiar with the arithmetic of complex numbers. Once a mask has been chosen, our method uses an evolutionary algorithm to perform a three-parameter search of a specified Mandelbrot set. This paper applies our techniques to the quadratic and cubic Mandelbrot sets.

Real analysis: a constructive approach

Preface. Acknowledgements. Introduction. 0 Preliminaries. 0.1 The Natural Numbers. 0.2 The Rationals. 1 The Real Numbers and Completeness. 1.0 Introduction. 1.1 Interval Arithmetic. 1.2 Families of Intersecting Intervals. 1.3 Fine Families. 1.4 Definition of the Reals. 1.5 Real Number Arithmetic. 1.6 Rational Approximations. 1.7 Real Intervals and Completeness. 1.8 Limits and Limiting Families. Appendix: The Goldbach Number and Trichotomy. 2 An Inverse Function Theorem and its Application. 2.0 Introduction. 2.1 Functions and Inverses. 2.2 An Inverse Function Theorem. 2.3 The Exponential Function. 2.4 Natural Logs and the Euler Number. 3 Limits. Sequences and Series. 3.1 Sequences and Convergence. 3.2 Limits of Functions. 3.3 Series of Numbers. Appendix I: Some Properties of Exp and Log. Appendix 11: Rearrangements of Series. 4 Uniform Continuity. 4.1 Definitions and Elementary Properties. 4.2 Limits and Extensions. Appendix I: Are there Non-Continuous Functions? Appendix XI: Continuity of Double-Sided Inverses. Appendix III: The Goldbach Function. 5 The Riemann Integral. 5.1 Definition and Existence. 5.2 Elementary Properties. 5.3 Extensions and Improper Integrals. 6 Differentiation. 6.1 Definitions and Basic Properties. 6.2 The Arithmetic of Differentiability. 6.3 Two Important Theorems. 6.4 Derivative Tools. 6.5 Integral Tools. 7 Sequences and Series of Functions. 7.1 Sequences of Functions. 7.2 Integrals and Derivatives of Sequences. 7.3 Power Series. 7.4 Taylor Series. 7.5 The Periodic Functions. Appendix: Binomial Issues. 8 The Complex Numbers and Fourier Series. 8.0 Introduction. 8.1 The Complex Numbers C. 8.2 Complex Functions and Vectors. 8.3 Fourier Series Theory. References. Index.

Review of Classical Mathematical Logic: The Semantic Foundations of Logic, by Richard L. Epstein, With contributions by Leslaw W. Szczerba

The main purpose of the book is a detailed exposition of methods used in semantical and deductive analysis of ordinary mathematical reasoning by means of classical mathematical logic. The book contains both a statement of modern mathematical logic and many its applications in various fields of mathematics. Introduction gives a brief review of the trends in mathematics in the 19th century which have resulted in a renewed interest in formal logic and its rapid development in the 20th century. Chapters I–XI contain a rather detailed exposition of basic facts in modern mathematical logic including classical propositional and predicate logics, their semantics and axiomatizations. Chapters XII–XVII present applications of mathematical logic to mathematics. In this connection formalizations of group theory, linear orderings, arithmetic of natural numbers, integers, rationals, and real numbers are considered. First-order and second-order arithmetic theories are discussed. Chapters XVIII and XIX written in collaboration with Lesław Szczerba give...

Caliper rule based ADC architectures

In this paper, a new kind of architectures for the full-flash or subranging Analogue to Digital Converters (ADC’s) is presented. To describe these architectures and explain the intrinsic conversion procedure, we refer to a technique, already presented by the author, which improves the accuracy of a measurement system. This technique is based on the arithmetic of the integer numbers in finite fields and include, as a particular case, the well known “caliper rule”. The said technique is recalled and explained by using some simple measurement examples, in particular length, time and mass measurements. Then the technique is applied to the full-flash and subranging ADC architectures, greatly reducing the number of the resistors necessary to generate the voltage reference scale. Indeed, the required resistors for a n-bits conversion word length are reduced from 2 n to about 3 ∗ 2 n / 2 . This reduction and the particular comparison method involved in the theory allow the realization of full-flash architectures having large word lengths.

A displayed inventory model with L–R fuzzy number

In this study, we formulate a multi-item displayed inventory model under shelf-space constraint in fuzzy environment. Here demand rate of an item is considered as a function of the displayed inventory level. The problem is formulated to maximize average profit. In real life situation, the goals and inventory parameters are may not precise. Such type of uncertainty may be characterized by fuzzy numbers. Here, the constraint goal and the inventory cost parameters are assumed to be triangular shaped fuzzy numbers with different types of left and right membership functions. The fuzzy numbers are then approximated to a nearest interval number. Using arithmetic of interval numbers, the problem is described as a multi-objective inventory problem. The problem is then solved by fuzzy geometric programming approach. Finally a numerical example is given to illustrate the problem.

Embedding infinitely parallel computation in Newtonian kinematics

First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore and classify the computational models that can be embedded in, and abstracted from, the physical systems specified by the physical theory T. We consider the embedding of arbitrary sets, functions, programs, and computers into designs for systems that can be specified in subtheories or fragments T of Newtonian kinematics in order to explore some of the physical assumptions of T that allows its systems to qualify as hyper-computers, i.e. physical models that compute sets and functions that cannot be computed in classical computability theory. In designing systems we work strictly within the chosen theory T and do not concern ourself with whether or not T is valid of the world today. We are interested in exploring the subtheory from a computational point of view and especially in restrictions on the assumptions of T that allow us to return from hyper-computation to classical computation. Secondly, we give a construction of an infinitely parallel machine that can decide all the arithmetical sets of natural numbers. We embed this hyper-computer as system in 3-dimensions obeying the laws of a fragment of Newtonian kinematics. In particular, the example shows that communication allowable in Newtonian kinematics is especially powerful. We conclude with further reflections and open problems.

Logic programming with infinite sets

Using the ideas from current investigations in Knowledge Representation we study the use of a class of logic programs for reasoning about infinite sets. Our programs reason about the codes for various infinite sets. Depending on the form of atoms allowed in the bodies of clauses we obtain a variety of completeness results for various classes of arithmetic sets of integers.

Noise-driven numerical irreversibility in molecular dynamics technique

It is widely believed that, in molecular dynamics (MD) simulations, round-off errors can cause numerical irreversibility since, in the standard MD, floating-point real number arithmetic is employed and round-off errors cannot be avoided. To investigate the characteristic of this numerical irreversibility, the ‘bit-reversible algorithm’, which is completely time-reversible and is free from any round-off error, is made use of as a test bed. Through this study, it is clearly demonstrated that, other than the extent of the stability of the system, the appearance of irreversibility is related to the ‘ quantity’ of the controlled noise. By means of the bit-reversible simulation added to the controlled noise of an appropriate ‘ quantity’, the characteristic of the numerical irreversibility in the standard MD is revealed.

Cardinality approach to fuzzy number arithmetic

To solve problems from the area of computing with words, arithmetic operations often have to be carried out on fuzzy numbers. The author proposes to call fuzzy numbers fuzzy sets of numbers (FSofN) to emphasize the fact that they are sets of many numbers (in the continuous case of infinitely many numbers) and not one, single number as the name fuzzy number suggests, because it occasionally leads to incorrect interpretations of their concept. To realize arithmetic operations on FSofN the standard extension principle is used. This principle was intended for possibilistic FSofN. However, in practical tasks many FSofN are of probabilistic character. If, for such FSofN, the standard extension principle is used, the results will be incorrect. In this paper a cardinality extension principle is proposed. It realizes arithmetic operations on probabilistic FSofN. This principle is proposed in 2 versions: as a normal and as a generalized version that enables context-dependent constraints to be taken into account. In addition, a new form of the possibilistic, constraint-extension principle of Klir is presented. In this paper, many examples are given that illustrate computations with both extension principles to make them less abstract and more user-friendly.

New Booth Modulo m Multipliers with Signed-Digit Number Arithmetic

New modulo m multipliers with a radix-two signed-digit (SD) number arithmetic is presented by using a modified Booth recoding method. To implement a modulo m multiplication, we usually generate modulo m partial products, then perform modulo m sum of them. In this paper, a new Booth recoding method is proposed to convert a radix-two SD number into a recoded SD (RSD) number in parallel. In the RSD number representation, there are no (1, 1)and (-1, -1) at any two-digit position. Thus, by using the RSD converted, the modulo m partial products can be cut from n into n/2 for an n × n modulo m multiplication. Parallel and serial modulo m multipliers have been designed by using the SD number arithmetic and the proposed Booth recoding. Compared to the former work, the area for VLSI implementation of the parallel modulo m multiplier is reduced to 80% from the original design, and the speed performance of the serial multiplier is improved up to twice by using the Booth recoding. The implementation method of the proposed Booth modulo m multipliers has been verified by a gate level simulation.

An extension to benefit/cost based fuzzy MCDM algorithm

This paper proposes a fuzzy multiple criteria decision making extension through the division of the summation of weighted normalized cost ratings over the summation of weighted normalized benefit ratings. The membership function of the final fuzzy evaluation value of each alternative can be developed via α -cut and interval arithmetic of fuzzy numbers. Propositions and corollaries will be provided to prove the correctness of the developed membership functions. A ranking method can then be used to determine the ordering of alternatives. A numerical example verifies the feasibility of the proposed model.

DEVELOPING A FUZZY MCDM MODEL VIA A BENEFIT/COST TRADEOFF CONCEPT

ABSTRACT A fuzzy multiple criteria decision-making model is suggested, where the ratings of alternatives versus criteria and the importance weights of criteria are assessed in linguistic terms represented by fuzzy numbers. The criteria are categorized into benefit and cost ones. The membership functions for the final fuzzy evaluation values of alternatives are developed by α-cuts and interval arithmetic of fuzzy numbers via a tradeoff concept between the summation of weighted normalized benefit ratings and that of weighted normalized cost ratings. A fuzzy number ranking approach is then used to defuzzify all the final fuzzy evaluation values for the ordering of alternatives. Some properties are discussed and an example demonstrates the feasibility of the proposed model.

The Arithmetic of Algebraic Numbers: An Elementary Approach

Let Q and R be the fields of rational and real numbers respectively. Recall that a real number r is algebraic over the rationals if there is a polynomial p with coefficients in Q that has r as a root, i.e., that has p(r) = 0. Any college freshman can understand that idea, but things get more challenging when one asks about arithmetic with algebraic numbers. For example, being the roots of x2− 3 and x2− 20 respectively, the real numbers r1 = √ 3 and s1 = 2 √ 5 are certainly algebraic over the rationals, but what about the numbers r1 + s1, r1s1 and r1 s1 ? As it happens, all three are algebraic over the rationals. For example, r1 +s1 is a root of x4−46x2 +289. But how was that polynomial constructed, and what rationalcoefficient-polynomials have r1s1 and r1 s1 as roots? Students who take a second modern algebra course will learn to use field extension theory to show that the required polynomials must exist. They will learn that whenever r and s 6= 0 are algebraic over Q, then the field Q(r,s) is an extension of Q of finite degree with the consequence that r + s, rs and s are indeed algebraic over Q (see [2, 3, 7]). However, one would hope that students would encounter more elementary solutions for such basic arithmetic questions. Furthermore, one might want to know how to construct rational-coefficient polynomials that have r+s, rs and s as roots and thereby obtain bounds on the minimum degrees of such polynomials.

A Modular Algorithm for Computing Cohomologies of Lie Algebras and Superalgebras

The paper describes an improved algorithm for computing cohomologies of Lie (super)algebras. The original algorithm developed earlier by the author of this paper is based on the decomposition of the entire cochain complex into minimal subcomplexes. The suggested improvement consists in the replacement of the arithmetic of rational or integer numbers by a more efficient arithmetic of modular fields and the use of the relationship dim Hk(\mathbb{F}p) ≥ dimHk(\mathbb{Q}) between the dimensions of cohomologies over an arbitrary modular field \mathbb{F}p e \mathbb{Z}/p\mathbb{Z} and the filed of rational numbers \mathbb{Q}. This inequality allows us to rapidly find subcomplexes for which dimHk(\mathbb{F}p) > 0 (the number of such subcomplexes is usually not great) using computations over an arbitrary \mathbb{F}p and, then, carry out all required computations over \mathbb{Q} in these subcomplexes.

Arithmetical definability and computational complexity

In this paper, we introduce and study some syntactical fragments of monadic second-order and first-order (P EANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an effective method for translating formulas between different logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.

Ranking alternatives via maximizing set and minimizing set based fuzzy MCDM approach

This work suggests a maximizing set and minimizing set based fuzzy multiple criteria decision‐making (MCDM) model, where criteria are classified into cost and benefit criteria. The final fuzzy evaluation value of each alternative is developed based on the concept of subtracting the summation of weighted normalized benefit ratings from that of weighted normalized cost ratings. Using interval arithmetic of fuzzy numbers can develop the membership functions for the final fuzzy evaluation values. Chen's maximizing set and minimizing set is then applied to defuzzify all the final fuzzy numbers for ranking alternatives. Formulas for the membership functions and ranking procedure of the final fuzzy numbers are clearly presented. The suggested method provides an extension to the fuzzy MCDM techniques available. A numerical example demonstrates the computational process of the proposed method.

On the effectiveness of cellular automata to add real numbers

This paper investigates the effectiveness of cellular automata to compute arithmetic functions over the set of real numbers. Since cellular automata (CA) have parallel computational ability in conjunction with the capacity to accept an infinite length input, the CA should be able to effectively compute real number arithmetic. This paper shows that typical representations of real numbers do not lend themselves to effective computation by a CA and that a nonstandard representation must be employed in order to accomplish this.

Arithmetical definability over finite structures

AbstractArithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform AC0 and FO(PLUS, TIMES). We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can first‐order define PLUS, that < and DIVIDES can first‐order define TIMES, and that < and COPRIME can first‐order define TIMES. The first result sharpens the equivalence FO(PLUS, TIMES) =FO(BIT) to FO(<, TIMES) = FO(BIT), answering a question raised by Barrington et al. about the Crane Beach Conjecture. Together with previous results on the Crane Beach Conjecture, our results imply that FO(PLUS) is strictly less expressive than FO(<, TIMES) = FO(<, DIVIDES) = FO(<,COPRIME). In more colorful language, one could say that, for parallel computation, multiplication is harder than addition.

Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class (respectively ). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a -complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation.