The double sign of a real division algebra of finite dimension greater than one

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_nFor any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_n$ of all n‐dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of $\mathscr {D}_n$.     

The arithmetic of cuts in models of arithmetic

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic.     

模糊一次方程求解的实际研究

传统的加法,减法,乘法和除法运算都属于自然运算,因此不能用于区间数（interval）的分析中,故需要进一步研究区间数的特殊的运算规律。本文介绍模糊集合理论中的区间数进行加、减、乘、除等特殊的算术运算规律为基础,主要介绍并提出模糊一次方程bx＋a=c的求解过程及方法,并给出其模糊方程的解。     

A flow-mode,self-steering,cellular multiplier-summation processor

Flow-mode or stream-processing digital systems have been proposed in which code, control and data are constantly moving so that multiple instructions are processed concurrently. We show the design of a flow-mode cellular array processor which can perform a number of two's complement fixed-point arithmetic operations. These operations are: three operand addition and/or subtraction, two operand multiplication and vector inner-product. Operand sizes are: 2N bit for addition and subtraction operands, andN bit for multiplication operands. Results are 2N bit. The network can simultaneously operate on 4N+2 datasets with any mix of the above operations being handled. The processor is based on the use of asynchronous cellular arrays. Given a continued flow of input datasets, the effective computation time is worst-case propagation time within one cell. A typical cell contains a 1-bit position 3-input full adder with associated input data storage. Thus the effective computation time is independent of the operand bit length.     

Helping Students Come to Grips With the Meaning of Division

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by defining two meanings for division). However, in the author's classes many students had relatively low scores (ranging from 60–80%) when trying to explain simple operations. Reported in this paper are ways to supplement the text that force students to address the actual meaning of division by stressing the relation between a “whole” and a “package,” and connect that meaning with previously learned operational definitions for area and volume.     

Fuzzy real algebra: Some results

In this paper, the possibility to perform easily most of the extended n-ary operations on fuzzy subsets of the real line is shown. A general algorithm is given. These results are particularized for usual operations such as addition, subtraction, multiplication, division, ‘max’ and ‘min’ operations for normalized convex fuzzy subsets of the real line, i.e. fuzzy numbers. A three parameters representation for fuzzy numbers is shown to be very convenient to perform usual operations. Lastly, interpretative comments about fuzzy real algebra are given and possible applications pointed out.     

模糊数的四则运算性质及其线性方程

讨论模糊数的加、减、乘、除的运算性质,提出模糊数线性方程的概念,并给出这种方程的一种解法。     

Multiplicative calculus and its applications

Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. In the period from 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In the present paper our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness.     

Some new Stein and Hardy type inequalities

We prove a generalization of the pointwise Stein inequality, considering two truncated versions. More generally than in the Stein inequality, we assume that the kernel is dominated by a radial function almost decreasing after the division by a power function with nonnegative exponent in the case of the truncation to the ball of the radius and almost increasing after the multiplication by a power function in the case of truncation to the exterior of this ball. We give some applications to a series of inequalities of Hardy type in norms of various function spaces, in particular, in the norm of variable exponent Lebesgue spaces L^{p( ·)}( \mathbbRⁿ ) {L^{p\left( \cdot \right)}}\left( {{{\mathbb{R}}^n}} \right) with weights. Bibliography: 40 titles.     

The inverses of an H-space

 A multiplication on an H-space X has a left inverse λ and a right inverse ρ. They are mutual inverses and λ = ρ if and only if . In this paper we investigate the order of λ. We give an example of a multiplication with , and prove that for any finite H-complex X there are finitely many left inverses of finite order. Conditions are given for there to be infinitely many multiplications on X with the same left inverse. We then give conditions for a left inverse to have infinite order. We apply these results to specific Lie groups. Received: 15 June 2001     

Some results on quasipolyhedral convexity

The objective of this paper is to analyze under what well-known operations the class of quasipolyhedral convex functions, which can be regarded as an extension of that of polyhedral convex functions, is closed. The operations that will be considered are those that preserve polyhedral convexity, such that the image and the inverse image under linear transformations, right scalar multiplication (including the case where λ=0⁺) and pointwise addition.      

The Arithmetical Hierarchy of Real Numbers

A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left (right) computable iff it is the supremum (infimum) of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable (multiple) sequences of rational numbers we introduce a non‐collapsing hierarchy {Σ_n, Π_n, Δ_n : n ∈ ℕ} of real numbers. We characterize the classes Σ₂, Π₂ and Δ₂ in various ways and give several interesting examples.     

Questions of cardinality of finite fuzzy sets

In this paper we focus our attention on finite fuzzy sets. A complete, simple and easily applicable cardinality theory for them is presented. Questions of equipotency and non-classically understood cardinal numbers of finite fuzzy sets are discussed in detail. Also, problems of arithmetical operations (addition, subtraction, multiplication, division, and exponentiation) on as well as ordering relation for those cardinals are carefully investigated.     

Fuzzified neural network based on fuzzy number operations

The fuzzified neural network based on fuzzy number operations is presented as a powerful modelling tool here. We systematically introduce ideas and concepts of a novel neural network based on fuzzy number operations. First we suggest how to compute the results of addition, subtraction, multiplication and division for two fuzzy numbers. Second we propose a learning algorithm, and present some ideas about the choice of fuzzy weights and fuzzy biases and a numerical scheme for the calculation of outputs of the fuzzified neural network. Finally, we show some results of computer simulations.     

Crystallization of space: Space-time fractals from fractal arithmetic

Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetic allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.     

Extension principles for interval-valued intuitionistic fuzzy sets and algebraic operations

The Atanassov’s intuitionistic fuzzy (IF) set theory has become a popular topic of investigation in the fuzzy set community. However, there is less investigation on the representation of level sets and extension principles for interval-valued intuitionistic fuzzy (IVIF) sets as well as algebraic operations. In this paper, firstly the representation theorem of IVIF sets is proposed by using the concept of level sets. Then, the extension principles of IVIF sets are developed based on the representation theorem. Finally, the addition, subtraction, multiplication and division operations over IVIF sets are defined based on the extension principle. The representation theorem and extension principles as well as algebraic operations form an important part of Atanassov’s IF set theory.     

On the Non‐Hierarchical Preconditioner Constructions

In this paper a new cyclic matrix representation of the H^–1/2 norm is presented. Its application as Schur complement preconditioning matrix requires only matrix‐vector multiplication. The computational cost of this matrix‐vector multiplication is O (N · log(N)) arithmetic operations, where N is the number of unknowns. The efficiency of the construction to elliptic problems has been verified by numerical tests. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Finite and Nonfinite Generation of Diagonal Acts

The diagonal right act of a semigroup S is the set S × S, with S acting by componentwise multiplication from the right. The diagonal left act and diagonal bi-act of S are defined analogously.

Necessary and sufficient conditions are found for the finite generation of the diagonal bi-acts of completely zero-simple semigroups and completely simple semigroups. It is also proved that various classes of semigroups do not have finitely generated or cyclic diagonal right, left, or bi-acts.     

Free left n-dinilpotent doppelsemigroups

A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain equations. Commutative dimonoids in the sense of Loday are examples of doppelsemigroups and two interassociative semigroups give rise to a doppelsemigroup. We introduce left (right) n-dinilpotent doppelsemigroups which are analogs of left (right) nilpotent semigroups of rank n considered by Schein. A free left (right) n-dinilpotent doppelsemigroup is constructed and the least left (right) n-dinilpotent congruence on a free doppelsemigroup is characterized. We also establish that the semigroups of the free left (right) n-dinilpotent doppelsemigroup are isomorphic and the automorphism group of the free left (right) n-dinilpotent doppelsemigroup is isomorphic to the symmetric group.     

The Taylor–Browder Spectrum on Prime C*-Algebras

We provide a formula for the Taylor–Browder spectrum of a pair (L _a , R _b ) of left and right multiplication operators acting on a prime C*-algebra with non-zero socle. We also compute ascent and descent for multiplication operators on a prime ring, characterise Browder elements in a prime C*-algebra and discuss upper semicontinuity for the Browder spectrum.