Abstract: This paper presents a characterization of a new family of operators. Namely, all non-decreasing, associative binary operators U: [0, 1]2 → [0, 1] with a left (or right) neutral element e ∈ [0, 1], and such that they satisfy an additional hypothesis on continuity which is called here left (or right) pseudocontinuity.

Abstract: Consider the set A = R ∪ {+∞} with the binary operations o1 = max and o2 = + and denote by An the set of vectors v = (v1,,vn) with entries in A Let the generalised sum u o1 v of two vectors denote the vector with entries uj o1 vj , and the product a o2 v of an element a ∈ A and a vector v ∈ An denote the vector with the entries a o2 vj With these operations, the set An provides the simplest example of an idempotent semimodule The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics

Abstract: A digital multiplication apparatus and method adopting a redundant binary arithmetic is provided. In this digital multiplication apparatus, when two numbers X and Y are multiplied using a radix-2k number system, a data converter data-converts the m-bit number Y into m/k-digit data D(=D m,k−1 D m/k−2 . . . D i . . . D i D 0 ). A partial product calculator converts each of the digits Di of the number Y converted by the data converter into a combination of the coefficients of a fundamental multiple, multiplies the combination by the number X, and outputs the product as a redundant binary partial product. A redundant binary adder sums the partial products for all of the digits of the converted number Y. A redundant binary (RB)-normal binary (NB) converter converts the redundant binary sum into a normal binary number and outputs the converted normal binary sum as the product of the two numbers. Therefore, even when the radix extends, the burden upon hardware can be minimized. Also, many systems having multipliers serving as important components can be more simply constructed.

Abstract: This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

Abstract: This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max,+) semiring and the tropical semiring . It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

Abstract: In Exact Real Arithmetic, real numbers are represented as potentially infinite streams of information units, called digits. In this paper, we work in the framework of Linear Fractional Transformations (LFT’s, also known as Mobius transformations) that provide an elegant approach to real number arithmetic (Gosper 1972, Vuillemin 1990, Nielsen and Kornerup 1995, Potts and Edalat 1996, Edalat and Potts 1997, Potts 1998b). Onedimensional LFT’s are used as digits and to implement basic unary functions, while two-dimensional LFT’s provide binary operations such as addition and multiplication, and can be combined to obtain infinite expression trees denoting transcendental functions. Peter Potts (1998a, 1998b) derived these expression trees from continued fraction expansions of the transcendental functions. In contrast, we show how to derive LFT expression trees from power series, which are available for a greater range of functions. In Section 2, we present the LFT approach in some detail. Section 3 contains the main results of the paper. We first derive an LFT expansion from apower series using Homer’s scheme (Section 3.1). The results are not very satisfactory. Thus, we show how LFT expansions may be modified using algebraic transformations (Section 3.2). A particular such transformation, presented in Section 3.3, yields satisfactory results for standard functions, as shown in the final examples section 4.

Abstract: There are only three irrationalities directly related to experimentally observed quasicrystals, namely, those which appear in extensions of rational numbers by O5, O2, O3. In this article, we demonstrate that the algebraically defined aperiodic point sets with precisely these three irrational numbers play an exceptional role. The exceptional role stems from the possibility of equivalent characterization of these point sets using one binary operation. PACS Nos.: 61.90+d, 61.50-f

Abstract: It is known that recognition of regular languages by finite monoids can be generalized to context-free languages and finite groupoids, which are finite sets closed under a binary operation. A loop is a groupoid with a neutral element and in which each element has a left and a right inverse. It has been shown that finite loops recognize exactly those regular languages that are open in the group topology. In this paper, we study the class of aperiodic loops, which are those loops that contain no nontrivial group. We show that this class is stable under various definitions, and we prove some closure properties. We also prove that aperiodic loops recognize only star-free open languages and give some examples. Finally, we show that the wreath product principle can be applied to groupoids, and we use it to prove a decomposition theorem for recognizers of regular open languages.

Abstract: This paper presents a graded hierarchy or chain of binary operations on the reals and the complex numbers. The operations are related distributively in the sense that any one of them distributes over the next lower operation in the chain. For one particular operation we explore specific properties and derive results, including several useful formulas and identities. Next, the operation is extended to the complex numbers and a new kind of derivative is defined based on this binary operation. Some basic formulae analogous to standard classical ones are proven for this new derivative. Finally, the derivative is generalized to the point that the new derivative, the classical one, and countably many others are seen to be special cases.

Abstract: We study properties of unary and binary operations on compact convex sets with respect to the Demyanov metric (D-metric). A class of D-regular parametric convex-valued maps is defined in terms of the D-metric. This class of variable convex sets is invariant under the arithmetic addition linear transformation, and also the intersection operation, if, additionally, the intersection is nonempty. The property of D-regularity is shown to be conserved under the Argmin operation for standard continuous parametric convex programs.

Abstract: A system is provided for finding the result of a binary operation performed on an array of values, along with its address, in a storage-efficient manner. The system is based on a binary tree structure having a pipeline of binary operators and corresponding multiplexers (103) and storage elements (105) to store the outputs of the multiplexers (103) in addition to the partial addresses of the outputs represented by the outputs of the binary operators. Each computation stage (100, 110, 120, 130) adds one more bit of address until the address is completely known at the last computation stage (130). The invention reduces the amount of storage required for keeping track of the addresses that are the result of the binary operation.

Abstract: This invention relates to intelligent componentry that can dynamically and incrementally produce result matrices from binary operations such as Blast. By dividing a dataset into same, new, changed and deleted subsets one can determine the minimum operations that need to be done in order to keep result matrices up-to-date with the changes in given datasets.

Abstract: The aim of this paper is a reduction algorithm for a basis b1, b2, b3 of a 3-dimensional lattice in Rn for fixed n ≥ 3 We give a definition of the reduced basis which is equivalent to that of the Minkowski reduced basis of a 3-dimensional lattice We prove that for b1, b2, b3 ∈ Zn, n = 3 and |b1|, |b2|, |b3| ≤ M, our algorithm takes O(log2M) binary operations, without using fast integer arithmetic, to reduce this basis and so to find the shortest vector in the lattice The definition and the algorithm can be extended to any dimension Elementary steps of our algorithm are rather different from those of the LLL-algorithm, which works in O(log3M) binary operations without using fast integer arithmetic

Abstract: We study pseudo MV-algebras introduced recently by Georgescu and Iorgulescu as a non-commutative generalization of MV-algebras. We introduce a partial binary operation which model the addition in pseudo MV-algebras. In the paper, we give basic properties of such an addition. We find conditions which entail the commutativity of pseudo MV-algebras, i.e., when they are classical MV-algebras. We study ideals and the conditions when a pseudo MV-algebra is representable. Finally, we introduce states and show how they are connected with the normal ideals.