Showing papers on "Binary operation published in 2004"
Book•
1 Jan 2004
TL;DR: This chapter discusses the construction of a modular number system, and some of the techniques used to achieve this goal, as well as some of those used in the design of the modern number system.
Abstract: Preface. List of Figures. List of Tables. About the Author. 1. Computer Number Systems. 1.1 Conventional Radix Number System. 1.2 Conversion of Radix Numbers. 1.3 Representation of Signed Numbers. 1.3.1 Sign-Magnitude. 1.3.2 Diminished Radix Complement. 1.3.3 Radix Complement. 1.4. Signed-Digit Number System. 1.5 Floating-Point Number Representation. 1.5.1 Normalization. 1.5.2 Bias. 1.6 Residue Number System. 1.7 Logarithmic Number System. References. Problems. 2. Addition and Subtraction. 2.1 Single-Bit Adders. 2.1.1 Logical Devices. 2.1.2 Single-Bit Half-Adder and Full-Adders. 2.2 Negation. 2.2.1 Negation in One's Complement System. 2.2.2 Negation in Two's Complement System. 2.3 Subtraction through Addition. 2.4 Overflow. 2.5 Ripple Carry Adders. 2.5.1 Two's Complement Addition. 2.5.2 One's Complement Addition. 2.5.3 Sign-Magnitude Addition. References. Problems. 3. High-Speed Adder. 3.1 Conditional-Sum Addition. 3.2 Carry-Completion Sensing Addition. 3.3 Carry-Lookahead Addition (CLA). 3.3.1 Carry-Lookahead Adder. 3.3.2 Block Carry Lookahead Adder. 3.4 Carry-Save Adders (CSA). 3.5 Bit-Partitioned Multiple Addition. References. Problems. 4. Sequential Multiplication. 4.1 Add-and-Shift Approach. 4.2 Indirect Multiplication Schemes. 4.2.1 Unsigned Number Multiplication. 4.2.2 Sign-Magnitude Number Multiplication. 4.2.3 One's Complement Number Multiplication. 4.2.4 Two's Complement Number Multiplication. 4.3 Robertson's Signed Number Multiplication. 4.4 Recoding Technique. 4.4.1 Non-overlapped Multiple Bit Scanning. 4.4.2 Overlapped Multiple Bit Scanning. 4.4.3 Booth's Algorithm. 4.4.4 Canonical Multiplier Recoding. References. Problems. 5. Parallel Multiplication. 5.1 Wallace Trees. 5.2 Unsigned Array Multiplier. 5.3 Two's Complement Array Multiplier. 5.3.1 Baugh-Wooley Two's Complement Multiplier. 5.3.2 Pezaris Two's Complement Multipliers. 5.4 Modular Structure of Large Multiplier. 5.4.1 Modular Structure. 5.4.2 Additive Multiply Modules. 5.4.3 Programmable Multiply Modules. References. Problems. 6. Sequential Division. 6.1 Subtract-and-Shift Approach. 6.2 Binary Restoring Division. 6.3 Binary Non-Restoring Division. 6.4 High-Radix Division. 6.4.1 High-Radix Non-Restoring Division. 6.4.2 SRT Division. 6.4.3 Modified SRT Division. 6.4.4 Robertson's High-Radix Division. 6.5 Convergence Division. 6.5.1 Convergence Division Methodologies. 6.5.2 Divider Implementing Convergence Division Algorithm. 6.6 Division by Divisor Reciprocation. References. Problems. 7. Fast Array Dividers. 7.1 Restoring Cellular Array Divider. 7.2 Non-Restoring Cellular Array Divider. 7.3 Carry-Lookahead Cellular Array Divider. References. Problems. 8. Floating Point Operations. 8.1 Floating Point Addition/Subtraction. 8.2 Floating Point Multiplication. 8.3 Floating Point Division. 8.4 Rounding. 8.5 Extra Bits. References. Problems. 9. Residue Number Operations. 9.1 RNS Addition, Subtraction and Multiplication. 9.2 Number Comparison and Overflow Detection. 9.2.1 Unsigned Number Comparison. 9.2.2 Overflow Detection. 9.2.3 Signed Numbers and Their Properties. 9.2.4 Multiplicative Inverse and the Parity Table. 9.3 Division Algorithm. 9.3.1 Unsigned Number Division. 9.3.2 Signed Number Division. 9.3.3 Multiplicative Division Algorithm. References. Problems. 10. Operations through Logarithms. 10.1 Multiplication and Addition in Logarithmic Systems. 10.2 Addition and Subtraction in Logarithmic Systems. 10.3 Realizing the Approximation. References. Problems. 11. Signed-Digit Number Operations. 11.1 Characteristics of SD Numbers. 11.2 Totally Parallel Addition/Subtraction. 11.3 Required and Allowed Values. 11.4 Multiplication and Division. References. Problems. Index.
79 citations
1 Jan 2004
TL;DR: In this article, the notion of decomposition order for partial commutative monoids was proposed and proved in weakly normed processes modulo bisimulation definable in ACP with linear communication.
Abstract: We discuss unique decomposition in partial commutative monoids. Inspired by a result from process theory, we propose the notion of decomposition order for partial commutative monoids, and prove that a partial commutative monoid has unique decomposition i it can be endowed with a decomposition order. We apply our result to establish that the commutative monoid of weakly normed processes modulo bisimulation definable in ACP " with linear communication, with parallel composition as binary operation, has unique decomposition. We also apply our result to establish that the partial commutative monoid associated with a well-founded commutative residual algebra has unique decomposition.
14 citations
25 Oct 2004
TL;DR: An extended class of Cellular Automata, in which arbitrary transition functions are allowed, are considered as a model for spatial dynamic simulation to create a formal basis for the CA composition methods.
Abstract: An extended class of Cellular Automata (CA), in which arbitrary transition functions are allowed, are considered as a model for spatial dynamic simulation. CA are represented as operators over a set of cellular arrays, where binary operations (superposition, addition and multiplication) are defined. The algebraic properties of the operations are studied. The aim of the investigation is to create a formal basis for the CA composition methods, which are also presented in brief.
5 citations
19 May 2004
TL;DR: A class of binary operators on [0, 1], which satisfy the weak associativity and some boundary conditions, are introduced, which is broader than the class of t-norms, and is included in that of Galois-operators on [1, 2], which, and of which, the right-residuals formGalois-connections on [ 0, 1].
Abstract: This report focuses on a property of conjunctive operators named "weak associativity," which is one of sufficient conditions for the syllogism in fuzzy logic. The authors introduce a class of binary operators on [0, 1], which satisfy the weak associativity and some boundary conditions. Our newly introduced class is broader than the class of t-norms, and is included in that of Galois-operators on [0, 1], which, and of which, the right-residuals form Galois-connections on [0, 1]. In addition, the authors show some methods to construct some weakly associative functions by using the generators of t-norms.
5 citations
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TL;DR: This work introduces a binary operation on strings (blocks) of elements from the set {0, 1, . . . ,m − 1} where m is an arbitrary integer greater than 1 and introduces the concept of similar, cyclic, and circular blocks.
Abstract: We introduce a binary operation on strings (blocks) of elements from the set {0, 1, . . . ,m − 1} where m is an arbitrary integer greater than 1. This operation is an extension of one introduced by Konrad Jacobs and Michael Keane in the 1960’s for blocks of 0’s and 1’s. We show that the extended operation is associative, introduce the concept of similar, cyclic, and circular blocks and provide a unique factorization theorem under this operation up to the similarity of the factors. We also give the conditions for commutativity of indecomposable blocks.
3 citations
Patent•
28 Oct 2004TL;DR: In this article, the equivalent of a hardware-based integer division operation is enabled via a reciprocal multiplication operation that is facilitated by a minimum combination of multiplication (and/or add) and shift operations.
Abstract: A method and apparatus to perform efficient software-based integer division. The equivalent of a hardware-based integer division operation is enabled via a reciprocal multiplication operation that is facilitated by a minimum combination of multiplication (and/or add) and shift operations. Properties and equations are derived for determining minimum multiplication and shift instructions to perform an integer division of a variable dividend and constant divisor using reciprocal multiplication. Computer functions are disclosed for determining parameters from which the minimum multiplication and shift instructions can be derived. Software/firmware is then coded employing the minimum multiplication and shift instructions to perform software-based integer division operations via reciprocal multiplication. In one embodiment, the integer division operations are employed to determine a minimum number of cells required to store the data in a packet or frame that is processed by a network processor.
3 citations
1 Jan 2004
TL;DR: In this paper, the authors deal with functional equations of the form f(x+y) = F(f(x)J(y)) (so called addition formulas) assuming that the given binary operation F is asso-ciative but its domain of definition is disconnected.
Abstract: We deal with functional equations of the form f(x+y) = F(f(x)J(y)) (so called addition formulas) assuming that the given binary operation F is asso ciative but its domain of definition is disconnected (admits "singularities"). The function
3 citations
1 Jan 2004
TL;DR: This talk will consider the role of theories in collegiate mathematics education research, propose characteristics that a theory might have, describe one particular example of a theory and consider the extent to which it possesses these characteristics.
Abstract: In this talk I will consider the role of theories in collegiate mathematics education research I will propose characteristics that a theory might have, describe one particular example of a theory and consider the extent to which it possesses these characteristics I will also describe how this theory has been used by its developers and others throughout the world to increase our understanding of the learning process and enhance student learning of post-secondary level mathematical concepts in Calculus, Discrete Mathematics, and Abstract Algebra Finally, I will give specific examples of the use of this theory in helping students learn certain concepts in abstract algebra
3 citations
16 Aug 2004
TL;DR: A new approach to the study of binary operations is shown after introducing the concept of external range and some results concerning the class of continuous binary operations with finite external range are given.
Abstract: In this paper, a new approach to the study of binary operations is shown. After introducing the concept of external range, some results concerning the class of continuous binary operations with finite external range are given.
2 citations
TL;DR: For quotiens of group algebras, we stady elements such that the multiplication by these elements generates a compact (left or right) multiplication operator as discussed by the authors, and the spectrum of such an algebra is analyzed in the case where all such multiplication operators are weakly compact.
Abstract: For quotiens of group algebras, we stady elements such that the multiplication by these elements generates a compact (left or right) multiplication operator. The spectrum of such an algebra is analyzed in the case where all such (left or right) multiplication operators are weakly compact. Bibliography: 33 titles.
2 citations
Journal Article•
TL;DR: In this paper, the notion of decomposition order for partial commutative monoid has been proposed and shown to have unique decomposition in weakly normed processes with linear communication, with parallel composition as binary operation.
Abstract: We discuss unique decomposition in partial commutative monoids Inspired by a
result from process theory, we propose the notion of decomposition order for partial
commutative monoids, and prove that a partial commutative monoid has unique decomposition
iff it can be endowed with a decomposition order We apply our result to
establish that the commutative monoid of weakly normed processes modulo bisimulation
definable in ACPe with linear communication, with parallel composition as binary
operation, has unique decomposition We also apply our result to establish that the
partial commutative monoid associated with a well-founded commutative residual algebra
has unique decomposition
1 Feb 2004
TL;DR: In this article, the authors studied the construction of closed meanders using Motzkin words with four letters, which are generated by applying binary operation on the set of Dyck words.
Abstract: We study the construction of closed meanders and systems of closed meanders, using Motzkin words with four letters. These words are generated by applying binary operation on the set of Dyck words. The procedure is based on the various kinds of intersection of the meandric curve with the horizontal line.
TL;DR: In this article, the concept of SNA-rings is introduced, which are non-associative structures on which are defined two binary operations one associative and other being non-Associative and addition distributes over multiplication both from the right and left.
Abstract: In this paper we introduce the concept of Smarandache non-associative rings, which we shortly denote as SNA-rings as derived from the general definition of a Smarandache Structure (i.e., a set A embedded with a week structure W such that a proper subset B in A is embedded with a stronger structure S). Till date the concept of SNA-rings are not studied or introduced in the Smarandache algebraic literature. The only non-associative structures found in Smarandache algebraic notions so far are Smarandache groupoids and Smarandache loops introduced in 2001 and 2002. But they are algebraic structures with only a single binary operation defined on them that is nonassociative. But SNA-rings are non-associative structures on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from the right and left. Further to understand the concept of SNA-rings one should be well versed with the concept of group rings, semigroup rings, loop rings and groupoid rings. The notion of groupoid rings is new and has been introduced in this paper. This concept of groupoid rings can alone provide examples of SNA-rings without unit since all other rings happens to be either associative or nonassociative rings with unit. We define SNA subrings, SNA ideals, SNA Moufang rings, SNA Bol rings, SNA commutative rings, SNA non-commutative rings and SNA alternative rings. Examples are given of each of these structures and some open problems are suggested at the end.
1 Jan 2004
TL;DR: In this paper, an algebraic approach to set theory is outlined and an axiom system for & is presented in L. In the algebraic set-up the operation &, and not the 2-relation, is a primitive notion and it is characterized by a number of simple and plausible conditions.
Abstract: In this paper an algebraic approach to set theory is outlined. Models of set theory are traditionally viewed as pairs (A,e), where A is a set (or a class), and e is a binary relation on A. In the algebraic approach models are defined as quadruples A = (A, ,&,0), where the reduct (A, ,0) is a partially ordered set (class) with zero and & is a binary operation defined on A. Intuitively, the elements of A are sets in the sense of A (A-sets, for short). is the inclusion relation in the sense of A and 0 is the empty A-set. The operation & plays a crucial role in this approach. If a,b 2 A, then a&b is the A-set obtained by adjoining the A-set b as a new element to the A-set a. Thus & corresponds to the set theoretic operation which, when performed on sets X and Y , yields the set X&Y := X [ {Y }. The 2-relation is then defined in terms of & by the condition: Y 2 X if and only if X&Y = X. In the algebraic set-up the operation &, and not the 2-relation, is a primitive notion and it is characterized by a number of simple and plausible conditions. Let L be the language having in its vocabulary a binary predicate , a binary operation symbol &, and a constant symbol 0 as the only non-logical symbols. In the paper an axiom system for & is presented in L. Moreover, taking into account the infinitistic character of models of set theory, some further axioms are provided, e.g. in the form of plausible induction principles. These infinitistic axioms are formulated as second-order conditions but elementary counterparts of these axioms (in the form of workable schemes in L) can also be investigated. Some of the known axioms of set theory as Regularity or Choice can be formulated as induction principles. Some space is devoted to the discussion of various induction principles, which come to light when one develops set theory from the algebraic perspective. The quadruples A = (A, ,&,0) which satisfy the list of these axioms are called set-theoretic domains. In the paper we investigate various classes of domains as well as a certain relaxation of this notion, viz. the notion of a set-theoretic proto-domain. Domains and proto-domains thus form the natural
Journal Article•
TL;DR: Some results regarding languages that are context-free or regular w.r.t. a non-deterministic binary operation on words that is left variable are presented, referring, in particular, to important instances of ”putting texts together” operations like concatenation and shuffle.
Abstract: We present some results regarding languages that are context-free or regular w.r.t. a non-deterministic binary operation on words that is left variable, only asking a few general properties like associativity. This could be part of a unified approach to intertextuality, the results referring, in particular, to important instances of ”putting texts together” operations like concatenation and shuffle (with all their variations - distributed, on trajectories etc.).
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TL;DR: Using only two basic building blocks and four combinators all standard designs can be described succinctly and rigorously and the rules of the algebra allow us to prove the circuits correct and to derive circuit designs in a systematic manner.
Abstract: A parallel prefix circuit takes n inputs x 1, x 2, ..., x n and produces the n outputs x 1, x 1 ∘ x 2, ..., x 1 ∘ x 2 ∘ ⋯ ∘ x n , where’∘’ is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerous applications ranging from fast integer addition over parallel sorting to convex hull problems. A parallel prefix circuit can be implemented in a variety of ways taking into account constraints on size, depth, or fan-out. Traditionally, implementations are either defined graphically or by enumerating the underlying graph. Both approaches have their pros and cons. A figure if well drawn conveys the possibly recursive structure of the scan but it is not amenable to formal manipulation. A description in form of a graph while rigorous obscures the structure of a scan and is equally hard to manipulate. In this paper we show that parallel prefix circuits enjoy a very pleasant algebra. Using only two basic building blocks and four combinators all standard designs can be described succinctly and rigorously. The rules of the algebra allow us to prove the circuits correct and to derive circuit designs in a systematic manner.
TL;DR: This paper compares four methods for the reliable propagation of uncertainty through calculations involving the binary operations of addition, multiplication, subtraction and division and shows that they converge to equivalent methods when they are restricted to cumulative distribution functions on the positive reals.
Posted Content•
TL;DR: In this paper, a non-commutative binary operation on matroids, called free product, was introduced, which respects matroid duality and has the property that, given only the cardinalities, an ordered pair of matroid may be recovered, up to isomorphism, from its free product.
Abstract: We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh's 1969 conjecture, which provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set.