Disjoint Sum Research Articles

Sums of powers of orthogonally additive polynomials

It is proved that for a given two positive integers N,n with N<n and N orthoregular homogeneous polynomials of the same degree acting between Archimedean vector lattices and pairwise independent in some sense, the sum of these polynomials each raised to the power of n and multiplied by an orthomorphism, is orthogonally additive if and only if all these polynomials multiplied by the corresponding orthomorphisms are disjointness preserving. It is also shown that given three positive integers N,n,m, an order bounded orthogonally additive polynomial with values in a Dedekind complete vector lattice is the sum of n-th powers of N order bounded disjointness preserving m-homogeneous polynomials if and only if it is representable as a disjoint sum of N disjointness preserving mn-homogeneous polynomials.

Novel Quantum Algorithms to Minimize Switching Functions Based on Graph Partitions

After Google reported its realization of quantum supremacy, Solving the classical problems with quantum computing is becoming a valuable research topic. Switching function minimization is an important problem in Electronic Design Automation (EDA) and logic synthesis, most of the solutions are based on heuristic algorithms with a classical computer, it is a good practice to solve this problem with a quantum processer. In this paper, we introduce a new hybrid classic quantum algorithm using Grover’s algorithm and symmetric functions to minimize small Disjoint Sum of Product (DSOP) and Sum of Product (SOP) for Boolean switching functions. Our method is based on graph partitions for arbitrary graphs to regular graphs, which can be solved by a Grover-based quantum searching algorithm we proposed. The Oracle for this quantum algorithm is built from Boolean symmetric functions and implemented with Lattice diagrams. It is shown analytically and verified by simulations on a quantum simulator that our methods can find all solutions to these problems.

On the disjoint sums of M-fuzzifying convex spaces

In this paper, we first extend the concept of the arity in crisp convex spaces to the case of fuzzification and give some related properties. From the view of arity and hull operator, we study the relations between the disjoint sum of M-fuzzifying convex spaces and its factor spaces. We also examine the additivity of the degree of separability (S0,S1,S2,S3,S4). Finally, we show that every factor space is M-fuzzifying JHC iff the corresponding disjoint sum space is JHC.

Operations on complex intuitionistic fuzzy soft lattice ordered group and CIFS-COPRAS method for equipment selection process

This paper introduces some new operations on complex intuitionistic fuzzy lattice ordered groups such as sum, product, bounded product, bounded difference and disjoint sum, and verifying its pertinent properties. The research exhibits the CIFS-COPRAS algorithm in a complex intuitionistic fuzzy soft set environment. This method was furthermore applied for the equipment selection process.

Disjoint Sum of Products by Orthogonalizing Difference-Building ⴱ

AbstractThe orthogonalization of Boolean functions in disjunctive form, that means a Boolean function formed by sum of products, is a classical problem in the Boolean algebra. In this work, the novel methodologyORTH[ⴱ] of orthogonalization which is an universally valid formula based on the combination technique »orthogonalizing difference-building ⴱ« is presented. Therefore, the technique ⴱ is used to transform Sum of Products into disjoint Sum of Products. The scope of orthogonalization will be solved by a novel formula in a mathematically easier way. By a further procedure step of sorting product terms, a minimized disjoint Sum of Products can be reached. Compared to other methods or heuristicsORTH[ⴱ] provides a faster computation time.

Holonomy saddles and supersymmetry

In gauge theories on a spacetime equipped with a circle, the holonomy variables, living in the Cartan torus, play special roles. With their periodic nature properly taken into account, we find that a supersymmetric gauge theory in $d$ dimensions tends to reduce in the small radius limit to a disjoint sum of multiple $(d-1)$ dimensional theories at distinct holonomies, called $H$-saddles. The phenomenon occurs regardless of the spacetime dimensions, and here we explore such $H$-saddles for $d=4$ $\cal N=1$ theories on $T^2$ fibred over $\Sigma_g$, in the limits of elongated $T^2$. This naturally generates novel relationships between 4d and 3d partition functions, including ones between 4d and 3d Witten indices, and also leads us to re-examine recent studies of the Cardy exponents and the Casimir energies and of their purported connections to the 4d anomalies.

Multiple Stuck-at Fault Testability Analysis of ROBDD Based Combinational Circuit Design

Automatic test pattern generation (ATPG) is the next step after synthesis in the process of chip manufacturing. The ATPG may not be successful in generating tests for all multiple stuck-at faults since the number of fault combinations is large. Hence a need arises for highly testable designs which have 100% fault efficiency under the multiple stuck-at fault(MSAF) model. In this paper we investigate the testability of ROBDD based 2×1 mux implemented combinational circuit design. We show that the ROBDD based 2×1 mux implemented circuit is fully testable under multiple stuck-at fault model. Principles of pseudoexhaustive testing and multiple stuck-at fault testing of two level AND-OR gates are applied to one sub-circuit(2×1 mux). We show that the composite test vector set derived for all 2×1 muxes is capable of detecting multiple stuck-at faults of the circuit as a whole. Algorithms to derive test set for multiple stuck-at faults are demonstrated. The multiple stuck-at fault test set is larger than the single stuck-at fault test set. We show that the multiple stuck-at fault test set can be derived from the Disjoint Sum of Product expression which allows test pattern generation at design time, eliminating the need of an ATPG after the synthesis stage.

Category Theoretic Properties of the A. R&amp;#233;nyi and C. Tsallis Entropies

The problem of embedding the Tsallis and R\'{e}nyi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the R\'{e}nyi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original R\'{e}nyi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and R\'{e}nyi entropies.

Multigames and strategies, coalgebraically

Coalgebraic games have been recently introduced as a generalization of Conway games and other notions of games arising in different contexts. Using coalgebraic methods, games can be viewed as elements of a final coalgebra for a suitable functor, and operations on games can be analyzed in terms of (generalized) coiteration schemata. Coalgebraic games are sequential in nature, i.e., at each step either the Left (L) or the Right (R) player moves (global polarization); moreover, only a single move can be performed at each step. Recently, in the context of Game Semantics, concurrent games have been introduced, where global polarization is abandoned, and multiple moves are allowed. In this paper, we introduce coalgebraic multigames, which are situated half-way between traditional sequential games and concurrent games: global polarization is still present, however multiple moves are possible at each step, i.e., a team of L/R players moves in parallel. Coalgebraic operations, such as sum and negation, can be naturally defined on multigames. Interestingly, sum on coalgebraic multigames turns out to be related to Conway's selective sum on games, rather than the usual (sequential) disjoint sum. Selective sum has a parallel nature, in that at each step the current player performs a move in at least one component of the sum game, while on disjoint sum the current player performs a move in exactly one component at each step. A presentation of strategies on coalgebraic games is given via a final coalgebra of a pair of mutually recursive functors, and a suitable notion of simulation. A monoidal closed category of coalgebraic multigames in the vein of a Joyal category of Conway games is then built. The relationship between coalgebraic multigames and games is then formalized via an equivalence of the multigame category and a monoidal closed category of coalgebraic games where tensor is selective sum.

POSETS ADMITTING THE LINEARITY OF ISOMETRIES

Abstract. In this paper, we deal with a characterization of the posetswith the property that every poset isometry of F nq fixing the origin is alinear map. We say such a poset to be admitting the linearity of isome-tries. We show that a poset P admits the linearity of isometries over F nq if and only if P is a disjoint sum of chains of cardinality 2 or 1 whenq= 2, or P is an anti-chain otherwise. 1. IntroductionLet F q be a finite field with q elements, and F nq the vector space of n-tuplesover F q . In 1995, Brualdi et al. [1] introduced a non-Hamming metric on F n which is associated to an arbitrary poset on [n] = {1,2,...,n}. It is calleda poset metric. The poset metric spaces have been extensively studied in[1, 3, 4, 5, 8]. We briefly introduce basic notions for poset metric on F nq .Let P = ([n],≤) be a poset on [n] of coordinate positions of vectors on F nq .A subset I of P is called an order ideal (or a down-set) if x ∈ I and y ≤ ximply y ∈ I. For an arbitrary subset A of P, we denote by hAi the smallestorder ideal of P containing A. The P-weight of a vector u = (u

Reliability assessment of power distribution systems using disjoint path-set algorithm

Finding the reliability expression of different substation configurations can help design a distribution system with the best overall reliability. This paper presents a computerized a nd implemented algorithm, based on Disjoint Sum of Product (DSOP) algorithm. The algorithm was synthesized and applied for the first time to the determination of reliability expression of a substation to determine reliability indices and costs of different substation arrangements. It deals with the implementation and synthesis of a new designed algorithm for DSOP implemented using C/C++, incorporating parallel problem solving capability and overcoming the disadvantage of Monte Carlo simulation which is the lengthy computational time to achieve satisfactory statistical convergence of reliability index values. The major highlight of this research being that the time consuming procedures of the DSOP solution generated for different substation arrangements using the proposed method is found to be significantly lower in comparison with the time consuming procedures of Monte Carlo-simulation solution or any other method used for the reliability evaluation of substations in the existing literature such as meta-heuristic and soft computing algorithms. This implementation gives the possibility of RBD simulation for different substation configurations in C/C++ using their path-set Boolean expressions mapped to probabilistic domain and result in simplest Sum of Disjoint Product which is on a one-to-one correspondence with reliability expression. This software tool is capable of handling and modeling a large, repairable system. Additionally, through its intuitive interface it can be easily used for industrial and commercial power systems. With simple Boolean expression for a configuration’s RBD inputted, users can define a power system utilizing a RBD and, through a fast and efficient built-in simulation engine, the required reliability expressions and indices can be obtained. Two case studies will be analyzed in this paper. The effects of different substation configurations on the reliability are analyzed and compared. Then, the reliability of a radial distribution system will be evaluated using DSOP solution. The failure results will be combined with a load flow scenario, and indices such as SAIDI, SAIFI will be determined.

Reliability analysis of breaker arrangements in distribution substations

Substations are the weakest link between the source of supply and the customer load points in a power system, because they comprise switching arrangements that would lead to loss of load. Finding the reliability expression of different substation configurations can help design a system with the best overall reliability. This paper presents a computerized and implemented algorithm, based on disjoint sum of product (DSOP) algorithm. The algorithm was synthesized and applied for the first time to the determination of reliability expression of a substation to determine reliability indices and costs of different substation arrangements. It deals with the implementation and synthesis of a new designed algorithm for DSOP implemented using C/C++, incorporating parallel problem solving capability and overcoming the disadvantage of Monte Carlo simulation which is the lengthy computational time to achieve satisfactory statistical convergence of reliability index values. The major highlight of this research being that the time consuming procedures of the DSOP solution generated for different substation arrangements using the proposed method is found to be significantly lower in comparison with the time consuming procedures of Monte Carlo-simulation solution or any other method used for the reliability evaluation of substations in the existing literature. This implementation gives the possibility of RBD simulation for different substation configurations in C/C++ using their path-set Boolean expressions mapped to probabilistic domain and result in simplest DSOP which is on a one-to-one correspondence with reliability expression. This software tool is capable of handling and modeling a large, repairable system. Additionally, through it’s intuitive interface it can be easily used for industrial and commercial power systems. With simple Boolean expression for a configuration’s RBD inputed, users can easily define a power system utilizing a RBD and, through a fast and efficient built-in simulation engine, required reliability expressions and indexes can be easily obtained.

The extended permutohedron on a transitive binary relation

For a given transitive binary relation e on a set E, the transitive closures of open (i.e., co-transitive in e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron one. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the generalized permutohedron on a partially ordered set on the one hand, and the bipartition lattice on a set on the other hand. We obtain a precise description of the completely join-irreducible (resp., meet-irreducible) elements of Reg(e) and the arrow relations between them. In particular, we prove that –Reg(e) is the Dedekind–MacNeille completion of the poset Clop(e);–Every open subset of e is a set-theoretical union of completely join-irreducible clopen subsets of e;–Clop(e) is a lattice iff every regular closed subset of e is clopen, iff e contains no “square” configuration, iff Reg(e)=Clop(e);–If e is finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is a bounded homomorphic image of a free lattice, iff e is a disjoint sum of antisymmetric transitive relations and two-element full relations. We illustrate our results by proving that, for n≥3, the congruence lattice of the lattice Bip(n) of all bipartitions of an n-element set is obtained by adding a new top element to a Boolean lattice with n⋅2n−1 atoms. We also determine the factors of the minimal subdirect decomposition of Bip(n), and we prove that if n≥3, then none of them embeds into Bip(n) as a sublattice.

Disjoint sum forms in reliability theory

The structure function f of a binary monotone system is assumed to be known and given in a disjunctive normal form, i.e. as the logical union of products of the indicator variables of the states of its subsystems. Based on this representation of f, an improved Abraham algorithm is proposed for generating the disjoint sum form of f. This form is the base for subsequent numerical reliability calculations. The approach is generalized to multivalued systems. Examples are discussed.

Delay testable logical circuit design

Specific physical defects of high performance circuits manifest themselves as path delay faults (PDFs). A pair (v1, v2) of test patterns is required to detect a PDF. A PDF is robust testable if there is a test pair on which the fault manifestation does not depend on delays of other circuit paths. A PDF is non robust testable if manifestation of the fault on any test pair is possible only when all other paths of the circuit are fault free. For high quality delay testing it is desirable to detect delay of any path regardless of delays of other paths. In this paper, synthesis of a circuit by covering each nonterminal node of the reduced ordered binary decision diagram (ROBDD) or free binary decision diagram (Free BDD) system with the original subcircuit from gates is suggested. Delay testability of such circuits is investigated. It is specified that the PDF of each path of the circuit is detected without injecting additional inputs into the circuit. It is well known that the BDD is a directed acyclic graph based on the Shannon decomposition in each nonterminal node v: 0 1 i i x x v i v i v f x f x f = = = ∨ . Here fv is the function corresponding to the node v, and the decomposition variable xi marks the node v. The dashed edge drops in the node corresponding to the function 0 i x v f = , and the bold edge drops in the node corresponding to the function 1 i x v f = . The BDD is called ordered (OBDD) if variables are encountered in the same order in all paths connecting the BDD root with the terminal node. The BDD is called reduced (RBDD) if it does not contain either isomorphic subgraphs or nonterminal nodes such that 0 1 i i x x v v f f = = = . An example of reduced and ordered BDD (ROBDD) is shown in Fig. 1. When deriving the Free BDD, a suitable decomposition variable is chosen for each nonterminal node v independently of other nonterminal nodes. That may cut the number of Free BDD of nonterminal nodes in comparison with the ROBDD (for the same function). Let 1 { ,..., } m F f f = be a system of Boolean functions describing combinational circuit behavior. Derive the ROBDD using the same order of variables for each Boolean function from F. Join isomorphic subgraphs in different ROBDDs. Combine different ROBDDs of 1-terminal nodes into one 1-terminal node and their 0-terminal nodes into one 0-terminal node. As a result, we obtain the graph with m roots and two terminal nodes. This graph represents a system of m Boolean functions. It is called the shared ROBDD. Without loss of generality, we will further consider systems with one function as an example. In Fig. 1, the ROBDD for one Boolean function is shown. The ROBDD represents the disjoint sum of products (DSoP) of the function f as follows:

On an Optimization Technique Using Binary Decision Diagram

Two-level logic minimization is a central problem in logic synthesis, and has applications in reliability analysis and automated reasoning. This paper represents a method of minimizing Boolean sum of products function with binary decision diagram and with disjoint sum of product minimization. Due to the symbolic representation of cubes for large problem instances, the method is orders of magnitude faster than previous enumerative techniques. But the quality of the approach largely depends on the variable ordering of the underlying BDD. The application of Binary Decision Diagrams (BDDs) as an efficient approach for the minimization of Disjoint Sums-of-Products (DSOPs). DSOPs are a starting point for several applications. The use of BDDs has the advantage of an implicit representation of terms. Due to this scheme the algorithm is faster than techniques working on explicit representations and the application to large circuits that could not be handled so far becomes possible. Theoretical studies on the influence of the BDDs to the search space are carried out. In experiments the proposed technique is compared to others. The results with respect to the size of the resulting DSOP are as good or better as those of the other techniques.

Association schemes and MacWilliams dualities for generalized Niederreiter–Rosenbloom–Tsfasman posets

Let $\mathbf{P}$ be a poset on the set $[m] \times [n]$, which is given as the disjoint sum of posets on `columns' of $[m]\times[n]$, and let $\check{\mathbf P}$ be the dual poset of $\mathbf{P}$. Then $\mathbf{P}$ is called a generalized Niederreiter&#

The [formula omitted]-discrepancy of a poset

Linear discrepancy and weak discrepancy have been studied as a measure of fairness in giving integer ranks to the points of a poset. In linear discrepancy, the points are totally ordered, while in weak discrepancy, ties in rank are permitted. In this paper we study the t -discrepancy of a poset, which can be viewed as a hybrid between linear and weak discrepancy, in which at most t points can receive the same rank. Interestingly, t -discrepancy is not a comparability invariant while both linear and weak discrepancy are. We show that for a poset P and positive integers t and k , the decision problem of determining whether the t -discrepancy of P is at most k is NP-complete in general; however, we give a polynomial time algorithm for computing the t -discrepancy of a semiorder. We also find the t -discrepancy for posets that are the disjoint sum of chains.

Topology of real cubic fourfolds

A solution of the problem of topological classification of real cubic fourfolds is given. It is proven that the real locus of a real non-singular cubic fourfold is diffeomorphic either to a connected sum ℝP4 #i(S2 × S2)#j(S1 × S3) or to a disjoint union ℝP4 ⊔ S4. La voie la plus courte et la meilleure entre deux vérités de domaine réel passe souvent par le domaine imaginaire.J. S. Hadamard, Essai sur la psychologie de l'invention dans le domainemathématique, Edition Gaithier-Villars, 1975, p. 114.

A topos foundation for theories of physics: IV. Categories of systems

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this program for a single system. In the present paper, we turn to considering a collection of systems; in particular, we are interested in the relation between the topos representation for a composite system and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category.