Exclusive Sum Research Articles

Sum Index, Difference Index and Exclusive Sum Number of Graphs

We consider two recent conjectures made by Harrington, Henninger-Voss, Karhadkar, Robinson and Wong concerning relationships between the sum index, difference index and exclusive sum number of graphs. One conjecture posits an exact relationship between the first two invariants; we show that in fact the predicted value may be arbitrarily far from the truth in either direction. In the process we establish some new bounds on both the sum and difference index. The other conjecture, that the exclusive sum number can exceed the sum index by an arbitrarily large amount, follows from known, but non-constructive, results; we give an explicit construction demonstrating it. Simultaneously with the first preprint of this paper appearing, Harrington et al. updated their preprint with two counterexamples to the first conjecture; however, their counterexamples only give a discrepancy of 1, and only in one direction. They therefore modified the conjecture from an equality to an inequality; our results show that this is still false in general.

Quark-hadron duality for heavy meson mixings in the \u2019t Hooft model

We study local quark-hadron duality and its violation for the {D}^0-{overline{D}}^0 , {B}_d^0-{overline{B}}_d^0 and {B}_s^0-{overline{B}}_s^0 mixings in the ’t Hooft model, offering a laboratory to test QCD in two-dimensional spacetime together with the large-Nc limit. With the ’t Hooft equation being numerically solved, the width difference is calculated as an exclusive sum over two-body decays. The obtained rate is compared to inclusive one that arises from four-quark operators to check the validity of the heavy quark expansion (HQE). In view of the observation in four-dimensions that the HQE prediction for the width difference in the {D}^0-{overline{D}}^0 mixing is four orders of magnitude smaller than the experimental data, in this work we investigate duality violation in the presence of the GIM mechanism. We show that the order of magnitude of the observable in the {D}^0-{overline{D}}^0 mixing is enhanced in the exclusive analysis relative to the inclusive counterpart, when the 4D-like phase space function is used for the inclusive analysis. By contrast, it is shown that for the {B}_d^0-{overline{B}}_d^0 and {B}_s^0-{overline{B}}_s^0 mixings, small yet non-negligible corrections to the inclusive result emerge, which are still consistent with what is currently indicated in four-dimensions.

Compression with Wildcards: From CNFs to Orthogonal DNFs by Imposing the Clauses One-by-One

Abstract We present a novel technique for converting a Boolean conjunctive normal form (CNF) into an orthogonal disjunctive normal form (DNF), aka exclusive sum of products. Our method (which will be pitted against a hardwired command from Mathematica) zooms in on the models of the CNF by imposing its clauses one by one. Clausal imposition invites parallelization, and wildcards beyond the common don’t-care symbol compress the output. The method is most efficient for few but large clauses. Generalizing clauses, one can in fact impose superclauses. By definition, superclauses are obtained from clauses by substituting each positive literal by an arbitrary conjunction of positive literals.

Cost Optimization Technique for Quantum Circuits

In this paper, an attempt is made to present a method of quantum cost minimization or optimization technique for quantum reversible circuits using proposed merger rules in Exclusive Sum of Product (ESOP) method. These modified ESOP methods are used to minimize the quantum circuits. We found that the quantum cost is drastically decreased than the previous ESOP method. It will be easy to find the quantum cost and quantum gate optimized quantum circuits implementation. It will also reduce quantum error while the quantum circuit is executed in real quantum processor.

Perfect Concurrent Fault Detection in CMOS Logic Circuits Using Parity Preservative Reversible Gates

Reversible logic has 100% fault observability meaning that a fault in any circuit node propagates to the output stage. In other words, reversible circuits are latent-fault-free. Our motivation is to incorporate this unique feature of reversible logic to design CMOS circuits having perfect or 100% Concurrent Error Detection (CED) capability. For this purpose, we propose a new fault preservative reversible gate library called Even Target - Mixed Polarity Multiple Control Toffoli (ET-MPMCT). By using ET-MPMCT, we ensure that the evenness/oddness of applied 1’s at input, is preserved at all levels of a circuit including output level unless there is a faulty node. A single fault always destroys the parity of input at the output. Our design strategy has two steps for a given function: 1) implement the function with our proposed reversible ET-MPMCT gate library; and 2) apply reversible-to-CMOS gate conversion. For the first step, we propose two approaches. For our first approach in step 1, we first need to have a reversible form of the given function if it is irreversible. Then, synthesize the reversible function using Mixed Polarity Multiple Control Toffoli (MPMCT) gate library by conventional reversible logic synthesis techniques. Finally, the synthesized circuit is converted to ET-MPMCT constructed circuit which is fault preservative. Our second approach, is an ESOP - Exclusive Sum of Products - based synthesis approach modified for our proposed fault preservative ET-MPMCT gate library. It does work with both irreversible and reversible functions. We synthesize our circuits with both approaches in step 1 and choose the circuit with lower number of reversible gates to be fed to step 2 of our design strategy. In second step of our design strategy, we convert our fault preservative reversible circuits into their CMOS counterparts. The performance of our designs is compared with other CED schemes in the literature in terms of area, power consumption, delay and detection rate. Simulations are done with Cadence Genus tool using TSMC 40nm technology. Clearly, results are in favor of our proposed techniques.

Characterisation of Graphs with Exclusive Sum Labelling

A sum graph G is a graph with a mapping of the vertex set of G onto a set of positive integers S in such a way that two vertices of G are adjacent if and only if the sum of their labels is an element of S. In an exclusive sum graph the integers of S that are the sum of two other integers of S form a set of integers that label a collection of isolated vertices associated with the graph G. A graph bears a k-exclusive sum labelling (abbreviated k-ESL), if the set of isolated vertices is of cardinality k.In this paper, observing that the property of having a k-ESL is hereditary, we provide a characterisation of graphs that have a k-exclusive sum labelling, for any k≥1, in terms of describing a universal graph for the property.

Some new formulae on Δ–optimum exclusive sum labeling of certain trees

All sum graphs are disconnected. The notions of sum labeling and sum graphs were introduced by Harary [1] and later extended to include all integers [2]. A mapping L is called a sum labeling a graph H(V (H), E(H)) if it is an injection from V(H) to a set of positive integers such that xy ∈E(H) if and only if there exists a vertex w ∈V(H) such that L(w)=L(x) + L(y). In this case w is called a working vertex. We call L as an exclusive sum labeling of a graph G if it is sum labeling of for some non negative integer r and G contains no working vertex. In general, a graph G will require some isolated vertices to be labeled exclusively. The least possible number of such isolated vertices is called exclusive sum number of G, denoted by ∈(G). An exclusive sum labeling of a graph G is said to be optimum if it labels G exclusively by using ∈(G) isolated vertices. In case ∈(G)=Δ(G), where D(G) denotes the maximum degree of vertices in G, the labeling is called Δ- optimum exclusive sum labeling. In this paper we developed some new formulae to find Δ- optimum exclusive sum labeling of certain trees.

Some Results on Exclusive Sum Labelings of Hypergraphs

We generalize the concept of exclusive sum labelings of graphs (cf. Miller et al., JCMCC 55:137---148, 2005) and determine the exclusive sum number for several classes of hypergraphs. The results disclose a significant difference between graphs (i.e. 2-uniform hypergraphs) and "genuine" hypergraphs.

REALISTIC OPINION AGGREGATION: LEHRER-WAGNER WITH A FINITE SET OF OPINION VALUES

AbstractAn allocation problem is a type of aggregation problem in which the values of individuals' opinions on some set of variables (canonically a set of mutually exclusive and exhaustive possibilities) sum to a constant. This paper shows that for realistic allocation problems, namely ones in which the set of possible opinion values is finite, the only universal aggregation methods that satisfy two commonly invoked conditions are the dictatorial ones. The two conditions are, first, that the aggregate opinion on any variable depends only on the individuals' opinions on that variable and not on their opinions on any other variable and, secondly, that the aggregate opinion minimally preserves unanimous individual opinion. Connections between this result and similar ones in the literature on judgment aggregation are explored and their significance assessed.

Young Child Homicide and Accidental Death Rates in the United States, 1940-2005: Classification Issues in Mutually Exclusive Events

Events that are mutually exclusive, collectively exhaustive, and have a constant sum must be negatively correlated. This study examined whether non-motor vehicle accidental deaths and homicides in young US children displayed a period of time when this rule governing mutually exclusive events was applicable. Homicide and non-motor vehicle accident mortality rates in boys and girls, aged 1 to 4 years old, in the United States from 1940 to 2005 were analyzed. Homicide mortality rates increased sharply in young boys when the accident mortality rate dropped to about 25/100,000; and in young girls, when the accident mortality dropped to about 18/100,000. This increase in child homicide mortality rates corresponded to a time period when the sum of homicide rates and non-motor vehicle accident rates in these children were relatively constant, making these rates of unnatural deaths mutually exclusive, collectively exhaustive, and having a nearly constant sum. Homicide rates in young US children were relatively stable both before and after this critical constant sum time period. These findings suggest that the increase in homicide rates in young US children appear to have reflected the necessary negative correlation between mutually exclusive, collectively exhaustive, and constant sum events, rather than an actual increase in societal violence directed against young children.

Some results on exclusive sum graphs

Let N(Z) denote the set of all positive integers (integers). The sum graph G+(S) of a finite subset S⊂N(Z) is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S⊂N(Z). A sum labelling S is called an exclusive sum labelling if u+v∈S∖V(G) for any edge uv∈E(G). We say that G is labeled exclusively. The least number r of isolated vertices such that G∪rK1 is an exclusive sum graph is called the exclusive sum number e(G) of graph G. In this paper, we discuss the exclusive sum number of disjoint union of two graphs and the exclusive sum number of some graph classes.

Exclusive Sum Labeling of Graphs: A Survey

All sum graphs are disconnected. In order for a connected graph to bear a sum labeling, the graph is considered in conjunction with a number of isolated vertices, the labels of which complete the sum labeling for the disjoint union. The smallest number of isolated vertices that must be added to a graph H to achieve a sum graph is called the sum number of H; it is denoted by σ(H). A sum labeling which realizes as a sum graph is called an optimal sum labeling of H.In this paper we survey a new type of labeling based on summation, the exclusive sum, labeling. A sum labeling L is called exclusive sum labeling with respect to a subgraph H of G if L is a sum labeling of G where H contains no working vertex. The exclusive sum number ∊(H) of a graph H is the smallest number r such that there exists an exclusive sum labeling L which realizes as a sum graph. A labeling L is an optimal exclusive sum labeling of a graph H if L is a sum labeling of H ≼ K∊(H) and H contains no working vertex.

Linear cofactor relationships in Boolean functions

This paper describes linear cofactor relationships (LCRs), which are defined as the exclusive sums of cofactors with respect to a pair of variables in Boolean functions. These relationships subsume classical symmetries and single-variable symmetries. The paper proposes an efficient algorithm to detect LCRs and discusses their potential applications in Boolean matching, minimization of decision diagrams, synthesis of regular layout-friendly logic circuits, and detection of support-reducing bound sets

Generalized partially-mixed-polarity Reed-Muller expansion and its fast computation

Generalized partially-mixed-polarity Reed-Muller (GPMPRM) expansion, a canonical subfamily of exclusive sum of products (ESOP), is presented. An efficient algorithm in two-dimensional data flow is proposed for computation of the GPMPRM forms. MCNC benchmark experimental results show that the minimal GPMPRM forms of these functions, on the average, have similar number of terms to their sum of products (SOP) counterparts while there are many functions for which the GPMPRM circuits are much smaller.