Super edge-magic deficiency of join-product graphs

A graph $G$ is called \textit{super edge-magic} if there exists a bijective function $f$ from $V(G) \cup E(G)$ to $\{1, 2, \ldots, |V(G) \cup E(G)|\}$ such that $f(V(G)) = \{1, 2, \ldots, |V(G)|\}$ and $f(x) + f(xy) + f(y)$ is a constant $k$ for every edge $xy$ of $G$. Furthermore, the \textit{super edge-magic deficiency} of a graph $G$ is either the minimum nonnegative integer $n$ such that $G \cup nK_1$ is super edge-magic or $+\infty$ if there exists no such integer. \emph{Join product} of two graphs is their graph union with additional edges that connect all vertices of the first graph to each vertex of the second graph. In this paper, we study the super edge-magic deficiencies of a wheel minus an edge and join products of a path, a star, and a cycle, respectively, with isolated vertices.Comment: 11 page

An Abstract Approach to Stratification in Linear Logic

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic

Strongly indexable graphs

AbstractA (p, q)-graph G = (V, E) is said to be strongly k-indexable if it admits a strong k-indexer viz., an injective function ƒ:V→{0, 1, 2, …, p − 1} such that ƒ(x)+ƒ(y)=ƒ+(xy)ϵ⨍+(E)={k, k + 1, k + 2, …, k + q − 1}.In the terms defined here, k will be omitted if it happens to be unity. We find that a strongly indexable graph has exactly one nontrivial component which is either a star or has a traingle. In any strongly k-indexable graph the minimum point degree is at most 3. Using this fact we show that there are exactly three strongly indexable regular graphs, viz. K2, K3 and K2xK3. If an eulerian (p, q)-graph is strongly indexable then q ϵ 0, 3(mod4)

Experimental investigation of cutting forces in high-feed milling of titanium alloy

Titanium super alloys are often used in the chemical and aerospace industries, especially because of financial savings. resulting primarily from cheaper operation of equipment. Machinability of titanium alloys is more difficult than that of other metals. In addition, the low thermal conductivity causes the alloy to stick to the cuffing edge of the cutting tool, thereby causing it to become dull faster. The article deals with the experimental evaluation of cutting forces and the design of suitable cutting parameters for the machining of the UNS R56260 titanium alloy with high-feed milling technology. Testing was carried out in climb and conventional milling under different cutting conditions. The cutting components of forces Fx, Fy, Fz were measured and evaluated. The results of the measurements were processed into a graphical form and suitable cutting conditions were designed in terms of the acting cutting forces.Web of Science141958

A Study of Optimal 4-bit Reversible Toffoli Circuits and Their Synthesis

Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! {\approx} 2^{44} functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesizes all optimal implementations. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of all optimal 4-bit permutations, synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions; we compose a list of the hardest permutations to synthesize, and show distribution of optimal circuits. We further illustrate that our proposed approach may be extended to accommodate physical constraints via reporting LNN-optimal reversible circuits. Our results have important implications in the design and optimization of reversible and quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.Comment: arXiv admin note: substantial text overlap with arXiv:1003.191