Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

EXPLORING GIFTED AND TALENTED MUSLIMS STUDENT’S PERFORMANCE USING GEOGEBRA IN TEACHING AND LEARNING MATHEMATICS

GeoGebra is a teaching tool that educators use in their lesson plans to improve the quality of teaching and learning. Instead of drawing on a sheet of paper, students can design a graph, adjust the actual graph shape, and examine the impact of changing graph pattern using GeoGebra in mathematics teaching and learning. Furthermore, students can keep all of their work materials in a structured manner for future reference. GeoGebra will make a school lecture more interesting, exciting, creative, and innovative. The goal of this study was to analyse the effects of GeoGebra software in Mathematics achievement in respect to quadratic functions among gifted and talented Muslims student’s at Kolej GENIUS Insan, Universiti Sains Islam Malaysia. The maximum or minimum point of quadratic function was determined by using GeoGebra software, and the characteristics of quadratic expressions in one variables was also identified. The results illustrate that the graph quadratic expression has the highest point or the lowest point based the values of coefficient a on the quadratic function. For the graph function with negative values of coefficient a on a quadratic function, there are highest values of coordinates x and y, which also known as maximum point, while the graph function with positive values of a on a quadratic function, there are lowest values of coordinates x and y, which also known as minimum point. When students utilize GeoGebra software, their performance in calculating the minimum and maximum points on quadratic functions improves. Keywords: Geogebra; Integrated naqli ‘aqli gifted education; Gifted muslims student; Mathematics achievement; Quadratic functions

Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

Basis Existence of Internal n-Graph Space

Graph of real valued continuous function with special addition and multiplication has already proven that is isomorphic to real number system. Furthermore, the graph of continous real valued function forms a field. The aim of this research was to generalize such concept to its n-tuple Cartesian Product and to prove that interchange of basis still able to be executed. The result of this research is n-tuple Cartesian Product of graph function forms a vector space over and interchange of basis still able to be executed

Control Digital System Function Set Defining While Debugging Tests Development

Digital control systems are considered, the functioning of which can be represented as a sequence of functions from a finite alphabet. For such systems projects debugging by simulation it is necessary to generate some set of tests for the applying on the simulated system to verify that it is functioning correctly. This paper is devoted to the development of test sets for function successions correctness. It is shown that on admissible function successions partly defined semigroup exists. There exists also word set on limited alphabet of functions, and they could be defined by some right liner grammar. Admissible successions are formally described by the graph of functions. Such a graph defines admissible functions for all digital system states. Digital system function set development is proposed in a way that admissible function successions could be defined as a graph. If the admissibility of two functions fulfillment one after another depends on previously fulfilled functions and the digital system internal state, then some functions should be divided into several subfunctions. The method of such a process is described. Developed graph of functions together with input interaction set for each digital system function define specification for digital system behavior. Proposed method is illustrated on the drawing machine control digital system functions development. The method of test set development on graph function is proposed.

On a weak form of weak quasi-continuity

A weak form of weak quasi-continuity, which we call subweak quasi-continuity, is introduced. It is shown that subweak quasi-continuity is strictly weaker than weak quasi-continuity. Subweak quasi-continuity is used to strengthen several results in the literature concerning weak quasi-continuity. Specifically, results concerning the graph, graph function, and restriction of a weakly quasi-continuous function are extended slightly. Also, a result concerning weakly quasi-continuous retractions is strengthened.

On the Edge Distribution of a Graph

We investigate a graph function which is related to the local density, the maximal cut and the least eigenvalue of a graph. In particular it enables us to prove the following assertions.Let p [ges ] 3 be an integer, c ∈ (0, 1/2) and G be a Kp-free graph on n vertices with e [les ] cn2 edges. There exists a positive constant α = α (c, p) such that:(a) some [lfloor ]n/2[rfloor ]-subset of V (G) induces at most (c-4 − α) n2 edges (this answers a question of Paul Erdős);(b) G can be made bipartite by the omission of at most (c-2 − α) n2 edges.

θ-regular spaces

In this paper we define a topological spaceXto beθ-regular if every filterbase inXwith a nonemptyθ-adherence has a nonempty adherence. It is shown that the class ofθ-regular topological spaces includes rim-compact topological spaces and thatθ-regularH(i)(Hausdorff) topological spaces are compact (regular). The concept ofθ-regularity is used to extend a closed graph theorem of Rose [1]. It is established that anr-subcontinuous closed graph function into aθ-regular topological space is continuous. Another sufficient condition for continuity of functions due to Rose [1] is also extended by introducing the concept of almost weak continuity which is weaker than both weak continuity of Levine and almost continuity of Husain. It is shown that an almost weakly continuous closed graph function into a strongly locally compact topological space is continuous.