Linear Ramsey Numbers for Bounded-Degree Hypergrahps

We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter, Jr., The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), pp. 239–243] showed in 1983. Our proof demonstrates the potential of a new regularity lemma by [Y. Ishigami, A simple regularization of hypergraphs, preprint, arXiv:math/0612838 (2006)].     

Hybrid Systems and Hybrid Computation 1st Part: Hybrid Systems

In the first part of this paper we will give a short historical survey of the field of hybrid systems, a precise definition of a hybrid system and some comments on the definition. In a second paper (Hybrid systems and hybrid computation – 2nd part: Hybrid computation) we will concentrate on a particular aspect of the theory closely related to scientific computation, that we have called hybrid computation.     

(Co)bordism groups in quantum super PDE's. III: Quantum super Yang–Mills equations

In this third part of a series of three papers devoted to the study of geometry of quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. I: quantum supermanifolds, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.007; (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.008], we apply our theory, developed in the first two parts, to quantum super Yang–Mills equations and quantum supergravity equations. For such equations we determine their integral bordism groups, and by using some surgery techniques, we obtain theorems of existence of global solutions, also with nontrivial topology, for Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity equations we prove existence of solutions of the type quantum black holes evaporation processes just by using an extension to quantum super PDEs of our theory of integral (co)bordism groups. Our proof is constructive, i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for the quantum super Yang–Mills equation, is given. Finally it is proved that quantum super PDE's contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum super PDE's are (nonlinear) generalizations of Dirac quantized superclassical PDE's. Applications of this result to free quantum super Yang–Mills equations are given.     

Comments on “Application of generalized differential transform method to multi-order fractional differential equations”, Vedat Suat Erturk,Shaher Momani,Zaid Odibat [Commun Nonlinear Sci Numer Simul 2008;13:1642–54]

In this note, we would like to point some similarities between the study [Erturk VS, Momani S, Odibat Z. Application of generalized differential transform method to multi-order fractional differential equations. Commun Nonlinear Sci Numer Simul. doi:10.1016/j.cnsns.2007.02.006] with the already existing one [Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method. Chaos Soliton Fract. 10.1016/j.chaos.2006.09.004].     

A note on the Dixon formula for a finite hypergeometric series

This is a note on the paper [4, B.M. Brown, M.S.P. Eastham, Extensions of the Kummer evaluation of a finite hypergeometric series, J. Comput. Appl. Math., to appear, doi:10.1016/j.cam.2005.06.018].     

Nonarchimedean Cantor set and string

We construct a nonarchimedean (or p-adic) analogue of the classical ternary Cantor set . In particular, we show that this nonarchimedean Cantor set is self-similar. Furthermore, we characterize as the subset of 3-adic integers whose elements contain only 0’s and 2’s in their 3-adic expansions and prove that is naturally homeomorphic to . Finally, from the point of view of the theory of fractal strings and their complex fractal dimensions [7, 8], the corresponding nonarchimedean Cantor string resembles the standard archimedean (or real) Cantor string perfectly. Dedicated to Vladimir Arnold, on the occasion of his jubilee     

The Bunce-Deddens algebras as crossed products by partial automorphisms

We describe both the Bunce-DeddensC ^*-algebras and their Toeplitz versions, as crossed products of commutativeC ^*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy of the set of natural numbers , fitted together in such a way that is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on . From this we deduce, by taking quotients, that the Bunce-DeddensC ^*-algebras are isomorphic to the (classical) crossed product of the algebra of continuous functions on the Cantor set by the odometer map.     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

Multivariate skew-normal distributions with applications in insurance

In this paper, we discuss the skew-normal distribution as an alternative to the classical normal one in the context of both risk measurement and capital allocation. As main risk measure, we consider the tail conditional expectation (TCE). Hence, we investigate an allocation formula based on the TCE, but we also consider Wang’s [Wang, S., 2002. A set of new methods and tools for enterprise risk capital management and portfolio optimization. Working paper. SCOR reinsurance company (www.casact.com/pubs/forum/02sforum/02sf043.pdf)] allocation formula.     

Hom complexes and homotopy in the category of graphs

In this extended abstract we develop a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product. We show that graph ×-homotopy is characterized by the topological properties of the so-called Hom complex, a functorial way to assign a poset to a pair of graphs. Along the way we establish some structural properties of Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. We end with a discussion of graph homotopies arising from other internal homs, including the construction of ‘A-theory’ associated to the cartesian product in the category of reflexive graphs. For proofs and further discussions we refer the reader to the full paper [Anton Dochtermann. Hom complexes and homotopy theory in the category of graphs. arXiv:math.CO/0605275].     

On the structures of generating iterated function systems of Cantor sets

A generating IFS of a Cantor set F is an IFS whose attractor is F. For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in under the open set condition (OSC). If dim_HF<1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study.     

Modified quasilinearization algorithm for optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.A modified quasilinearization algorithm is developed. Its main property is the descent property in the performance indexR, the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of the scaling factor (or stepsize) in the system of variations. The stepsize is determined by a one-dimensional search on the performance indexR. Since the first variation R is negative, the decrease inR is guaranteed if is sufficiently small. Convergence to the solution is achieved whenR becomes smaller than some preselected value.Here, the state inequality constraint is handled in a direct manner. A predetermined number and sequence of subarcs is assumed and, for the time interval for which the trajectory of the system lies on the state boundary, the control is determined so that the state boundary is satisfied. The state boundary and the entrance conditions are assumed to be linear inx and , and the modified quasilinearization algorithm is constructed in such a way that the state inequality constraint is satisfied at each iteration and along all of the subarcs composing the trajectory.At first glance, the assumed linearity of the state boundary and the entrance conditions appears to be a limitation to the theory. Actually, this is not the case. The reason is that every constrained minimization problem can be brought to the present form through the introduction of additional state variables.In order to start the algorithm, some nominal functionsx(t),u(t), and nominal multipliers (t), (t), , must be chosen. In a real problem, the selection of the nominal functions can be made on the basis of physical considerations. Concerning the nominal multipliers, no useful guidelines have been available thus far. In this paper, an auxiliary minimization algorithm for selecting the multipliers optimally is presented: the performance indexR is minimized with respect to (t), (t), , . Since the functionalR is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear and, therefore, can be solved without difficulty.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Dr. R. R. Iyer and Mr. A. K. Aggarwal for helpful discussions as well as analytical and numerical assistance. This paper is a condensation of the investigations described in Refs. 1–2.     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

On representation and aggregation of social evaluations in computational trust and reputation models

Interest for computational trust and reputation models is on the rise. One of the most important aspects of these models is how they deal with information received from other individuals. More generally, the critical choice is how to represent and how to aggregate social evaluations. In this article, we make an analysis of the current approaches of representation and aggregation of social evaluations under the guidelines of a set of basic requirements. Then we present two different proposals of dealing with uncertainty in the context of the Repage system [J. Sabater, M. Paolucci, R. Conte, Repage: Reputation and image among limited autonomous partners, Journal of Artificial Societies and Social Simulation 9 (2). URL http://jasss.soc.surrey.ac.uk/9/2/3.html], a computational module for management of reputational information based on a cognitive model of imAGE, REPutation and their interplay already developed by the authors. We finally discuss these two proposals in the context of several examples.     

On the topological structure of univoque sets

Erd?s, Horváth and Joó discovered some years ago that for some real numbers 1<q<2 there exists only one sequence ci of zeroes and ones such that ∑ciq−i=1. Subsequently, the set U of these numbers was characterized algebraically in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions 1=∑q−ni and related problems, Bull. Soc. Math. France 118 (1990) 377-390] and [V. Komornik, P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar. 44 (2) (2002) 195-216]. We establish an analogous characterization of the closure of U. This allows us to clarify the topological structure of these sets: is a countable dense set of , so the latter set is perfect. Moreover, since U is known to have zero Lebesgue measure, is a Cantor set.     

On Cantor cubes

Some decision making models are discussed from the point of view of neurophysiology and quantum mechanics. The main feature of these models is that a straight line segment is replaced by the Cantor set. In this direction, many interesting results have been obtained by methods of number theory, p-adic analysis, and the theory of dynamical systems. Some generalizations of existing models are also discussed, which are formulated in terms of the so-called Cantor cubes, that is, Cartesian products of infinitely many standard two-point spaces D (as is known, the Cantor cube $D^{\aleph _0 }$ is homeomorphic to the Cantor set). This approach involves difficulties caused by the nonmetrizability and nonseparability of the Cantor cubes D ^m for m > ?₀ and nonseparable for m > c, respectively.     

On the rate of convergence to a stable limit law. II

Summary In [5] Feller states the following assertion: As soon as we leave the domain of applicability of the central limit theorem we find ourselves on practically unknown terrain; the problems receive an entirely new aspect and no systematic tools have as yet been developed for treating the theory, After the appearance of Volume II [6], many papers were published on stable distributions. The reader will find several references in [8].In the same paper [5] Feller states the following assertions: The theory becomes the simpler the fewer moments are finite. This assertion was stated in a paper in which he proved a law of the iterated logarithm for the heavy tail of stable distributions. (See also Chapter 8 in [8].) We shall show that the second assertion also holds in the study of the rate of convergence to a stable limit distribution. In several examples we shall compare our results with the results of Christoph [1–3] and Dubinskaite [4].Department of Mathematics, University of Leiden, Holland. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 26, No. 3, pp. 482–487, July–September, 1986.     

On a packet scheduling problem for smart antennas and polyhedra defined by circular-ones matrices

In [E. Amaldi, A. Capone, F. Malucelli, Circular Arc Models and Algorithms for Packet Scheduling in Smart Antennas, IV ALIO/EURO Workshop on Applied Combinatorial Optimization, see: http://www-di.inf.puc-rio.br/~celso/artigos/pucon.ps, E. Amaldi, A. Capone, F. Malucelli, Discrete models and algorithms for packet scheduling in smart antennas, 2nd Cologne Twente Workshop on Graphs and Combinatorial Optimization] E. Amaldi et al. posed a combinatorial optimization problem that arises when scheduling packets in a smart antenna. The objective is to partition the set of users so as to minimize the number of time slots needed to transmit all the given packets. Here we will present a polynomial time algorithm for solving this packet scheduling problem. More generally, the algorithm solves an integer decomposition problem for polyhedra determined by a circular-ones constraint matrix, which might make it interesting also for other cyclic scheduling problems.