Commutative lmc algebras with discrete spectrum

SupposeE is a topological algebra with non-empty spectrum Gel'fand spaceM(E) andE M(E)) the algebra of allC-valued continuous functions onM(E). EndowE M(E) with the topologies “c”, “e” of compact resp. equicontinuous convergence. ThenE M (E) characterizes all unital (commutative) semisimple complete lmc algebras with discrete spectrum, while all unital uniform complete lmc algebras with dispersed spectrum are of the formE(E)). The first result may fail if completeness is dropped. The second one fails if “e” is replaced by “c” even ifE is complete. The part of the work due to the 2nd author was carried out during her 3-month visit at the Institute of Mathematics, Univ. of Münster (Germany) in Spring of 1995. The warm hospitality from Professor Dr. G. Maltese and the financial support from a DAAD grant are thankfully acknowledged.     

Quasi-parabolic analytic transformations of <Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>. Parabolic manifolds

In [Rong, F., Quasi-parabolic analytic transformations of C ⁿ , J. Math. Anal. Appl. 343 (2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of C ⁿ . Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.     

Optimal buffer size for a stochastic processing network in heavy traffic

We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b>0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A “control cost” related to the dynamically controlled service rate, a “congestion cost” which depends on the queue length and a “rejection penalty” for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b ^*>0. When the buffer size b>0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005)). To obtain a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we analyze a family of ordinary differential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b ^*>0. A.P. Weerasinghe’s research supported by US Army Research Office grant W911NF0510032.     

Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the s-dimensional discrepancy of nonlinear congruential pseudorandom numbers over the residue ring \Bbb Z_M{\Bbb Z}_M modulo M for an “almost squarefree” integer M. It is useful to recall that almost all integers are of this type. Moreover, if the generator is associated with a permutation polynomial over \Bbb Z_M{\Bbb Z}_M we obtain a stronger bound “on average” over all initial values. This bound is new even in the case when M = p is prime.     

Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

The Splitting Problem for Coalgebras: A Direct Approach

In this note we give a new and elementary proof of a result of Năstăsescu and Torrecillas (J. Algebra, 281:144–149, 2004) stating that a coalgebra C is finite dimensional if and only if the rational part of any right module M over the dual algebra is a direct summand in M (the splitting problem for coalgebras). Research supported by a CNCSIS BD-type grant, and by the bilateral project BWS04/04 “New Techniques in Hopf Algebra Theory and Graded Ring Theory” of the Flemish and Romanian governments.     

Linear forms and quadratic uniformity for functions on ℤ<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript>

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b], [GW10c], we gave a complete answer to this question for groups of the form G = F _p ⁿ , provided p is not too small. In ℤ _N , substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm || ·||_U³{\left\| \cdot \right\|_{{U^3}}} requires a more sophisticated “local” formulation. When N is prime, ℤ _N is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a]. Moreover, we obtain a doubly exponential bound.     

Structural stability and robustness to bounded rationality for non-compact cases

We study the model M consisting of “general games” with noncompact action space, together with an associated abstract rationality function. We prove that M is structurally stable and robust to ϵ-equilibria for “almost all” parameters. As applications, we investigate structural stability and robustness to bounded rationality for noncooperative games, multiobjective optimizations and fixed point problems satisfying existence and some continuity conditions. Specifically, we introduce concrete rationality functions for such three kinds of problems with both payoffs and strategy sets, objective functions and domain spaces, and correspondence and domain spaces as parameters, respectively, and show the generic structural stability and robustness to bounded rationality for the corresponding model Ms.     

How to Play M 13?

A puzzle called “M ₁₃” J. H. Conway has described recently is explained. We report an implementation of the puzzle in the programming language Java. The program allows the human user to “play M ₁₃” interactively (and to cheat by solving it automatically). The program is an example on how to bring to life a nice piece of discrete mathematics. In this sense it presents not only a didactical way of seeing “mathematics at work”, but also displays the stabilizer chain method developed by C. Sims to solve group theoretic puzzles, the most famous of which being Rubik's cube.     

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let {M³_i}\{M^{3}_{i}\} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0 . Suppose that all unit metric balls in M³_iM^{3}_{i} have very small volume, at most v _i →0 as i→∞, and suppose that either M³_iM^{3}_{i} is closed or has possibly convex incompressible toral boundary. Then M³_iM^{3}_{i} must be a graph manifold for sufficiently large i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.     

On the Consistency of the Quantum-Like Representation Algorithm for Hyperbolic Interference

Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]–[28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations. The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric, hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric interference was recently proved [29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning. For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H ^b|a and H ^a|b (in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary equivalent (in the sense of hyperbolic Hilbert space).     

Stabilization of flows through porous media

In this article, we study the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed layer having an external source (with an injection rate b), and above by a free surface moving under the influence of gravity. The flow is governed by Darcy’s law. If b(c) = 0 for some c > 0 then the system admits (u, f) ≡ (c, c) as an equilibrium solution. We shall prove that the stability properties of this equilibrium are determined by the slope of b in c : The equilibrium is unstable if b′(c) < 0, whereas b′(c) > 0 implies exponential stability. Zhaoyong Feng: He is grateful to the DFG for financial support through the Graduiertenkolleg 615 “Interaction of Modeling, Computation Methods and Software Concepts for Scientific-Technological Problems”.     

Notes on δ-Koszul Algebras

This paper is a continuous work of δ-Koszul algebras, which were first introduced by Green and Marcos in 2005 (see Green and Marcos, Commun Algebra 33(6):1753–1764, 2005). Let K^d(A)\mathcal{K}^{\delta}(A) be the category of δ-Koszul modules. It is proved that K^d(A)\mathcal{K}^{\delta}(A) preserves kernels of epimorphisms if and only if the “minimal Horseshoe Lemma” (“MHL” for short) holds. Further, a special class of δ-Koszul algebras named periodic δ -algebras are introduced, which have close connection with Koszul algebras and provide answers to the questions raised by Green and Marcos (Commun Algebra 33(6):1753–1764, 2005). Finally, we construct new periodic δ-algebras from the given ones in terms of one-point extension and sum-extension.     

The Discrete and Classical Dirichlet Problem

Let W ì \mathbbR^d{\Omega \subset \mathbb{R}^d} be some bounded domain with reasonable boundary and let f be a continuous function on the complement Ω ^c . We can construct an unique continuous function u that is harmonique on Ω and u = f on Ω ^c . Similarly, u _d is the unique function on the lattice points such that for each lattice point of Ω satisfies the “average” property with respect to its nearest neighbours and u _d = f on Ω ^c . In this paper when ∂Ω is Lipschitz I give a “best possible” estimate of ||u − u _d ||_∞.     

Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the s-dimensional discrepancy of nonlinear congruential pseudorandom numbers over the residue ring modulo M for an “almost squarefree” integer M. It is useful to recall that almost all integers are of this type. Moreover, if the generator is associated with a permutation polynomial over we obtain a stronger bound “on average” over all initial values. This bound is new even in the case when M = p is prime.     

On the Parametric Cubic Spline Approach for Solving Second Order Boundary Value Problems

In this article some comments on the paper “parametric cubic spline approach to the solution of a system of second order boundary value problems” in (Khan and Aziz, J. Optim. Theory Appl. 118:45–54, 2003) are given. This paper concerns with a numerical method for solving a second order boundary value problem associated with obstacle, unilateral and contact problems. Corrections are given for the convergence analysis of the numerical method and the computational experiments.     

Chern-Simons invariant and conformal embedding of a 3-manifold

This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and A ： M → SO（3） is the ＂difference＂ map. An interesting phenomenon is that the ＂jumps＂ of the Chern-Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS（e＋） and CS（e_）, where e＋ and e_ are the ＂left＂ and ＂right＂ quaternionic frames on M3 induced from an immersion M^3 → E^4, respectively. Consequently we find many metrics on S^3 （Berger spheres） so that they can not be conformally embedded in E^4.     

Two-dimensionalism and the “Knowing Which” Requirement

Two-dimensional semantics aims to eliminate the puzzle of necessary a posteriori and contingent a priori truths. Recently many argue that even assuming two-dimensional semantics we are left with the puzzle of necessary and a posteriori propositions. Stephen Yablo (Pacific Philosophical Quarterly, 81, 98–122, 2000) and Penelope Mackie (Analysis, 62(3), 225–236, 2002) argue that a plausible sense of “knowing which” lets us know the object of such a proposition, and yet its necessity is “hidden” and thus a posteriori. This paper answers this objection; I argue that given two-dimensional semantics you cannot know a necessary proposition without knowing that it is true.

Statistical Causality,Extremal Measures and Weak Solutions of Stochastic Differential Equations with Driving Semimartingales

In this paper we consider different concepts of causality in filtered probability spaces. Especially, we consider a generalization of a causality relationship “G is a cause of J within H ” which was first given by Mykland (1986) and which is based on Granger’s definition of causality (Granger, Econometrica 37:424–438, 1969). Then we apply this concept on weak solutions of stochastic differential equations with driving semimartingales. We also show that the given causality concept is closely connected to the concept of extremality of measures and links Granger’s causality with the concept of adapted distribution. Finally, the concept of causality is applied on solution of martingale problem.