On the rate of convergence in the strong law of large numbers for martingales

The aim of this note is to establish the Baum–Katz type rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759–763; Baum–Katz–Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489–1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910–913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.     

Conjugate duality in problems of constrained utility maximization

We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler–Lagrange and transversality relations, which are then used to establish existence of optimal portfolios in terms of solutions of the dual problem. The approach is completely synthetic, and does not require the rather difficult a priori hypothesis of a fictitious complete market for unconstrained optimization, which has been the standard approach for synthesizing optimal portfolios in problems of utility maximization with trading constraints. It also complements a duality synthesis of Rogers [Lecture Notes in Mathematics, No. LNM-1814, Springer-Verlag, New York, 2003, pp. 95–131] and Klein and Rogers [Math. Finance, 17 (2007), pp. 225–247] for general problems of utility maximization with market imperfections.     

Characterization of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection

The subdual latticial cones in Hilbert spaces are characterized by the isotonicity of a generalization of the positive part mapping which can be expressed in terms of the metric projection only. Although Németh characterized the positive cone of Hilbert lattices with the metric projection and ordering only [A.B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123 (2) (2003), pp. 295–299.], this has been done for the first time for subdual latticial cones in this article. We also note that the normal generating pointed closed convex cones for which the projection onto the cone is isotone are subdual latticial cones, but there are subdual latticial cones for which the metric projection onto the cone is not isotone [G. Isac, A.B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46 (6) (1986), pp. 568–576; G. Isac, A.B. Néemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1) (1990), pp. 53–62; G. Isac, A.B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7 (4) (1990), pp. 773–802; G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153 (1) (1990), pp. 258–275; G. Isac, A.B. Németh, Isotone projection cones in Eucliden spaces, Ann. Sci. Math Québec 16 (1) (1992), pp. 35–52].     

A note on arbitrage,approximate arbitrage and the fundamental theorem of asset pricing

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017     

Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing

By using previous results of Djafari Rouhani for non-expansive sequences in Refs (Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale University, Part I (1981), pp. 1–76; Djafari Rouhani, J. Math. Anal. Appl. 147 (1990), pp. 465–476; Djafari Rouhani, J. Math. Anal. Appl. 151 (1990), pp. 226–235), we study the existence and asymptotic behaviour of solutions to first-order as well as second-order difference equations of monotone type with periodic forcing. In the first-order case, our result extends to general maximal monotone operators, the discrete analogue of a result of Baillon and Haraux (Rat. Mech. Anal. 67 (1977), 101–109) proved for subdifferential operators. In the second-order case, our results extend among other things, previous results of Apreutesei (J. Math. Anal. Appl. 288 (2003), 833–851) to the non-homogeneous case, and show the asymptotic convergence of every bounded solution to a periodic solution.     

Viscosity methods for common solutions for equilibrium and hierarchical fixed point problems

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence of Krasnosel'ski?–Mann iteration for hierarchical fixed point problems, Inverse Probl. 24 (2008), 015015 8 pp.], S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006), pp. 506–515], Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, preprint.], Combettes and Hirstoaga [Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), pp. 117–136] and Plubtieng and Pumbaeang [A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), pp. 455–469.].     

On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

In 1832, Jakob Steiner (Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander, Reimer (Berlin)) asked for a characterization of those planar graphs which are combinatorially equivalent to polyhedra inscribed in the sphere. This question was answered in the 1990s by Igor Rivin (Ann. of Math.143(1996), 51–70), as a byproduct of his classification of ideal polyhedra in hyperbolic 3-space. Rivin also proposed a more direct approach to these results in Rivin (Ann. of Math.139 (1994), 553–580). In this paper, we prove a combinatorial result (Theorem 6) which enables one to complete the program of Rivin (Ann. Math.139(1994), 553–580).     

Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature.     

A note on extrema of linear combinations of elementary symmetric functions

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed.     

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.     

Contribution of copositive formulations to the graph partitioning problem

This article provides analysis of several copositive formulations of the graph partitioning problem and semidefinite relaxations based on them. We prove that the copositive formulations based on results from Burer [S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479–495] and the author of the paper [J. Povh, Semidefinite approximations for quadratic programs over orthogonal matrices. J. Global Optim. 48 (2010), pp. 447–463] are equivalent and that they both imply semidefinite relaxations which are stronger than the Donath–Hoffman eigenvalue lower bound [W.E. Donath and A.J. Hoffman, Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 (1973), pp. 420–425] and the projected semidefinite lower bound from Wolkowicz and Zhao [H. Wolkowicz and Q. Zhao, Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97 (1999), pp. 461–479].     

Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.     

Near-optimal mean–variance controls under two-time-scale formulations and applications

Although the mean–variance control was initially formulated for financial portfolio management problems in which one wants to maximize the expected return and control the risk, our motivations stem from highway vehicle platoon controls that aim to maximize highway utility while ensuring zero accident. This paper develops near-optimal mean–variance controls of switching diffusion systems. To reduce the computational complexity, with motivations from earlier work on singularly perturbed Markovian systems [Sethi and Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhäuser, Boston, MA, 1994; Yin and Zhang, Continuous-Time Markov Chains and Applications: A Singular Pertubation Approach, Springer-Verlag, New York, 1998 and Yin et al., Ann. Appl. Probab. 10 (2000), pp. 549–572], we use a two-time-scale formulation to treat the underlying system, which is represented by the use of a small parameter. As the small parameter goes to 0, we obtain a limit problem. Using the limit problem as a guide, we construct controls for the original problem, and show that the control so constructed is nearly optimal.     

Estimates for domains of local invertibility of diffeomorphisms

Using a novel Wintner-type formulation of the classical Peano's existence theorem [Math. Ann. 37 (1890), 182-228], we enhance Wazewski's result on invertibility of maps defined on closed balls [Ann. Soc. Pol. Math. 20 (1947), 81-125] securing the size of the domain of invertibility that agrees with the bounds derived by John [Comm. Pure Appl. Math. 21 (1968), 77-110] and Sotomayor [Z. Angew. Math. Phys. 41 (1990), 306-310].

     

Moderate deviations for Markovian occupation times

In this paper, we establish the moderate deviations for occupation times of Markov processes under the conditions given in Darling–Kac (1957. Trans. Amer. Math. Soc. 84, 444–458). When applied to the law of the iterated logarithm, our results generalize those obtained in Marcus–Rosen (1994a. Ann. Probab. 22, 626–658; 1994b. Ann. Inst. Henri Poincaré Probab. Statist. 30, 467–499) for Levy processes and random walks, and those obtained recently by the author (1999. Ann. Probab. 27, 1324–1346) for Harris recurrent Markov chains.     

Mild solutions of local non-Lipschitz neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching

In this article, we initiate a study on neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching in real separable Hilbert spaces. Our goal here is to derive the existence and uniqueness of mild solutions to equations of this class under local non-Lipschitz condition proposed by Taniguchi [J. Math. Anal. Appl. 340:(2009)197–208] by means of stopping time technique and Banach fixed-point theorem. The results obtained here generalize the main results from Luo and Taniguchi [Stoch. Dyn. 9:(2009)135–152] and Jiang and Shen [Comput. Math. Appl. 61:(2011)1590–1594]. Finally, an example is worked out to illustrate the obtained results.     

Geodesic G-invex sets and semistrictly geodesic η-preinvex functions

In this article, we introduce semistrictly geodesic η-preinvex functions on Riemannian manifolds, geodesic G-invex sets and study their properties. Our results extend and improve the results of Yang and Li [X. Yang and D. Li, Semistrictly preinvex functions, J. Math. Anal. Appl. 258 (2001), pp. 287–308]. Example is constructed in support of our defnition.     

Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term

New oscillation criteria for the second-order Emden–Fowler delay differential equation with a sublinear neutral term are presented. An essential feature of our results is that oscillation of the studied equation is ensured via only one condition. Furthermore, as opposed to the results by Agarwal et al. (Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1861–1875), Li and Rogovchenko (Math. Nachr. 288 (2015), no. 10, 1150–1162; Monatsh. Math. 184 (2017), no. 3, 489–500), and Xu (Monatsh. Math. 150 (2007), no. 2, 157–171), new criteria can be applied to Emden–Fowler delay differential equations with noncanonical operators and a sublinear neutral term. Our results essentially improve, extend, and simplify some known ones reported in the literature. The results are illustrated with examples.     

A structural characterization of real k-potent matrices

Some well-known characterizations of nonnegative k-potent matrices have been obtained by Flor [P. Flor, On groups of nonnegative matrices, Compositio Math. 21 (1969), pp. 376–382.] and Jeter and Pye [M. Jeter and W. Pye, Nonnegative (s,?t)-potent matrices, Linear Algebra Appl. 45 (1982), pp. 109–121.]. In this article, we obtain a structural characterization of a real k-potent matrix A, provided that (sgn(A)) ^k+1 is unambiguously defined, regardless of whether A is nonnegative or not.