Weak convergence theorem for Lipschizian pseudocontraction semigroups in Banach spaces

The purpose of this paper is to study the weak convergence problems of the irnplicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [Nonlinear Anal., 68, 2977-2983 （2008）], Chen, et ah [J. Math. Anal. Appl., 314, 701 709 （2006）], Xu and Ori [Numer. Funct. Anal. Optim, 22, 767-773 （2001）] and Osilike [J. Math. Anal. Appl., 294, 73-81 （2004）]. Keywords     

Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].     

On relations of vector optimization results with C 1,1 data

In this article we prove that some of the sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Luc [Appl. Math., 51, 5-36 （2006）] generalize （strictly） those presented by Guerraggio, Luc [J. Optim. Theory Appl., 109, 615-629 （2001）]. While the former paper shows examples for which the conditions given there are effective but the ones from the latter paper fail, it does not prove that generally the conditions it proposes are stronger. In the present note we complete this comparison with the lacking proof.     

Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.     

Remarks on an overdetermined boundary value problem

We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006).     

Minimax signal detection forl q -ellipsoids withl p -balls removed

The paper is close to the investigations of M. Ermakov [Theory Probab. Appl.,35, 667–679 (1990)] and Yu. Ingster [Zap. Nauchn. Semin. LOMI,184, 152–168 (1990)]. On the basis of the latter, the case q>p, p∈[0, 1], is considered in detail. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 126–136, 1993. Translated by the author.     

Correction of CT burst array errors in the generalized-lee-RT spaces

In [Jain, S.： Array codes in the generalized-Lee-RT-pseudo-metric （the GLRTP-metric）, to appear in Algebra Colloq.], Jain introduced a new pseudo-metric on the space Matm×s（Zq）, the module space of all m × s matrices with entries from the finite ring Zq, generalized the classical Lee metric [Lee, C. Y.： Some properties of non-binary error correcting codes. IEEE Trans. Inform. Theory, IT-4, 77- 82 （1958）] and array RT-metric [Rosenbloom, M. Y., Tsfasman, M. A.： Codes for m-metric. Prob. Inf. Transm., 33, 45-52 （1997）] and named this pseudo-metric as the Generalized-Lee-RT-Pseudo-Metric （or the GLRTP-Metric）. In this paper, we obtain some lower bounds for two-dimensional array codes correcting CT burst array errors [Jain, S.： CT bursts from classical to array coding. Discrete Math., 308-309, 1489-1499 （2008）] with weight constraints under the GLRTP-metric.     

On construction of asymptotic expansions for Pólya distributions

In the paper [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dokl. Akad. Nauk SSSR,246, No. 3 (1979)], asymptotic formulas for multidimensional Pólya distributions, including both polynomial and multidimensional hypergeometric distributions, were deduced. The derivation of those formulas was rather bulky and constituted the paper [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dep. VINITI, No. 3155-79, Moscow (1979)]. The aim of the present paper is to realize another approach to constructing asymptotic expansions for multidimensional distributions of more general structure generated by plans of Pólya random walks [Ya. P. Lumel'skii,Dokl. Akad. Nauk SSSR,209, No. 6 (1973)]. This allows us not only to prove the results of [Ya. P. Lumel'skii and Ya. M. L'vovskii,Dep. VINITI, No. 3155-79, Moscow (1979)] but to derive new asymptotic formulas as well. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 125–136, Perm, 1993.     

Circulant Double Coverings of a Circulant Graph of Valency Five

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory. A covering of G is called circulant if its covering graph is circulant. Recently, the authors [Discrete Math., 277, 73-85 （2004）1 enumerated the isomorphism classes of circulant double coverings of a certain type, called a typical covering, and showed that no double covering of a circulant graph of valency three is circulant. Also, in [Graphs and Combinatorics, 21,386 400 （2005）], the isomorphism classes of circulant double coverings of a circulant graph of valency four are enumerated. In this paper, the isomorphism classes of circulant double coverings of a circulant graph of valency five are enumerated.     

On testing extreme value conditions

Applications of univariate extreme value theory rely on certain as- sumptions. Recently, two methods for testing these extreme value conditions are derived by [Dietrich, D., de Haan, L., Hüsler, J., Extremes 5: 71–85, (2002)] and [Drees, H., de Haan, L., Li, D., J. Stat. Plan. Inference, 136: 3498–3538, (2006)]. In this paper we compare the two tests by simulations and investigate the effect of a possible weight function by choosing a parameter, the test error and the power of each test. The conclusions are useful for extreme value applications.     

Chromatic uniqueness of certain complete tripartite graphs

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k ²/4 + i + 1.     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.     

On methods for the solution of integral equations of the theory of plasticity based on the concept of slip

We have proposed a modification of the methods for solving the system of integral equations [M. Ya. Leonov and N. Yu. Shvaiko, “Complex plane deformation,” Dokl. Akad. Nauk SSSR, 159, No. 2, 1007–1010 (1964); N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005)]. These equations describe the development of plane plastic deformation for simple and complex loading processes. A characteristic feature of these equations lies in the presence of unknown functions both under the integral sign and in the integration limits. We have written analytical solutions for monotone deformation and in a small neighborhood of an angular point of the loading trajectory. For arbitrary piecewise smooth trajectories, we have reduced this problem to the Cauchy problem for a first-order differential equation with known initial conditions. The results obtained simplify significantly the construction of constitutive equations [(s)\dot]_mn ~ [(e)\dot]_mn {\dot{\sigma }_{mn}} \sim {\dot{\varepsilon }_{mn}} and their use in applied problems of the theory of plasticity as compared with [N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005); N. Yu. Shvaiko, Complex Loading and Problems of Stability [in Russian], Izd. DGU, Dnepropetrovsk (1989)].     

Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.      

On two problems concerning end extensions

We study problems of Clote and Paris, concerning the existence of end extensions of models of Σ _n -collection. We continue the study of the notion of ‘Γ-fullness’, begun by Wilkie and Paris (Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North- Holland, Amsterdam, 1989) and introduce and study a generalization of it, to be used in connection with the existence of Σ _n -elementary end extensions (instead of plain end extensions). We obtain (a) alternative proofs of results (Adamowicz in Fund. Math. 136, 133–145, 1990) and (Wilkie and Paris in Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North-Holland, Amsterdam, 1989) related to the problem of Paris and (b) a partial solution to the problem of Clote. Research partially supported by INTAS grant 2000-447 and grant 70/4/5633 of the University of Athens Research Secretariat.     

Halpern Iteration in CAT（k） Spaces

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.     

A formula for deficiency (L-andR-tests)

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

A new class of lattice paths and partitions with <Emphasis Type="Italic">n</Emphasis> copies of <Emphasis Type="Italic">n</Emphasis>

Agarwal and Bressoud (Pacific J. Math. 136(2) (1989) 209–228) defined a class of weighted lattice paths and interpreted several q-series combinatorially. Using the same class of lattice paths, Agarwal (Utilitas Math. 53 (1998) 71–80; ARS Combinatoria 76 (2005) 151–160) provided combinatorial interpretations for several more q-series. In this paper, a new class of weighted lattice paths, which we call associated lattice paths is introduced. It is shown that these new lattice paths can also be used for giving combinatorial meaning to certain q-series. However, the main advantage of our associated lattice paths is that they provide a graphical representation for partitions with n + t copies of n introduced and studied by Agarwal (Partitions with n copies of n, Lecture Notes in Math., No. 1234 (Berlin/New York: Springer-Verlag) (1985) 1–4) and Agarwal and Andrews (J. Combin. Theory A45(1) (1987) 40–49).     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

We prove C ¹ regularity of c-convex weak Alexandrov solutions of a Monge–Ampère type equation in dimension two, assuming only a bound from above on the Monge–Ampère measure. The Monge–Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C ¹ regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampère equation det D ² u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].