Selection principles in hyperspaces with generalized Vietoris topologies

We continue investigations of ?ech closure spaces and their hyperspaces started in [M. Mrševi?, M. Jeli?, Selection principles and hyperspace topologies in closure spaces, J. Korean Math. Soc. 43 (2006) 1099-1114] and [M. Mrševi?, M. Jeli?, Selection principles, γ-sets and αi-properties in ?ech closure spaces, Topology Appl., in press], focusing on generalized upper and lower Vietoris topologies.     

Fractal oscillations for a class of second order linear differential equations of Euler type

The s-dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied. The main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of Euler type. Next, it will be shown that the dimensional number s only depends on a positive real parameter α appearing in a singular term of the main equation. It continues some recent results on the rectifiable and unrectifiable oscillations given in Paši? [M. Paši?, Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl. 335 (2007) 724-738] and Wong [J.S.W. Wong, On rectifiable oscillation of Euler type second order linear differential equations, Electron. J. Qual. Theory Differ. Equ. 20 (2007) 1-12].     

Jensen's inequality for filtration consistent nonlinear expectation without domination condition

In this paper, the general filtration consistent nonlinear expectation defined on the integrable variable space is considered, based on the results in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27]. Under a natural continuous assumption for the nonlinear expectation, which weakens the domination assumption in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27], the author obtains the necessary and sufficient conditions under which Jensen's inequality for filtration consistent nonlinear expectation holds in general, respectively on scalar function and bivariate function. These two results generalize the known results on Jensen's inequality for g-expectation in [Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 1, C. R. Acad. Sci. Paris Ser. I 337 (11) (2003) 725-730; Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 2, C. R. Acad. Sci. Paris Ser. I 337 (12) (2003) 797-800; L. Jiang, On Jensen's inequality of bivariate function for g-expectation, J. Shandong Univ. 38 (5) (2003) 13-22 (in Chinese); L. Jiang, Z. Chen, On Jensen's inequality for g-expectation, Chinese Ann. Math. Ser. B 25 (3) (2004) 401-412; L. Jiang, Jensen's inequality for backward stochastic differential equation, Chinese Ann. Math. Ser. B 27 (5) (2006) 553-564; S. Fan, Jensen's inequality for g-expectation on convex (concave) function, Chinese Ann. Math. Ser. A 27 (5) (2006) 635-644 (in Chinese)].     

On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

It is well known that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold [Kowalski, J. Reine Angew. Math. 250 (1971) 124-129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1-29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.     

Integral means of analytic functions

For analytic functions f(z) and g(z) which satisfy the subordination f(z)?g(z), J.E. Littlewood [Proc. London Math. Soc. 23 (1925) 481-519] has shown some interesting results for integral means of f(z) and g(z). The object of the present paper is to derive some applications of integral means by J.E. Littlewood. We also show interesting examples for our theorems.     

The ordering of unicyclic graphs with the smallest algebraic connectivity

Fielder [M. Fielder, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305] has turned out that G is connected if and only if its algebraic connectivity a(G)>0. In 1998, Fallat and Kirkland [S.M. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48-74] posed a conjecture: if G is a connected graph on n vertices with girth g≥3, then a(G)≥a(C_n,g) and that equality holds if and only if G is isomorphic to C_n,g. In 2007, Guo [J.M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711] gave an affirmatively answer for the conjecture. In this paper, we determine the second and the third smallest algebraic connectivity among all unicyclic graphs with vertices.     

On zeros and fixed points of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+1)−f(z). A number of results are proved concerning the existences of zeros and fixed points of g(z) or g(z)/f(z) which expand results of Bergweiler and Langley [W. Bergweiler, J.K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007) 133-147].     

A note on domination, girth and minimum degree

Let G be a graph of order n, minimum degree δ?2, girth g?5 and domination number γ. In 1990 Brigham and Dutton [Bounds on the domination number of a graph, Q. J. Math., Oxf. II. Ser. 41 (1990) 269-275] proved that γ?⌈n/2-g/6⌉. This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131-141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195-212] who for i∈{1,2} determined a finite set of graphs G_i such that γ?⌈n/2-g/6-(3i+3)/6⌉ unless G is a cycle or G∈G_i.Our main result is that for every i∈N there is a finite set of graphs G_i such that γ?n/2-g/6-i unless G is a cycle or G∈G_i. Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.     

Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple

Our aim is to describe the semicentre of the enveloping algebra of a parabolic subalgebra p of a semisimple finite dimensional complex Lie algebra g. Whilst [F. Fauquant-Millet, A. Joseph, Transformation Groups 6 (2) (2001) 125-142] describes explicitly the semicentre of the quantized enveloping algebra associated to p, specialization at q=1 gives only part of the required classical semicentre, even when p is a Borel. Similarly the graded of a polynomial subalgebra of the Hopf dual of the enveloping algebra of g, associated to the Kostant filtration, gives a lower bound on the required semicentre. Then by a method developed from [A. Joseph, Amer. J. Math. 99 (6) (1977) 1151-1165; J. Algebra 48 (1977) 241-289] we obtain an upper bound. Finally when g is a product of simple Lie algebras of type An or Cn we show that these bounds coincide and conclude that in this case the semicentre of the enveloping algebra of p is a polynomial algebra.     

A note on polar decomposition based Geršgorin-type sets

Let B∈C^n×n denote a finite-dimensional square complex matrix. In [L. Smithies, R.S. Varga, Singular value decomposition Geršgorin sets, J. Linear Algebra Appl. 417 (2004) 370-380; N. Fontes, J. Kover, L. Smithies, R.S. Varga, Singular value decomposition normally estimated Geršgorin sets, Electron. Trans. Numer. Anal. 26 (2007) 320-329], Professor Varga and I introduced Geršgorin-type sets which were developed from singular value decompositions (SVDs) of B. In this note, our work is extended by introducing the polar SV-Geršgorin set, Γ^PSV(B). The set Γ^PSV(B) is a union of n closed discs in C, whose centers and radii are defined in terms of the entries of a polar decomposition B=Q|B|. The set of eigenvalues of B, σ(B), is contained in Γ^PSV(B).     

Enumerating typical abelian prime-fold coverings of a circulant graph

Enumerating the isomorphism classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory (see [R. Feng, J.H. Kwak, J. Kim, J. Lee, Isomorphism classes of concrete graph coverings, SIAM J. Discrete Math. 11 (1998) 265-272; R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73-85; R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196-207; SIAM J. Discrete Math. 21 (2007) 548-550 (erratum); M. Hofmeister, Graph covering projections arising from finite vector spaces over finite fields, Discrete Math. 143 (1995) 87-97; M. Hofmeister, Enumeration of concrete regular covering projections, SIAM J. Discrete Math. 8 (1995) 51-61; M. Hofmeister, A note on counting connected graph covering projections, SIAM J. Discrete Math. 11 (1998) 286-292; J.H. Kwak, J. Chun, J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273-285; J.H. Kwak, J. Lee, Isomorphism classes of graph bundles, Canad. J. Math. XLII (1990) 747-761]). A covering is called abelian (or circulant, respectively) if its covering graph is a Cayley graph on an abelian (or a cyclic, respectively) group. A covering p from a Cayley graph onto another Cay (Q,Y) is called typical if the map p:A→Q on the vertex sets is a group epimorphism. Recently, the isomorphism classes of connected typical circulant r-fold coverings of a circulant graph are enumerated in [R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73-85] for r=2 and in [R. Feng, J.H. Kwak, Y.S. Kwon, Enumerating typical circulant covering projections onto a circulant graph, SIAM J. Discrete Math. 19 (2005) 196-207; SIAM J. Discrete Math. 21 (2007) 548-550 (erratum)] for any r. As a continuation of these works, we enumerate in this paper the isomorphism classes of typical abelian prime-fold coverings of a circulant graph.     

Invariant subspaces for operator semigroups with commutators of rank at most one

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of ST−TS is at most 1 for all {S,T}⊂S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997) 443-456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladi?, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998) 452-465].     

Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces

Recently, Agarwal, Cho, Li and Huang [R.P. Agarwal, Y.J. Cho, J. Li, N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 272 (2002) 435-447] introduced the new iterative procedures with errors for approximating the common fixed point of a couple of quasi-contractive mappings and showed the stability of these iterative procedures with errors in Banach spaces. In this paper, we introduce a new concept of a couple of q-contractive-like mappings (q>1) in a Banach space and apply these iterative procedures with errors for approximating the common fixed point of the couple of q-contractive-like mappings. The results established in this paper improve, extend and unify the corresponding ones of Agarwal, Cho, Li and Huang [R.P. Agarwal, Y.J. Cho, J. Li, N.J. Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 272 (2002) 435-447], Chidume [C.E. Chidume, Approximation of fixed points of quasi-contractive mappings in Lp spaces, Indian J. Pure Appl. Math. 22 (1991) 273-386], Chidume and Osilike [C.E. Chidume, M.O. Osilike, Fixed points iterations for quasi-contractive maps in uniformly smooth Banach spaces, Bull. Korean Math. Soc. 30 (1993) 201-212], Liu [Q.H. Liu, On Naimpally and Singh's open questions, J. Math. Anal. Appl. 124 (1987) 157-164; Q.H. Liu, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math. Anal. Appl. 146 (1990) 301-305], Osilike [M.O. Osilike, A stable iteration procedure for quasi-contractive maps, Indian J. Pure Appl. Math. 27 (1996) 25-34; M.O. Osilike, Stability of the Ishikawa iteration method for quasi-contractive maps, Indian J. Pure Appl. Math. 28 (1997) 1251-1265] and many others in the literature.     

Further summation formulae related to generalized harmonic numbers

By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ(n), Trans. Amer. Math. Soc. 347 (1995) 1391-1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15-25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51-70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103-118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ(m) with m=5,6,7, including several known ones as examples.     

Edgeworth expansion for an estimator of the adjustment coefficient

We establish an Edgeworth expansion for an estimator of the adjustment coefficient R, directly related to the geometric-type estimator for general exponential tail coefficients, proposed in [Brito, M., Freitas, A.C.M., 2003. Limiting behaviour of a geometric-type estimator for tail indices. Insurance Math. Econom. 33, 211-226].Using the first term of the expansion, we construct improved confidence bounds for R. The accuracy of the approximation is illustrated using an example from insurance (cf. [Schultze, J., Steinebach, J., 1996. On least squares estimates of an exponential tail coefficient. Statist. Dec. 14, 353-372]).     

Gradient regularity for elliptic equations in the Heisenberg group

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485-544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C1,α-regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math., vol. 370, 2005, pp. 17-23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli, I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21], the a priori estimates found are shown to imply the suitable local Calderón-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [T. Iwaniec, Projections onto gradient fields and Lp-estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293-312; E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107-1134], and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.     

Some common fixed point theorems for Banach operator pairs with applications in best approximation

The ordered pair (T,I) of two self-maps of a metric space (X,d) is called a Banach operator pair if the set F(I) of fixed points of I is T-invariant i.e. T(F(I))⊆F(I). Some common fixed point theorems for a Banach operator pair and the existence of common fixed points of best approximation are presented in this paper. The results prove, generalize and extend some results of Al-Thagafi [M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318-323], Carbone [A. Carbone, Applications of fixed point theorems, Jnanabha 19 (1989) 149-155], Chen and Li [J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximations, J. Math. Anal. Appl. 336 (2007) 1466-1475], Habiniak [L. Habiniak, Fixed point theorems and invariant approximation, J. Approx. Theory 56 (1989) 241-244], Jungck and Sessa [G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon. 42 (1995) 249-252], Sahab, Khan and Sessa [S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349-351], Shahzad [N. Shahzad, Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39-45] and of few others.     

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].     

Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumptions

In this paper, the equivalence of the strong convergence between the modified Mann and Ishikawa iterations with errors in two different schemes by Xu [Y.G. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91-101] and Liu [L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114-125] respectively is proven for the generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumption. Our results generalize the recent results of the papers [Zhenyu Huang, F. Bu, The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl. 325 (1) (2007) 586-594; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289 (2004) 266-278; B.E. Rhoades, S.M. Soltuz, The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219-228] by extending to the most general class of the generalized strongly successively Φ-pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u₁, x₁ in uniformly smooth Banach spaces.     

Rigidity and the Lower Bound Theorem for Doubly Cohen–Macaulay Complexes

We prove that for d≥3, the 1-skeleton of any (d?1)-dimensional doubly Cohen–Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121–125, 1971; Barnette, D. Pac. J. Math. 46:349–354, 1973) hold for every 2-CM complex of dimension ≥2 (see Kalai, G. Invent. Math. 88:125–151, 1987). Moreover, the initial part (g ₀,g ₁,g ₂) of the g-vector of a 2-CM complex (of dimension ≥3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305–1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.