The Genuine Bernstein–Durrmeyer Operators Revisited

We present some direct and inverse results for the approximation by the genuine Bernstein–Durrmeyer operators U _n . We also consider iterates of U _n and discuss some convergence results towards the corresponding semigroup. Using the eigenstructure of the operators U _n we give new proofs of some known qualitative results and obtain new quantitative estimates concerning the convergence of iterates towards the semigroup.     

Influence of permutizers of subgroups on the structure of finite groups

A group G is said to satisfy the maximal permutizer condition if the permutizer of any maximal subgroup M of G in G, P_G(M), is G. In this paper, we characterize the supersolubility of finite groups by using the maximal permutizer condition. We also get some results for when both G/N and N are supersoluble, which implies that G is supersoluble. Our results unify or generalize some known results.     

Existence and uniqueness for several non-linear elliptic problems arising in lubrication theory

We consider the non-linear elliptic equation ∇·(A(x,p)∇p)=∇·b(x,p) with positive Dirichlet boundary data. The coefficients A and b are taken from models used in lubrication theory, and in particular are not defined for negative values of p. We prove some general existence and uniqueness results for a family of models, which extend related results in the literature. These results allow us to prove existence, uniqueness and positivity of the solution to advanced compressible lubrication models such as the kinetic-based Fukui-Kaneko model and the second-order-slip model. We also consider a spring-like model of compliant-foil compressible bearing, and weaken some hypotheses of previous results on more classical models such as the standard Reynolds model and the first-order-slip model.     

The bondage numbers of graphs with small crossing numbers

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. Kang and Yuan proved b(G)?8 for every connected planar graph G. Fischermann, Rautenbach and Volkmann obtained some further results for connected planar graphs. In this paper, we generalize their results to connected graphs with small crossing numbers.     

Second order symmetric duality in nondifferentiable multiobjective fractional programming with cone convex functions

In the present paper, we consider Mond-Weir type nondifferentiable second order fractional symmetric dual programs over arbitrary cones and derive duality results under second order K?F-convexity/K?F-pseudoconvexity assumptions. Our results generalize several known results in the literature.     

Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces

The existence of common fixed points is established for three mappings where T is either generalized (f,g)-nonexpansive or asymptotically (f,g)-nonexpansive on a nonempty subset of a Banach space. As applications, the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained. Our results unify and substantially improve several recent results existing in the current literature.     

Positive solution to the nonlinear abstract Hammerstein equation and applications to {\phi} -Laplacian BVPs

We provide in this paper existence results for positive solution to the abstract Hammerstein equation NFu = u where N : E → E is a completely continuous operator, F : C → C is a continuous and bounded map and C is a cone in the Banach space E. The obtained results are used to prove existence results for positive solution to ${\phi}$ -Laplacian boundary value problem.     

Arbitrary decompositions into open and closed trails

The problem of arbitrary decomposition of a graph G into closed trails i.e. a decomposition into closed trails of prescribed lengths summing up to the size of the graph G was first considered in the case of the complete graph G=K_n (for odd n) in connection with vertex-distinguishing coloring of the union of cycles.Next, the same problem was investigated for other families of graphs.In this paper we consider a more general problem: arbitrary decomposition of a graph into open and closed trails. Our results are based on and generalize known results on decomposition of a graph into closed trails. Our results also generalize some results concerning decomposition of a graph into open trails. We here emphasize that the known results on the closed case are basic ingredients for the proof of the open and closed case.     

Friendship decompositions of graphs

We investigate the maximum number of elements in an optimal t-friendship decomposition of graphs of order n. Asymptotic results will be obtained for all fixed t?4 and for t=2,3 exact results will be derived.     

Composition of entire functions with finite iterated order

In this paper, we investigate the growth of two composite entire functions of finite iterated order and a series of comparative growths of logp+qT(r,f(g)) (p,q∈N) with logpT(r,f) and logqT(r,g). At the end of this paper, we apply some growth results into the factorization of the solutions of linear differential equation. We achieve some results which are the improvements and extensions of the previous results.     

Weak convergence of linear forms in D[0, 1]

Convergence in probability of the linear forms Σ_k=1^∞a_nkX_k is obtained in the space D[0, 1], where (X_k) are random elements in D[0, 1] and (a_nk) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements (X_n) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D[0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.     

New Kloosterman sums identities over F2m for all m

Recently, Shin and Sung found new identities for Kloosterman sums over F_2^m with odd m. They posed the question whether similar results could be obtained for even m. In this paper, we will give a positive answer to this question. We will present new results that hold for any m and include as special cases the results of Shin and Sung in the case where m is odd.     

Collective fixed points and maximal elements with applications to abstract economies

In this paper, we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is Φ-condensing. As an application of our collective fixed points theorem, we derive the coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of L_S-majorized multivalued maps whose product is Φ-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not L_S-majorized but their product is Φ-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.     

Periodic solutions of non-autonomous second order systems with (q(t), p(t))-Laplacian

Using the least action principle in critical point theory we obtain some existence results of periodic solutions for (q(t), p(t))-Laplacian systems which generalize some existence results.     

Matrices and the linear complementarity problem

The paper is a collection of results on the linear complementarity problem (q, M). The results are stated in terms of the matrix M.     

Some results for the q-Bernoulli and q-Euler polynomials

We show some results for the q-Bernoulli and q-Euler polynomials. The formulas in series of the Carlitz's q-Stirling numbers of the second kind are also considered. The q-analogues of well-known formulas are derived from these results.     

On the behavior of the oscillatory solutions of first or second order delay differential equations

Some new results are given concerning the behavior of the oscillatory solutions of first or second order delay differential equations. These results establish that all oscillatory solutions x of a first or second order delay differential equation satisfy x(t)=O(v(t)) as t→∞, where v is a nonoscillatory solution of a corresponding first or second order linear delay differential equation. Some applications of the results obtained are also presented.     

Syzygies of surfaces of general type

We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type X embedded by adjoint line bundles L _r = K + r B, where B is a base point free, ample line bundle. Our main results determine the r for which L _r has N _p property. In corollaries, we will relate the bounds on r to the regularity of B. Examples in the last section show that several results are optimal.