Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

Decompositions of some weaker forms of continuity

The aim of this paper is to give decompositions of some weaker forms of continuity using the concepts of classes B ₁, B ₂, B ₃, αA and αC introduced by ourselves.     

2K2 vertex-set partition into nonempty parts

A graph is 2K₂-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C, D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. Determining whether an arbitrary graph is 2K₂-partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We show that for C₄-free graphs, circular-arc graphs, spiders, P₄-sparse graphs, and bipartite graphs the 2K₂-partition problem can be solved in polynomial time.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

On the cut polytope

The cut polytopeP _C (G) of a graphG=(V, E) is the convex hull of the incidence vectors of all edge sets of cuts ofG. We show some classes of facet-defining inequalities ofP _C (G). We describe three methods with which new facet-defining inequalities ofP _C (G) can be constructed from known ones. In particular, we show that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented. We characterize the facet defining inequalities ofP _C (G) ifG is not contractible toK ₅. We give a simple characterization of adjacency inP _C (G) and prove that for complete graphs this polytope has diameter one and thatP _C (G) has the Hirsch property. A relationship betweenP _C (G) and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied.     

Stable n-pointed trees of projective lines

Stable n-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves: Let C be a totally degenerate stable curve of genus g ≥ 2 over a field k. This means that C is a connected projective curve of arithmetic genus g satisfyingo

1. (a) every irreducible component of C is a rational curve over κ.
2. (b) every singular point of C is a κ-rational ordinary double point.
3. (c) every nonsingular component L of C meets C−L in at least three points. It is always possible to find g singular points P₁,..., P_g on C such that the blow up C of C at P₁,..., P_g is a connected projective curve with the following properties:o
   1. (i) every irreducible component of C is isomorphic to P_k¹
   2. (ii) the components of C intersect in ordinary κ-rational double points
   3. (iii) the intersection graph of C is a tree.

The morphism φ : C → C is an isomorphism outside 2g regular points Q₁, Q₁′, Q_g, Q_g′ and identifies Q_i with Q_j′. is uniquely determined by the g pairs of regular κ-rational points (Q_i, Q_i′). A curve C satisfying (i)-(iii) together with n κ-rational regular points on it is called a n-pointed tree of projective lines. C is stable if on every component there are at least three points which are either singular or marked. The object of this paper is the classification of stable n-pointed trees. We prove in particular the existence of a fine moduli space B_n of stable n-pointed trees. The discussion above shows that there is a surjective map πB_2g → D_g of B_2g onto the closed subscheme D_g of the coarse moduli scheme M_g of stable curves of genus g corresponding to the totally degenerate curves. By the universal property of M_g, π is a (finite) morphism. π factors through B_2g = B_2g mod the action of the group of pair preserving permutations of 2g elements (a group of order 2^gg, isomorphic to a wreath product of S_g and ℤ/2ℤThe induced morphism π: B_2g → D_g is an isomorphism on the open subscheme of irreducible curves in D_g, but in general there may be nonequivalent choices of g singular points on a totally degenerated curve for the above construction, so π has nontrivial fibres. In particular, π is not the quotient map for a group action on B_2g. This leads to the idea of constructing a Teichmüller space for totally degenerate curves whose irreducible components are isomorphic to B_2g and on which a discontinuous group acts such that the quotient is precisely D_g; π will then be the restriction of this quotient map to a single irreducible component. This theory will be developped in a subsequent paper.In this paper we only consider stable n-pointed trees and their moduli theory. In 4 1 we introduce the abstract cross ratio of four points (not necessarily on the same projective line) and show that for a field κ the κ-valued points in the projective variety B_n of cross ratios are in 1 − 1 correspondence with the isomorphy classes of stable n-pointed trees of projective lines over κ. We also describe the structure of the subvarieties B(T, ψ) of stable n-pointed trees with fixed combinatorial type.We generalize our notion in 4 2 to stable n-pointed trees of projective lines over an arbitrary noetherian base scheme S and show how the cross ratios for the fibres fit together to morphisms on S. This section is closely related to [Kn], but it is more elementary since we deal with a special case.4 3 contains the main result of the paper: the canonical projection B_{n + 1} → B_n is the universal family of stable n-pointed trees. As a by-product of the proof we find that B_n is a smooth projective scheme of relative dimension 2n - 3 over ℤ. We also compare B_n to the fibre product B_n−1 × _Bn-2 B_{n − 1} and investigate the singularities of the latter.In 4 4 we prove that the Picard group of B_n is free of rank 2ⁿ⁻1−(n+1)−n(n−3)/2.We also give a method to compute the Betti numbers of the complex manifold B_n(ℂ).In 4 5 we compare B_n to the quotient Q_n: = ℙ_ssⁿ/PGL₂ of semi-stable points in ℙ₁ⁿ for the action of fractional linear transformations in every component. This orbit space has been studied in greater detail by several authors, see [GIT], [MS], [G]. It turns out that B_n is a blow-up of Q_n, and we describe the blow-up in several steps where at each stage the obtained space is interpreted as a solution to a certain moduli problem.     

Theoremes de congruence pour triplets de points dans les varietes de GrassmannG 2(ℝ N )

La classe di congruenza di una terna {A, B, C} inG ₂(? ⁿ ), sotto l’azione del gruppo ortogonale 0(n, ?), dipende da 9 parametri al più. Possiamo associare in modo naturale a questa terna, una terna di proiettori ortogonali {P _A ,P _B ,P _C }. Utilizzando la teoria algebrica degli invarianti, viene dato un sistema completo di invarianti ortogonali composto da tracce di opportune combinazioni diP _A ,P _B , eP _C che consentono di determinare in modo unico la classe di congruenza della terna di partenza.     

Carleson Measures for Besov-Sobolev Spaces with Applications in the Unit Ball of C

This paper is devoted to give the connections between Carleson measures for Besov-Sobolev spaces B_p^σ (B) and p-Carleson measure in the unit ball of Cⁿ. As applications, we characterize the Riemann-Stieltjes operators and multipliers acting on B_p^σ (B) spaces by means of Carleson measures for B_p^σ (B).     

On rational invariants of a set of symmetric matrices

Some identities resulting from the Cayley-Hamilton theorem are derived. Some applications include: (a) for k = 1,2,…,n ? 1 a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with n(n+3)/2 generators consisting of elements symmetric with respect to the permutaion group P_n; (c) it is shown that any rational invariant of (g+2) symmetric operators A,B,C₁,C₂,…, C_g can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), (A,C₂),…, (A,C_g, (A,B+C₁), (A,B+C₂),…,(A,B+C_g).     

On the efficiency of Newton and Broyden numerical methods in the investigation of the regular polygon problem of (N + 1) bodies

Numerical methods of finding the roots of a system of non-linear algebraic equations are treated in this paper. This paper attempts to give an answer to the selection of the most efficient method in a complex problem of Celestial Dynamics, the so-called ring problem of (N + 1) bodies. We apply Newton and Broyden’s method to these problems and we investigate, by means of their use, the planar equilibrium points, the five equilibrium zones, which are symbolized by A₁, A₂, B, C₂, and C₁ (by order of appearance from the center O to the periphery of the imaginary circle on which the primaries lie) [T.J. Kalvouridis, A planar case of the N + 1 body problem: the ring problem. Astrophys. Space Sci. 260 (3) (1999) 309-325], and the attracting regions of the system. The efficiency of these methods is studied through a comparative process. The obtained results are demonstrated in figures and are discussed.     

Stability for semilinear elliptic variational inequalities depending on the gradient

In this work we give a result concerning the continuous dependence on the data for weak solutions of a class of semilinear elliptic variational inequalities (P_n) with a nonlinear term depending on the gradient of the solution. This paper can be seen as the second part of the work Matzeu and Servadei (2010) [9], in the sense that here we give a stability result for the C^1,α-weak solutions of problem (P_n) found in Matzeu and Servadei (2010) [9] through variational techniques. To be precise, we show that the solutions of (P_n), found with the arguments of Matzeu and Servadei (2010) [9], converge to a solution of the limiting problem (P), under suitable convergence assumptions on the data.     

A renorming in some Banach spaces with applications to fixed point theory

We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk(⋅)} which satisfy some special conditions. We define an equivalent norm ?⋅? on X such that if C is a convex bounded closed subset of (X,?⋅?) which is τ-relatively sequentially compact, then every nonexpansive mapping T:C→C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier-Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G=T, we recover P.K. Lin's renorming in the sequence space ?₁. Moreover, we give new norms in ?₁ with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L₁(μ) can be renormed to have the FPP.     

Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions

The present paper is devoted to an optimal control problem given by hyperbolic discrete (P _D ) and differential inclusions (P _C ) of generalized Darboux type and ordinary discrete inclusions. The results are extended to non-convex problems. An approach concerning necessary and sufficient conditions for optimality is proposed. In order to formulate sufficient conditions of optimality for problem (P _C ) the approximation method is used. Formulation of these conditions is based on locally adjoint mappings. Moreover for construction of adjoint partial differential inclusions the equivalence theorems of locally adjoint mappings are proved. One example with homogeneous boundary conditions is considered.     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G ? B _p (3) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

Viscosity approximation methods for nonexpansive mappings and inverse-strongly monotone mappings in Hilbert spaces

Let C be a closed convex subset of a Hilbert space H. Let f is a contraction on C. Let S be a nonexpansive mapping of C into itself and A be an α-inverse-strongly monotone mapping of C into H. Assuming that F(S)∩VI(C,A)≠φ, and x ₀=x∈C, in this paper we introduce the iterative process x _n+1=α _n f(x _n )+β _n x _n +γ _n (μ Sx _n +(1?μ)(P _C (I?λ _n A)y _n )), where y _n =P _C (I?λ _n A)x _n . We prove that {x _n } and {y _n } converge strongly to the same point z∈F(S)∩VI(C,A). As its application, we give a strong convergence theorem for nonexpansive mapping and strictly pseudo-contractive mapping in a Hilbert space.     

A polynomial algorithm for some preemptive multiprocessor task scheduling problems

In this paper we consider a problem of preemptive scheduling of multiprocessor tasks on dedicated processors in order to minimize the sum of completion times. Using a standard notation, our problem can be denoted as P ∣ fix_j, pmtn ∣ ∑C_j. We give a polynomial-time algorithm to solve P ∣ fix_j, G = {P₄, dart}-free, pmtn ∣ ∑C_j problem. This result generalizes the following problems: P2 ∣ fix_j, pmtn ∣ ∑C_j, P ∣ ∣fix_j∣ ∈ {1, m}, pmtn ∣ ∑C_j and P4 ∣ fix_j = 2, pmtn ∣ ∑C_j.     

Spectra of generalized Bethe trees attached to a path

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P_m be a path of m vertices. Let {B_i:1?i?m} be a set of generalized Bethe trees. Let P_m{B_i:1?i?m} be the tree obtained from P_m and the trees B₁,B₂,…,B_m by identifying the root vertex of B_i with the i-th vertex of P_m. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P_m{B_i:1?i?m}. In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B₁=B₂=…=B_m.     

Global inverse spectral problems for rational matrix functions

Given rational matrix functions ψ₁(λ) = I_m + C₁(λI_n1 − A₁)⁻¹B₁ and ψ₂(λ) = I_m + C₂(λI_n2 − A₂)⁻¹B₂ which are analytic and invertible on the unit circle, we characterize in terms of the operators A₁,B₁,C₁,A₂,B₂,C₂ when there exists a single rational matrix function W(λ) = I_m + C(λI_n − A)⁻¹B such that WH²_m^⊥ = ψ ₁H²_m^⊥and WH²_m = ψ₂H²_m. When this is the case, we give explicit formulae for A,B,C in terms of A₁,B₁,C₁,A₂,B₂,C₂. Applications include Wiener-Hopf factorization, J- inner-outer factorization, and coprime factorization. The results on J-inner-outer factorization have application to a model reduction problem for discrete time linear systems.