Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R ^k →Y is a C ²-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R ¹ that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ₀)∈X×R ^k and we describe the solution set of the studied equation in a small neighbourhood of this point.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

A geometric theory of harmonic and semi-conformal maps

We describe for any Riemannian manifold M a certain scheme M _L , lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M _L ; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M _L .     

Exact Convergence Rate for the Distributions of GI/M/c/K Queue as K Tends to Infinity

We obtain the exact convergence rate of the stationary distribution ^(K) of the embedded Markov chain in GI/M/c/K queue to the stationary distribution of the embedded Markov chain in GI/M/c queue as K. Similar result for the time-stationary distributions of queue size is also included. These generalize Choi and Kim's results of the case c=1 by nontrivial ways. Our results also strengthen the Simonot's results [5].     

On the lattice of deductive systems of a BL-algebra

For a BL-algebra A we denote by Ds(A) the lattice of all deductive systems of A. The aim of this paper is to put in evidence new characterizations for the meet-irreducible elements on Ds(A). Hyperarchimedean BL-algebras, too, are characterized.     

Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

Fast formula of a reliability ofm-consecutive-k-out-of-n:F system with cyclek

Anm-consecutive-k-out-of-n:F system is a system ofn linearly arranged components which fails if and only if at leastm non-overlapping sequences ofk components fail, when there arek distinct components with failure probabilitiesq _i fori=1,...,k and where the failure probability of thej-th component (j=rk+i (1 ≤i ≤k) isq _j =q _i , we call this system by anm-consecutive-k-out-of-n:F system with cycle (or period)k. In this paper we give a formula of the failure probability ofm-consecutive-k-out-of-n:F system with cyclek via the failure probability of consecutive-k-out-of-n:F system.     

A Method of Deducing L-Polyhedra for n-Lattices

We suggest a method for selecting an L-simplex in an L-polyhedron of an n-lattice in Euclidean space. By taking into account the specific form of the condition that a simplex in the lattice is an L-simplex and by considering a simplex selected from an L-polyhedron, we present a new method for describing all types of L-polyhedra in lattices of given dimension n. We apply the method to deduce all types of L-polyhedra in n-dimensional lattices for n=2,3,4, which are already known from previous results.     

Recursive constructions of -polynomials over

This paper presents procedures for constructing irreducible polynomials over GF(2^s) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N-polynomial F₀(x)GF(2^s) of degree n, polynomials F_k(x)GF(2^s) of degrees n2^k are constructed by iteratively applying the transformation x→x+x^-1, and their roots are shown to form a normal basis of GF(2^sn2k) over GF(2^s). In addition, the sequences are shown to be trace compatible, i.e., the trace map T_{GF(2^sn2k+1)/GF(2^sn2k)} fromGF(2^sn2k+1) onto GF(2^sn2k) maps the roots of F_k+1(x) onto those of F_k(x).     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

A unified theory of R-continuity

We introduce and investigate R-M-continuous functions defined between sets satisfying some minimal conditions. The functions enable us to formulate a unified theory of modifications of R-continuity [22]: R-irresoluteness [6], R-preirresoluteness [7].      

Infinitesimally homogeneous submanifolds of euclidean spaces

For a class of submanifolds of ^N, the infinitesimally homogeneous ones, the second fundamental form and itss-times iterated derivativessk+1 at any fixed point determine the immersion uniquely. The integerk>0 will be called the extrinsic Singer invariant. Any infinitesimally homogeneous submanifoldM (which is not necessarily complete) is an open part of a globally homogeneous (complete) submanifold. Indeed, the infinitesimal data at any pointp, determine, canonically, a Lie subgroupG of the isometry group of ^N , whose orbit atp is a complete submanifold that extendsM. Work partially supported by the GNSAGA of CNR and by the MURST of Italy     

Nonadaptive search problem with sets of equal sum

Consider the set A={1,2,3,…,2 ⁿ }, n≥3 and let x∈ A be unknown element. For given natural number S we are allowed to ask whether x belongs to a subset B of A such that the sum of the elements of B equals S. We investigate for which S it is possible to find x using a nonadaptive search.     

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B _4,4,4ℍ³. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B _4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B _4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. This work is partially supported by the Sonderforschungsbereich 288.     

Orthogonal factorizations of digraphs

Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g ⁻, g ⁺) and ƒ = (ƒ ⁻, ƒ ⁺) be pairs of positive integer-valued functions defined on V(G) such that g ⁻(x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ ƒ ⁺(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g ⁻(x) ⩽ id _H (x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ od _H (x) ⩽ ƒ ⁺(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let = {F ₁, F ₂,…, F _m} and H be a factorization and a subdigraph of G, respectively. is called k-orthogonal to H if each F _i , 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g ⁻(x), g ⁺(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.      

On super edge-antimagic total labelings of

A graph G of order p and size q is called (a,d)-edge-antimagic total if there exists a bijective function f:V(G)E(G)→{1,2,…,p+q} such that the edge-weights w(uv)=f(u)+f(v)+f(uv), uvE(G), form an arithmetic sequence with first term a and common difference d. The graph G is said to be super (a,d)-edge-antimagic total if the vertex labels are 1,2,…,p. In this paper we study super (a,d)-edge-antimagic properties of mK_n, that is, of the graph formed by the disjoint union of m copies of K_n.     

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which is co-finite in ind A, and derive some consequences. Dedicated to Stanislaw Balcerzyk on the occation of his 70th birthday