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© *Hindawi* *Publishing* *Corp*.

, 2004

"... The existence of an N-sequence in a continuum is a common obstruction that implies non-smoothness, noncontractibility, nonselectibility, and nonexistence of any mean. The aim of the present paper is to investigate if some variants of the concept of an N-sequence also keep these properties. In partic ..."

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The existence of an N-sequence in a continuum is a common obstruction that implies non-smoothness, noncontractibility, nonselectibility, and nonexistence of any mean. The aim of the present paper is to investigate if some variants of the concept of an N-sequence also keep these properties. In particular, mapping properties of bend sets are studied. 2000 Mathematics Subject Classification: 54B20, 54C65, 54E40, 54F15. All considered spaces are assumed to be metric and all mappings are continuous. The symbol N stands for the set of all positive integers. Given a space X and its subspaces A and B with A ⊂ B, we denote by clB(A) and bdB(A) the closure and the boundary of A with respect to B, respectively. A continuum means a compact connected space. A 1-dimensional continuum is called a curve. A continuum is said to be hereditarily unicoherent provided that the inter-section of every two of its subcontinua is connected. A dendroid means an arcwise connected and hereditarily unicoherent continuum. A ramification point in a dendroid X means a vertex of a simple triod contained in X. A fan denotes a dendroid having exactly one ramification point.

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© *Hindawi* *Publishing* *Corp*.

, 2004

"... The existence of an N-sequence in a continuum is a common obstruction that implies nonsmoothness, noncontractibility, nonselectibility, and nonexistence of any mean. The aim of the present paper is to investigate if some variants of the concept of an N-sequence also keep these properties. In particu ..."

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The existence of an N-sequence in a continuum is a common obstruction that implies nonsmoothness, noncontractibility, nonselectibility, and nonexistence of any mean. The aim of the present paper is to investigate if some variants of the concept of an N-sequence also keep these properties. In particular, mapping properties of bend sets are studied. 2000 Mathematics Subject Classification: 54B20, 54C65, 54E40, 54F15. All considered spaces are assumed to be metric and all mappings are continuous. The symbol N stands for the set of all positive integers. Given a space X and its subspaces A and B with A ⊂ B, wedenotebyclB(A) and bdB(A) the closure and the boundary of A with respect to B, respectively. A continuum means a compact connected space. A 1-dimensional continuum is called a curve. A continuum is said to be hereditarily unicoherent provided that the intersection of every two of its subcontinua is connected. A dendroid means an arcwise connected and hereditarily unicoherent continuum. A ramification point in a dendroid X means a vertex of a simple triod contained in X. Afan denotes a dendroid having exactly one ramification point.

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© *Hindawi* *Publishing* *Corp*. NETTED MATRICES

, 2002

"... We prove that powers of 4-netted matrices (the entries satisfy a four-term recurrence δai,j = αai−1,j + βai−1,j−1 + γai,j−1) preserve the property of nettedness: the entries of the eth power satisfy δea (e) (e) (e) = αea +βea +γea (e) i,j−1, i,j i−1,j i−1,j−1 where the coefficients are all instances ..."

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We prove that powers of 4-netted matrices (the entries satisfy a four-term recurrence δai,j = αai−1,j + βai−1,j−1 + γai,j−1) preserve the property of nettedness: the entries of the eth power satisfy δea (e) (e) (e) = αea +βea +γea (e) i,j−1, i,j i−1,j i−1,j−1 where the coefficients are all instances of the same sequence xe+1 = (β + δ)xe − (βδ+αγ)xe−1. Also, we find a matrix Qn(a,b) and a vector v such that Qn(a,b) e · v acts as a shifting on the general second-order recurrence sequence with parameters a, b. The shifting action of Qn(a,b) generalizes the known property) e ·(1,0) t = (Fe−1,Fe) t. Finally, we prove some results about congruences sat-

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© *Hindawi* *Publishing* *Corp*. QUARTIC EXERCISES

, 2002

"... A correspondence between quartic étale algebras over a field and quadratic étale extensions of cubic étale algebras is set up and investigated. The basic constructions are laid out in general for sets with a profinite group action and for torsors, and translated in terms of étale algebras and Galois ..."

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A correspondence between quartic étale algebras over a field and quadratic étale extensions of cubic étale algebras is set up and investigated. The basic constructions are laid out in general for sets with a profinite group action and for torsors, and translated in terms of étale algebras and Galois algebras. A parametrization of cyclic quartic algebras is given. 2000 Mathematics Subject Classification: 12F10, 12G05, 13B40. 1. Introduction. It

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© *Hindawi* *Publishing* *Corp*. THE DIOPHANTINE EQUATION

, 2002

"... We discuss, with the aid of arithmetical properties of the ring of the Gaussian integers, the solvability of the Diophantine equation ax 2 + 2bxy − 4ay 2 =±1, where a and b are nonnegative integers. The discussion is relative to the solution of Pell’s equation v 2 −(4a 2 +b 2)w 2 =−4. 2000 Mathemati ..."

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We discuss, with the aid of arithmetical properties of the ring of the Gaussian integers, the solvability of the Diophantine equation ax 2 + 2bxy − 4ay 2 =±1, where a and b are nonnegative integers. The discussion is relative to the solution of Pell’s equation v 2 −(4a 2 +b 2)w 2 =−4. 2000 Mathematics Subject Classification: 11D09. 1. Introduction. The

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© *Hindawi* *Publishing* *Corp*. ON SAKAGUCHI FUNCTIONS

, 2002

"... Let Ss(α) (0 ≤ α<1/2) be the class of functions f(z) = z+·· · which are analytic in the unit disk and satisfy there Re{zf ′ (z)/(f (z) − f(−z))}>α. In the present paper, we find the sharp lower bound on Re{(f (z) − f(−z))/z} and investigate two subclasses S0(α) and T0(α) of Ss(α). We derive ..."

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Let Ss(α) (0 ≤ α<1/2) be the class of functions f(z) = z+·· · which are analytic in the unit disk and satisfy there Re{zf ′ (z)/(f (z) − f(−z))}>α. In the present paper, we find the sharp lower bound on Re{(f (z) − f(−z))/z} and investigate two subclasses S0(α) and T0(α) of Ss(α). We derive sharp distortion inequalities and some properties of the partial sums for functions in the classes S0(α) and T0(α). 2000 Mathematics Subject Classification: 30C45. 1. Introduction. Let A be the class of functions f(z) = z +·· · which are analytic in the unit disk E ={z: |z | < 1}. WedenotebyS the subclass of A consisting of functions which are univalent in E. A function f(z) ∈ A is said

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© *Hindawi* *Publishing* *Corp*. ON AN INEQUALITY OF DIANANDA

, 2002

"... We consider certain refinements of the arithmetic and geometric means. The results generalize an inequality of P. Diananda. 2000 Mathematics Subject Classification: 26D15. 1. Introduction. Let Pn,r (x ..."

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We consider certain refinements of the arithmetic and geometric means. The results generalize an inequality of P. Diananda. 2000 Mathematics Subject Classification: 26D15. 1. Introduction. Let Pn,r (x

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© *Hindawi* *Publishing* *Corp*. QUASIREDUCIBLE OPERATORS

, 2002

"... We introduce the concept of quasireducible operators. Basic properties and illustrative examples are considered in some detail in order to situate the class of quasireducible operators in its due place. In particular, it is shown that every quasinormal operator is quasireducible. The following resul ..."

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We introduce the concept of quasireducible operators. Basic properties and illustrative examples are considered in some detail in order to situate the class of quasireducible operators in its due place. In particular, it is shown that every quasinormal operator is quasireducible. The following result links this class with the invariant subspace problem: essentially normal quasireducible operators have a nontrivial invariant subspace, which implies that quasireducible hyponormal operators have a nontrivial invariant subspace. The paper ends with some open questions on the characterization of the class of all quasireducible operators. 2000 Mathematics Subject Classification: 47A15, 47B20. 1. Introduction. Let

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© *Hindawi* *Publishing* *Corp*. BOEHMIANS ON MANIFOLDS

, 2000

"... Abstract. The construction of Boehmians on amanifold requires a commutative convolu-tion structure. We present such constructions in two specific cases: anN-dimensional torus and an N-dimensional sphere. Then we formulate conditions under which a construction of Boehmians on a manifold is possible. ..."

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Abstract. The construction of Boehmians on amanifold requires a commutative convolu-tion structure. We present such constructions in two specific cases: anN-dimensional torus and an N-dimensional sphere. Then we formulate conditions under which a construction of Boehmians on a manifold is possible.

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© *Hindawi* *Publishing* *Corp*. NEIGHBORHOOD SPACES

, 2002

"... Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or clo-sure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory an ..."

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Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or clo-sure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more general p-stacks. 2000 Mathematics Subject Classification: 54A05, 54A10. 1. Introduction. In his original definition of topology, Hausdorff [7] assigned to each point x in a set X a system of neighborhoods subject to certain axioms. We will define a neighborhood structure ν on X in the same way, using axioms weaker than those of Hausdorff, but just strong enough to include both pretopological and closure spaces as special cases of neighborhood spaces. Of particular significance is