Bridgeland’s stabilities on abelian surfaces

In this paper, we shall study the structure of walls for Bridgeland’s stability conditions on abelian surfaces. In particular, we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the Picard number 1.     

Whitney’s formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney’s formula to curves on an oriented punctured surface Σ _{m, n} , obtaining a family of identities indexed by elements of π ₁(Σ _{m, n} ). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σ _{m, n} . Similar formulas are obtained for non-based curves.     

Stable and unstable minimal surfaces and Hopf’s problem

Hopf’s well-known conjecture is considered, which states that there exists no metric of strictly positive curvature on the topological product S² × S² of two 2-spheres. Three theorems are proved.     

A generalization of Bonnet’s theorem on Darboux surfaces

The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to n-dimensional hypersurfaces of Euclidean spaces.     

Donagi–Morrison’s examples on 2-elementary K3 surfaces

Let X be an algebraic K3 surface, and let L be a base point free and big line bundle on X. If X admits a map of degree 2 to the projective plane branched over a smooth sextic and L is the pullback of the line bundle O_\mathbbP²(3),{\mathcal{O}_{\mathbb{P}^{2}}(3),} then the gonality of the smooth curves of the complete linear system |L| is not constant. The polarized K3 surface (X, L) consisting of the K3 surface X and the line bundle L is called Donagi–Morrison’s example. In this paper, we give a necessary and sufficient condition for the polarized K3 surface (X, L) consisting of a 2-elementary K3 surface X and an ample line bundle L to be Donagi–Morrison’s example.     

Born–Infeld solitons,maximal surfaces,and Ramanujan’s identities

We show that a Born–Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born–Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan’s identities and the Weierstrass–Enneper representation of maximal surfaces, we derive further non-trivial identities.     

The bondage number of graphs on topological surfaces and Teschner’s conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of graphs. Also, we present stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner’s Conjecture in affirmative for almost all graphs. As an auxiliary result, we show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.     

Green’s Conjecture for curves on rational surfaces with an anticanonical pencil

Green’s Conjecture is proved for smooth curves $C$ lying on a rational surface $S$ with an anticanonical pencil, under some mild hypotheses on the line bundle $L=\mathcal{O }_S(C)$ . Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system $\vert L\vert $ is also obtained.     

On the parameter dependence of the real cubic surfaces in arnold’s form

We consider the topological classification of cubic surfaces which are obtained as intersection of the sphere with the algebraic variety defined by the zeroes of a homogeneous cubic polynomial in Arnold’s normal form. This classification is based on the parameters appearing in this normal form, obtaining a correspondence between the parameters of the surface and its topological type. General classifications of cubic surfaces are made in the projective space ℙ³(ℝ), but our method, based on a very simple combinatorial procedure is easier to implement in . We split the cubic surfaces parameter space into ten equivalence classes.     

Analysis of Atkinson’s variable transformation for numerical integration over smooth surfaces in ℝ3

Summary Recently, a variable transformation for integrals over smooth surfaces in ³ was introduced in a paper by Atkinson. This interesting transformation, which includes a grading parameter that can be fixed by the user, makes it possible to compute these integrals numerically via the product trapezoidal rule in an efficient manner. Some analysis of the approximations thus produced was provided by Atkinson, who also stated some conjectures concerning the unusually fast convergence of his quadrature formulas observed for certain values of the grading parameter. In a recent report by Atkinson and Sommariva, this analysis is continued for the case in which the integral is over the surface of a sphere and the integrand is smooth over this surface, and optimal results are given for special values of the grading parameter. In the present work, we give a complete analysis of Atkinsons method over arbitrary smooth surfaces that are homeomorphic to the surface of the unit sphere. We obtain optimal results that explain the actual rates of convergence, and we achieve this for all values of the grading parameter.     

Reich’s problem concerning Halpern’s convergence

We discuss Halpern’s convergence for nonexpansive mappings in Hilbert spaces. We prove that one of the conditions in [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel), 58 (1992), 486–491] is the weakest sufficient condition among the conditions known to us. We also improve a necessary condition, which is close to Wittmann’s. This is one step to solve the problem raised by Reich in 1974 and 1983. Received: 15 July 2008     

Warfield’s Duality and Malcev’s Matrix

In this work, we investigate relations between Malcev’s matrices of a torsion-free group G of finite rank and Malcev’s matrices of groups Hom(R,G) and Hom(G,R), where G is a locally free group and R is a torsion-free group of rank 1.     

Bergmann’s Dilemma and Internalism’s Escape

Michael Bergmann has argued that internalist accounts of justification face an insoluble dilemma. This paper begins with an explanation of Bergmann??s dilemma. Next, I review some recent attempts to answer the dilemma, which I argue are insufficient to overcome it. The solution I propose presents an internalist account of justification through direct acquaintance. My thesis is that direct acquaintance can provide subjective epistemic assurance without falling prey to the quagmire of difficulties that Bergmann alleges all internalist accounts of justification cannot surmount.     

Generalizations of Sherman’s Inequality Via Fink’s Identity and Green’s Function

Ukrainian Mathematical Journal - New generalizations of Sherman’s inequality for n-convex functions are obtained with the help of Fink’s identity and Green’s function. By using...     

Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

On Andrunakievich’s chain and Koethe’s problem

In 1969 Andrunakievich asked whether one gets a ring without nonzero nil left ideals from an arbitrary ring R by factoring out the ideal A(R) which is the sum of all nil left ideals of R. Recently, it was shown that this problem is equivalent to Koethe’s problem. In this context one may consider the chain of ideals, which starts with A ₁(R) = A(R) ⊆ A ₂(R), where A ₂(R)/A ₁(R) = A(R/A ₁(R)), and extends by repeating this process. We study the properties of this chain and show that, assuming a negative solution of Koethe’s problem, this chain can terminate at any given ordinal number.     

Kantorovich’s majorants principle for Newton’s method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA. The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).     

A note on Stone’s,Baire’s,Ky Fan’s and Dugundji’s theorem in tvs-cone metric spaces

Recently, Ayse Sonmez [A. Sonmez, On paracompactness in cone metric spaces, Appl. Math. Lett. 23 (2010) 494–497] proved that a cone metric space is paracompact when the underlying cone is normal. Also, very recently, Kieu Phuong Chi and Tran Van An [K.P. Chi, T. Van An, Dugundji’s theorem for cone metric spaces, Appl. Math. Lett. (2010) doi:10.1016/j.aml.2010.10.034] proved Dugundji’s extension theorem for the normal cone metric space. The aim of this paper is to prove this in the frame of the tvs-cone spaces in which the cone does not need to be normal. Examples are given to illustrate the results.     

Editor’s Letter

Departments

Editor’s Letter