Epireflective subcategories of TOP, T 2 UNIF, UNIF, closed under epimorphic images, or being algebraic

The epireflective subcategories of Top, that are closed under epimorphic (or bimorphic) images, are { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} and Top. The epireflective subcategories of T2Unif, closed under epimorphic images, are: { X∣ | X| ≤ 1 } , { X∣ X is compact T2} , { X∣ covering character of X is ≤ λ0} (where λ0 is an infinite cardinal), and T2Unif. The epireflective subcategories of Unif, closed under epimorphic (or bimorphic) images, are: { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} , { X∣ covering character of X is ≤ λ0} (where λ0 is an infinite cardinal), and Unif. The epireflective subcategories of Top, that are algebraic categories, are { X∣ | X| ≤ 1 } , and { X∣ X is indiscrete}. The subcategories of Unif, closed under products and closed subspaces and being varietal, are { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} , { X∣ X is compact T2}. The subcategories of Unif, closed under products and closed subspaces and being algebraic, are { X∣ X is indiscrete} , and all epireflective subcategories of { X∣ X is compact T2}. Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of T3 spaces, closed for products, closed subspaces and surjective images. © 2016, Akadémiai Kiadó, Budapest, Hungary

Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra

Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a C*-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a C*-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions. © 2016, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic

Weak and strong structures and the T3.5 property for generalized topological spaces

We investigate weak and strong structures for generalized topological spaces, among others products, sums, subspaces, quotients, and the complete lattice of generalized topologies on a given set. Also we introduce T3.5 generalized topological spaces and give a necessary and sufficient condition for a generalized topological space to be a T3.5 space: they are exactly the subspaces of powers of a certain natural generalized topology on [0,1]. For spaces with at least two points here we can have even dense subspaces. Also, T3.5 generalized topological spaces are exactly the dense subspaces of compact T4 generalized topological spaces. We show that normality is productive for generalized topological spaces. For compact generalized topological spaces we prove the analogue of the Tychonoff product theorem. We prove that also Lindelöfness (and κ-compactness) is productive for generalized topological spaces. On any ordered set we introduce a generalized topology and determine the continuous maps between two such generalized topological spaces: for | X| , | Y| ≧ 2 they are the monotonous maps continuous between the respective order topologies. We investigate the relation of sums and subspaces of generalized topological spaces to ways of defining generalized topological spaces. © 2016, Akadémiai Kiadó, Budapest, Hungary

Kvantitatív Makyoh-topográfia = Quantitative Makyoh topography

A korszerű félvezető-technológiában alapvető fontosságú a szeletek felületi morfológiájának, az ideális síkjellegtől való eltérésének a minősítése. A korszerű nagy átmérőjű szeletek megjelenésével a síkjelleg problémája és így a megfelelő minősítési eljárás szükségessége fokozottabban jelentkezik. A jelen pályázat témája egy, a Távol-Keletről származó ősi, "mágikus" tulajdonságú tükör elvén alapuló optikai vizsgálati módszer, a Makyoh-topográfia alkalmassá tétele igényes metrológiai célokra. A kutatás során új koncepciójú, nagy méretű minták vizsgálatára alkalmas mérési összeállításokat valósítottunk meg. Tanulmányoztuk a felületi domborzat visszanyerésére szolgáló eljárások érzékenységét és pontosságát, valamint a leképezés alapvető tulajdonságait. A kidolgozott mérési eljárást számos félvezető-technológiai és egyéb kutatásban alkalmaztuk. Lépéseket tettünk a mérési eljárás gazdasági hasznosítása érdekében. | The assessment of the surface morphology and flatness of the wafers is a key issue in modern semiconductor technology. The need for a proper flatness characterisation method became even more important with the advent of today's large-diameter wafers. The aim of the present project is to make Makyoh topography, an optical characterisation tool based on an ancient 'magic' mirror of Far-East origin suitable for advanced metrological purposes. During our research, we have constructed novel measurement set-ups suitable for the study of large-diameter samples. We have studied the sensivity and accuracy of the numerical methods for the reconstruction of the surface topography and investigated the basic characteristics of the imaging mechanism. The developed methods have been applied in semicondutor technolgy research as well as in other areas. We have taken steps forward the industrial exploitation

BALL CHARACTERIZATIONS IN SPACES OF CONSTANT CURVATURE

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C-+(2) with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d (sic) 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points - for S-d of radius less than pi/2 - and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls