Neither first countable nor Cech-complete spaces are maximal Tychonoff connected

A connected Tychonoff space is called maximal Tychonoff connected if there is no strictly finer Tychonoff connected topology on . We show that if is a connected Tychonoff space and locally separable spaces, locally \v{C}ech-complete spaces, first countable spaces, then is not maximal Tychonoff connected. This result is new even in the cases where is compact or metrizable.

     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.     

Kinetics of Polydomain Ordering at Second-Order Phase Transitions (by the Example of the AuCu<Subscript>3</Subscript> Alloy)

Kinetics of polydomain spinodal ordering is studied in alloys of AuCu₃ type. We introduce four non-conserved long-range order parameters whose sum, however, is conserved and, using the statistical approach, follow the temporal evolution of their random spatial distribution after a rapid temperature quench. A system of nonlinear differential equations for correlators of second and third order is derived. Asymptotical analysis of this system allows to investigate the scaling regime, which develops on the late stages of evolution and to extract additional information concerning the rate of decrease of the specific volume of disordered regions and the rate of decrease of the average thickness of antiphase boundaries. Comparison of these results to experimental data is given. The quench below the spinodal and the onset of long-range order may be separated by the incubation time, whose origin is different from that in first-order phase transitions. Numerical integration of equations for correlators shows also, that it is possible to prepare a sample in such a way that its further evolution will go with formation of transient kinetically slowed polydomain structures different from the final L1₂ structure.     

Size of monochromatic components in local edge colorings

An edge coloring of a graph is a local r coloring if the edges incident to any vertex are colored with at most r distinct colors. We determine the size of the largest monochromatic component that must occur in any local r coloring of a complete graph or a complete bipartite graph.     

Weaker connected and weaker nowhere locally compact topologies for metrizable and similar spaces

We prove that any metrizable non-compact space has a weaker metrizable nowhere locally compact topology. As a consequence, any metrizable non-compact space has a weaker Hausdorff connected topology. The same is established for any Hausdorff space X with a σ-locally finite base whose weight w(X) is a successor cardinal.     

Hydrothermal Synthesis,Crystal Structure,and Luminescent Properties of Two Zinc(II) and Cadmium(II) 3D Metal‐­Organic Frameworks

Two zinc(II) and cadmium(II) metal‐organic frameworks with mixed ligands, {[Zn₂(biim‐4)₂(TDC)₂] · 2.5H₂O}_n ( 1 ) and {[Cd₂(biim‐4)₂(TDC)₂ · 2H₂O]}_n ( 2 ) [biim‐4 = 1,1′‐(1,4‐butanediyl)bis(imidazole); H₂TDC = thiophene‐2,5‐dicarboxylic acid], were hydrothermally synthesized. Both of them are characterized by elemental analysis, IR spectroscopy, and single crystal X‐ray diffraction. In 1 , the four‐connected Zn^II nodes are connected by four linear ligands extending into a 3D network, which further integrates a fivefold interpenetrating diamond 3D topological network and the free water molecules distribute in void space, whereas in 2 , the Cd^II ions are in a distorted octahedral arrangement linked by TDC^2– and biim‐4 ligands to construct a 3D framework. In topology analysis, C11 and C14 are simplified as 3‐connected nodes and the 3D framework displays a (3,5)‐connected net. Furthermore, the thermal and photoluminescent properties of 1 and 2 were also studied.     

Ricci-positive metrics on connected sums of projective spaces

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed Ricci positive metrics on arbitrary connected sums of complex projective planes. In this paper, we revisit and extend Perelman's techniques to construct Ricci positive metrics on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces in every dimension.     

On the Numerical Range of Rational Matrix Functions

In this paper we investigate bounds of the numerical range of the derivative of a rational matrix function. Moreover, some results on connectedness are presented.     

Global Minimization of a Multivariate Polynomial using Matrix Methods

The problem of minimizing a polynomial function in several variables over R ⁿ is considered and an algorithm is given. When the polynomial has a minimum the algorithm returns the global minimal value and finds at least one point in every connected component of the set of minimizers. A characterization of such points is given. When the polynomial does not have a minimum the algorithm computes its infimum. No assumption is made on the polynomial.