Eutactic quantum codes

We consider sets of quantum observables corresponding to eutactic stars. Eutactic stars are systems of vectors which are the lower dimensional ``shadow'' image, the orthogonal view, of higher dimensional orthonormal bases. Although these vector systems are not comeasurable, they represent redundant coordinate bases with remarkable properties. One application is quantum secret sharing.Comment: 6 page

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

Density functional theory for hard-sphere mixtures: the White-Bear version Mark II

In the spirit of the White-Bear version of fundamental measure theory we derive a new density functional for hard-sphere mixtures which is based on a recent mixture extension of the Carnahan-Starling equation of state. In addition to the capability to predict inhomogeneous density distributions very accurately, like the original White-Bear version, the new functional improves upon consistency with an exact scaled-particle theory relation in the case of the pure fluid. We examine consistency in detail within the context of morphological thermodynamics. Interestingly, for the pure fluid the degree of consistency of the new version is not only higher than for the original White-Bear version but also higher than for Rosenfeld's original fundamental measure theory.Comment: 16 pages, 3 figures; minor changes; J. Phys.: Condens. Matter, accepte

Morphological characterization of shocked porous material

Morphological measures are introduced to probe the complex procedure of shock wave reaction on porous material. They characterize the geometry and topology of the pixelized map of a state variable like the temperature. Relevance of them to thermodynamical properties of material is revealed and various experimental conditions are simulated. Numerical results indicate that, the shock wave reaction results in a complicated sequence of compressions and rarefactions in porous material. The increasing rate of the total fractional white area $A$ roughly gives the velocity $D$ of a compressive-wave-series. When a velocity $D$ is mentioned, the corresponding threshold contour-level of the state variable, like the temperature, should also be stated. When the threshold contour-level increases, $D$ becomes smaller. The area $A$ increases parabolically with time $t$ during the initial period. The $A(t)$ curve goes back to be linear in the following three cases: (i) when the porosity $\delta$ approaches 1, (ii) when the initial shock becomes stronger, (iii) when the contour-level approaches the minimum value of the state variable. The area with high-temperature may continue to increase even after the early compressive-waves have arrived at the downstream free surface and some rarefactive-waves have come back into the target body. In the case of energetic material ... (see the full text)Comment: 3 figures in JPG forma

Multi-interaction mean-field renormalization group

We present an extension of the previously proposed mean-field renormalization method to model Hamiltonians which are characterized by more than just one type of interaction. The method rests on scaling assumptions about the magnetization of different sublattices of the given lattice and it generates as many flow equations as coupling constants without arbitrary truncations on the renormalized Hamiltonian. We obtain good results for the test case of Ising systems with an additional second-neighbor coupling in two and three dimensions. An application of the method is also done to a morphological model of interacting surfaces introduced recenlty by Likos, Mecke and Wagner [J. Chem. Phys. {\bf{102}}, 9350 (1995)]. PACS: 64.60.Ak, 64.60.Fr, 05.70.JkComment: Tex file and three macros appended at the end. Five figures available upon request to: [email protected], Fax: [+]39-40-224-60

Improper colourings inspired by Hadwiger’s conjecture

Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree