The complexity of tropical matrix factorization

The tropical arithmetic operations on R are defined by a⊕b=min{a,b}$a\oplus b=\min \{a,b\}$ and a⊗b=a+b$a\otimes b=a+b$. Let A be a tropical matrix and k   a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices B∈R^m×k$B\in {\mathbf{R}}^{m\times k}$ and C∈R^k×n$C\in {\mathbf{R}}^{k\times n}$ satisfying B⊗C=A$B\otimes C=A$. We show that the TMF problem is NP-hard for every k≥7$k\ge 7$ fixed in advance, thus resolving a problem proposed by Barvinok in 1993.     

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b$ax+b$-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.     

Fractal regularity results on optimal irrigation patterns

In this paper the problem of the regularity, i.e. fractal behaviour, of the minima of the branched transport problem is addressed. We show that, under suitable conditions on the irrigated measure, the minima present a fractal regularity, that is on a given branch of length l the number of branches bifurcating from it whose length is comparable with ε   can be estimated both from above and below by l/ε$l/\epsilon$.     

Some stable homology calculations and Occam's razor for Hodge structures

Motivated by motivic zeta function calculations, Vakil and Wood in [10] made several conjectures regarding the topology of subspaces of symmetric products. The purpose of this note is to prove two of these conjectures and disprove a strengthening of one of them.     

Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory

We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions. We prove that the interface is close to a hypersurface with mean curvature zero when no Lagrange multiplier is present, and with locally constant mean curvature in general. The energy density of the limiting measure has integer multiplicity almost everywhere modulo division by a surface energy constant. Received March 16, 1999 / Accepted June 11, 1999     

Piecewise linear concave dynamical systems appearing in the microscopic traffic modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are uniquely defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue is introduced, and the ergodic theory is used to define a growth rate. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the average growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of roads with a junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions.     

Cup and cap products in intersection (co)homology

We construct cup and cap products in intersection (co)homology with field coefficients. The existence of the cap product allows us to give a new proof of Poincaré duality in intersection (co)homology which is similar in spirit to the usual proof for ordinary (co)homology of manifolds.     

Comparative study of kinetics isomerization of substituted polyacetylene (Cl,F and I): Ab initio and DFT calculations

Ab initio and DFT methods were used to investigate the interconversions of substituted polyacetylene conformers C₁₀H₆X₆ (X=F, Cl and I) in the vapour phase. The rates of this geometrical isomerization have been calculated and the Arrhenius parameters evaluated. In the case of unsubstituted polyacetylene as the reference, the B3LYP Arrhenius parameters obtained are A₁=2.99 × 10¹⁷ s^–1 and E_a=17.30 kcal mol^–1. The values of the equilibrium constant for the reaction have also been determined at various temperatures between 300 and 500 K and the value of the energies change calculated. The results also suggest that the straightforward kinetics characterizing the majority of substituted polyacetylene isomerizations above 300 K. The isomerization energies are positive and the barrier heights ΔE_barrier are expected to be sensitive for the magnitude of halogens effects. According to geometries features the Cis → Trans isomerization in the gas phase occurs by a rotational mechanism.     

The spectral factorization problem for multivariable distributed parameter systems

This paper studies the solution of the spectral factorization problem for multivariable distributed parameter systems with an impulse response having an infinite number of delayed impulses. A coercivity criterion for the existence of an invertible spectral factor is given for the cases that the delays are a) arbitrary (not necessarily commensurate) and b) equally spaced (commensurate); for the latter case the criterion is applied to a system consisting of two parallel transmission lines without distortion. In all cases, it is essentially shown that, under the given criterion, the spectral density matrix has a spectral factor whenever this is true for its singular atomic part, i.e. its series of delayed impulses (with almost periodic symbol). Finally, a small-gain type sufficient condition is studied for the existence of spectral factors with arbitrary delays. The latter condition is meaningful from the system theoretic point of view, since it guarantees feedback stability robustness with respect to small delays in the feedback loop. Moreover its proof contains constructive elements.     

Epitaxy of Binary Compounds and Alloys

The present work aims at constructing a theoretical framework within which to address the issues of morphological instabilities (one-dimensional step bunching and two-dimensional step meandering), alloying, and phase segregation in binary systems in the context of (physical or chemical) vapor deposition. The length scale of interest, although nanoscopic, is sufficiently large that the steps on a vicinal surface can be viewed as smooth curves and, correspondingly, the theory is a continuum one. In a departure from theories inaugurated by Burton, Cabrera, and Frank [The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. Roy. Soc. A 243 (1951) 299–358] the steps are endowed with a thermodynamic structure whose main ingredients are a step free-energy density and edge species chemical potentials. Moreover, crystal anisotropy, with its altering of the dynamics of steps and the associated morphological instabilities, is accounted for – in a manner consistent with the second law – both in the thermodynamic and kinetic properties of terraces and, more importantly, of steps. Additionally, in contrast with most of the literature on the subject (cf. [J. Krug, Introduction to step dynamics and step instabilities. In: A. Voigt (ed.) Multiscale Modeling in Epitaxial Growth. Birkhäusser, Berlin (2005)]), adsorption–desorption along the steps, bulk atomic diffusion, and chemical reactions (both on the terraces and along the step edges) are incorporated and coupled to the other mechanisms, e.g., terrace adatom diffusion and step attachment–detachment kinetics, whose interplay governs the evolution of steps on vicinal surfaces. Importantly, aided by the concept of configurational forces for which a separate balance law is postulated Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin Heidelberg New York (2000)]), the proposed theory allows the steps to evolve away from local equilibrium thus contributing to a general treatment of the dynamics of steps. Finally, a specialization to the epitaxy of binary compounds and alloys is afforded, yielding a generalization of the classical Gibbs–Thomson relation in the former and novel evolution equations for an individual step in the latter.