Pentaheptite modifications of the graphite sheet

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-packed to hexagonal-close-packed packings in 3D. Symmetries realizable for these pentaheptite layers are three strip groups (periodic in one dimension), p1a1, p112, and p111, and five Fedorov groups (periodic in two dimensions), cmm, pgg, pg, p2, and p1. All can be constructed by simultaneous rotation of the central bonds of pyrene tilings of the graphite sheet. The unique lattice of cmm symmetry corresponds to the previously proposed pentaheptite carbon metal. Analogous pentagon-heptagon tilings on other surfaces including the torus, Klein bottle, and cylinder, face-regular tilings of pentagons and b-gons, and a full characterization of tilings involving isolated pairs and/or triples of pentagons are presented. The Kelvin paradigm of a continuum of structures arising from propagation of two original motifs has many potential applications in 2D and 3D.     

Odd Systems of Vectors and Related Lattices

We consider uniform odd systems, i.e. sets of vectors of constant odd norm with odd inner product, and the lattice L(V) linearly generated by a uniform odd system V of odd norm 2t+1. If uu p (mod 4) for all u V, one has v² p (mod 4) if v² is odd and v² 0 (mod 4) if v² is even, for any vector v L(V). The vectors of even norm form a double even sublattice L₀(V) of L(V), i.e. is an even lattice. The closure of V, i.e. all vectors of L(V) of norm 2t+1, are minimal vectors of L(V) for t=1, and they are almost always minimal for t=2. For such t, the convex hull of vectors of the closure of V is an L-polytope of L₀V and the contact polytope of L(V). As an example, we consider closed uniform odd systems of norm 5 spanning equiangular lines.     

Orthogonal systems

An equidistant permutation array (E.P.A.)A(r, v) is av × r array in which every row is a permutation of the integers 1, 2, ,r such that any two distinct rows have precisely columns in common. In this paper we introduce the concept of orthogonality for E.P.A.s. A special case of this is the well known idea of a set of pairwise orthogonal latin squares. We show that a set of these arrays is equivalent to a particular type of resolvable (r, )-design. It is also shown that the cardinality of such a set is bounded byr – with the upper bound being obtained only if = 0. A brief survey of related orthogonal systems is included. In particular, sets of pairwise orthogonal symmetric latin squares, sets of orthogonal Steiner systems and sets of orthogonal skeins.     

Codes in platonic, Archimedean, Catalan, and related polyhedra: a model for maximum addition patterns in chemical cages

The notion of d-code is extended to general polyhedra by defining maximum sets of vertices with pairwise separation > or =d. Codes are enumerated and classified by symmetry for all regular and semiregular polyhedra and their duals. Partial results are also given for the series of medials of Archimedean polyhedra. In chemistry, d-codes give a model for maximal addition to or substitution in polyhedral frameworks by bulky groups. Some illustrative applications from the chemistry of fullerenes and boranes are described.     

Fullerenes as tilings of surfaces

If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Klein-bottle fullerenes, none. Klein-bottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Klein-bottle, and toroidal cases, respectively.     

Aspects of stochastic resonance in reaction–diffusion systems: The nonequilibrium-potential approach

We analyze several aspects of the phenomenon of stochastic resonance in reaction–diffusion systems, exploiting the nonequilibrium potential's framework. The generalization of this formalism (sketched in the appendix) to extended systems is first carried out in the context of a simplified scalar model, for which stationary patterns can be found analytically. We first show how system-size stochastic resonance arises naturally in this framework, and then how the phenomenon of array-enhanced stochastic resonance can be further enhanced by letting the diffusion coefficient depend on the field. A yet less trivial generalization is exemplified by a stylized version of the FitzHugh–Nagumo system, a paradigm of the activator–inhibitor class. After discussing for this system the second aspect enumerated above, we derive from it–through an adiabatic-like elimination of the inhibitor field–an effective scalar model that includes a nonlocal contribution. Studying the role played by the range of the nonlocal kernel and its effect on stochastic resonance, we find an optimal range that maximizes the system's response.     

Formation and synchronization of autocatalytic noise-sustained structures under Poiseuille flow

The formation and synchronization of 2D noise-sustained structures are investigated for Gray–Scott kinetics in packed-bed reactors under Poiseuille flows, when identical systems are submitted to independent spatiotemporal Gaussian white noise sources. A finite-wavelength instability is theoretically predicted and numerically confirmed for uncoupled reactors. In particular, noise-sustained structures that flow with viscous boundary conditions are numerically observed above threshold. When the systems are coupled in master–slave configuration, the numerical simulations show that the slave system replicates to a very high degree of precision the convective patterns arising in the master one due to the selective amplification of noise. The nature of the synchronization and the stability of the synchronization manifold are elucidated.     

Correction to: Distance between vertices of lattice polytopes

Optimization Letters - The erratum mostly concerns Table 4 and Figure 6 where two polytopes were misrepresented in the original version.     

Fluctuation theorems from non-equilibrium Onsager-Machlup theory for a Brownian particle in a time-dependent harmonic potential

We discuss the case of a Brownian particle which is harmonically bound and multiplicatively forced-namely bound by V(x,t)=1/2 a(t)x ² where a(t)is externally controlled-as another instance that provides a generalization of Onsager-Machlup’s theory to non-equilibrium states, thus allowing establishment of several fluctuation theorems. In particular, we outline the derivation of a fluctuation theorem for work, through the calculation of the work probability distribution as a functional integral over stochastic trajectories.      

On t-distance sets of (0, ±1)-vectors

We consider sets of (0, +1)-vectors in R ⁿ, having exactly s non-zero positions. In some cases we give best or nearly best possible bounds for the maximal number of such vectors if all the pairwise scalar products belong to a fixed set D of integers. The investigated cases include D={ -d, d}, which corresponds to equiangular lines.