Bound on <Emphasis Type="Italic">m</Emphasis>-restricted Edge Connectivity

An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restrict edge connectivity λm is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let θ(X) denote the number of edges with one end in X and the other not in X and ξm=min{θ(X) ;X is a connected vertex-induced subgraph of order m}.It is proved in this paper that if G has girth at least m/2 2,then λm≤ξm.The upper bound of λm is sharp.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

Planar Line Processes for Void and Density Statistics in Thin Stochastic Fibre Networks

Using results for the distribution of perimeters of random polygons arising from random lines in a plane, we obtain new analytic approximations to the distributions of areas and local line densities for random polygons and compute various limiting properties of random polygons. Using simulation, we show that the lengths of adjacent sides of polygons generated by random line processes in the plane are correlated with ρ=0.616±0.001.     

Fagnano Orbits of Polygonal Dual Billiards

Given a convex n-gon P, a Fagnano periodic orbit of the respective dual billiard map is an n-gon Q whose sides are bisected by the vertices of P. For which polygons P does the ratio Area Q/Area P have the minimal value? The answer is shown to be: for affine-regular polygons.     

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p^m+2)-regular graphs of girth five and order 2p^2m, where p?5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f(k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f(k), k?16.     

Improved convex upper bound via conditional comonotonicity

Comonotonicity provides a convenient convex upper bound for a sum of random variables with arbitrary dependence structure. Improved convex upper bound was introduced via conditioning by Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168]. In this paper, we unify these results in a more general context using the concept of conditional comonotonicity. We also construct an approximating sequence of convex upper bounds with nice convergence properties.     

The derivation of explicit solutions to Wannier–Stark resonances for electrons on coupled chains

Electrons on infinite coupled chains with nearest neighbour couplings under uniform electric and magnetic fields can be expressed as conditionally solvable systems, which concerns both discrete coordinate and wavenumber representations. We then have to account for multiparameter extra-conditions relying on the single chain phase of the system, which amounts to perform a suitable selection of parameters. The implicit plots provided by such conditions exhibit both regular and irregular patterns. This results in the onset of a finite number of Wannier–Stark resonances, now by performing rescalings needed. However, this time the resonance width is sensitive to the quantum number characterizing the Stark-ladder. Bound-state limits, rescalings and approximations proceeding irrespective of the wavenumber have also been presented.     

Inversion of the Kirkwood–Buff theory of solutions: application to binary systems

Kirkwood–Buff (K–B) integrals play an important role in characterizing the properties and interactions of various liquid mixtures. However, there exists no method to calculate directly the K–B integrals from the easily available experimental data of ultrasonic velocity and density. An attempt has been made to evaluate the same using these initial parameters. A statistical mechanical theory advanced by Arakawa et al. has been used here, in combination with a semi-empirical formula to compute K–B parameters in the whole concentration range for six binary systems at atmospheric pressure and at 298.15 K.     

Korn's inequality and Donati's theorem for the conformal Killing operator on pseudo-Euclidean space

We prove the Korn's inequality for the conformal Killing operator on pseudo-Euclidean space Rp,q, and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which is a pseudo-Euclidean conformal generalization of Donati's theorem for Euclidean Killing operator.     

Phase Transitions for the Growth Rate of Linear Stochastic Evolutions

We consider a discrete-time stochastic growth model on d-dimensional lattice. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the “total number of particles” in this framework. The main results are roughly as follows: If d≥3 and the system is “not too random”, then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, d=1,2, or the system is “random enough”, then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the model with proper normalization. Supported in part by JSPS Grant-in-Aid for Scientific Research, Kiban (C) 17540112.