Hypergraphs and the Clar problem in hexagonal systems

We show that for a hypergraph H that is separable and has the Helly property, the perfect matchings of H are the strongly stable sets of the line graph of H. Also, we show that the hypergraph generated by a hexagonal system is separable and has the Helly property. Finally, we note that the Clar problem of a hexagonal system is a minimum cardinality perfect matching problem of the hypergraph generated by the hexagonal system. Hence, the Clar problem of a hexagonal system is a minimum cardinality strongly stable set problem in the line graph of the hypergraph generated by the hexagonal system.     

Fractal sets determined by dilation-stable processes

Let be a dilation-stable process on . We determine a Hausdorff measure function (a) such that the fractal set X[0,1]={X(t):0t1} has positive finite -measure. We also investigate the packing measure of X[0,1].     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.     

Characterizations of some classes of relatively regular operators

A bounded linear operator A on a Banach space is called relatively regular, if there is a bounded linear operator B such that ABA=A. In this case B is called a g₁-inverse of A. In this paper we characterize some classes of relatively regular operators A via the set {B₁-B₂:B₁ and B₂ are g₁-inverses of A}.     

The PI index of product graphs

The Padmakar–Ivan index of a graph G is the sum over all edges uv of G of number of edges which are not equidistant from u and v. In this work, an exact expression for the PI index of the Cartesian product of bipartite graphs is computed. Using this formula, the PI indices of C₄ nanotubes and nanotori are computed.     

Coclass and cohomology

Recently, it was proved by Leedham-Green and others that with a finite number of exceptions, every p-group of coclass r is a quotient of one of only a finite number of p-adic uniserial space groups. In this paper we use that structure to demonstrate that there are only finitely many isomorphism classes of cohomology rings of 2-groups of coclass r with coefficients in any fixed field k of characteristic 2. In addition, there is experimental evidence indicating that in many cases successive quotients of the uniserial space groups have isomorphic cohomology rings.     

Buchsbaum homogeneous algebras with minimal multiplicity

In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.     

Nonexistence of a code for

In this paper, we shall prove that there is no [3q⁴-q³-q²-3q-1,5,3q⁴-4q³-2q+1]_q code over the finite field for q11. Thus, we conclude the nonexistence of a [g_q(5,d),5,d]_q code for 3q⁴-4q³-2q+1d3q⁴-4q³-q.     

On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem

where 0<ε1 is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0,1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.     

On association schemes all elements of which have valency 1 or 2

In the present note, we investigate schemes S in which each element s satisfies n_s2 and n_s^*s≠2. We show that such a scheme is schurian. More precisely, we show that it is isomorphic to G//t, where G is a finite group and t an involution of G weakly closed in C_G(t).

Groups G with an involution t weakly closed in C_G(t) have been described in Glauberman's Z^*-Theorem [G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420] with the help of the largest normal subgroup of G having odd order.     

Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions

We describe high-precision computations of the Pearcey integral Pe(x,y) for real x and y by means of Hadamard expansions. Numerical results for (x,y) situated in different regions of the x,y-plane are given to illustrate the levels of precision that can be achieved. Particular emphasis is given to computation in the neighbourhood of the two cusped curves associated with Pe(x,y) across which there is either a coalescence of saddles or a Stokes phenomenon.     

The last subconstituent of the Hemmeter graph

We prove that when q is any odd prime power, the distance-2 graph on the set of vertices at maximal distance D from any fixed vertex of the Hemmeter graph Hem_D(q) is isomorphic to the graph Quad_D-1(q) of quadratic forms on .     

Edge-coloring of multigraphs

We introduce a monotone invariant π(G) on graphs and show that it is an upper bound of the chromatic index of graphs. Moreover, there exist polynomial time algorithms for computing π(G) and for coloring edges of a multigraph G by π(G) colors. This generalizes the classical edge-coloring theorems of Shannon and Vizing.     

Isovariant maps from free -manifolds to representation spheres

The isovariant version of Borsuk–Ulam type theorems has been studied by Wasserman and the first author. In this paper, first we consider the relation between the existence of C_n-isovariant maps from free C_n-manifolds to representation spheres and Borsuk–Ulam type inequalities for their dimensions. Our main result classifies the C_n-isovariant maps by C_n-isovariant homotopy types when a Borsuk–Ulam type inequality holds. For proving it, we use the multidegree of a C_n-equivariant map developed by the first author.     

A new generalized p-value for ANOVA under heteroscedasticity

For the problem of comparing the means of k populations with unequal population variances, a new generalized test variable is defined and the generalized p-value based on this generalized test variable is given. It is shown that the proposed generalized p-value is invariant under the group of scale transformations. Numerical results show that the proposed generalized p-value test performs better than a generalized F-test.     

Existence of an elliptic system involving Pucci operator

We study the existence of solutions for the nonlinear elliptic system

where Ω is a bounded domain, f₁ is superlinear and f₂ is sublinear at zero and infinity, h₁ and h₂ are perturbation terms. We will show that the system has at least two semi-trivial solutions (u,0), (0,v) and a nontrivial solution (u^*,v^*).     

Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms

We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has, in every column, blocks of consecutive 1s, each corresponding to a shift. We focus on three types of shift scheduling problems classified according to the column structure in the constraint matrix: columns of consecutive 1s, columns of cyclical 1s, and columns of k consecutive blocks. In particular, the complexity of the cyclical scheduling problem, where the matrix satisfies the property of cyclical 1s in each column, was noted recently by Hochbaum and Tucker to be open. They further showed that the unit demand case is polynomially solvable. Here we extend this result to the uniform requirements case, and provide a 2-approximation algorithm for the non-uniform case. We also establish that the cyclical scheduling problem’s complexity is equivalent to that of the exact matching problem—a problem the complexity status of which is known to be randomized polynomial (RP). We then investigate the three types of shift scheduling problems and show that, while the consecutive ones version is polynomial and the k-block columns version is NP-hard (for k≥2), for the k-blocks problem we give a simple k-approximation algorithm, which is the first approximation algorithm determined for the problem.     

Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

Imaginary automorphisms on real hyperelliptic curves

A real hyperelliptic curve X is said to be Gaussian if there is an automorphism such that , where [-1] denotes the hyperelliptic involution on X. Gaussian curves arise naturally in several contexts, for example when one studies real Jacobians. In the present paper, we study the properties of Gaussian curves and we describe their moduli spaces.     

Limit shape of convex lattice polygons with minimal perimeter

Let n be a positive integer and · any norm in . Denote by B the unit ball of · and the class of convex lattice polygons with n vertices and least ·-perimeter. We prove that after suitable normalization, all members of tend to a fixed convex body, as n→∞.