Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

A construction of non-isomorphic amorphous association schemes from pseudo-cyclic association schemes

As a continuation of my paper [T. Fujisaki, A construction of amorphous association scheme from a pseudo-cyclic association scheme, Discrete Math. 285(1–3) (2004) 307–311], we show a construction of amorphous association scheme which is a fusion scheme of a direct product of two pseudo-cyclic association schemes with same first eigenmatrix. By using this construction, we can get at most three amorphous association scheme. We prove that if two pseudo-cyclic association scheme are non-isomorphic, then these three amorphous association schemes are mutually non-isomorphic.     

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.     

Ideal ramp schemes and related combinatorial objects

In 1996, Jackson and Martin (Jackson and Martin, 1996) proved that a strong ideal ramp scheme is equivalent to an orthogonal array. However, there was no good characterization of ideal ramp schemes that are not strong. Here we show the equivalence of ideal ramp schemes to a new variant of orthogonal arrays that we term augmented orthogonal arrays. We give some constructions for these new kinds of arrays, and, as a consequence, we also provide parameter situations where ideal ramp schemes exist but strong ideal ramp schemes do not exist.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

Algebraic structure of association schemes of prime order

Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.     

The G-biliaison class of symmetric determinantal schemes

We consider a family of schemes, that are defined by minors of a homogeneous symmetric matrix with polynomial entries. We assume that they have maximal possible codimension, given the size of the matrix and of the minors that define them. We show that these schemes are G-bilinked to a linear variety of the same dimension. In particular, they can be obtained from a linear variety by a finite sequence of ascending G-biliaisons on some determinantal schemes. We describe the biliaisons explicitly in the proof of Theorem 2.3. In particular, it follows that these schemes are glicci.     

Investigations on association schemes with elementary abelian thin residue

In this paper, we give a new class of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of order $p^2$. We then study the automorphism groups of these schemes and determine whether these schemes are schurian.     

A normal form and schemes of quadratic forms

We present a solution of the problem of construction of a normal diagonal form for quadratic forms over a local principal ideal ring R = 2R with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over R with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of p-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 161–178, 2007.     

Regular population monotonic allocation schemes

The paper introduces a refinement of the notion of population monotonic allocation scheme, called regular population monotonic allocation scheme (regularpmas). This refinement is based on economic situations in which players may have to select new partners from a set of potential players and in which there exist certain capacity constraints. A sufficient condition for the existence of a regularpmas is given. For the class of games with regularpmas, we prove that the core coincides with the Davis and Maschler and the Mas-Colell bargaining sets.     

Asymptotic preserving HLL schemes

This work concerns the derivation of HLL schemes to approximate the solutions of systems of conservation laws supplemented by source terms. Such a system contains many models such as the Euler equations with high friction or the M1 model for radiative transfer. The main difficulty arising from these models comes from a particular asymptotic behavior. Indeed, in the limit of some suitable parameter, the system tends to a diffusion equation. This article is devoted to derive HLL methods able to approximate the associated transport regime but also to restore the suitable asymptotic diffusive regime. To access such an issue, a free parameter is introduced into the source term. This free parameter will be a useful correction to satisfy the expected diffusion equation at the discrete level. The derivation of the HLL scheme for hyperbolic systems with source terms comes from a modification of the HLL scheme for the associated homogeneous hyperbolic system. The resulting numerical procedure is robust as the source term discretization preserves the physical admissible states. The scheme is applied to several models of physical interest. The numerical asymptotic behavior is analyzed and an asymptotic preserving property is systematically exhibited. The scheme is illustrated with numerical experiments. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1396–1422, 2011     

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with . A sufficient condition on a cyclotomic scheme that guarantee the inclusion where is a finite field with elements, is given. I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1.     

Gorenstein categories and Tate cohomology on projective schemes

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov–Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category ??????(X) of quasi‐coherent sheaves on X is such a category and so has these features. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

SYMPLECTIC SCHEMES FOR NONAUTONOMOUS HAMILTONIAN SYSTEM

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Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

Distance and routing labeling schemes for non-positively curved plane graphs

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently (e.g., in constant or logarithmic time) by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes are schemes that label the vertices of a graph with short labels in such a way that given the label of a source vertex and the label of a destination, it is possible to compute efficiently (e.g., in constant or logarithmic time) the port number of the edge from the source that heads in the direction of the destination. In this paper we show that the three major classes of non-positively curved plane graphs enjoy such distance and routing labeling schemes using O(log²n) bit labels on n-vertex graphs. In constructing these labeling schemes interesting metric properties of those graphs are employed.     

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semi-linear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. The convergence of the approximate solution toward a weak solution is established in the cases of first and second order accurate MUSCL relaxed methods.

     

On Macaulayfication of Noetherian schemes

The Macaulayfication of a Noetherian scheme  is a birational proper morphism from a Cohen-Macaulay scheme to . In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension or . In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic . Our method, however, to construct a Macaulayfication is independent of the characteristic.