A Constructive Approach to Lattice Rule Canonical Forms

The rank and invariants of a general lattice rule are conventionally defined in terms of the group-theoretic properties of the rule. Here we give a constructive definition of the rank and invariants using integer matrices. This underpins a nonabstract algorithm set in matrix algebra for obtaining the Sylow p-decomposition of a lattice rule. This approach is particularly useful when it is not known whether the form in which the lattice rule is specified is canonical or even repetitive. A new set of necessary and sufficient conditions for recognizing a canonical form is given.     

Triangular canonical forms for lattice rules of prime-power order

In this paper we develop a theory of -cycle representations for -dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a -matrix consisting of the nontrivial invariants. Among these is a family of triangular forms, which, besides being canonical, have the defining property that their -matrix is a column permuted version of a unit upper triangular matrix. Triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra. Our main result is to define a unique canonical form for prime-power rules. This ultratriangular form is a triangular form, is easy to recognize, and may be derived in a straightforward manner.

     

The Number of Lattice Rules of Specified Upper Class and Rank

The upper class of a lattice rule is a convenient entity for classification and other purposes. The rank of a lattice rule is a basic characteristic, also used for classification. By introducing a rank proportionality factor and obtaining certain recurrence relations, we show how many lattice rules of each rank exist in any prime upper class. The Sylow p-decomposition may be used to obtain corresponding results for any upper class.     

Canonical forms of some special matrices useful in statistics

In experimental situations wheren two or three level factors are involved andn observations are taken, then theD-optimal first order saturated design is ann ×n matrix with elements ±1 or 0, ±1 with the maximum determinant. Canonical forms are useful for the specification of the non-isomorphicD-optimal designs. In this paper, we study canonical forms such as the Smith normal form, the first, second and the Jordan canonical form ofD-optimal designs. Numerical algorithms for the computation of these forms are described and some numerical examples are also given.     

Modular forms of orthogonal type and Jacobi theta-series

In this paper, we study Jacobi forms of half-integral index for any even integral positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A ₁=〈2〉). We construct Jacobi forms of singular (respectively, critical) weight in all dimensions n≥8 (respectively, n≥9). We give the Jacobi lifting for Jacobi forms of half-integral indices and we obtain an additive lifting construction of new reflective modular forms which are natural generalizations to O(2,n) (n=4, 5 and 6) of the Igusa modular form Δ ₅.     

On the Influence of the Indices of Normalizers of Sylow Subgroups on the Structure of a Finite p-Soluble Group

We study the dependence of the structure of finite p-soluble groups on the indices of normalizers of Sylow subgroups. We obtain estimates for the p-length of these groups, and for small values of indices we find the nilpotent length of a soluble group.     

Generalized Arf invariants in algebraic L-theory

The difference between the quadratic L-groups L_*(A) and the symmetric L-groups L^*(A) of a ring with involution A is detected by generalized Arf invariants. The special case A=Z[x] gives a complete set of invariants for the Cappell UNil-groups UNil_*(Z;Z,Z) for the infinite dihedral group D_∞=Z₂*Z₂, extending the results of Connolly and Ranicki [Adv. Math. 195 (2005) 205-258], Connolly and Davis [Geom. Topol. 8 (2004) 1043-1078, e-print http://arXiv.org/abs/math/0306054].     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

Galois structure of modular forms of even weight

We calculate the equivariant Euler characteristics of powers of the canonical sheaf on certain modular curves over Z which have a tame action of a finite abelian group. As a consequence, we obtain information on the Galois module structure of modular forms of even weight having Fourier coefficients in certain ideals of rings of cyclotomic algebraic integers.     

On finite groups with prime-power order <Emphasis Type="Italic">S</Emphasis>-quasinormally embedded subgroups

A subgroup H of a finite group G is said to be S-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. In this paper we investigate the structure of finite groups that have some S-quasinormally embedded subgroups of prime-power order, and new criteria for p-nilpotency are obtained.     

Separation of variables of a generalized porous medium equation with nonlinear source

This paper considers a general form of the porous medium equation with nonlinear source term: u_t=(D(u)u_xⁿ)_x+F(u), n≠1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions to the resulting equations are constructed, and their behavior are also investigated.     

Nerve complexes and moment-angle spaces of convex polytopes

We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope P to the nerve K _P of the cover of the boundary ∂P by facets. We develop a theory of nerve complexes and apply it to the moment-angle spaces Z _P of convex polytopes P. In the case of a polytope P with m facets, its moment-angle space Z _P is defined by the canonical embedding in the cone ℝ_≥ ^m . It is well-known that the space Z _P is homeomorphic to the polyhedral product (D ², S ¹)∂P* if the polytope P is simple. We show that the homotopy equivalence Z_P @ (D² ,S¹ )^K_P \mathcal{Z}_P \simeq (D^2 ,S^1 )^{K_P } holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known f-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, f-polynomials) coincide, while the new invariants are different.     

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.     

Decomposability and embeddability of discretely Henselian division algebras

LetF be a discretely Henselian field of rank one, with residue fieldk a number field, and letD/F be anF-division algebra. We conduct an exhaustive study of the decomposability of an arbitraryD. Specifically, we prove the following:D has a semiramified (SR)F-division subalgebra if and only ifD has a totally ramified (TR) subfield. However, there may be TR subfields not contained in any SR subalgebra. IfD has prime-power index, thenD is decomposable if and only ifD properly contains a SR division subalgebra. Equivalently,D has a decomposable Sylow factor if and only if ii(D ^⊗n )≠1/n i(D) for somen dividing the period ofD, that is, if and only if the index fails to mimic the behavior of the period ofD. There exists indecomposableD with prime-power periodp ² and indexp ³. Every proper division subalgebra ofD is indecomposable. Conversely, every indecomposableF-division algebra ofp-power index embeds properly in someD ofp-power index if and only ifk does not have a certain strengthened form of class field theory’s Special Case. Semiramified division algebras and division algebras of odd index always properly embed. Finally, these results apply to an extent overk(t), and we prove that there exist indecomposablek(t)-division algebras of periodp ² and indexp ³, solving an open problem of Saltman. Dedicated to the memory of Amitsur Research supported in part by NSF Grant DMS-9100148.     

Tiling space with the aid of the holomorph

The notion of a factorization of a group is generalized and a method is presented for obtaining new factorizations from old ones. The results are applied to obtain new fillings of the lattice spaces Z, Z ⊕ Z and the cube.     

Random weights, robust lattice rules and the geometry of the cbcrc algorithm

In this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube [0,1] ^s from a weighted reproducing kernel Hilbert space. We assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications: (i) It is usually not known in practice how to choose the weights. Thus by assuming that the weights are random variables, we obtain robust constructions (with respect to the weights) of lattice rules. This, to some extend, removes the necessity to carefully choose the weights. (ii) In practice it is convenient to use the same lattice rule for many different integrands. The best choice of weights for each integrand may vary to some degree, hence considering the weights random variables does justice to how lattice rules are used in applications. In this paper the worst-case error is therefore a random variable depending on random weights. We show how one can construct lattice rules which perform well for weights taken from a set with large measure. Such lattice rules are therefore robust with respect to certain changes in the weights. The construction algorithm uses the component-by-component (cbc) idea based on two criteria, one using the mean of the worst case error and the second criterion using a bound on the variance of the worst-case error. We call the new algorithm the cbc2c (component-by-component with 2 constraints) algorithm. We also study a generalized version which uses r constraints which we call the cbcrc (component-by-component with r constraints) algorithm. We show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights ?? ⁽¹⁾, . . . , ?? ^(r). Thus, in applications, instead of finding one set of weights, it is enough to find a convex polytope in which the optimal weights lie. The price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule. Thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules. Numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights.     

Valuative invariants for polymatroids

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant G introduced by the first author, are valuative. In this paper we construct the Z-modules of all Z-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the Z-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that G is universal for valuative invariants.     

Dirichlet spectrum and heat content

Let M be a complete Riemannian manifold and D⊂M a smoothly bounded domain with compact closure. We use Brownian motion to study the relationship between the Dirichlet spectrum of D and the heat content asymptotics of D. Central to our investigation is a sequence of invariants associated to D defined using exit time moments. We prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions, that our invariants determine the heat content asymptotics associated to the manifold, and that when the manifold is a generic domain in Euclidean space, the invariants determine the Dirichlet spectrum.     

Subalgebras of matrix algebras generated by companion matrices

Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈M_n(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice M_n(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=M_n(R), in which case a presentation for M_n(R) in terms of C and D is given.