Matroid optimization with the interleaving of two ordered sets

We study the structure of the minimum weight base of a matroid M = (E, I) the order of whose element set E is determined by the interleaving of two ordered subsets of E, R and W. The results imply an interesting application in economics, and are useful for the rapid recomputation of the minimum weight base when the order of E is successively modified by changing the interleaving of R and W. As a special case of the main result, the following parametric problem is efficiently solved: For M = (E, I) a matroid with weighted element set E, and R a subset of E, find for all feasible values of q, the minimum weight base of M containing exactly q elements of R. This parametric problem is a weighted matroid intersection problem and hence can be solved by known matroid intersection algorithms. The approach in this paper is different, and vastly improves the efficiency of the solution, as well as determining structural information about the bases.     

Injective hulls with distinct ring structures

It is well known from Osofsky’s work that the injective hull E(R_R) of a ring R need not have a ring structure compatible with its R-module scalar multiplication. A closely related question is: if E(R_R) has a ring structure and its multiplication extends its R-module scalar multiplication, must the ring structure be unique? In this paper, we utilize the properties of Morita duality to explicitly describe an injective hull of a ring R with R=Q(R) (where Q(R) is the maximal right ring of quotients of R) such that every injective hull of R_R has (possibly infinitely many) distinct compatible ring structures which are mutually ring isomorphic and quasi-Frobenius. Further, these rings have the property that the ring structures for E(R_R) also are ring structures on E(_RR).     

Bounded point evaluations and capacity

Let E be a compact set in the complex plane with positive Lebesgue measure, and denote by R^p(E), p ? 1, the closure in the L^p(E) norm of the rational functions with poles off E. A point z?E is said to be a bounded point evaluation for R^p(E) if the map z   ?(z), defined for the rational functions, can be extended to a bounded linear functional on R^p(E). For p < 2 there are no other bounded point evaluations for R^p(E) than the interior points of E, but for p ? 2 there may be bounded point evaluations on the boundary, ∂E. We give a condition, in terms of capacity, which is necessary and sufficient for a point on ∂E to be a bounded point evaluation for R^p(E), 2 < p < ∞, and close to necessary and sufficient when p = 2. We also treat bounded point derivations, and the corresponding problems for L_p-spaces of analytic functions on open sets.     

Theory of connectivity: a systematic formulation of boundary element methods

A theory of connectivity recently developed by the author is applied to construct a systematic formulation of boundary element methods. The concept of complete connectivity condition is shown to supply an alternative to boundary integral equations. The general problem of connecting solutions defined in neighbouring regions R and E is shown to lead to complete connectivity conditions which permit the formulation of three kinds of variational principles; they involve, respectively, R ∪ E, R and the common boundary between R and E, only.     

Self-similar sets with optimal coverings and packings

We prove that if a self-similar set E in Rn with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the Hs-density on the class of arbitrary subsets of Rn and on the class of Euclidean balls are attained, and the inverses of these values give the exact values of the Hausdorff and spherical Hausdorff measure of E. We also show that a ball of minimal density exists, and the inverse density of this ball gives the exact packing measure of E. Lastly, we show that these elements of optimal densities allow us to construct an optimal almost covering of E by arbitrary subsets of Rn, an optimal almost covering of E by balls and an optimal packing of E.     

Visible lattice points in the sphere

The number of visible (primitive) lattice points in the sphere of radius R is well approximated by . We consider an integral expression involving the error term E^∗(R), which leads to E^∗(R)=Ω(R(logR)^1/2). This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E^∗(R). This approach to prove Ω-results seems to be new and could be useful in other problems.     

Ratliff-Rush closures of ideals with respect to a Noetherian module

Let R be a commutative Noetherian ring, E a non-zero finitely generated R-module and I a E-proper ideal of R. The purpose of this paper is to provide some new characterizations of when all powers of I are Ratliff-Rush closed with respect to E and to answer a question raised by W. Heinzer et al. in (The Ratliff-Rush Ideals in a Noetherian Ring: A Survey, in Methods in Module Theory, Dekker, New York, 1992, pp. 149-159).     

Canonical decompositions of symmetric submodular systems

Let E be a finite set, R be the set of real numbers and f: 2^E→R be a symmetric submodular function. The pair (E,f) is called a symmetric submodular system. We examine the structures of symmetric submodular systems and provide a decomposition theory of symmetric submodular systems. The theory is a generalization of the decomposition theory of 2-connected graphs developed by Tutte and can be applied to any (symmetric) submodular systems.     

Henriksen?s contributions to residue class rings of analytic and entire functions

The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.Let A(R) denote the ring of analytic functions over reals R and E(K) the ring of entire functions over R or complex numbers C. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real-closed field that is η₁-set, while if m is a maximal ideal of E(K), then E(K)/m is isomorphic to one of these latter two fields or to complex numbers.     

Pointwise estimates for marginals of convex bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E⊂Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c>0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.     

On the asymptotic properties of the Rees powers of a module

Let R be a commutative ring and G a free R-module with finite rank e>0. For any R-submodule E⊂G one may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components E_n, n≥0, of the image, that we shall call the n-th Rees powers of E (with respect to the embedding E⊂G). In this work we prove some asymptotic properties of the R-modules E_n, n≥0, which extend well known similar ones for the case of ideals, among them Burch’s inequality for the analytic spread.     

Reduced convex bodies in the plane

A convex bodyR of Euclideand-spaceE ^d is called reduced if there is no convex body properly contained inR of thickness equal to the thickness Δ(R) ofR. The paper presents basic properties of reduced bodies inE ². Particularly, it is shown that the diameter of a reduced bodyR?E ² is not greater than √2Δ(R), and that the perimeter is at most (2+½π)Δ(R). Both the estimates are the best possible.     

Doubling measures on Whitney modification sets

A Whitney modification set in R is a set of the form E∪F(W), where E is a nonempty closed set in R, W is a Whitney decomposition of R?E, and F(W) consists of centers of intervals in W. We prove that a measure on a Whitney modification set is doubling if and only if it is the Whitney modification of a doubling measure on R.     

A commutative analogue of the group ring

Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded Z-algebra R_G. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element.In the case when G is an elementary Abelian p-group it turns out that R_G is closely related to the symmetric algebra over F_p of the dual of G. We intend in subsequent papers to explore the close relationship between G and R_G in the case of a general (possibly non-Abelian) group G.Here we show that the Krull dimension of R_G is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1.     

Canonical configurations

Given a finite configuration E in Rⁿ and an arbitrary coloring of Rⁿ (possibly with an infinite number of colors) it is shown that there exists E′ homothetic to E such that E′ is colored in a “special” way. The proof is based on the recent theorem of W. Deuber, R. L. Graham, H. J. Prömel, and B. Voigt, J. Combin. Theory Ser. A.34 (1983), 331–339.     

Linear transformations on matrices: the invariance of sets of ranks

Let R_E denote the set of all m × n matrices over an algebraically closed field F whose ranks lie in the set E, where E is a subset of {1,2,…,m}. Let T be a linear transformation which maps R_E into itself. Under some restrictions on E, or when T is nonsingular, there are nonsingular matrices U and V such that T(A) = UAV for every m × n matrix A.     

Weighted analogues of capacity,transfinite diameter,and Chebyshev constant

For an arbitrary closed subsetE of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant ofE with respect to an admissible weightw onE are introduced. For thew-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an extremal measure whose support is the “smallest” compact set where the sup norm of weighted polynomials “live.” The introduction of a weightw has the advantage that the classical quantities mentioned in the title can be considered for unbounded setsE. Some of the theorems presented are generalizations of the authors' previous results for the case whenE?R.     

Convergence parameter associated with a Markov chain and a family of functions

The proposed definition of convergence parameter R(W) corresponding to a Markov chain X with a measurable state space (E,?) and any nonempty setW of bounded below measurable functions f: E → ? is wider than the well-known definition of convergence parameter R in the sense of Tweedie or Nummelin. Very often, R(W) < ∞, and there exists a set playing the role of the absorbing set inNummelin’s definition ofR. Special attention is paid to the case in whichE is locally compact, X is a Feller chain on E, and W coincides with the family ? ₀ ⁺ of all compactly supported continuous functions f ≥ 0 (f ? 0). In particular, certain conditions for R(? ₀ ⁺ )^?1 to coincide with the norm of an appropriate modification of the chain transition operator are found.     

Trilinear products and comtrans algebra representations

Comtrans algebras are modules over a commutative ring R equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. The standard construction of a comtrans algebra uses the ternary commutator and translator of a trilinear product. If 6 is invertible in R, then each comtrans algebra arises in this standard way from the so-called bogus product.Consider a vector space E of dimension n over a field R. While the dimension of the space of all trilinear products on E is n⁴, the dimension of the space of all comtrans algebras on E is less, namely . The paper determines which trilinear products may be represented as linear combinations of the commutator and translator of a comtrans algebra. For R not of characteristic 3, the necessary and sufficient condition for such a representation is the strong alternativityxxy+xyx+yxx=0 of the trilinear product xyz. For R also not of characteristic 2, it is shown that the representation may be given by the bogus product. A suitable representation for the characteristic 2 case is also obtained.     

About a construction and some analysis of time inhomogeneous diffusions on monotonely moving domains

We construct and analyze in a very general way time inhomogeneous (possibly also degenerate or reflected) diffusions in monotonely moving domains E⊂R×R^d, i.e. if E_t?{x∈R^d|(t,x)∈E}, t∈R, then either E_s⊂E_t, ∀s?t, or E_s⊃E_t, ∀s?t, s,t∈R. Our major tool is a further developed L²(E,m)-analysis with well chosen reference measure m. Among few examples of completely different kinds, such as e.g. singular diffusions with reflection on moving Lipschitz domains in R^d, non-conservative and exponential time scale diffusions, degenerate time inhomogeneous diffusions, we present an application to what we name skew Bessel process on γ. Here γ is either a monotonic function or a continuous Sobolev function. These diffusions form a natural generalization of the classical Bessel processes and skew Brownian motions, where the local time refers to the constant function γ≡0.