Topological ergodic decompositions and applications to products of powers of a minimal transformation

For a minimal distal flow (X, T) and a positive integern, let be the largest distal factor of ordern. The existence of a denseG _δ subset ω ofX is shown, such that forx ∈ ω the orbit closure of (x,x,...,x) ∈ X ⁿ⁺¹ under τ =T ×T ² ... ×T ⁿ⁺¹ is π-saturated. In fact, an analogous statement for a general minimal flow is proved in terms of its PI-tower. On the way we get some topological “ergodic” decomposition theorems.     

On the surjectivity of some trace maps

LetK be a commutative ring with a unit element 1. Let Γ be a finite group acting onK via a mapt: Γ→Aut(K). For every subgroupH≤Γ define tr _H :K→K ^H by tr _h (x)=Σ_σ∈H σ(x). We proveTheorem: tr_Γ is surjective onto K ^Γ if and only if tr _P is surjective onto K ^P for every (cyclic) prime order subgroup P of Γ. This is false for certain non-commutative ringsK.     

Orthogonal polynomials for exponential weights on

Let I=[0,d), where d is finite or infinite. Let , where and Q is continuous and increasing on I, with limit ∞ at d. We study the orthonormal polynomials associated with the weight , obtaining bounds on the orthonormal polynomials, zeros, and Christoffel functions. In addition, we obtain restricted range inequalities.     

The vapour pressures over saturated aqueous solutions of dl-2-aminobutyric acid, 4-aminobutyric acid,sodium-d-gluconate,sodium hippurate,and potassium magnesium-l-aspartate

Vapour pressures of water over saturated solutions of dl-2-aminobutyric acid, 4-aminobutyric acid, sodium-d-gluconate, sodium hippurate, and potassium magnesium-l-aspartate were determined over the (278 to 322) K temperature range. The determined vapour pressures were used to obtain the water activities, the molar enthalpies of vaporization, and the osmotic coefficients of sodium-d-gluconate.     

The intersection of a matroid and a simplicial complex

A classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a ``covering" parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case of a matroidal version of Ryser's conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter--the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on ``independent systems of representatives" (which are the special case in which the matroid is a partition matroid). In particular, we define a notion of -matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph is -matroidally colorable.

The methods used are mostly topological.

     

Effective location models for sorting recyclables in public management

The recycling of urban solid wastes is a critical point for the “closing supply chains” of many products, mainly when their value cannot be completely recovered after use. In addition to environmental aspects, the process of recycling involves technical, economic, social and political challenges for public management. For most of the urban solid waste, the management of the end-of-life depends on selective collection to start the recycling process. For this reason, an efficient selective collection has become a mainstream tool in the Brazilian National Solid Waste Policy. In this paper, we study effective models that might support the location planning of sorting centers in a medium-sized Brazilian city that has been discussing waste management policies over the past few years. The main goal of this work is to provide an optimal location planning design for recycling urban solid wastes that fall within the financial budget agreed between the municipal government and the National Bank for Economic and Social Development. Moreover, facility planning involves deciding on the best sites for locating sorting centers along the four-year period as well as finding ways to meet the demand for collecting recyclable materials, given that economic factors, consumer behavior and environmental awareness are inherently uncertain future outcomes. To deal with these issues, we propose a deterministic version of the classical capacity facility location problem, and both a two-stage recourse formulation and risk-averse models to reduce the variability of the second-stage costs. Numerical results suggest that it is possible to improve the current selective collection, as well as hedge against data uncertainty by using stochastic and risk-averse optimization models.     

Relative weak mixing is generic

A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergolic extensions of a fixed, but arbitrary,aperiodic transformation T_0. We then use a result of Ornstein and Weiss to extend this relative theorem to the general(countable) amenable group.     

The Max-Flow Min-Cut theorem for countable networks

We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are “orthogonal” to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff's first law also for ends.     

Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

On Time Discretizations for Spectral Methods

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.