Optimal Recovery of Isotropic Classes of rth Differentiable Functions on ℝ d

We consider the optimal recovery problem of isotropic classes of r-th differentiable multivariate functions defined on ? ^d , and obtain some asymptotically optimal results. It turns out that this optimal recovery problem is intimately related to the optimal covering problem of ? ^d by equal balls in discrete geometry.     

Prediction of liquid–liquid equilibrium for ternary systems containing ionic liquids with the tetrafluoroborate anion using ASOG

Liquid–liquid equilibrium (LLE) data for different systems involving ionic liquids are essential for design, optimization and operation of separation processes, such as recovery of valuable products and remotion of polluting agents in effluents. In this work, the ASOG model for the activity coefficient is used to predict LLE data for 32 ternary systems at 101.3 kPa and several temperatures; all the systems are formed by ionic liquids including the tetrafluoroborate anion plus alkanes, alkenes, cycloalkanes, alkanols, ketones, carboxylic acids, esters and aromatics. New group interaction parameters were determined by using a modified Simplex method, minimizing a composition-based objective function. The results, in terms of mean deviation between the experimental and calculated compositions, are satisfactory, with rms deviations of about 4%.     

On regular 4-coverings and their application for lattice coverings in normed planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquel’s six-circles theorem from the Euclidean plane to all strictly convex normed planes.     

New universal primers facilitate Pyrosequencing

A cost-driving factor in Pyrosequencing is the need for single-stranded PCR products that are usually obtained by biotin-labeling of one primer. We designed new universal primers that allow the introduction of biotin during the specific PCR at either the forward or the reverse primer in a single reaction. When converting five human single nucleotide polymorphism assays from the standard format into the universal format, we obtained pyrograms of similar good quality. Although the universal nonhuman sequences are unlikely to form loop structures, we mostly failed to establish assays without using the proprietary software from Biotage AB. Nevertheless, our universal primer pair adds flexibility to the process of assay design due to optional strand selection and contributes to the reduction of genotyping costs.     

The lotto numbers L(n,3,p,2)

An (n,k,p,t)‐lotto design is an n‐set N and a set of k‐subsets of N (called blocks) such that for each p‐subset P of N, there is a block for which . The lotto number L(n,k,p,t) is the smallest number of blocks in an (n,k,p,t)‐lotto design. The numbers C(n,k,t) = L(n,k,t,t) are called covering numbers. It is easy to show that, for n ≥ k(p ? 1), For k = 3, we prove that equality holds if one of the following holds:

• (i) n is large, in particular
• (ii)
• (iii) 2 ≤ p ≤ 6.

© 2006 Wiley Periodicals, Inc. J Combin Designs 14: 333–350, 2006     

Regular graphs are not universal fixers

For any permutation π of the vertex set of a graph G, the graph πG is obtained from two copies G^′ and G^″ of G by joining u∈V(G^′) and v∈V(G^″) if and only if v=π(u). Denote the domination number of G by γ(G). For all permutations π of V(G), γ(G)≤γ(πG)≤2γ(G). If γ(πG)=γ(G) for all π, then G is called a universal fixer. We prove that regular graphs and graphs with γ=4 are not universal fixers.     

MIP reformulations of the probabilistic set covering problem

In this paper, we address the following probabilistic version (PSC) of the set covering problem: where A is a 0-1 matrix, is a random 0-1 vector and is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of are pairwise independent, the distribution function of is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second. Anureet Saxena’s research was supported by the National Science Foundation through grant #DMI-0352885 and by the Office of Naval Research through contract N00014-03-1-0133. Vineet Goyal’s research was supported in part by NSF grant CCF-0430751 and ITR grant CCR-0122581.     

On openness and surjectivity of lifted frame homomorphisms

Given a completely regular frame L, let, as usual, βL, λL and υL denote, respectively, the Stone-?ech compactification, the universal Lindelöfication and the Hewitt realcompactification of L. Let γ denote any of the functors β, λ or υ. It is well known that any frame homomorphism h:L→M has a unique “lift” to a frame homomorphism hγ:γL→γM such that σM⋅hγ=h⋅σL, where the σ-maps are effected by join. We find a condition on h such that if h satisfies it, then h is open iff its lift hγ is open. Furthermore, the same condition ensures that hγ is nearly open iff h is nearly open. This latter result is, in fact, a special case of a more general phenomenon. In the last part of the paper we investigate when hυ is surjective. The instances when hβ or hλ is surjective are known. It turns out that the surjectivity of the lifted map hυ:υL→υM captures Blair's notion of υ-embedding in the sense that a subspace S of a Tychonoff space X is υ-embedded iff the lifted map υ(Oi):υ(OX)→υ(OS) is surjective, where i:S→X is the subspace embedding.     

Group actions and covering maps in the uniform category

In Rips complexes and covers in the uniform category (Brodskiy et al. [4]) we define, following James (1990) [9], covering maps of uniform spaces and introduce, inspired by Berestovskii and Plaut (2007) [2], the concept of generalized uniform covering maps. In this paper we investigate when these covering maps are induced by group actions which allows us to relate our covering maps to those in Berestovskii and Plaut (2007) [2]. Also, as an application of our results we present an exposition of Prajs' (2002) [16] homogeneous curve that is path-connected but not locally connected.