An algorithmic Friedman-Pippenger theorem on tree embeddings and applications

An (n, d)-expander is a graph G = (V, E) such that for every X subset of V with vertical bar X vertical bar <= 2n - 2 we have vertical bar Gamma(G)(X) vertical bar >= (d + 1) vertical bar X vertical bar. A tree T is small if it has at most n vertices and has maximum degree at most d. Friedman and Pippenger (1987) proved that any ( n; d)- expander contains every small tree. However, their elegant proof does not seem to yield an efficient algorithm for obtaining the tree. In this paper, we give an alternative result that does admit a polynomial time algorithm for finding the immersion of any small tree in subgraphs G of (N, D, lambda)-graphs Lambda, as long as G contains a positive fraction of the edges of Lambda and lambda/D is small enough. In several applications of the Friedman-Pippenger theorem, including the ones in the original paper of those authors, the (n, d)-expander G is a subgraph of an (N, D, lambda)-graph as above. Therefore, our result suffices to provide efficient algorithms for such previously non-constructive applications. As an example, we discuss a recent result of Alon, Krivelevich, and Sudakov (2007) concerning embedding nearly spanning bounded degree trees, the proof of which makes use of the Friedman-Pippenger theorem. We shall also show a construction inspired on Wigderson-Zuckerman expander graphs for which any sufficiently dense subgraph contains all trees of sizes and maximum degrees achieving essentially optimal parameters. Our algorithmic approach is based on a reduction of the tree embedding problem to a certain on-line matching problem for bipartite graphs, solved by Aggarwal et al. (1996).

On the resilience of long cycles in random graphs

In this paper we determine the local and global resilience of random graphs G(n,p) (p >> n(-1)) with respect to the property of containing a cycle of length at least (1 - alpha)n. Roughly speaking, given alpha > 0, we determine the smallest r(g) (G, alpha) with the property that almost surely every subgraph of G = G(n,p) having more than r(g) (G, alpha)vertical bar E(G)vertical bar edges contains a cycle of length at least (1 - alpha)n (global resilience). We also obtain, for alpha < 1/2, the smallest r(l) (G, alpha) such that any H subset of G having deg(H) (v) larger than r(l) (G, alpha) deg(G) (v) for all v is an element of V(G) contains a cycle of length at least (1 - alpha)n (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.

Gibbs random graphs on point processes

Consider a discrete locally finite subset Gamma of R(d) and the cornplete graph (Gamma, E), with vertices Gamma and edges E. We consider Gibbs measures on the set of sub-graphs with vertices Gamma and edges E` subset of E. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when Gamma is sampled from a homogeneous Poisson process; and (b) for a fixed Gamma with sufficiently sparse points. (c) 2010 American Institute of Physics. [doi:10.1063/1.3514605]

Induced Modules for Affine Lie Algebras

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra P of an affine Lie algebra G, our main result establishes the equivalence between a certain category of P-induced G-modules and the category of weight P-modules with injective action of the central element of G. In particular, the induction functor preserves irreducible modules. If P is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra P(ps), P subset of P(ps). The structure of P-induced modules in this case is fully determined by the structure of P(ps)-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. Konig, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63].

ON THE THREE-DIMENSIONAL BLASCHKE-LEBESGUE PROBLEM

The width of a closed convex subset of n-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n >= 3. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n = 3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant and therefore are either spherical caps or pieces of tubes (canal surfaces).

Can we make a Bohmian electron reach the speed of light, at least for one instant?

In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light-a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem, Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current (psi) over bar gamma(mu)psi is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function psi : R(4) -> C(4) that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of C(4) along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions psi : R(4) -> C(4), Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation. (C) 2010 American Institute of Physics. [doi:10.1063/1.3520529]

Extremal problems on sum-free sets and coverings in tridimensional spaces

Given a prime power q, define c (q) as the minimum cardinality of a subset H of F 3 q which satisfies the following property: every vector in this space di ff ers in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c (q)

Modeling associations between genetic markers using Bayesian networks

Motivation: Understanding the patterns of association between polymorphisms at different loci in a population ( linkage disequilibrium, LD) is of fundamental importance in various genetic studies. Many coefficients were proposed for measuring the degree of LD, but they provide only a static view of the current LD structure. Generative models (GMs) were proposed to go beyond these measures, giving not only a description of the actual LD structure but also a tool to help understanding the process that generated such structure. GMs based in coalescent theory have been the most appealing because they link LD to evolutionary factors. Nevertheless, the inference and parameter estimation of such models is still computationally challenging. Results: We present a more practical method to build GM that describe LD. The method is based on learning weighted Bayesian network structures from haplotype data, extracting equivalence structure classes and using them to model LD. The results obtained in public data from the HapMap database showed that the method is a promising tool for modeling LD. The associations represented by the learned models are correlated with the traditional measure of LD D`. The method was able to represent LD blocks found by standard tools. The granularity of the association blocks and the readability of the models can be controlled in the method. The results suggest that the causality information gained by our method can be useful to tell about the conservability of the genetic markers and to guide the selection of subset of representative markers.

Singular Perturbation for the Discounted Continuous Control of Piecewise Deterministic Markov Processes

This paper deals with the expected discounted continuous control of piecewise deterministic Markov processes (PDMP`s) using a singular perturbation approach for dealing with rapidly oscillating parameters. The state space of the PDMP is written as the product of a finite set and a subset of the Euclidean space a""e (n) . The discrete part of the state, called the regime, characterizes the mode of operation of the physical system under consideration, and is supposed to have a fast (associated to a small parameter epsilon > 0) and a slow behavior. By using a similar approach as developed in Yin and Zhang (Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Applications of Mathematics, vol. 37, Springer, New York, 1998, Chaps. 1 and 3) the idea in this paper is to reduce the number of regimes by considering an averaged model in which the regimes within the same class are aggregated through the quasi-stationary distribution so that the different states in this class are replaced by a single one. The main goal is to show that the value function of the control problem for the system driven by the perturbed Markov chain converges to the value function of this limit control problem as epsilon goes to zero. This convergence is obtained by, roughly speaking, showing that the infimum and supremum limits of the value functions satisfy two optimality inequalities as epsilon goes to zero. This enables us to show the result by invoking a uniqueness argument, without needing any kind of Lipschitz continuity condition.

Extension of the core map of common bean with EST-SSR, RGA, AFLP, and putative functional markers

Microsatellites and gene-derived markers are still underrepresented in the core molecular linkage map of common bean compared to other types of markers. In order to increase the density of the core map, a set of new markers were developed and mapped onto the RIL population derived from the `BAT93` x `Jalo EEP558` cross. The EST-SSR markers were first characterized using a set of 24 bean inbred lines. On average, the polymorphism information content was 0.40 and the mean number of alleles per locus was 2.7. In addition, AFLP and RGA markers based on the NBS-profiling method were developed and a subset of the mapped RGA was sequenced. With the integration of 282 new markers into the common bean core map, we were able to place markers with putative known function in some existing gaps including regions with QTL for resistance to anthracnose and rust. The distribution of the markers over 11 linkage groups is discussed and a newer version of the common bean core linkage map is proposed.

Culture-Independent Assessment of Rhizobiales-Related Alphaproteobacteria and the Diversity of Methylobacterium in the Rhizosphere and Rhizoplane of Transgenic Eucalyptus

The rhizosphere is an ecosystem exploited by a variety of organisms involved in plant health and environmental sustainability. Abiotic factors influence microorganism-plant interactions, but the microbial community is also affected by expression of heterologous genes from host plants. In the present work, we assessed the community shifts of Alphaproteobacteria phylogenetically related to the Rhizobiales order (Rhizobiales-like community) in rhizoplane and rhizosphere soils of wild-type and transgenic eucalyptus. A greenhouse experiment was performed and the bacterial communities associated with two wild-type (WT17 and WT18) and four transgenic (TR-9, TR-15, TR-22, and TR-23) eucalyptus plant lines were evaluated. The culture-independent approach consisted of the quantification, by real-time polymerase chain reaction (PCR), of a targeted subset of Alphaproteobacteria and the assessment of its diversity using PCR-denaturing gradient gel electrophoresis (DGGE) and 16S rRNA gene clone libraries. Real-time quantification revealed a lesser density of the targeted community in TR-9 and TR-15 plants and diversity analysis by principal components analysis, based on PCR-DGGE, revealed differences between bacterial communities, not only between transgenic and nontransgenic plants, but also among wild-type plants. The comparison between clone libraries obtained from the transgenic plant TR-15 and wild-type WT17 revealed distinct bacterial communities associated with these plants. In addition, a culturable approach was used to quantify the Methylobacterium spp. in the samples where the identification of isolates, based on 16S rRNA gene sequences, showed similarities to the species Methylobacterium nodulans, Methylobacterium isbiliense, Methylobacterium variable, Methylobacterium fujisawaense, and Methylobacterium radiotolerans. Colonies classified into this genus were not isolated from the rhizosphere but brought in culture from rhizoplane samples, except for one line of the transgenic plants (TR-15). In general, the data suggested that, in most cases, shifts in bacterial communities due to cultivation of transgenic plants are similar to those observed when different wild-type cultivars are compared, although shifts directly correlated to transgenic plant cultivation may be found.

Two physically, functionally, and developmentally distinct peritoneal macrophage subsets

The peritoneal cavity (PerC) is a unique compartment within which a variety of immune cells reside, and from which macrophages (Mempty set) are commonly drawn for functional studies. Here we define two Mempty set subsets that coexist in PerC in adult mice. One, provisionally called the large peritoneal Mempty set (LPM), contains approximately 90% of the PerC Mempty set in unstimulated animals but disappears rapidly from PerC following lipopolysaccharide (LPS) or thioglycolate stimulation. These cells express high levels of the canonical Mempty set surface markers, CD11b and F4/80. The second subset, referred to as small peritoneal Mempty set (SPM), expresses substantially lower levels of CD11b and F4/80 but expresses high levels of MHC-II, which is not expressed on LPM. SPM, which predominates in PerC after LPS or thioglycolate stimulation, does not derive from LPM. Instead, it derives from blood monocytes that rapidly enter the PerC after stimulation and differentiate to mature SPM within 2 to 4 d. Both subsets show clear phagocytic activity and both produce nitric oxide (NO) in response to LPS stimulation in vivo. However, their responses to LPS show key differences: in vitro, LPS stimulates LPM, but not SPM, to produce NO; in vivo, LPS stimulates both subsets to produce NO, albeit with different response patterns. These findings extend current models of Mempty set heterogeneity and shed new light on PerC Mempty set diversity, development, and function. Thus, they introduce a new context for interpreting (and reinterpreting) data from ex vivo studies with PerC Mempty set.

Rational Design and 3D-Pharmacophore Mapping of 5 `-Thiourea-Substituted alpha-Thymidine Analogues as Mycobacterial TMPK Inhibitors

Thymidine monophosphate kinase (TMPK) has emerged as an attractive target for developing inhibitors of Mycobacterium tuberculosis growth. In this study the receptor-independent (RI) 4D-QSAR formalism has been used to develop QSAR models and corresponding 3D-pharmacophores for a set of 5`-thiourea-substituted alpha-thymidine inhibitors. Models were developed for the entire training set and for a subset of the training set consisting of the most potent inhibitors. The optimized (RI) 4D-QSAR models are statistically significant (r(2) = 0.90, q(2) = 0.83 entire set, r(2) = 0.86, q(2) = 0.80 high potency subset) and also possess good predictivity based on test set predictions. The most and least potent inhibitors, in their respective postulated active conformations derived from the models, were docked in the active site of the TMPK crystallographic structure. There is a solid consistency between the 3D-pharmacophore sites defined by the QSAR models and interactions with binding site residues. This model identifies new regions of the inhibitors that contain pharmacophore sites, such as the sugar-pyrimidine ring structure and the region of the 5`-arylthiourea moiety. These new regions of the ligands can be further explored and possibly exploited to identify new, novel, and, perhaps, better antituberculosis inhibitors of TMPKmt. Furthermore, the 3D-pharmacophores defined by these models can be used as a starting point for future receptor-dependent antituberculosis drug design as well as to elucidate candidate sites for substituent addition to optimize ADMET properties of analog inhibitors.

Process Capability Indexes Determination on Tabletting Performance during the Process Validation for Metamizol Tablets

The aim of the present study was to provide a numerical measure, through the process capability indexes (PCIs), C(p) and C(pk), on whether or not the manufacturing process can be considered capable of producing metamizol (500 mg) tablets. They were also used as statistical tool in order to prove the consistency of the tabletting process, making sure that the tablet weight and the content uniformity of metamizol are able to comply with the preset requirements. Besides that, the ANOVA, the t-test and the test for equal variances were applied to this study, allowing additional knowledge of the tabletting phase. Therefore, the proposed statistical approach intended to assure more safety, precision and accuracy on the process validation analysis.

Optimization of preservative system in ophthalmic suspension with dexamethasone and polymyxin B

A simplex-lattice statistical project was employed to study an optimization method for a preservative system in an ophthalmic suspension of dexametasone and polymyxin B. The assay matrix generated 17 formulas which were differentiated by the preservatives and EDTA (disodium ethylene diamine-tetraacetate), being the independent variable: X-1 = chlorhexidine digluconate (0.010 % w/v); X-2 = phenylethanol (0.500 % w/v); X-3 = EDTA (0.100 % w/v). The dependent variable was the Dvalue obtained from the microbial challenge of the formulas and calculated when the microbial killing process was modeled by an exponential function. The analysis of the dependent variable, performed using the software Design Expert/W, originated cubic equations with terms derived from stepwise adjustment method for the challenging microorganisms: Pseudomonas aeruginosa, Burkholderia cepacia, Staphylococcus aureus, Candida albicans and Aspergillus niger. Besides the mathematical expressions, the response surfaces and the contour graphics were obtained for each assay. The contour graphs obtained were overlaid in order to permit the identification of a region containing the most adequate formulas (graphic strategy), having as representatives: X-1 = 0.10 ( 0.001 % w/v); X-2 = 0.80 (0.400 % w/v); X-3 = 0.10 (0.010 % w/v). Additionally, in order to minimize responses (Dvalue), a numerical strategy corresponding to the use of the desirability function was used, which resulted in the following independent variables combinations: X-1 = 0.25 (0.0025 % w/v); X-2 = 0.75 (0.375 % w/v); X-3 = 0. These formulas, derived from the two strategies (graphic and numerical), were submitted to microbial challenge, and the experimental Dvalue obtained was compared to the theoretical Dvalue calculated from the cubic equation. Both Dvalues were similar to all the assays except that related to Staphylococcus aureus. This microorganism, as well as Pseudomonas aeruginosa, presented intense susceptibility to the formulas independently from the preservative and EDTA concentrations. Both formulas derived from graphic and numerical strategies attained the recommended criteria adopted by the official method. It was concluded that the model proposed allowed the optimization of the formulas in their preservation aspect.