Discrete concavity and the half-plane property

Murota et al. have recently developed a theory of discrete convex analysis which concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over a field of generalized Puiseux series) with prescribed non-vanishing properties. This family contains several of the most studied M-concave functions in the literature. In the language of tropical geometry we study the tropicalization of the space of polynomials with the half-plane property, and show that it is strictly contained in the space of M-concave functions. We also provide a short proof of Speyer's hive theorem which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices.Comment: 14 pages. The proof of Theorem 4 is corrected

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Ideal hierarchical secret sharing schemes

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.Peer ReviewedPostprint (author's final draft

The matricial relaxation of a linear matrix inequality

Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x_j. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L_1(X) and L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^* L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensaetze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form {X | L(X) PsD}. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in the source file, or see http://srag.fmf.uni-lj.si

A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex $N$-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex $N$-fold integer minimization problems for which our approach provides polynomial time solution algorithms.Comment: 19 pages, 1 figur

A niche-mimicking polymer hydrogel-based approach to identify molecular targets for tackling human pancreatic cancer stem cells.

BACKGROUND: Pancreatic adenocarcinoma (PAAD) is one of the most fatal human cancers, but effective therapies remain to be established. Cancer stem cells (CSCs) are highly resistant to anti-cancer drugs and a deeper understanding of their microenvironmental niche has been considered important to provide understanding and solutions to cancer eradication. However, as the CSC niche is composed of a wide variety of biological and physicochemical factors, the development of multidisciplinary tools that recapitulate their complex features is indispensable. Synthetic polymers have been studied as attractive biomaterials due to their tunable biofunctionalities, while hydrogelation technique further renders upon them a diversity of physical properties, making them an attractive tool for analysis of the CSC niche. METHODS: To develop innovative materials that recapitulate the CSC niche in pancreatic cancers, we performed polymer microarray analysis to identify niche-mimicking scaffolds that preferentially supported the growth of CSCs. The niche-mimicking activity of the identified polymers was further optimized by polyethylene glycol (PEG)-based hydrogelation. To reveal the biological mechanisms behind the activity of the optimized hydrogels towards CSCs, proteins binding onto the hydrogel were analyzed by liquid chromatography with tandem mass spectrometry (LC-MS/MS), and the potential therapeutic targets were validated by looking at gene expression and patients' outcome in the TCGA database. RESULTS: PA531, a heteropolymer composed of 2-methoxyethyl methacrylate (MEMA) and 2-(diethylamino)ethyl methacrylate (DEAEMA) (5.5:4.5) that specifically supports the growth and maintenance of CSCs was identified by polymer microarray screening using the human PAAD cell line KLM1. The polymer PA531 was converted into five hydrogels (PA531-HG1 to HG5) and developed to give an optimized scaffold with the highest CSC niche-mimicking activities. From this polymer that recapitulated CSC binding and control, the proteins fetuin-B and angiotensinogen were identified as candidate target molecules with clinical significance due to the correlation between gene expression levels and prognosis in PAAD patients and the proteins associated with the niche-mimicking polymer. CONCLUSION: This study screened for biofunctional polymers suitable for recapitulation of the pancreatic CSC niche and one hydrogel with high niche-mimicking abilities was successfully fabricated. Two soluble factors with clinical significance were identified as potential candidates for biomarkers and therapeutic targets in pancreatic cancers. Such a biomaterial-based approach could be a new platform in drug discovery and therapy development against CSCs, via targeting of their niche

A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions

On the validity of the reduced Salpeter equation

We adapt a general method to solve both the full and reduced Salpeter equations and systematically explore the conditions under which these two equations give equivalent results in meson dynamics. The effects of constituent mass, angular momentum state, type of interaction, and the nature of confinement are all considered in an effort to clearly delineate the range of validity of the reduced Salpeter approximations. We find that for $J\not{\hspace*{-1.0mm}=}0$ the solutions are strikingly similar for all constituent masses. For zero angular momentum states the full and reduced Salpeter equations give different results for small quark mass especially with a large additive constant coordinate space potential. We also show that $\frac{1}{m}$ corrections to heavy-light energy levels can be accurately computed with the reduced equation.Comment: Latex (uses epsf macro), 24 pages of text, 12 postscript figures included. Slightly revised version, to appear in Phys. Rev.

Liver fatty acid-binding protein binds monoacylglycerol in vitro and in mouse liver cytosol

Liver fatty acid-binding protein (LFABP; FABP1) is expressed both in liver and intestinal mucosa. Mice null for LFABP were recently shown to have altered metabolism of not only fatty acids but also monoacylglycerol, the two major products of dietary triacylglycerol hydrolysis (Lagakos, W. S., Gajda, A. M., Agellon, L., Binas, B., Choi, V., Mandap, B., Russnak, T., Zhou, Y. X., and Storch, J. (2011) Am. J. Physiol. Gastrointest. Liver Physiol. 300, G803-G814). Nevertheless, the binding and transport of monoacylglycerol (MG) by LFABP are uncertain, with conflicting reports in the literature as to whether this single chain amphiphile is in fact bound by LFABP. In the present studies, gel filtration chromatography of liver cytosol from LFABP-/- mice shows the absence of the low molecular weight peak of radiolabeled monoolein present in the fractions that contain LFABP in cytosol from wild type mice, indicating that LFABP binds sn-2 MG in vivo. Furthermore, solution-state NMRspectroscopy demonstrates two molecules of sn-2 monoolein bound in the LFABP binding pocket in positions similar to those found for oleate binding. Equilibrium binding affinities are ~2-fold lower for MG compared with fatty acid. Finally, kinetic studies examining the transfer of a fluorescent MG analog show that the rate of transfer of MG is 7-fold faster from LFABP to phospholipid membranes than from membranes to membranes and occurs by an aqueous diffusion mechanism. These results provide strong support for monoacylglycerol as a physiological ligand for LFABP and further suggest that LFABP functions in the efficient intracellular transport of MG.Instituto de Investigaciones Bioquímicas de La Plat