Rainbow sets in the intersection of two matroids

Given sets $F_1, \ldots ,F_n$, a {\em partial rainbow function} is a partial choice function of the sets $F_i$. A {\em partial rainbow set} is the range of a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if $M$ and $N$ are matroids on the same ground set, and $F_1, \ldots ,F_n$ are pairwise disjoint sets of size $n$ belonging to $M \cap N$, then there exists a rainbow set of size $n-1$ belonging to $M \cap N$. Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\sqrt{n}$

LP duality in infinite hypergraphs

AbstractIn any graph there exist a fractional cover and a fractional matching satisfying the complementary slackness conditions of linear programming. The proof uses a Gallai-Edmonds decomposition result for infinite graphs. We consider also the same problem for infinite hypergraphs, in particular in the case that the edges of the hypergraph are intervals on the real line. We prove an extension of a theorem of Gallai to the infinite case

Reflectance Spectra Analysis Algorithms for the Characterization of Deposits and Condensed Traces on Surfaces

Identification of particulate matter and liquid spills contaminations is essential for many applications, such as forensics, agriculture, security, and environmental protection. For example, toxic industrial compounds deposition in the form of aerosols, or other residual contaminations, pose a secondary, long-lasting health concern due to resuspension and secondary evaporation. This chapter explores several approaches for employing diffuse reflectance spectroscopy in the mid-IR and SWIR to identify particles and films of materials in field conditions. Since the behavior of thin films and particles is more complex compared to absorption spectroscopy of pure compounds, due to the interactions with background materials, the use of physical models combined with statistically-based algorithms for material classification, provides a reliable and practical solution and will be presented

Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes

3-D struktura serumske paraoksonaze 1 objašnjava njezinu aktivnost, stabilnost, topljivost i kristalizaciju

Serum paraoxonases (PONs) exhibit a wide range of physiologically important hydrolytic activities, including drug metabolism and detoxification of nerve gases. PON1 and PON3 reside on high-density lipoprotein (HDL) (the “good cholesterol”), and are involved in the alleviation of atherosclerosis. Members of the PON family have been identified not only in mammals and other vertebrates, but also in invertebrates. We earlier described the first crystal structure of a PON family member, a directly-evolved variant of PON1, at 2.2 Å resolution. PON1 is a 6-bladed beta-propeller with a unique active-site lid which is also involved in binding to HDL. The 3-D structure, taken together with directed evolution studies, permitted analysis of mutations which enhanced the stability, solubility and crystallizability of this PON1 variant. The structure permits a detailed description of PON1’s active site and suggests possible mechanisms for its catalytic activity on certain substrates.Serumske paraoksonaze (PONs) imaju široki raspon fiziološki važnih hidrolitičkih aktivnosti uključujući metabolizam lijekova i detoksikaciju nervnih plinova. PON1 i PON3 smještene su na lipoproteinima visoke gustoće (engl. high-density lipoprotein; HDL - “dobri kolesterol”) i uključene su u ublažavanje ateroskleroze. Članovi skupine PON identificirani su ne samo u sisavaca i drugih kralježnjaka već i kod beskralješnjaka. Prije smo opisali prvu kristalnu strukturu člana PON skupine, direktno razrađenu varijantu PON1 pri rezoluciji 2,2 Å. PON1 je beta-propeler sa šest lopatica s jedinstvenim poklopcem aktivnog mjesta, koji je tako|er uključen u vezanje na HDL. 3-D struktura, gledana zajedno s direktnim razvojnim istraživanjima, omogućila je analizu mutacija koje povećavaju stabilnost, topljivost i kristalizaciju te PON1 varijante. Struktura dopušta detaljan opis aktivnog mjesta PON1 i sugerira moguće mehanizme za njezinu katalitičku aktivnost prema odre|enim supstratima

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices {\DH} groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science