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.     

Constructing near spanning trees with few local inspections

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded‐degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph consisting of edges (for a prespecified constant ), where the decision for different edges should be consistent with the same subgraph . Can this task be performed by inspecting only a constant number of edges in G ? Our main results are:

• We show that if every t‐vertex subgraph of G has expansion then one can (deterministically) construct a sparse spanning subgraph of G using few inspections. To this end we analyze a “local” version of a famous minimum‐weight spanning tree algorithm.
• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3‐regular graphs of high girth, in which every t‐vertex subgraph has expansion . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

© 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 183–200, 2017     

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V _i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V _i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s. The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion. The research of the third author was supported by the Sacta-Rashi Foundation.     

Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Codes of length 2 ^m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of all have an rth order spectral null at zero frequency. Establishing the connection between and the parity-check matrix of Reed-Muller codes, the minimum distance of is obtained along with upper bounds on the redundancy of . An efficient algorithm is presented for encoding unconstrained binary sequences into .     

Studies of Hindered Rotation and Magnetic Anisotropy by 1H, 13C and 19F NMR. The Diels-Alder Adduct of N- Pentafluorophenylmaleimide and Phencyclone: A Model for Drugs.

The potent Diels-Alder diene, phencyclone, 1, reacts with N-pentafluorophenylmaleimide, 2, to form an adduct, 3, characterized by ¹H, ¹³C, and ¹⁹F NMR at 300, 75 and 282 MHz, respectively. The one-dimensional (1D) and two-dimensional (2D) ¹H and ¹³C NMR spectra of 3 at ambient temperatures imply a slow exchange limit (SEL) regime with respect to rotation of the unsubstituted bridgehead phenyl groups about severely hindered C(sp²)-C(sp³) bonds. Major non-bonded interactions are expected between the ortho protons of the C₆H₅ groups and H-1, 8 of the phenanthrenoid moiety of 3. ¹⁹F 1D and 2D (COSY) NMR spectra show that the SEL regime also obtains for rotation about the N-C₆F₅ bond of 3, with five separate fluorine signals seen, consistent with a preferred conformation in which the C₆F₅ may lie roughly perpendicular to the plane of the pyrrolidinedione moiety, and may be in the mirror symmetry plane of 3. The results are considered relevant to hindered aryl rotations in numerous Pharmaceuticals. Selected spectral data for 2 and precursors are also presented.     

A Review of the Integration of Science and Mathematics: Implications for Further Research

Building on the earlier analysis by Berlin (1991) , this paper reviews various studies on integrating mathematics and science in the 1990s and provides some implications for further research. The areas identified for further exploration include comparison of the nature of mathematics and science, epistemological debates in mathematics and in science education, the bases used to emphasize science over mathematics or vice versa, empirical evidence of effectiveness of integration, connections between teacher education programs for integration and teachers' subsequent classroom teaching practices, perceptions of integration on the part of teacher educators, contextual difficulties in implementing integrated approaches and possible solutions, and rationales of integrating mathematics and science through technology. In order to help all students become scientifically literate, which most reform documents call for, more focused attention on integration of curriculum and instruction is necessary.     

Solution equilibria of aromatic dinitroso compounds: a combined NMR and DFT study

Solution-state nitroso monomer-azodioxide equilibria and conformational freedom of several aromatic dinitroso derivatives, differing in the spacer group between the aromatic rings, were studied by one- and two-dimensional variable temperature ¹H NMR spectroscopy and by quantum chemical calculations. The proton signals of nitroso monomer-azodioxide mixtures revealed by low-temperature NMR were assigned and validated using B3LYP-D3/6-311+G(2d,p)/SMD level of theory. In almost all cases, a preference towards the formation of only one azodioxy isomer of aromatic dinitroso compounds was found, which was assigned to Z-dimer according to computational data. Nevertheless, the computed small energy difference between the Z- and E-isomer could not account for the extreme preference for Z-dimer formation, indicating an influence of entropic or solvent effects. The formation of shorter oligomers in solution was excluded based on integrated ¹H NMR signal intensities. The experimental results indicated an average dimerization Gibbs energy of about ??5 kJ/mol at 223 K and were found to be in very good correlation with dimerization energies obtained by solution-phase optimization.

     

Quantification of midazolam,morphine and metabolites in plasma using 96‐well solid‐phase extraction and ultra‐performance liquid chromatography–tandem mass spectrometry

Currently, pharmacokinetic–pharmacodynamic studies of sedatives and analgesics are performed in neonates and children to find suitable dose regimens. As a result, sensitive assays using only small volumes of blood are necessary to determine drug and metabolite concentrations. We developed an ultra‐performance liquid chromatographic method with tandem mass spectrometry detection for quantification of midazolam, 1‐hydroxymidazolam, hydroxymidazolamglucuronide, morphine, morphine‐3‐glucuronide and morphine‐6‐glucuronide in 100 μL of plasma. Cleanup consisted of 96 wells micro‐solid phase extraction, before reversed‐phase chromatographic separation (ultra‐performance liquid chromatography) and selective detection using electrospray ionization tandem mass spectrometry. Separate solid‐phase extraction methods were necessary to quantify morphine, midazolam and their metabolites because of each group's physicochemical properties. Standard curves were linear over a large dynamic range with adequate limits of quantitation. Intra‐ and interrun accuracy and precision were within 85–115% (of nominal concentration using a fresh calibration curve) and 15% (coefficient of variation, CV) respectively. Recoveries were >80% for all analytes, with interbatch CVs (as a measure of matrix effects) of less than 15% over six batches of plasma. Stability in plasma and extracts was sufficient, allowing large autosampler loads. Runtime was 3.00 min per sample for each method. The combination of 96‐well micro‐SPE and UPLC‐MS/MS allows reliable quantification of morphine, midazolam and their major metabolites in 100 μL of plasma. Copyright © 2010 John Wiley & Sons, Ltd.