Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial‐Meshulam model for random simplicial complexes on n vertices, where each d‐cell is added independently with probability p to the complete ‐skeleton. Under the assumption , we prove that the spectral gap between the smallest eigenvalues and the remaining eigenvalues is with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi‐Komlós‐type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 506–537, 2017     

Confirmation of drug delivery after liver chemoembolization: direct tissue doxorubicin measurement by UHPLC‐MS‐MS

Because liver cancer is rarely suitable for surgery, transcatheter arterial chemoembolization (TACE) is used for palliative therapy. In this procedure, an emulsion of doxorubicin in iodized oil is injected directly into liver tumors through a catheter positioned within the artery supplying blood flow to the tumor. At present, there is limited understanding of factors affecting the delivery and dispersion of doxorubicin within treated tumors during TACE. This study addresses the development and application of an ultrahigh‐pressure liquid chromatography–tandem mass spectrometry (UHPLC‐MS‐MS) method for rapid confirmation of drug delivery after TACE in a rabbit VX2 liver cancer model. Doxorubicin levels in liver tumors were measured using UHPLC‐MS‐MS and compared with computed tomography measured levels of iodized oil, a metric used clinically to indicate drug delivery. We found that tissue drug levels determined using UHPLC‐MS‐MS did not correlate with the regional iodized oil concentration (vehicle) within tumors following TACE, suggesting that chemotherapeutic drugs like doxorubicin spread throughout tumors, and that lack of iodized oil staining in portions of a tumor does not necessarily indicate inadequate therapy during TACE. Copyright © 2012 John Wiley & Sons, Ltd.     

Enantioselective Sulfoxidations Catalyzedby Horseradish Peroxidase, ManganesePeroxidase, and Myeloperoxidase

Summary.  Horseradish peroxidase (HRP), myeloperoxidase (MPO), and manganese peroxidase (MnP) have been shown to catalyze the asymmetric sulfoxidation of thioanisole. When H₂O₂ was added stepwise to MPO, a maximal yield of 78% was obtained at pH 5 (ee 23%), whereas an optimum in the enantiomeric excess (32%, (R)-sulfoxide) was found at pH 6 (60% yield). For MnP a yield of 18% and a high enantiomeric excess of 91% of the (S)-sulfoxide were obtained at pH 5 and a yield of 36% and an ee of 87% at pH 7.0. Optimization of the conversion catalyzed by horseradish peroxidase at pH 7.0 by controlled continuous addition of hydrogen peroxide during turnover and monitoring the presence of native enzyme as well as of intermediates I, II, and III led to the formation of the sulfoxide in high yield (100%) and moderate enantioselectivity (60%, (S)-sulfoxide). Received November 18, 1999. Accepted January 21, 2000     

The wavelet dimension function is the trace function of a shift-invariant system

In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.

     

Optimal fractional matchings and covers in infinite hypergraphs: Existence and duality

We study fractional matchings and covers in infinite hypergraphs, paying particular attention to the following questions: Do fractional matchings (resp. covers) of maximal (resp. minimal) size exist? Is there equality between the supremum of the sizes of fractional matchings and the infimum of the sizes of fractional covers? (This is called weak duality.) Are there a fractional matching and a fractional cover that satisfy the complementary slackness conditions of linear programming? (This is called strong duality.) In general, the answers to all these questions are negative, but for certain classes of infinite hypergraphs (classified according to edge cardinalities and vertex degrees) we obtain positive results. We also consider the question of the existence of optimal fractional matchings and covers that assume rational values.