Overview of differential equations with non-standard growth

Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.     

Analysis of acetylcholine and choline in microdialysis samples by liquid chromatography/tandem mass spectrometry

A sensitive liquid chromatography/electrospray ionisation tandem mass spectrometric (LC/ESI-MS/MS) method was developed for the analysis of acetylcholine and choline in microdialysis samples. A Ringer's solution that contains high (150 mM) concentrations of inorganic salts was used to extract acetylcholine and choline from a rat or mouse brain. The separation of acetylcholine, choline, an internal standard acetyl-beta-methylcholine, endogenous compounds and inorganic cations was achieved with hydrophilic interaction chromatography using a diol column. The eluent consisted of 20 mM ammonium formate (pH 3.3) and acetonitrile (20:80) which is favourable for the ESI process. Limits of detection (signal-to-noise (S/N) ratio = 3) of 0.02 nM (0.2 fmol) for acetylcholine and 1 nM (10 fmol) for choline were observed using standards diluted in Ringer's solution. A good linearity was obtained from the limit of quantitation: 0.1 nM (S/N ratio = 10) to 50 nM (r = 0.999) for acetylcholine and within the concentration range of 100-3500 nM (r = 0.998) for choline. The between-day repeatability of the method was good; RSD was 3.1% at 1 nM level of acetylcholine and 3.5% at 1000 nM level of choline. The recoveries for addition of 1 or 2.5 nM acetylcholine and 0.2 or 1 microM choline in microdialysis balancing samples were between 93 and 101% indicating that no suppressing endogenous compounds were co-eluting with acetylcholine or choline. The developed method was applied to the analysis of microdialysis balancing samples collected from rat and mouse brains.     

Steiner triple systems of order 19 and 21 with subsystems of order 7

Steiner triple systems (STSs) with subsystems of order 7 are classified. For order 19, this classification is complete, but for order 21 it is restricted to Wilson-type systems, which contain three subsystems of order 7 on disjoint point sets. The classified STSs of order 21 are tested for resolvability; none of them is doubly resolvable.     

There are exactly five biplanes with k = 11

A biplane is a 2‐(k(k ? 1)/2 + 1,k,2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3‐(57, 12, 2) design, no 112₁₁ symmetric configuration, and no (324, 57, 0, 12) strongly regular graph. The five biplanes have 16 residual designs, which by the Hall–Connor theorem constitute a complete classification of the 2‐(45, 9, 2) designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 117–127, 2008     

The Steiner triple systems of order 19

Using an orderly algorithm, the Steiner triple systems of order are classified; there are pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch configurations it contains are recorded; of the designs are anti-Pasch. There are three main parts of the classification: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the final triple systems. Isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on Pasch configurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters .

     

Spectral and network methods in the analysis of correlation matrices of stock returns

Correlation matrices inferred from stock return time series contain information on the behaviour of the market, especially on clusters of highly correlating stocks. Here we study a subset of New York Stock Exchange (NYSE) traded stocks and compare three different methods of analysis: (i) spectral analysis, i.e. investigation of the eigenvalue-eigenvector pairs of the correlation matrix, (ii) asset trees, obtained by constructing the maximal spanning tree of the correlation matrix, and (iii) asset graphs, which are networks in which the strongest correlations are depicted as edges. We illustrate and discuss the localisation of the most significant modes of fluctuation, i.e. eigenvectors corresponding to the largest eigenvalues, on the asset trees and graphs.     

Queueing systems fed by many exponential on-off sources: an infinite-intersection approach

In queueing theory, an important class of events can be written as ‘infinite intersections’. For instance, in a queue with constant service rate c, busy periods starting at 0 and exceeding L > 0 are determined by the intersection of the events , i.e., queue Q _t is empty at 0 and for all t∊ [0, L] the amount of traffic A _t arriving in [0,t) exceeds the server capacity. Also the event of exceeding some predefined threshold in a tandem queue, or a priority queue, can be written in terms of this kind of infinite intersections. This paper studies the probability of such infinite intersections in queueing systems fed by a large number n of i.i.d. traffic sources (the so-called ‘many-sources regime’). If the sources are of the exponential on-off type, and the queueing resources are scaled proportional to n, the probabilities under consideration decay exponentially; we explicitly characterize the corresponding decay rate. The techniques used stem from large deviations theory (particularly sample-path large deviations). M. Mandjes is also with Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, the Netherlands, and EURANDOM, Eindhoven, the Netherlands. Work done while P. Mannersalo was on leave at CWI. MSC 2000: 60F10 (Large deviations), 60K25 (Queueing theory)