Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

Functional inequality characterizing nonnegative concave functions in (0, ∞) k

Summary In the present note we prove that every functionf: (0, ) [0, ) satisfying the inequalityaf(s) + bf(t) f(as + bt), s, t > 0, for somea andb such that 0 <a < 1 <a + b must be of the formf(t) = f(1)t, (t > 0). This improves our recent result in [2], where the inequality is assumed to hold for alls, t 0, and gives a positive answer to the question raised there.An analogue for functions of several real variables of the above result characterizes concave functions. Conjugate functions and some relations to Hölder's and Minkowski's inequalities are mentioned.     

Bootstrap,wild bootstrap,and asymptotic normality

Summary We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.This work has been supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 Stochastische Mathematische Modelle     

Problème inverse de la théorie du potentiel logarithmique pour les lemniscates

The paper is related to the question of uniqueness in the inverse logarithmic potential problem. This question is to find the conditions on which two domains D ₁ and D ₂ producing the same external potential must coincide. Assuming the general hypothesis of regularity and an additional condition of connectivity of (D₁D₂)_c, we prove a theorem of uniqueness in the case when one of the domains is a lemniscate. The main tool is one lemma for Cauchy's potential due to M. Sakai. We give a simple proof of its extension to Newtonian potential.

     

Chern forms,chern numbers and curvature conditions on a Kähler-Einstein surface

In this paper we considered curvature conditions on a Kähler-Einstein surface of general type. In particular we showed that it has negative holomorphic sectional curvature if theL ²-norm of (3C ₂ ?C ₁ ² )/C ₁ ² is sufficiently small, whereC ₁ andC ₂ are the first and second Chern classes of the surfaces. This generalizes a result of Yau on the uniformization of Kähler-Einstein surfaces of general type and with 3C ₂ ?C ₁ ² = 0. Also in the process, we obtain a necessary condition in terms of an inequality between Chern numbers for a Kähler-Einstein metric to have negative holomorphic sectional curvature.     

Factorial regular representation of groups in complete graphs

A necessary and sufficient condition for the group of isomorphisms involved in a factorization of a complete graph into isomorphic factors is established.     

Quantitative and Qualitative Analysis of the Anti-Proliferative Potential of the Pyrazole Scaffold in the Design of Anticancer Agents

The current work presents an objective overview of the impact of one important heterocyclic structure, the pyrazole ring, in the development of anti-proliferative drugs. A set of 1551 pyrazole derivatives were extracted from the National Cancer Institute (NCI) database, together with their growth inhibition effects (GI%) on the NCI’s panel of 60 cancer cell lines. The structures of these derivatives were analyzed based on the compounds’ averages of GI% values across NCI-60 cell lines and the averages of the values for the outlier cells. The distribution and the architecture of the Bemis–Murcko skeletons were analyzed, highlighting the impact of certain scaffold structures on the anti-proliferative effect’s potency and selectivity. The drug-likeness, chemical reactivity and promiscuity risks of the compounds were predicted using AMDETlab. The pyrazole ring proved to be a versatile scaffold for the design of anticancer drugs if properly substituted and if connected with other cyclic structures. The 1,3-diphenyl-pyrazole emerged as a useful scaffold for potent and targeted anticancer candidates.     

Synthesis and Systematic Study on the Effect of Different PEG Units on Stability of PEGylated,Integrin-αvβ6-Specific A20FMDV2 Analogues in Rat Serum and Human Plasma

A20FMDV2 is a 20-mer peptide that exhibits high selectivity and affinity for the tumour-related αvβ6 integrin that can compete with extracellular ligands for the crucial RGD binding site, playing a role as a promising αvβ6-specific inhibitor for anti-cancer therapies. Unfortunately, the clinical value of A20FMDV2 is limited by its poor half-life in blood caused by rapid renal excretion and its reported high susceptibility to serum proteases. The incorporation of poly (ethylene glycol) chains, coined PEGylation, is a well-established approach to improve the pharmacokinetic properties of drug molecules. Here, we report a systematic study on the incorporation of a varying number of ethylene glycol units (1–20) into the A20FMDV2 peptide to establish the effects of PEGylation size on the peptide stability in both rat serum and human plasma. In addition, the effect of acetyl and propionyl PEGylation handles on peptide stability is also described. Selected peptide analogues were assessed for integrin-αvβ6-targeted binding, showing good specificity and activity in vitro. Stability studies in rat serum established that all of the PEGylated peptides displayed good stability, and an A20FMDV2 peptide containing twenty ethylene glycol units (PEG₂₀) was the most stable. Surprisingly, the stability testing in human plasma identified shorter PEGs (PEG₂ and PEG₅) as more resistant to degradation than longer PEGs, a trend which was also observed with affinity binding to integrin αvβ6.