Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length

We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.     

Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network

Recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators [R. Curtu, A. Shpiro, N. Rubin, J. Rinzel, Mechanisms for frequency control in neuronal competition models, SIAM J. Appl. Dyn. Syst. 7 (2) (2008) 609-649] showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics. Several types of behavior were identified such as: fusion, equivalent to a steady state of identical activity levels for both neural units; oscillations due to either an escape or a release mechanism; and a winner-take-all state of bistability. The model consists of two neural populations interacting through reciprocal inhibition, each endowed with a slow negative-feedback process in the form of spike frequency adaptation. In this paper we report the occurrence of another complex oscillatory pattern, the mixed-mode oscillations (MMOs). They exist in the model at the transition between the relaxation oscillator dynamical regime and the winner-take-all regime. The system distinguishes itself from other neuronal models where MMOs were found by the following interesting feature: there is no autocatalysis involved (as in the examples of voltage-gated persistent inward currents and/or intrapopulation recurrent excitation) and therefore the two cells in the network are not intrinsic oscillators; the oscillations are instead a combined result of the mutual inhibition and the adaptation. We prove that the MMOs are due to a singular Hopf bifurcation point situated in close distance to the transition point to the winner-take-all case. We also show that in the vicinity of the singular Hopf other types of bifurcations exist and we construct numerically the corresponding diagrams.     

An evaluation of infrared microwindows for ozone retrievals using the Eureka Bruker 125HR Fourier transform spectrometer

A Bruker 125HR Fourier transform spectrometer was installed at the Polar Environment Atmospheric Research Laboratory (PEARL) at Eureka, Nunavut, Canada in the summer of 2006 to study atmospheric composition. Using the optimal estimation method, typically over a limited spectral region called a microwindow, information about the vertical distribution of trace gas species that have absorption bands in the mid-infrared spectral range can be retrieved. Total and partial columns can also be determined to show the temporal evolution of the target gas. For ozone in particular, retrievals have been performed using several of its many mid-infrared absorption features, resulting in a lack of consistency in the literature in the microwindows chosen for retrievals. This work focuses on the optimization of the ozone retrieval, assessing a set of 22 microwindows between 780 and 3052 cm⁻¹ to determine which is best suited to conditions at Eureka. The 1000-1004.5 cm⁻¹ spectral interval is shown to be the most sensitive to both the stratosphere and troposphere. This microwindow gives the highest number of degrees of freedom for signal (∼7 for total column), and the smallest total error (4.3%) compared with 21 other spectral regions. Retrievals performed with this microwindow agree well with results obtained from other instruments on-site. Total column ozone measured by the Bruker 125HR in this microwindow agreed to 2% with two other Fourier transform spectrometers, to 0.7% with a Brewer spectrophotometer, to 8% with a SAOZ UV-VIS spectrometer, and to 7% with ozone sondes.     

Thermal behaviour of new biological active cadmium mixed ligands complexes

This paper reports the investigation on the thermal stability of new complexes with mixed ligands of the type [Cd(NN)(C₃H₃O₂)₂(H₂O)_m]·nH₂O [(1) NN: 1,10-phenantroline, m = 1, n = 0; (2) NN: 2,2′-bipyridine, m = 0, n = 1.5 and (C₃H₃O₂): acrylate anion]. The IR data indicate a bidentate coordination mode for both heterocyclic amine and acrylate. The in vitro qualitative and quantitative antimicrobial activity assays showed that the complexes exhibited variable antimicrobial activity against planktonic as well as biofilm embedded Gram-negative (Escherichia coli, Klebsiella sp., Proteus sp., Salmonella sp., Shigella sp., Acinetobacter boumani, Pseudomonas aeruginosa), Gram-positive (Bacillus subtilis, Staphylococcus aureus) and fungal (Candida albicans) strains, reference and isolated ones from the hospital environment. The thermal behaviour steps were investigated in synthetic air flow. The thermal transformations are complex processes according to TG and DTA curves including dehydration, amine as well as acrylate thermolysis. The final products of decomposition are the most stable metal oxides.     

Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

The properties of matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators.A new, direct proof is given to the conjecture of Durán and Grünbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type.Commutative classes of orthogonal polynomials (corresponding to weights that are self-adjoint but not positive semidefinite) are found, which satisfy all the properties usually associated to orthogonal polynomials, and are not of scalar type.     

Nonlinear differential problems with first- or second-order differences in Hilbert spaces

In a real Hilbert space, we investigate the existence, uniqueness and asymptotic properties of the generalized solutions of some nonlinear differential systems with first- or second-order differences, subject to extreme conditions and initial data. Some generalizations of Minty’s theorem for the characterization of maximal monotone operators are also presented.     

Micropolarity and microviscosity of Pluronics L62 and L64 core–shell aggregates in water at various concentrations and additives examined by absorption and fluorescence probes

The microstructure of aggregates formed in aqueous solutions of the triblock copolymers poly(ethylene oxide) (PEO)–poly(propylene oxide)–PEO, Pluronics L62 and L64, and the effect of additives (n-butanol, n-hexanol, and o-xylene) on the local polarity and viscosity were investigated using several absorption and fluorescence probes with different hydrophilic/hydrophobic trade-offs. The absorption probe was 2,2,6,6-tetramethylpiperidine-1-oxyl and the fluorescence probes pyrene (P), 1-anilinonaphthalene-8-sulfonic acid (ANS), 1,10-bis(1-pyrene) decane (PD), and N-[5-(dimethylamino)naphthalene-1-sulfonyl]hexadecylamine (Dansyl). The specific absorption and fluorescence parameters, sensitive to changes in micropolarity and microviscosity, were related to the hydration calibration curves carried out in homogeneous tetraethylene glycol/water mixtures. Thus, the effective local hydration of the molecular probe solubilized in the guest aggregate was quantified, and at the same time, the probe location is established with respect of the corona and core aggregate. The ANS and PD probes evidence differences in microviscosity and track the effect of the block copolymer structure and additive concentration on microviscosity.     

On a notion of unilateral slope for the Mumford-Shah functional

In this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional.     

Cost allocation protocols for supply contract design in network situations

The class of Construct and Charge (CC-) rules for minimum cost spanning tree (mcst) situations is considered. CC-rules are defined starting from the notion of charge systems, which specify particular allocation protocols rooted on the Kruskal algorithm for computing an mcst. These protocols can be easily implemented in practical network situations (for instance, in supply transportation networks), are flexible to changes in the network situation and meet the requirement of continuous monitoring by the agents involved. Special charge systems, that we call conservative, lead to a subclass of CC-rules that coincides with the class of obligation rules for mcst situations. The authors thank two anonymous referees both for detailed remarks and for interesting general comments on a previous version of the paper. Stef Tijs and Rodica Branzei are indebted to Daniel Granot for useful discussions on the topics treated in this paper and his hospitality during our research visit at British Columbia University at Vancouver in July–August 2003.