The Lavrentiev Phenomenon in Nonlinear Elasticity

In 1985 J.M. Ball and V.J. Mizel raised the question of whether there exist nonlinearly elastic materials possessing a physically natural stored energy density, i.e., one which is independent of an observer's coordinate frame (objective) and is invariant under the group of orthogonal linear transformations of space (isotropic), as well as physically reasonable boundary value problems for such materials such that the infimum of the total stored energy for those continuous deformations of the material meeting the boundary condition (admissible deformations) which belong to a Sobolev space W ^{1 p} 2 for some p ₂>1 is strictly greater than its infimum for those admissible continuous deformations belonging to some Sobolev space W ¹ p ₁, p ₁<p ₂, despite the density of W ^{1 p} 2 in W ¹ p ₁. The question was motivated by M. Lavrentiev's demonstration in 1926 of the presence of such a gap for a 1-dimensional variational boundary value problem on a bounded interval whose smooth integrand satisfied the conditions of Tonelli's existence theorem (as well as the development of improved versions in the 1980's). The present article describes a positive response to the question raised in 1985. Namely, we provide examples of nonlinearly elastic materials in 2-dimensions and physically reasonable boundary value problems for these materials in which a positive gap exists between the infimum of the total stored energy over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 2 and its infimum over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 1, with p ₁<p ₂. The physical and computational significance of such results is also discussed.     

Measurements of Zeta Potential on Corroding,High-Purity Iron Particles: Influence of Corrosion Inhibitors

Adsorption of cetyl trimethyl ammonium bromide (CTAB), and two commercial inhibitor base chemicals; an oleic imidazoline salt (OI) and a phosphate ester (PE), onto high purity, corroding iron particles was studied by zeta potential measurements in a 0.1 Wt% sodium chloride (NaCl) solution under 1 bar CO₂ at 22°C. The particles were exposed to the inhibitor compounds for 24 hours before measurements were done. The results show that the measured zeta potential in the absence of inhibitor is zero at both pH 4.0 ± 0.2 and pH 5.8 ± 0.2. It is concluded that this might be caused by the electrochemical reactions occurring at the steel surface when placed in an electrical field. When adding inhibitor, which slows the electrochemical reactions at the steel surface, the zeta potential moves away from zero and an adsorption isotherm is obtained for all three inhibitors. The measured potential is probably a mixed potential where the apparent potential measured is a combination of the potential at the shear plane and a contribution form the electrochemical reactions occurring on the surface.     

Dopamine-assisted rapid fabrication of nanoscale protein arrays by colloidal lithography

The development of cost-effective methodologies for the precise nanometer-scale positioning of biomolecules permits the low-cost production of various biofunctional devices for a range of biomedical and nanotechnological applications. By combining colloidal lithography and the mussel-inspired multifunctional polydopamine coating, we present a novel parallel benchtop method that allows rapid nanoscale patterning of proteins without the need for electrically powered equipment in the fabrication process. The PDA-immobilized binary nanopattern consisting of BSA surrounded by PLL-g-PEG is fabricated over a large area, and the integrity of the pattern is confirmed using AFM and FM.     

On Transience Conditions for Markov Chains and Random Walks

We prove a new transience criterion for Markov chains on an arbitrary state space and give a corollary for real-valued chains. We show by example that in the case of a homogeneous random walk with infinite mean the proposed sufficient conditions are close to those necessary. We give a new proof of the well-known criterion for finiteness of the supremum of a random walk.     

Convergence rates in monotone separable stochastic networks

We study bounds on the rate of convergence to the stationary distribution in monotone separable networks which are represented in terms of stochastic recursive sequences. Monotonicity properties of this subclass of Markov chains allow us to formulate conditions in terms of marginal network characteristics. Two particular examples, generalized Jackson networks and multiserver queues, are considered.     

Kinetic stabilization of an oligomeric protein by a single ligand binding event

Protein native state stabilization imposed by small molecule binding is an attractive strategy to prevent the misfolding and misassembly processes associated with amyloid diseases. Transthyretin (TTR) amyloidogenesis requires rate-limiting tetramer dissociation before misassembly of a partially denatured monomer ensues. Selective stabilization of the native TTR tetramer over the dissociative transition state by small molecule binding to both thyroxine binding sites raises the kinetic barrier of tetramer dissociation, preventing amyloidogenesis. Assessing the amyloidogenicity of a TTR tetramer having only one amyloidogenesis inhibitor (I) bound is challenging because the two small molecule binding constants are generally not distinct enough to allow for the exclusive formation of TTR.I in solution to the exclusion of TTR.I(2) and unliganded TTR. Herein, we report a method to tether one fibril formation inhibitor to TTR by disulfide bond formation. Occupancy of only one of the two thyroxine binding sites is sufficient to inhibit tetramer dissociation in 6.0 M urea and amyloidogenesis under acidic conditions by imposing kinetic stabilization on the entire tetramer. The sufficiency of single occupancy for stabilizing the native state of TTR provides the incentive to search for compounds displaying striking negative binding cooperativity (e.g., K(d1) in nanomolar range and K(d2) in the micromolar to millimolar range), enabling lower doses of inhibitor to be employed in the clinic, mitigating potential side effects.     

Partial regularity for subquadratic parabolic systems with continuous coefficients

We establish the partial regularity of solutions to quasilinear parabolic systems with elliptic part that grows subquadratically. More precisely, it is shown that there is an open subset with full measure, of the solution’s domain, on which the solution is H?lder continuous. A key feature in this article is that we only require the coefficients of the system to be continuous with respect to the first two arguments. To prove the result, we use the A-caloric approximation method and an intrinsic scaling. To accommodate the subquadratic growth an adaptation of the A-caloric approximation lemma is also provided.     

A decision support model for establishing an air taxi service: a case study

Recent availability of relatively cheap small jet aircraft creates opportunities for a new air transport business: Air taxi, an on-demand service in which travellers call in one or a few days in advance to book transportation. In this paper, we present a methodology and simulation study supporting important strategic decisions, like for instance determining the required number of aircraft, for a company planning to establish an air taxi service in Norway. The methodology is based on a module simulating incoming bookings, built around a heuristic for solving the underlying dial-a-flight problem. The heuristic includes a separate method for solving the important subproblem of determining the best distribution of waiting time along a single aircraft schedule. The methodology has proved to provide reliable decision support to the company.