Combinatorics,graph theory,and computing

Journal of Algebraic Combinatorics -     

Flow auctions

Flow goods (like electricity) are sold through auctions in a dynamic framework. An important design question is the frequency of such auctions. We use a simple dynamic auction model in continuous time to answer this question. We focus on the relationship between the persistency of bidders’ valuations and the optimal choice of frequency. If the seller focuses on the equilibrium in which bidders follow a repeated static Nash strategy, then the frequency of auctions should typically increase when persistency declines. However, accounting for the fact that bidders can follow different equilibria that are collusive in nature, the comparative statics are reversed, forcing the seller to reduce the frequency when bidders’ valuations are less persistent. The argument builds on the fact that high frequency auctions are more conducive to collusion among bidders.     

The continuous grey pattern problem

A new location problem is formulated and solved. It is the continuous version of the grey pattern problem which is a special case of the Quadratic Assignment Problem. The problem is a minimization of a convex function subject to non-convex constraints and has infinitely many optimal solutions. We propose several mathematical programming formulations that are suitable for a multi-start heuristic algorithm. In addition to solving these formulations by the Solver in Excel and Mathematica, a special Nelder–Mead algorithm is proposed. This special algorithm provided the best results. One suggested modification may improve the performance of the Nelder–Mead algorithm for other optimization problems as well.     

Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner

A computational technique which is based on a numerical-asymptotic expansion matching for computing the local singular behavior of a viscous flow around a sharp right-angle expansion corner is presented. Moffatt's (1964) asymptotic solution is extended and a matching with a time-marching finite-difference scheme of the Navier--Stokes equations is formulated. Local mesh refinement around the corner is required to meet the validity of the asymptotic solution. Flows in an expanding channel with expansion ratio D/d=3 at various Reynolds numbers 1≤Re≤700 are simulated. The results are compared with those from a standard finite-difference scheme that uses second-order forward/backward differences near the corner. It is found that the results of the standard scheme converge toward those of the present technique as the level of local refinement near the corner is increased. The time-dependent parameters of the first two terms of the asymptotic solution at the steady-state solution are also described for various cases of Re and D/d. It is demonstrated that the present method enhances the accuracy of the simulations and requires less refinements near the corners to achieve converged numerical results. Received 14 August 2000 and accepted 25 October 2001     

A variational problem determined by probability measures

ABSTRACT

An optimization problem of maximizing an integral of a function over a family of probability measures is considered. The problem is a generalization of a well-studied variational problem in mathematical economics, concerning optimal allocations. The specific generalization that we examine arises also in the limit of singularly perturbed optimal control problems. We examine the mathematical problem and allude to the singular perturbation motivation.     

Skin Microbiome Compositional Changes in Atopic Dermatitis Accompany Dead Sea Climatotherapy

Dead Sea climatotherapy (DSC) is a well‐established therapeutic modality for the treatment of several diseases, including atopic dermatitis. Skin microbiome studies have shown that skin microbiome diversity is anticorrelated with both atopic dermatitis severity and concurrent Staphylococcus aureus overgrowth. This study aimed to determine whether DSC induces skin microbiome changes concurrent with clinical improvements in atopic dermatitis. We sampled 35 atopic dermatitis patients and ten healthy controls on both the antecubital and popliteal fossa. High‐resolution microbial community profiling was attained by sequencing multiple regions of the 16S rRNA gene. Dysbiosis was observed in both lesional and nonlesional sites, which was partially attenuated following treatment. Severe AD skin underwent the most significant community shifts, and Staphylococcus epidermidis, Streptococcus mitis and Micrococcus luteus relative abundance were significantly affected by Dead Sea climatotherapy. Our study highlights the temporal shifts of the AD skin microbiome induced by Dead Sea climatotherapy and offers potential explanations for the success of climatotherapy on a variety of skin diseases, including AD.     

Experimental Measurements of Stress and Chemical Controls on the Evolution of Fracture Permeability

We explore how fracture permeability in confined tight carbonates evolves due to flow of reactive fluids. Core plugs of the Capitan Massive Limestone are saw-cut to form a smooth axial fracture that is subsequently roughened to control the fracture surface topography. Either distilled water or distilled water–ammonium chloride solutions are circulated through these plugs, where fracture roughness, inlet fluid pH, and confining stresses are controlled. Throughout the experiment we measure the fluid flow rate and chemical composition of the effluent fluid. Mass balance, conducted on the effluent fluid mass and on dissolved mineral components, independently constrains the mineral mass removal. We use an idealized lumped parameter model of asperity supported fractures undergoing simultaneous stress corrosion cracking-induced diffusion and free-face dissolution to infer theoretical rates of aperture loss or gain. This model incorporates the roles of confining stress, fracture contact area, and composition and reactivity of the permeating fluid while identifying zones of diffusion-dominated mass transfer within the fracture. These theoretical rates of aperture strain are compared to those inferred from the experimentally determined permeability evolution and permeating fluid mineral mass balance. By measuring in regimes of both increasing and decreasing permeability we quantitatively constrain the transition between fracture-gaping and fracture-closing modes of behavior. We parameterize this transition in permeability evolution by the ratio of mechanically to chemically controlled dissolved mass fluxes. The transition from regimes of closing to regimes of gaping occurs at unity ( $\chi \approx 1$ ) when stress and chemically driven mass fluxes are theoretically equal.     

On the complexity of the exchange algorithm for minimax optimization problems

We present an exchange algorithm for the solution of minimax optimization problems involving convex functions. For a certain class of functions, the complexity of this algorithm is shown to be either linear in the number of functions, or at least squared in that number.