Nonlocal Singular Problems and Applications to MEMS

We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechanicalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory

Constructing Lifshitz solutions from AdS

Under general assumptions, we show that a gravitational theory in d+1 dimensions admitting an AdS solution can be reduced to a d-dimensional theory containing a Lifshitz solution with dynamical exponent z=2. Working in a d=4, N=2 supergravity setup, we prove that if the AdS background is N=2 supersymmetric, then the Lifshitz geometry preserves 1/4 of the supercharges, and we construct the corresponding Killing spinors. We illustrate these results in examples from supersymmetric consistent truncations of type IIB supergravity, enhancing the class of known 4-dimensional Lifshitz solutions of string theory. As a byproduct, we find a new AdS4 x S1 x T(1,1) solution of type IIB.Comment: 29 pages, no figures; v2 minor corrections, a reference adde

The Controversy of Myopia as a Risk Factor for Glaucoma: a Mathematical Approach

poster abstractPurpose: to quantify how individual variations in anatomical parameters often associated with myopia (e.g. longer ocular axial length (OAL), reduced scleral thickness (ST), lamina cribrosa diameter (LCD) and thickness (LCT)) affect retinal blood flow (RBF) and its sensitivity to ocular perfusion pressure (OPP). Methods: A mathematical model is used to calculate RBF through central retinal artery (CRA), arterioles, capillaries, venules, and central retinal vein (CRV). The flow is time-dependent, driven by systemic pressure and regulated by variable resistances to account for nonlinear effects due to (1) autoregulation (AR), and (2) lamina cribrosa effect on CRA and CRV. The latter is a nonlinear function of intraocular pressure (IOP), cerebrospinal fluid pressure (CSF) and OAL, ST, LCD, and LCT. RBF is computed as the solution of a system of five non-linear ordinary differential equations. The system is solved for different OPP values, obtained by varying independently IOP and mean arterial pressure (MAP), with and without AR. Results: Four representative eyes are compared: Eye 1 (OAL=24mm, ST=1mm, LCD=3mm, LCT=0.4mm), Eye 2 (OAL=28mm, ST=1mm, LCD=3mm, LCT=0.4mm), Eye 3 (OAL=24mm, ST=0.7mm, LCD=2mm, LCT=0.2mm), Eye 4 (OAL=28mm, ST=0.7mm, LCD=2mm, LCT=0.2mm). The model predicts that the cardiac cycle RBF average (RBFav) for eyes with smaller LCD and LCT is notably less than in normal eyes when IOP is elevated and without AR (c). Without AR and reduced MAP, the four eyes show similar RBFav reductions (d). With AR, anatomical changes do not induce notable changes in RBFav, (a) and (b). Conclusions: Reduced LCD and LCT, often associated with myopia, seem to affect RBFav more than elevated OAL. RBFav reductions magnify when AR is impaired, and this might reduce IOP safe levels for eyes with reduced LCD and LCT. These findings suggest that a combination of anatomical and vascular factors might cause certain myopic eyes to be at higher risk for glaucomatous damage than others

Control of macrophytes by grass carp (ctenopharyngodon idella) in a Waikato drain, New Zealand

Hornwort (Ceratophyllum demersum L.) and other aquatic macrophytes have historically been mechanically removed from the Rangiriri drain and Churchill East drain to maintain drain efficiency. As an alternative control method for the high plant biomass that accumulates at the end of summer, the effect of stocking diploid grass carp (Ctenopharyngodon idella L.) on the aquatic vegetation was evaluated in these Waikato drainage systems. At the start of the trial, both drains had a low diversity of aquatic macrophytes, and of the nine species (including the emergents), seven were exotic. Two months after grass carp were released to Churchill East drain (the treated drain) the four submerged and floating macrophyte species became scarce in the main drain. Over the same period, these species increased in biomass in Rangiriri drain (the untreated drain), where hornwort became dense and surface-reaching and remained so for the duration of the trial. However, grass carp did not control submerged vegetation in smaller side drains or the shallow, upper parts of the main drain, or the marginal sprawling species and emergent species. The cost of leasing the grass carp was similar to the cost of clearing the drains mechanically, but grass carp provided continuous weed control. However, subsequent to this trial, 62 dead grass carp were found in Churchill East drain in February 2001, and weed cover subsequently increased. This illustrates that grass carp management in New Zealand agricultural drains can be problematic due to periodic fish kills

de Sitter Supersymmetry Revisited

We present the basic $\mathcal{N} =1$ superconformal field theories in four-dimensional de Sitter space-time, namely the non-abelian super Yang-Mills theory and the chiral multiplet theory with gauge interactions or cubic superpotential. These theories have eight supercharges and are invariant under the full $SO(4,2)$ group of conformal symmetries, which includes the de Sitter isometry group $SO(4,1)$ as a subgroup. The theories are ghost-free and the anti-commutator $\sum_\alpha\{Q_\alpha, Q^{\alpha\dagger}\}$ is positive. SUSY Ward identities uniquely select the Bunch-Davies vacuum state. This vacuum state is invariant under superconformal transformations, despite the fact that de Sitter space has non-zero Hawking temperature. The $\mathcal{N}=1$ theories are classically invariant under the $SU(2,2|1)$ superconformal group, but this symmetry is broken by radiative corrections. However, no such difficulty is expected in the $\mathcal{N}=4$ theory, which is presented in appendix B.Comment: 21 pages, 2 figure

Heterotic Flux Attractors

We find attractor equations describing moduli stabilization for heterotic compactifications with generic SU(3)-structure. Complex structure and K\"ahler moduli are treated on equal footing by using SU(3)xSU(3)-structure at intermediate steps. All independent vacuum data, including VEVs of the stabilized moduli, is encoded in a pair of generating functions that depend on fluxes alone. We work out an explicit example that illustrates our methods.Comment: 37 pages, references and clarifications adde

Hypermoduli Stabilization, Flux Attractors, and Generating Functions

We study stabilization of hypermoduli with emphasis on the effects of generalized fluxes. We find a class of no-scale vacua described by ISD conditions even in the presence of geometric flux. The associated flux attractor equations can be integrated by a generating function with the property that the hypermoduli are determined by a simple extremization principle. We work out several orbifold examples where all vector moduli and many hypermoduli are stabilized, with VEVs given explicitly in terms of fluxes.Comment: 45 pages, no figures; Version submitted to JHE

On BPS bounds in D=4 N=2 gauged supergravity II: general matter couplings and black hole masses

We continue the analysis of BPS bounds started in arXiv:1110.2688, extending it to the full class of N=2 gauged supergravity theories with arbitrary vector and hypermultiplets. We derive the general form of the asymptotic charges for asymptotically flat (M_4), anti-de Sitter (AdS_4), and magnetic anti-de Sitter (mAdS_4) spacetimes. Some particular examples from black hole physics are given to explicitly demonstrate how AdS and mAdS masses differ when solutions with non-trivial scalar profiles are considered.Comment: 21 pages; v2 added reference, published version; v3 minor correction

Consistent reduction of charged D3-D7 systems

We provide a consistent reduction to five dimensions of the system of D3-branes at Calabi-Yau singularities coupled to D7-branes with world-volume gauge flux. The D3-branes source the dual to would-be conformal quiver theories. The D7-branes, which are homogeneously distributed in their transverse directions, are dual to massless matter in the fundamental representation at finite (baryon) density. We provide the five-dimensional action and equations of motion, and discuss a few sub-truncations. The reduction can be used in the study of transport properties and stability of D3-D7 charged systems.Comment: 23 pages. v2: references added and minor change

A special road to AdS vacua

We apply the techniques of special Kaehler geometry to investigate AdS_4 vacua of general N=2 gauged supergravities underlying flux compactifications of type II theories. We formulate the scalar potential and its extremization conditions in terms of a triplet of prepotentials P_x and their special Kaehler covariant derivatives only, in a form that recalls the potential and the attractor equations of N=2 black holes. We propose a system of first order equations for the P_x which generalize the supersymmetry conditions and yield non-supersymmetric vacua. Special geometry allows us to recast these equations in algebraic form, and we find an infinite class of new N=0 and N=1 AdS_4 solutions, displaying a rich pattern of non-trivial charges associated with NSNS and RR fluxes. Finally, by explicit evaluation of the entropy function on the solutions, we derive a U-duality invariant expression for the cosmological constant and the central charges of the dual CFT's.Comment: 41 pages; v2, v3: minor improvements, references added, published versio