Convexity, C-convexity and Pseudoconvexity

Николай М. Николов - Разгледани са характеризации на различни понятия за изпъкналост, като тези понятия са сравнени.

Does Atkinson-Wilcox Expansion Converges for any Convex Domain?

2000 Mathematics Subject Classification: 35C10, 35C20, 35P25, 47A40, 58D30, 81U40.

A Schwarz lemma for Kahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of Kahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kahler affine metrics of Yau s Schwarz lemma for volume forms. By a theorem of Cheng and Yau, there is a canonical Kahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an n -dimensional cone, a rescaling of the canonical potential is an n -normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kahler space.

The N-terminal domain of a K+ channel β subunit increases the rate of C-type inactivation from the cytoplasmic side of the channel

Voltage-gated K+ channels are complexes of membrane-bound, ion-conducting α and cytoplasmic ancillary (β) subunits. The primary physiologic effect of coexpression of α and β subunits is to increase the intrinsic rate of inactivation of the α subunit. For one β subunit, Kvβ1.1, inactivation is enhanced through an N-type mechanism. A second β subunit, Kvβ1.2, has been shown to increase inactivation, but through a distinct mechanism. Here we show that the degree of enhancement of Kvβ1.2 inactivation is dependent on the amino acid composition in the pore mouth of the α subunit and the concentration of extracellular K+. Experimental conditions that promote C-type inactivation also enhance the stimulation of inactivation by Kvβ1.2, showing that this β subunit directly stimulates C-type inactivation. Chimeric constructs containing just the nonconserved N-terminal region of Kvβ1.2 fused with an α subunit behave in a similar fashion to coexpressed Kvβ1.2 and α subunit. This shows that it is the N-terminal domain of Kvβ1.2 that mediates the increase in C-type inactivation from the cytoplasmic side of the pore. We propose a model whereby the N terminus of Kvβ1.2 acts as a weakly binding “ball” domain that associates with the intracellular vestibule of the α subunit to effect a conformational change leading to enhancement of C-type inactivation.

On the compactness of isometry groups in complex analysis

We prove that the group of continuous isometries for the Kobayashi or Caratheodory metrics of a strongly convex domain in C-n is compact unless the domain is biholomorphic to the ball. A key ingredient, proved using differential geometric ideas, is that a continuous isometry between a strongly convex domain and the ball has to be biholomorphic or anti-biholomorphic. Combining this with a metric version of Pinchuk's rescaling technique gives the main result.

COMPLEX GEODESICS, THEIR BOUNDARY REGULARITY, AND A HARDY-LITTLEWOOD-TYPE LEMMA

We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to partial derivative D. This example suggests that continuity at the boundary of the complex geodesics of a convex domain Omega (sic) C-n, n >= 2, is affected by the extent to which partial derivative Omega curves or bends at each boundary point. We provide a sufficient condition to this effect (on C-1-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy-Littlewood-type lemma that might be of independent interest.

Convergence of bimodules over maximal abelian selfadjoint algebras

We prove a continuity result for the map sending a masa-bimodule to its support. We characterise the convergence of a net of weakly closed convex hulls of bilattices in terms of the convergence of the corresponding supports, and establish a lower-semicontinuity result for the map sending a support to the corresponding masa-bimodule.

Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

The Fundamental Gap For Hyperbolic Triangles

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Problems for P-Monge-Ampere Equations

2010 Mathematics Subject Classification: 35A23, 35B51, 35J96, 35P30, 47J20, 52A40.

Efficient algorithms for dynamic probabilistic safety assessment. An application to convex damage domain.

The design of nuclear power plant has to follow a number of regulations aimed at limiting the risks inherent in this type of installation. The goal is to prevent and to limit the consequences of any possible incident that might threaten the public or the environment. To verify that the safety requirements are met a safety assessment process is followed. Safety analysis is as key component of a safety assessment, which incorporates both probabilistic and deterministic approaches. The deterministic approach attempts to ensure that the various situations, and in particular accidents, that are considered to be plausible, have been taken into account, and that the monitoring systems and engineered safety and safeguard systems will be capable of ensuring the safety goals. On the other hand, probabilistic safety analysis tries to demonstrate that the safety requirements are met for potential accidents both within and beyond the design basis, thus identifying vulnerabilities not necessarily accessible through deterministic safety analysis alone. Probabilistic safety assessment (PSA) methodology is widely used in the nuclear industry and is especially effective in comprehensive assessment of the measures needed to prevent accidents with small probability but severe consequences. Still, the trend towards a risk informed regulation (RIR) demanded a more extended use of risk assessment techniques with a significant need to further extend PSA’s scope and quality. Here is where the theory of stimulated dynamics (TSD) intervenes, as it is the mathematical foundation of the integrated safety assessment (ISA) methodology developed by the CSN(Consejo de Seguridad Nuclear) branch of Modelling and Simulation (MOSI). Such methodology attempts to extend classical PSA including accident dynamic analysis, an assessment of the damage associated to the transients and a computation of the damage frequency. The application of this ISA methodology requires a computational framework called SCAIS (Simulation Code System for Integrated Safety Assessment). SCAIS provides accident dynamic analysis support through simulation of nuclear accident sequences and operating procedures. Furthermore, it includes probabilistic quantification of fault trees and sequences; and integration and statistic treatment of risk metrics. SCAIS comprehensively implies an intensive use of code coupling techniques to join typical thermal hydraulic analysis, severe accident and probability calculation codes. The integration of accident simulation in the risk assessment process and thus requiring the use of complex nuclear plant models is what makes it so powerful, yet at the cost of an enormous increase in complexity. As the complexity of the process is primarily focused on such accident simulation codes, the question of whether it is possible to reduce the number of required simulation arises, which will be the focus of the present work. This document presents the work done on the investigation of more efficient techniques applied to the process of risk assessment inside the mentioned ISA methodology. Therefore such techniques will have the primary goal of decreasing the number of simulation needed for an adequate estimation of the damage probability. As the methodology and tools are relatively recent, there is not much work done inside this line of investigation, making it a quite difficult but necessary task, and because of time limitations the scope of the work had to be reduced. Therefore, some assumptions were made to work in simplified scenarios best suited for an initial approximation to the problem. The following section tries to explain in detail the process followed to design and test the developed techniques. Then, the next section introduces the general concepts and formulae of the TSD theory which are at the core of the risk assessment process. Afterwards a description of the simulation framework requirements and design is given. Followed by an introduction to the developed techniques, giving full detail of its mathematical background and its procedures. Later, the test case used is described and result from the application of the techniques is shown. Finally the conclusions are presented and future lines of work are exposed.

Performance Analysis of Reconstruction Techniques for Frequency-Domain Optical-Coherence Tomography

We address the issue of noise robustness of reconstruction techniques for frequency-domain optical-coherence tomography (FDOCT). We consider three reconstruction techniques: Fourier, iterative phase recovery, and cepstral techniques. We characterize the reconstructions in terms of their statistical bias and variance and obtain approximate analytical expressions under the assumption of small noise. We also perform Monte Carlo analyses and show that the experimental results are in agreement with the theoretical predictions. It turns out that the iterative and cepstral techniques yield reconstructions with a smaller bias than the Fourier method. The three techniques, however, have identical variance profiles, and their consistency increases linearly as a function of the signal-to-noise ratio.

Conservation law with the flux function discontinuous in the space variable- II - Convex-concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex-concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine-Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine-Hugoniot condition by a generalized Rankine-Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L-1. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented. (C) 2006 Elsevier B.V. All rights reserved.

Dependence of the large-scale vortex instability on latitude, stratification and domain size

In an earlier study, we reported on the excitation of large-scale vortices in Cartesian hydrodynamical convection models subject to rapid enough rotation. In that study, the conditions for the onset of the instability were investigated in terms of the Reynolds (Re) and Coriolis (Co) numbers in models located at the stellar North pole. In this study, we extend our investigation to varying domain sizes, increasing stratification, and place the box at different latitudes. The effect of the increasing box size is to increase the sizes of the generated structures, so that the principal vortex always fills roughly half of the computational domain. The instability becomes stronger in the sense that the temperature anomaly and change in the radial velocity are observed to be enhanced. The model with the smallest box size is found to be stable against the instability, suggesting that a sufficient scale separation between the convective eddies and the scale of the domain is required for the instability to work. The instability can be seen upto the colatitude of 30 degrees, above which value the flow becomes dominated by other types of mean flows. The instability can also be seen in a model with larger stratification. Unlike the weakly stratified cases, the temperature anomaly caused by the vortex structures is seen to depend on depth.