On degeneracy and invariances of random fields paths with applications in Gaussian process modelling

We study pathwise invariances and degeneracies of random fields with motivating applications in Gaussian process modelling. The key idea is that a number of structural properties one may wish to impose a priori on functions boil down to degeneracy properties under well-chosen linear operators. We first show in a second order set-up that almost sure degeneracy of random field paths under some class of linear operators defined in terms of signed measures can be controlled through the two first moments. A special focus is then put on the Gaussian case, where these results are revisited and extended to further linear operators thanks to state-of-the-art representations. Several degeneracy properties are tackled, including random fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process modelling. In a first numerical experiment, it is shown that dedicated kernels can be used to infer an axis of symmetry. Our second numerical experiment deals with conditional simulations of a solution to the heat equation, and it is found that adapted kernels notably enable improved predictions of non-linear functionals of the field such as its maximum.

On the Structure of Spatial Branching Processes

The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching processes are built up from single-line processes, whereas the regular ones are mixtures of left-tail trivial processes with a Poisson family structure.

Some Contributions to the Class of Two-Sex Branching Processes Depending on the Number of Couples in the Population

2000 Mathematics Subject Classification: 60J80

A Note on the Extinction Problem for Controlled Multitype Branching Processes

2000 Mathematics Subject Classification: 60J80, 60J10.

An extension of Lorentz's almost convergence and applications in Banach spaces

2000 Mathematics Subject Classification: Primary 40C99, 46B99.

SYSTEMS OF DIFFERENTIAL EQUATIONSTHAT ARE COMPETITIVE OR COOPERATIVE II:CONVERGENCE ALMOST EVERYWHERE*MORRIS W. HIRSCH

A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main results in this article are the following. A cooperative system cannot have nonconstant attracting periodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbit converges for almost every point having compact forward orbit closure. In a cooperative system in 2 dimensions, every solution is eventually monotone.  Applications are made to generalizations of positive feedback loops.

Total factor productivity growth and convergence in the petroleum industry : empirical analysis testing for convexity

While economic theory acknowledges that some features of technology (e.g., indivisibilities, economies of scale and specialization) can fundamentally violate the traditional convexity assumption, almost all empirical studies accept the convexity property on faith. In this contribution, we apply two alternative flexible production technologies to measure total factor productivity growth and test the significance of the convexity axiom using a nonparametric test of closeness between unknown distributions. Based on unique field level data on the petroleum industry, the empirical results reveal significant differences, indicating that this production technology is most likely non-convex. Furthermore, we also show the impact of convexity on answers to traditional convergence questions in the productivity growth literature.

Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.

IMAGE DENOISING USING OPTIMALLY WEIGHTED BILATERAL FILTERS: A SURE AND FAST APPROACH

The bilateral filter is known to be quite effective in denoising images corrupted with small dosages of additive Gaussian noise. The denoising performance of the filter, however, is known to degrade quickly with the increase in noise level. Several adaptations of the filter have been proposed in the literature to address this shortcoming, but often at a substantial computational overhead. In this paper, we report a simple pre-processing step that can substantially improve the denoising performance of the bilateral filter, at almost no additional cost. The modified filter is designed to be robust at large noise levels, and often tends to perform poorly below a certain noise threshold. To get the best of the original and the modified filter, we propose to combine them in a weighted fashion, where the weights are chosen to minimize (a surrogate of) the oracle mean-squared-error (MSE). The optimally-weighted filter is thus guaranteed to perform better than either of the component filters in terms of the MSE, at all noise levels. We also provide a fast algorithm for the weighted filtering. Visual and quantitative denoising results on standard test images are reported which demonstrate that the improvement over the original filter is significant both visually and in terms of PSNR. Moreover, the denoising performance of the optimally-weighted bilateral filter is competitive with the computation-intensive non-local means filter.

Crystal Coulomb energies: I. Organic donor-acceptor complexes--a review. II. Classical Ewald calculations of the Coulomb binding energy of four donor-acceptor crystals and Wurster's blue perchlorate. III. A rapid-convergence quantum-mechanical formalism for crystal electronic energies

Chapter I

Theories for organic donor-acceptor (DA) complexes in solution and in the solid state are reviewed, and compared with the available experimental data. As shown by McConnell et al. (Proc. Natl. Acad. Sci. U.S., 53, 46-50 (1965)), the DA crystals fall into two classes, the holoionic class with a fully or almost fully ionic ground state, and the nonionic class with little or no ionic character. If the total lattice binding energy 2ε₁ (per DA pair) gained in ionizing a DA lattice exceeds the cost 2ε_o of ionizing each DA pair, ε₁ + ε_o less than 0, then the lattice is holoionic. The charge-transfer (CT) band in crystals and in solution can be explained, following Mulliken, by a second-order mixing of states, or by any theory that makes the CT transition strongly allowed, and yet due to a small change in the ground state of the non-interacting components D and A (or D⁺ and A^-). The magnetic properties of the DA crystals are discussed.

Chapter II

A computer program, EWALD, was written to calculate by the Ewald fast-convergence method the crystal Coulomb binding energy E_C due to classical monopole-monopole interactions for crystals of any symmetry. The precision of E_C values obtained is high: the uncertainties, estimated by the effect on E_C of changing the Ewald convergence parameter η, ranged from ± 0.00002 eV to ± 0.01 eV in the worst case. The charge distribution for organic ions was idealized as fractional point charges localized at the crystallographic atomic positions: these charges were chosen from available theoretical and experimental estimates. The uncertainty in E_C due to different charge distribution models is typically ± 0.1 eV (± 3%): thus, even the simple Hückel model can give decent results.

E_C for Wurster's Blue Perchl orate is -4.1 eV/molecule: the crystal is stable under the binding provided by direct Coulomb interactions. E_C for N-Methylphenazinium Tetracyanoquino- dimethanide is 0.1 eV: exchange Coulomb interactions, which cannot be estimated classically, must provide the necessary binding.

EWALD was also used to test the McConnell classification of DA crystals. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine: 7,7,8,8-Tetracyanoquinodimethan) E_C = -4.0 eV while 2ε_o = 4.6₅ eV: clearly, exchange forces must provide the balance. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine:para-Chloranil) E_C = -4.4 eV, while 2ε_o = 5.0 eV: again EC falls short of 2ε₁. As a Gedankenexperiment, two nonionic crystals were assumed to be ionized: for (1:1)-(Hexamethyl- benzene:para-Chloranil) E_C = -4.5 eV, 2ε_o = 6.6 eV; for (1:1)- (Napthalene:Tetracyanoethylene) E_C = -4.3 eV, 2ε_o = 6.5 eV. Thus, exchange energies in these nonionic crystals must not exceed 1 eV.

Chapter III

A rapid-convergence quantum-mechanical formalism is derived to calculate the electronic energy of an arbitrary molecular (or molecular-ion) crystal: this provides estimates of crystal binding energies which include the exchange Coulomb inter- actions. Previously obtained LCAO-MO wavefunctions for the isolated molecule(s) ("unit cell spin-orbitals") provide the starting-point. Bloch's theorem is used to construct "crystal spin-orbitals". Overlap between the unit cell orbitals localized in different unit cells is neglected, or is eliminated by Löwdin orthogonalization. Then simple formulas for the total kinetic energy Q^(XT)_λ, nuclear attraction [λ/λ]^XT, direct Coulomb [λλ/λ'λ']^XT and exchange Coulomb [λλ'/λ'λ]^XT integrals are obtained, and direct-space brute-force expansions in atomic wavefunctions are given. Fourier series are obtained for [λ/λ]^XT, [λλ/λ'λ']^XT, and [λλ/λ'λ]^XT with the help of the convolution theorem; the Fourier coefficients require the evaluation of Silverstone's two-center Fourier transform integrals. If the short-range interactions are calculated by brute-force integrations in direct space, and the long-range effects are summed in Fourier space, then rapid convergence is possible for [λ/λ]^XT, [λλ/λ'λ']^XT and [λλ'/λ'λ]^XT. This is achieved, as in the Ewald method, by modifying each atomic wavefunction by a "Gaussian convergence acceleration factor", and evaluating separately in direct and in Fourier space appropriate portions of [λ/λ]^XT, etc., where some of the portions contain the Gaussian factor.

The ocean’s role in setting the mean position of the Inter-Tropical Convergence Zone

Through study of observations and coupled climate simulations, it is argued that the mean position of the Inter-Tropical Convergence Zone (ITCZ) north of the equator is a consequence of a northwards heat transport across the equator by ocean circulation. Observations suggest that the hemispheric net radiative forcing of climate at the top of the atmosphere is almost perfectly symmetric about the equator, and so the total (atmosphere plus ocean) heat transport across the equator is small (order 0.2 PW northwards). Due to the Atlantic ocean’s meridional overturning circulation, however, the ocean carries significantly more heat northwards across the equator (order 0.4 PW) than does the coupled system. There are two primary consequences. First, atmospheric heat transport is southwards across the equator to compensate (0.2 PW southwards), resulting in the ITCZ being displaced north of the equator. Second, the atmosphere, and indeed the ocean, is slightly warmer (by perhaps 2 °C) in the northern hemisphere than in the southern hemisphere. This leads to the northern hemisphere emitting slightly more outgoing longwave radiation than the southern hemisphere by virtue of its relative warmth, supporting the small northward heat transport by the coupled system across the equator. To conclude, the coupled nature of the problem is illustrated through study of atmosphere–ocean–ice simulations in the idealized setting of an aquaplanet, resolving the key processes at work.

A note on convergence of Peck-Shell and Green-Lin mechanisms in the Diamond-Dybvig model

We study the effects of population size in the Peck-Shell analysis of bank runs. We find that a contract featuring equal-treatment for almost all depositors of the same type approximates the optimum. Because the approximation also satisfies Green-Lin incentive constraints, when the planner discloses positions in the queue, welfare in these alternative specifications are sandwiched. Disclosure, however, it is not needed since our approximating contract is not subject to runs.

Convergence in Brazil: recent trends and long run prospects

This paper applies to the analysis of the interstate income distribution in BraziI a set of techniques that have been widely used in the current empirical literature on growth and convergence. Usual measures of dispersion in the interstate income distribution (the coefficient of variation and Theil' s index) suggest that cr-convergence was an unequivoca1 feature of the regional growth experience in BraziI, between 1970 and 1986. After 1986, the process of convergence seems, however, to have sIowed down almost to a halt. A standard growth modeI is shown to fit the regional data well and to expIain a substantial amount of the variation in growth rates, providing estimates of the speed of (conditional) J3-convergence of approximateIy 3% p.a .. Different estimates of the long run distribution implied by the recent growth trends point towards further reductions in the interstate income inequality, but also suggest that the relative per capita incomes of a significant number of states and the number of ''very poor" and "poor" states were, in 1995, already quite c10se to their steady-state values.

Network convergence and QoS for future multimedia services in the VISION project

The emerging use of real-time 3D-based multimedia applications imposes strict quality of service (QoS) requirements on both access and core networks. These requirements and their impact to provide end-to-end 3D videoconferencing services have been studied within the Spanish-funded VISION project, where different scenarios were implemented showing an agile stereoscopic video call that might be offered to the general public in the near future. In view of the requirements, we designed an integrated access and core converged network architecture which provides the requested QoS to end-to-end IP sessions. Novel functional blocks are proposed to control core optical networks, the functionality of the standard ones is redefined, and the signaling improved to better meet the requirements of future multimedia services. An experimental test-bed to assess the feasibility of the solution was also deployed. In such test-bed, set-up and release of end-to-end sessions meeting specific QoS requirements are shown and the impact of QoS degradation in terms of the user perceived quality degradation is quantified. In addition, scalability results show that the proposed signaling architecture is able to cope with large number of requests introducing almost negligible delay.

Some Bounds for Almost Absorbing Birth and Death Processes with Catastrophes

2000 Mathematics Subject Classification: 60J27.