An error analysis of probabilistic fibre tracking methods: average curves optimization

Fibre tractography using diffusion tensor imaging is a promising method for estimating the pathways of white matter tracts in the human brain. The success of fibre tracking methods ultimately depends upon the accuracy of the fibre tracking algorithms and the quality of the data. Uncertainty and its representation have an important role to play in fibre tractography methods to infer useful information from real world noisy diffusion weighted data. Probabilistic fibre tracking approaches have received considerable interest recently for resolving orientational uncertainties. In this study, an average curves approach was used to investigate the impact of SNR and tensor field geometry on the accuracy of three different types of probabilistic tracking algorithms. The accuracy was assessed using simulated data and a range of tract geometries. The average curves representations were employed to represent the optimal fibre path of probabilistic tracking curves. The results are compared with streamline tracking on both simulated and in vivo data

A finite element method for fully nonlinear elliptic problems

We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems including the Monge-Amp\`ere equation and Pucci's equation.Comment: 22 pages, 31 figure

Bivariate spline interpolation with optimal approximation order

Let be a triangulation of some polygonal domain f c R2 and let S9 (A) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [71 and [181

Local RBF approximation for scattered data fitting with bivariate splines

In this paper we continue our earlier research [4] aimed at developing effcient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, signicantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given

Investigation of vegetation as a precondition for improving the management of a national nature park (on the example of Dzharylhatskyi NP)

The protected area of Dzharylhatskyi National Park is only 3% of the total area. This is not enough to preserve dynamic ecosystems that are very vulnerable to natural factors and almost impossible to restore following damage by anthropogenic factors. In the investigated area there are territories with high potential for conservation, which are characterized by the presence of species from the Red Data Book of Ukraine and International Red Lists, plant communities from the Green Data Book of Ukraine and biotopes of global significance. The proposed ten sites should receive protected status, which will bring the protected area up to the necessary minimum to preserve the rare ecosystems of the park from tourism and economic pressure, which have intensified in recent years. For the period of research in 2014–2018, 14 types of sozophytes were identified and confirmed, two of them are the highly localised species endemic to Dzharylhach Island: Molinia euxina Pobed. and Poacynum russanovii (Pobed.) Mavrodiev, A. Laktionov et Y. Alexeev. The syntaxonomic structure of the sozologically valuable coenoses is represented by two basal communities, two subassociations and 8 associations belonging to 9 alliances, 9 orders and 9 classes. Out of them, we provisionally propose the new following groups: ass. Apero maritimi-Chrysopogonetum grylli nom. prov., subass. Apero maritimi-Chrysopogonetum grylli, Stipetum borysthenicae nom. prov., subass. Cladietum marisci, Caricetum extensae nom. prov., BC Molinia euxina [Molinion caeruleae] nom. prov. The proposed sites represent 8 biotopes from Annex I of the Habitat Directive, which imposes obligations for their conservation at the world level. We carried out an analysis of the Ukrainian legislative acts, their correlation with international requirements and the zoning of the NPP “Dzharylhatskyi” in relation to these requirements and recommendations. Thus, the urgent need to expand the boundaries of the protected area of the Dzharylhatskyi National Nature Park by including distinguished protected tracts has been confirmed

Jahn-Teller systems from a cavity QED perspective

Jahn-Teller systems and the Jahn-Teller effect are discussed in terms of cavity QED models. By expressing the field modes in a quadrature representation, it is shown that certain setups of a two-level system interacting with a bimodal cavity is described by the Jahn-Teller $E\times\epsilon$ Hamiltonian. We identify the corresponding adiabatic potential surfaces and the conical intersection. The effects of a non-zero geometrical Berry phase, governed by encircling the conical intersection, are studied in detail both theoretically and numerically. The numerical analysis is carried out by applying a wave packet propagation method, more commonly used in molecular or chemical physics, and analytic expressions for the characteristic time scales are presented. It is found that the collapse-revival structure is greatly influenced by the geometrical phase and as a consequence, the field intensities contain direct information about this phase. We also mention the link between the Jahn-Teller effect and the Dicke phase transition in cavity QED.Comment: 10 pages, 6 figures. Replaced with final versio

Part of the D - dimensional Spiked harmonic oscillator spectra

The pseudoperturbative shifted - l expansion technique PSLET [5,20] is generalized for states with arbitrary number of nodal zeros. Interdimensional degeneracies, emerging from the isomorphism between angular momentum and dimensionality of the central force Schrodinger equation, are used to construct part of the D - dimensional spiked harmonic oscillator bound - states. PSLET results are found to compare excellenly with those from direct numerical integration and generalized variational methods [1,2].Comment: Latex file, 20 pages, to appear in J. Phys. A: Math. & Ge