Towards the computer-aided diagnosis of dementia based on the geometric and network connectivity of structural MRI data

We present an intuitive geometric approach for analysing the structure and fragility of T1-weighted structural MRI scans of human brains. Apart from computing characteristics like the surface area and volume of regions of the brain that consist of highly active voxels, we also employ Network Theory in order to test how close these regions are to breaking apart. This analysis is used in an attempt to automatically classify subjects into three categories: Alzheimer’s disease, mild cognitive impairment and healthy controls, for the CADDementia Challenge.

Geometric polarimetry − part I: spinors and wave states

A new formal approach for representation of polarization states of coherent and partially coherent electromagnetic plane waves is presented. Its basis is a purely geometric construction for the normalised complex-analytic coherent wave as a generating line in the sphere of wave directions, and whose Stokes vector is determined by the intersection with the conjugate generating line. The Poincare sphere is now located in physical space, simply a coordination of the wave sphere, its axis aligned with the wave vector. Algebraically, the generators representing coherent states are represented by spinors, and this is made consistent with the spinor-tensor representation of electromagnetic theory by means of an explicit reference spinor we call the phase flag. As a faithful unified geometric representation, the new model provides improved formal tools for resolving many of the geometric difficulties and ambiguities that arise in the traditional formalism.

Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited

This report is a review of Darwin`s classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.

Semiconductor nanoparticle modeling via density functional theory

In this work, a 2.0 nm nanoparticle (low limit synthesized system) is compared to possible simplified models: passivated clusters, small (1.3 nm) nanoparticles and sets of plane surfaces. Our density functional theory results suggest that even when geometric aspects are properly described by the simplifications considered, electronic properties might be very different, especially when edge atoms are not properly taken into account in the nanoparticle`s modeling. In addition, we propose a protocol that might help future theoretical descriptions of nanoparticles.

A general treatment of geometric phases and dynamical invariants

Based only on the parallel-transport condition, we present a general method to compute Abelian or non-Abelian geometric phases acquired by the basis states of pure or mixed density operators, which also holds for nonadiabatic and noncyclic evolution. Two interesting features of the non-Abelian geometric phase obtained by our method stand out: i) it is a generalization of Wilczek and Zee`s non-Abelian holonomy, in that it describes nonadiabatic evolution where the basis states are parallelly transported between distinct degenerate subspaces, and ii) the non-Abelian character of our geometric phase relies on the transitional evolution of the basis states, even in the nondegenerate case. We apply our formalism to a two-level system evolving nonadiabatically under spontaneous decay to emphasize the non- Abelian nature of the geometric phase induced by the reservoir. We also show, through the generalized invariant theory, that our general approach encompasses previous results in the literature. Copyright (c) EPLA, 2008.

SIGMA THEORY AND TWISTED CONJUGACY CLASSES

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut(G) such that for phi is an element of H the Reidemeister number R(phi) is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP(infinity); groups G/G `` of finite Prufer rank; groups G of type FP(2) without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group. Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups. In some cases, such as with the generalized Thompson`s groups F(n,0) and their finite direct products, H = Aut(G).

Optimal growth in a two-sector model without discounting: a geometric investigation

We characterize optimal policy in a two-sector growth model with xed coeÆcients and with no discounting. The model is a specialization to a single type of machine of a general vintage capital model originally formulated by Robinson, Solow and Srinivasan, and its simplicity is not mirrored in its rich dynamics, and which seem to have been missed in earlier work. Our results are obtained by viewing the model as a specific instance of the general theory of resource allocation as initiated originally by Ramsey and von Neumann and brought to completion by McKenzie. In addition to the more recent literature on chaotic dynamics, we relate our results to the older literature on optimal growth with one state variable: speci cally, to the one-sector setting of Ramsey, Cass and Koopmans, as well as to the two-sector setting of Srinivasan and Uzawa. The analysis is purely geometric, and from a methodological point of view, our work can be seen as an argument, at least in part, for the rehabilitation of geometric methods as an engine of analysis.

Generic Bifurcations of Planar Filippov Systems via Geometric Singular Perturbations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Applications of Subleading-Color Amplitudes in N=4 SYM Theory

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Geometrical Lagrangian for a supersymmetric Yang-Mills theory on the group manifold

Perhaps one of the main features of Einstein's General Theory of Relativity is that spacetime is not flat itself but curved. Nowadays, however, many of the unifying theories like superstrings on even alternative gravity theories such as teleparalell geometric theories assume flat spacetime for their calculations. This article, an extended account of an earlier author's contribution, it is assumed a curved group manifold as a geometrical background from which a Lagrangian for a supersymmetric N = 2, d = 5 Yang-Mills - SYM, N = 2, d = 5 - is built up. The spacetime is a hypersurface embedded in this geometrical scenario, and the geometrical action here obtained can be readily coupled to the five-dimensional supergravity action. The essential idea that underlies this work has its roots in the Einstein-Cartan formulation of gravity and in the 'group manifold approach to gravity and supergravity theories'. The group SYM, N = 2, d = 5, turns out to be the direct product of supergravity and a general gauge group g: G = g circle times <(SU(2, 2/1))over bar>.

Theoretical investigation of geometric configurations and vibrational spectra in citric acid complexes

The performance of advanced electronic ceramics is directly related to the synthesis route employed. Sol-gel methods are widely used for this purpose. However, the physicochemical intermediate steps are still not well understood. Better understanding and control of these processes can improve the final quality of samples. In this work, we studied theoretically the formation of metal complexes between citric acid and lithium or barium metal cations with different citric acid/metal proportions, using Density Functional Theory electronic structure calculations. Infrared and Raman scattering spectra were simulated for the more stable geometric configurations. Using this methodology, we identified some features of complexes formed in the synthesis process. Our results show that the complexes can be distinguished by changes in the bands assigned to C=O, COH-, and COO- group vibrations. An estimate of the most stable complexes is made based on total energy.

Encapsulation of small magnetic clusters in fullerene cages: A density functional theory investigation within van der Waals corrections

The encapsulation of magnetic transition-metal (TM) clusters inside carbon cages (fullerenes, nanotubes) has been of great interest due to the wide range of applications, which spread from medical sensors in magnetic resonance imaging to photonic crystals. Several theoretical studies have been reported; however, our atomistic understanding of the physical properties of encapsulated magnetic TM 3d clusters is far from satisfactory. In this work, we will report general trends, derived from density functional theory within the generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof (PBE), for the encapsulation properties of the TMm@C-n (TM = Fe, Co, Ni; m = 2-6, n = 60,70,80,90) systems. Furthermore, to understand the role of the van der Waals corrections to the physical properties, we employed the empirical Grimme's correction (PBE + D2). We found that both PBE and PBE + D2 functionals yield almost the same geometric parameters, magnetic and electronic properties, however, PBE + D2 strongly enhances the encapsulation energy. We found that the center of mass of the TMm clusters is displaced towards the inside C-n surfaces, except for large TMm clusters (m = 5 and 6). For few cases, e. g., Co-4 and Fe-4, the encapsulation changes the putative lowest-energy structure compared to the isolated TMm clusters. We identified few physical parameters that play an important role in the sign and magnitude of the encapsulation energy, namely, cluster size, fullerene equatorial diameter, shape, curvature of the inside C-n surface, number of TM atoms that bind directly to the inside C-n surface, and the van der Waals correction. The total magnetic moment of encapsulated TMm clusters decreases compared with the isolated TMm clusters, which is expected due to the hybridization of the d-p states, and strongly depends on the size and shape of the fullerene cages.

Ethanol and Water Adsorption on Close-Packed 3d, 4d, and 5d Transition-Metal Surfaces: A Density Functional Theory Investigation with van der Waals Correction

Nowadays, there is a great interest in the economic success of direct ethanol fuel cells; however, our atomistic understanding of the designing of stable and low-cost catalysts for the steam reforming of ethanol is still far from satisfactory, in particular due to the large number of undesirable intermediates. In this study, we will report a first-principles investigation of the adsorption properties of ethanol and water at low coverage on close-packed transition-metal (TM) surfaces, namely, Fe(110), Co(0001), Ni(111), Cu(111), Ru(0001), Rh(111), Pd(111), Ag(111), Os(0001), Ir(111), Pt(111), and Au(111), employing density functional theory (DFT) calculations. We employed the generalized gradient approximation with the formulation proposed by Perdew, Burke, and Erzenholf (PBE) to the exchange correlation functional and the empirical correction proposed by S. Grimme (DFT+D3) for the van der Waals correction. We found that both adsorbates binds preferentially near or on the on top sites of the TM surfaces through the 0 atoms. The PBE adsorption energies of ethanol and water decreases almost linearly with the increased occupation of the 4d and 5d d-band, while there is a deviation for the 3d systems. The van der Waals correction affects the linear behavior and increases the adsorption energy for both adsorbates, which is expected as the van der Waals energy due to the correlation effects is strongly underestimated by DFT-PBE for weak interacting systems. The geometric parameters for water/TM are not affected by the van der Waals correction, i.e., both DFT and DFT+D3 yield an almost parallel orientation for water on the TM surfaces; however, DFT+D3 changes drastically the ethanol orientation. For example, DFT yields an almost perpendicular orientation of the C-C bond to the TM surface, while the C-C bond is almost parallel to the surface using DFT +D3 for all systems, except for ethanol/Fe(110). Thus, the van der Waals correction decreases the distance of the C atoms to the TM surfaces, which might contribute to break the C-C bond. The work function decreases upon the adsorption of ethanol and water, and both follow the same trends, however, with different magnitude (larger for ethanol/TM) due to the weak binding of water to the surface. The electron density increases mainly in the region between the topmost layer and the adsorbates, which explains the reduction of the substrate work function.

Adsorption of NO on the Rh-13, Pd-13, Ir-13, and Pt-13 Clusters: A Density Functional Theory Investigation

The adsorption of NO on transition-metal (TM) surfaces has been widely studied by experimental and theoretical techniques; however, our atomistic understanding of the interaction of nitrogen monoxide (NO) with small TM clusters is far from satisfactory, which compromises a deep understanding of real catalyst devices. In this study, we report a density functional theory study of the adsorption properties of NO on the TM13 (TM = Rh, Pd, Ir, Pt) clusters employing the projected augmented wave method. We found that the interaction of NO with TM13 is much more complex than that for NO/TM(111). In particular, for low symmetry TM13 clusters, there is a strong rearrangement of the electronic charge density upon NO adsorption and, as a consequence, the adsorption energy shows a very complex dependence even for adsorption sites with the same local effective coordination. We found a strong enhancement of the binding energy of NO to the TM13 clusters compared with the TM(111) surfaces, as the antibonding NO states are not occupied for NO/TM13, and the general relationship based on the d-band model between adsorption energy and the center of gravity of the occupied d-states does not hold for the studied TM13 clusters, in particular, for clusters with low symmetry. In contrast with the adsorption energy trends, the geometric NO/TM13 parameters and the vibrational N-O frequencies for different coordination sites follow the same trend as for the respective TM(111) surfaces, while the changes in the frequencies between different surfaces and TM13 clusters reflect the strong NO-TM13 interaction.

Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.