Conformal Wasserstein distances: comparing surfaces in polynomial time

We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global M\"{o}bius transformations. We present in detail the case where the surfaces to compare are disk-like; we also sketch how the approach can be generalized to other types of surfaces.Comment: 23 pages, 3 figure

Quantum Field Theories on Algebraic Curves. I. Additive bosons

Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed filed k of constants of characteristic zero, define algebraic analogs of additive multi-valued functions on X and prove corresponding generalized residue theorem. Using the representation theory of the global Heisenberg and lattice Lie algebras, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.Comment: 31 pages, published version. Invariant formulation added, multiplicative section remove

Hamiltonian structure and quantization of 2+1 dimensional gravity coupled to particles

It is shown that the reduced particle dynamics of 2+1 dimensional gravity in the maximally slicing gauge has hamiltonian form. This is proved directly for the two body problem and for the three body problem by using the Garnier equations for isomonodromic transformations. For a number of particles greater than three the existence of the hamiltonian is shown to be a consequence of a conjecture by Polyakov which connects the auxiliary parameters of the fuchsian differential equation which solves the SU(1,1) Riemann-Hilbert problem, to the Liouville action of the conformal factor which describes the space-metric. We give the exact diffeomorphism which transforms the expression of the spinning cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing gauge. It is explicitly shown that the boundary term in the action, written in hamiltonian form gives the hamiltonian for the reduced particle dynamics. The quantum mechanical translation of the two particle hamiltonian gives rise to the logarithm of the Laplace-Beltrami operator on a cone whose angular deficit is given by the total energy of the system irrespective of the masses of the particles thus proving at the quantum level a conjecture by 't Hooft on the two particle dynamics. The quantum mechanical Green's function for the two body problem is given.Comment: 34 pages LaTe

An SU(N) Mott insulator of an atomic Fermi gas realized by large-spin Pomeranchuk cooling

The Hubbard model, containing only the minimum ingredients of nearest neighbor hopping and on-site interaction for correlated electrons, has succeeded in accounting for diverse phenomena observed in solid-state materials. One of the interesting extensions is to enlarge its spin symmetry to SU(N>2), which is closely related to systems with orbital degeneracy. Here we report a successful formation of the SU(6) symmetric Mott insulator state with an atomic Fermi gas of ytterbium (173Yb) in a three-dimensional optical lattice. Besides the suppression of compressibility and the existence of charge excitation gap which characterize a Mott insulating phase, we reveal an important difference between the cases of SU(6) and SU(2) in the achievable temperature as the consequence of different entropy carried by an isolated spin. This is analogous to Pomeranchuk cooling in solid 3He and will be helpful for investigating exotic quantum phases of SU(N) Hubbard system at extremely low temperatures.Comment: 20 pages, 6 figures, to appear in Nature Physic

Accessory parameters for Liouville theory on the torus

We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.Comment: 30 pages, LaTex; typos corrected, discussion of eq.(74) improve

Pattern Classification of Large-Scale Functional Brain Networks: Identification of Informative Neuroimaging Markers for Epilepsy

The accurate prediction of general neuropsychiatric disorders, on an individual basis, using resting-state functional magnetic resonance imaging (fMRI) is a challenging task of great clinical significance. Despite the progress to chart the differences between the healthy controls and patients at the group level, the pattern classification of functional brain networks across individuals is still less developed. In this paper we identify two novel neuroimaging measures that prove to be strongly predictive neuroimaging markers in pattern classification between healthy controls and general epileptic patients. These measures characterize two important aspects of the functional brain network in a quantitative manner: (i) coordinated operation among spatially distributed brain regions, and (ii) the asymmetry of bilaterally homologous brain regions, in terms of their global patterns of functional connectivity. This second measure offers a unique understanding of brain asymmetry at the network level, and, to the best of our knowledge, has not been previously used in pattern classification of functional brain networks. Using modern pattern-recognition approaches like sparse regression and support vector machine, we have achieved a cross-validated classification accuracy of 83.9% (specificity: 82.5%; sensitivity: 85%) across individuals from a large dataset consisting of 180 healthy controls and epileptic patients. We identified significantly changed functional pathways and subnetworks in epileptic patients that underlie the pathophysiological mechanism of the impaired cognitive functions. Specifically, we find that the asymmetry of brain operation for epileptic patients is markedly enhanced in temporal lobe and limbic system, in comparison with healthy individuals. The present study indicates that with specifically designed informative neuroimaging markers, resting-state fMRI can serve as a most promising tool for clinical diagnosis, and also shed light onto the physiology behind complex neuropsychiatric disorders. The systematic approaches we present here are expected to have wider applications in general neuropsychiatric disorders

Aging brain from a network science perspective: Something to be positive about?

To better understand age differences in brain function and behavior, the current study applied network science to model functional interactions between brain regions. We observed a shift in network topology whereby for older adults subcortical and cerebellar structures overlapping with the Salience network had more connectivity to the rest of the brain, coupled with fragmentation of large-scale cortical networks such as the Default and Fronto-Parietal networks. Additionally, greater integration of the dorsal medial thalamus and red nucleus in the Salience network was associated with greater satisfaction with life for older adults, which is consistent with theoretical predictions of age-related increases in emotion regulation that are thought to help maintain well-being and life satisfaction in late adulthood. In regard to cognitive abilities, greater ventral medial prefrontal cortex coherence with its topological neighbors in the Default Network was associated with faster processing speed. Results suggest that large-scale organizing properties of the brain differ with normal aging, and this perspective may offer novel insight into understanding age-related differences in cognitive function and well-being. © 2013 Voss et al

Resting-State Multi-Spectrum Functional Connectivity Networks for Identification of MCI Patients

In this paper, a high-dimensional pattern classification framework, based on functional associations between brain regions during resting-state, is proposed to accurately identify MCI individuals from subjects who experience normal aging. The proposed technique employs multi-spectrum networks to characterize the complex yet subtle blood oxygenation level dependent (BOLD) signal changes caused by pathological attacks. The utilization of multi-spectrum networks in identifying MCI individuals is motivated by the inherent frequency-specific properties of BOLD spectrum. It is believed that frequency specific information extracted from different spectra may delineate the complex yet subtle variations of BOLD signals more effectively. In the proposed technique, regional mean time series of each region-of-interest (ROI) is band-pass filtered ( Hz) before it is decomposed into five frequency sub-bands. Five connectivity networks are constructed, one from each frequency sub-band. Clustering coefficient of each ROI in relation to the other ROIs are extracted as features for classification. Classification accuracy was evaluated via leave-one-out cross-validation to ensure generalization of performance. The classification accuracy obtained by this approach is 86.5%, which is an increase of at least 18.9% from the conventional full-spectrum methods. A cross-validation estimation of the generalization performance shows an area of 0.863 under the receiver operating characteristic (ROC) curve, indicating good diagnostic power. It was also found that, based on the selected features, portions of the prefrontal cortex, orbitofrontal cortex, temporal lobe, and parietal lobe regions provided the most discriminant information for classification, in line with results reported in previous studies. Analysis on individual frequency sub-bands demonstrated that different sub-bands contribute differently to classification, providing extra evidence regarding frequency-specific distribution of BOLD signals. Our MCI classification framework, which allows accurate early detection of functional brain abnormalities, makes an important positive contribution to the treatment management of potential AD patients