Distribution of Modular Symbols in ℍ3

We introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if K is a quadratic imaginary number field of class number one and OK is its ring of integers, then for certain congruence subgroups of PSL2(OK)⁠, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and second moments of the distribution

Supersymmetric Canonical Commutation Relations

We present unitarily represented supersymmetric canonical commutation relations which are subsequently used to canonically quantize massive and massless chiral,antichiral and vector fields. The massless fields, especially the vector one, show new facets which do not appear in the non superymmetric case. Our tool is the supersymmetric positivity induced by the Hilbert-Krein structure of the superspace.Comment: 14 page

Supersymmetric Distributions, Hilbert Spaces of Supersymmetric Functions and Quantum Fields

The recently investigated Hilbert-Krein and other positivity structures of the superspace are considered in the framework of superdistributions. These tools are applied to problems raised by the rigorous supersymmetric quantum field theory.Comment: 24 page

Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic nn-space

We provide a new and simple automorphic method using Eisenstein series tostudy the equidistribution of modular symbols modulo primes, which we apply toprove an average version of a conjecture of Mazur and Rubin. More precisely, weprove that modular symbols corresponding to a Hecke basis of weight 2 cuspforms are asymptotically jointly equidistributed mod $p$ while we allowrestrictions on the location of the cusps. As an application, we obtain aresidual equidistribution result for Dedekind sums. Furthermore, we calculatethe variance of the distribution and show a surprising bias with connections toperturbation theory. Additionally, we prove the full conjecture in someparticular cases using a connection to Eisenstein congruences. Finally, ourmethods generalise to equidistribution results for cohomology classes of finitevolume quotients of $n$-dimensional hyperbolic space.<br

Van der Waerden calculus with commuting spinor variables and the Hilbert-Krein structure of the superspace

Working with anticommuting Weyl(or Mayorana) spinors in the framework of the van der Waerden calculus is standard in supersymmetry. The natural frame for rigorous supersymmetric quantum field theory makes use of operator-valued superdistributions defined on supersymmetric test functions. In turn this makes necessary a van der Waerden calculus in which the Grassmann variables anticommute but the fermionic components are commutative instead of being anticommutative. We work out such a calculus in view of applications to the rigorous conceptual problems of the N=1 supersymmetric quantum field theory.Comment: 14 page

Natural Metric for Quantum Information Theory

We study in detail a very natural metric for quantum states. This new proposal has two basic ingredients: entropy and purification. The metric for two mixed states is defined as the square root of the entropy of the average of representative purifications of those states. Some basic properties are analyzed and its relation with other distances is investigated. As an illustrative application, the proposed metric is evaluated for 1-qubit mixed states.Comment: v2: enlarged; presented at ISIT 2008 (Toronto

Unstable states in QED of strong magnetic fields

We question the use of stable asymptotic scattering states in QED of strong magnetic fields. To correctly describe excited Landau states and photons above the pair creation threshold the asymptotic fields are chosen as generalized Licht fields. In this way the off-shell behavior of unstable particles is automatically taken into account, and the resonant divergences that occur in scattering cross sections in the presence of a strong external magnetic field are avoided. While in a limiting case the conventional electron propagator with Breit-Wigner form is obtained, in this formalism it is also possible to calculate $S$-matrix elements with external unstable particles.Comment: Revtex, 7 pages. To appear in Phys. Rev. D53(2

Graph Theoretic Analysis of Brain Connectomics in Multiple Sclerosis: Reliability and Relationship to Cognition

Research suggests that disruption of brain networks might explain cognitive deficits in multiple sclerosis (MS). The reliability and effectiveness of graph-theoretic network metrics as measures of cognitive performance were tested in 37 people with MS and 23 controls. Specifically, relationships to cognitive performance (linear regression against the Paced Auditory Serial Addition Test [PASAT-3], Symbol Digit Modalities Test [SDMT] and Attention Network Test [ANT]) and one-month reliability (using the intra-class correlation coefficient [ICC]) of network metrics were measured using both resting-state functional and diffusion MRI data. Cognitive impairment was directly related to measures of brain network segregation and inversely related to network integration (prediction of PASAT-3 by small-worldness, modularity, characteristic path length, R2=0.55; prediction of SDMT by small-worldness, global efficiency and characteristic path length, R2=0.60). Reliability of the measures over one month in a subset of 9 participants was mostly rated as good (ICC>0.6) for both controls and MS patients in both functional and diffusion data but was highly dependent on the chosen parcellation and graph density, with the 0.2-0.5 density range being the most reliable. This suggests that disrupted network organisation predicts cognitive impairment in MS and its measurement is reliable over a 1-month period. These new findings support the hypothesis of network disruption as a major determinant of cognitive deficits in MS and the future possibility of the application of derived metrics as surrogate outcomes in trials of therapies for cognitive impairment

Constrained Statistical Modelling of Knee Flexion from Multi-Pose Magnetic Resonance Imaging

© 1982-2012 IEEE.Reconstruction of the anterior cruciate ligament (ACL) through arthroscopy is one of the most common procedures in orthopaedics. It requires accurate alignment and drilling of the tibial and femoral tunnels through which the ligament graft is attached. Although commercial computer-Assisted navigation systems exist to guide the placement of these tunnels, most of them are limited to a fixed pose without due consideration of dynamic factors involved in different knee flexion angles. This paper presents a new model for intraoperative guidance of arthroscopic ACL reconstruction with reduced error particularly in the ligament attachment area. The method uses 3D preoperative data at different flexion angles to build a subject-specific statistical model of knee pose. To circumvent the problem of limited training samples and ensure physically meaningful pose instantiation, homogeneous transformations between different poses and local-deformation finite element modelling are used to enlarge the training set. Subsequently, an anatomical geodesic flexion analysis is performed to extract the subject-specific flexion characteristics. The advantages of the method were also tested by detailed comparison to standard Principal Component Analysis (PCA), nonlinear PCA without training set enlargement, and other state-of-The-Art articulated joint modelling methods. The method yielded sub-millimetre accuracy, demonstrating its potential clinical value