Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters

We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,\rho)$, where $W$ is a compact closed oriented 3-manifold, $\rho$ is a flat principal bundle over $W$ with structural group PSL(2,\mc), and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms}, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely computed as state sums over the decorated hyperbolic ideal tetrahedra of the {\it idealization} of any fixed {\it \Dd-triangulation}; the \Dd-triangulations are simplicial 1-cocycle descriptions of $(W,\rho)$ in which the link is realized as a Hamiltonian subcomplex. We also discuss how to set the Volume Conjecture for the coloured Jones invariants $J_N(L)$ of hyperbolic knots $L$ in $S^3$ in the framework of the general QHI theory.Comment: 49 pages, 17 figures. Together with our paper `Classical And Quantum Dilogarithmic Invariants Of Flat PSL(2,C)-Bundles Over 3-Manifolds' avalaible on the same ArXiv, this develops with full details the results announced in math.GT/021106

Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations

An outstanding challenge for models of non-perturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry. In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the Causal Dynamical Triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravity-matter models in a high- and low-temperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12, and analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart from providing evidence for a simplification of the model's analytic structure due to the dynamical nature of the geometry, the technique introduced can shed further light on criteria \`a la Harris and Luck for the influence of random geometry on the critical properties of matter systems.Comment: 40 pages, 15 figures, 13 table

Asymptotic safety in higher-derivative gravity

We study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques. The non-perturbative contributions to the $\beta$-functions shift the known perturbative ultraviolet fixed point into a non-trivial fixed point with three UV-attractive and one UV-repulsive eigendirections, consistent with the asymptotic safety conjecture of gravity. The implication of this transition on the unitarity problem, typically haunting higher-derivative gravity theories, is discussed.Comment: 8 pages; 1 figure; revised versio

Effectiveness of continence promotion for older women via community organisations: A cluster randomised trial

This is an Open Access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 3.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/3.0/Objectives: The primary objective of this cluster randomised controlled trial was to compare the effectiveness of the three experimental continence promotion interventions against a control intervention on urinary symptom improvement in older women with untreated incontinence recruited from community organisations. A second objective was to determine whether changes in incontinence-related knowledge and new uptake of risk-modifying behaviours explain these improvements. Setting: 71 community organisations across the UK. Participants: 259 women aged 60 years and older with untreated incontinence entered the trial; 88% completed the 3-month follow-up. Interventions: The three active interventions consisted of a single 60 min group workshop on (1) continence education (20 clusters, 64 women); (2) evidence-based self-management (17 clusters, 70 women); or (3) combined continence education and self-management (17 clusters, 61 women). The control intervention was a single 60 min educational group workshop on memory loss, polypharmacy and osteoporosis (17 clusters, 64 women). Primary and secondary outcome measures: The primary outcome was self-reported improvement in incontinence 3 months postintervention at the level of the individual. The secondary outcome was change in the International Consultation on Incontinence Questionnaire (ICIQ) from baseline to 3-month follow-up. Changes in incontinence-related knowledge and behaviours were also assessed. Results: The highest rate of urinary symptom improvement occurred in the combined intervention group (66% vs 11% of the control group, prevalence difference 55%, 95% CI 43% to 67%, intracluster correlation 0). 30% versus 6% of participants reported significant improvement respectively (prevalence difference 23%, 95% CI 10% to 36%, intracluster correlation 0). The number-needed-to-treat was 2 to achieve any improvement in incontinence symptoms, and 5 to attain significant improvement. Compared to controls, participants in the combined intervention reported an adjusted mean 2.05 point (95% CI 0.87 to 3.24) greater improvement on the ICIQ from baseline to 3-month follow-up. Changes in knowledge and self-reported risk-reduction behaviours paralleled rates of improvement in all intervention arms. Conclusions: Continence education combined with evidence-based self-management improves symptoms of incontinence among untreated older women. Community organisations represent an untapped vector for delivering effective continence promotion interventions.Canadian Institutes of Health Research – Institute on Aging and the Economic and Social Research Council (UK

Self-reported gait unsteadiness in mildly impaired neurological patients: an objective assessment through statistical gait analysis

Background Self-reported gait unsteadiness is often a problem in neurological patients without any clinical evidence of ataxia, because it leads to reduced activity and limitations in function. However, in the literature there are only a few papers that address this disorder. The aim of this study is to identify objectively subclinical abnormal gait strategies in these patients. Methods Eleven patients affected by self-reported unsteadiness during gait (4 TBI and 7 MS) and ten healthy subjects underwent gait analysis while walking back and forth on a 15-m long corridor. Time-distance parameters, ankle sagittal motion, and muscular activity during gait were acquired by a wearable gait analysis system (Step32, DemItalia, Italy) on a high number of successive strides in the same walk and statistically processed. Both self-selected gait speed and high speed were tested under relatively unconstrained conditions. Non-parametric statistical analysis (Mann-Whitney, Wilcoxon tests) was carried out on the means of the data of the two examined groups. Results The main findings, with data adjusted for velocity of progression, show that increased double support and reduced velocity of progression are the main parameters to discriminate patients with self-reported unsteadiness from healthy controls. Muscular intervals of activation showed a significant increase in the activity duration of the Rectus Femoris and Tibialis Anterior in patients with respect to the control group at high speed. Conclusions Patients with a subjective sensation of instability, not clinically documented, walk with altered strategies, especially at high gait speed. This is thought to depend on the mechanisms of postural control and coordination. The gait anomalies detected might explain the symptoms reported by the patients and allow for a more focused treatment design. The wearable gait analysis system used for long distance statistical walking assessment was able to detect subtle differences in functional performance monitoring, otherwise not detectable by common clinical examination

Alexander quandle lower bounds for link genera

We denote by Q_F the family of the Alexander quandle structures supported by finite fields. For every k-component oriented link L, every partition P of L into h:=|P| sublinks, and every labelling z of such a partition by the natural numbers z_1,...,z_n, the number of X-colorings of any diagram of (L,z) is a well-defined invariant of (L,P), of the form q^(a_X(L,P,z)+1) for some natural number a_X(L,P,z). Letting X and z vary in Q_F and among the labellings of P, we define a derived invariant A_Q(L,P)=sup a_X(L,P,z). If P_M is such that |P_M|=k, we show that A_Q(L,P_M) is a lower bound for t(L), where t(L) is the tunnel number of L. If P is a "boundary partition" of L and g(L,P) denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the L_j's, then we show that A_Q(L,P) is at most 2g(L,P)+2k-|P|-1. We set A_Q(L):=A_Q(L,P_m), where |P_m|=1. By elaborating on a suitable version of a result by Inoue, we show that when L=K is a knot then A_Q(K) is bounded above by A(K), where A(K) is the breadth of the Alexander polynomial of K. However, for every g we exhibit examples of genus-g knots having the same Alexander polynomial but different quandle invariants A_Q. Moreover, in such examples A_Q provides sharp lower bounds for the genera of the knots. On the other hand, A_Q(L) can give better lower bounds on the genus than A(L), when L has at least two components. We show that in order to compute A_Q(L) it is enough to consider only colorings with respect to the constant labelling z=1. In the case when L=K is a knot, if either A_Q(K)=A(K) or A_Q(K) provides a sharp lower bound for the knot genus, or if A_Q(K)=1, then A_Q(K) can be realized by means of the proper subfamily of quandles X=(F_p,*), where p varies among the odd prime numbers.Comment: 36 pages; 16 figure

Classical Teichmuller theory and (2+1) gravity

We consider classical Teichmuller theory and the geodesic flow on the cotangent bundle of the Teichmuller space. We show that the corresponding orbits provide a canonical description of certain (2+1) gravity systems in which a set of point-like particles evolve in universes with topology S_g x R where S_g is a Riemann surface of genus g >1. We construct an explicit York's slicing presentation of the associated spacetimes, we give an interpretation of the asymptotic states in terms of measured foliations and discuss the structure of the phase spaces

On simultaneous diagonalization of one Hermitian and one symmetric form

AbstractIt is remarked that if A, B ϵ Mn(C), A = At, and B̄ = Bt, B positive definite, there exists a nonsingular matrix U such that (1) ŪtBU = I and (2) UtAU is a diagonal matrix with nonnegative entries. Some related actions of the real orthogonal group and equations involving the unitary group are studied