Modular classes of Poisson-Nijenhuis Lie algebroids

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.Comment: To appear in Letters in Mathematical Physic

On localization in holomorphic equivariant cohomology

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.Comment: 16 pages. Completely rewritten, new title. v3: Minor changes in the exposition. v4: final version to appear in Centr. Eur. J. Mat

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e. mod(E)=0. Further, relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and that of a Poisson map.Comment: 20 page

Poisson sigma model on the sphere

We evaluate the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kahler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the full quantum result.Comment: 38 page

Codimension one symplectic foliations and regular Poisson structures

Original manuscript June 21, 2011In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9]

Regularization as quantization in reducible representations of CCR

A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincar\'e covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to R\'enyi's $\alpha$. It is shown that the ``Shannon limit" $\alpha\to 1$ plays here a role of correspondence principle with the standard regularized formalism.Comment: minor extensions, version submitted for publicatio

‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal

Automatic regularization by quantization in reducible representations of CCR: Point-form quantum optics with classical sources

Electromagnetic fields are quantized in manifestly covariant way by means of a class of reducible representations of CCR. $A_a(x)$ transforms as a Hermitian four-vector field in Minkowski four-position space (no change of gauge), but in momentum space it splits into spin-1 massless photons (optics) and two massless scalars (similar to dark matter). Unitary dynamics is given by point-form interaction picture, with minimal-coupling Hamiltonian constructed from fields that are free on the null-cone boundary of the Milne universe. SL(2,C) transformations and dynamics are represented unitarily in positive-norm Hilbert space describing $N$ four-dimensional oscillators. Vacuum is a Bose-Einstein condensate of the $N$-oscillator gas. Both the form of $A_a(x)$ and its transformation properties are determined by an analogue of the twistor equation. The same equation guarantees that the subspace of vacuum states is, as a whole, Poincar\'e invariant. The formalism is tested on quantum fields produced by pointlike classical sources. Photon statistics is well defined even for pointlike charges, with UV/IR regularizations occurring automatically as a consequence of the formalism. The probabilities are not Poissonian but of a R\'enyi type with $\alpha=1-1/N$. The average number of photons occurring in Bremsstrahlung splits into two parts: The one due to acceleration, and the one that remains nonzero even if motion is inertial. Classical Maxwell electrodynamics is reconstructed from coherent-state averaged solutions of Heisenberg equations. Static pointlike charges polarize vacuum and produce effective charge densities and fields whose form is sensitive to both the choice of representation of CCR and the corresponding vacuum state.Comment: 2 eps figures; in v2 notation in Eq. (39) and above Eq. (38) is correcte

Opportunities, challenges and systems requirements for developing post-abortion family planning services: Perceptions of service stakeholders in China

Post-abortion family planning (PAFP) has been proposed as a key strategy to decrease unintended pregnancy and repeat induced abortions. However, the accessibility and quality of PAFP services remain a challenge in many countries including China where more than 10 million unintended pregnancies occur each year. Most of these unwanted pregnancies end in repeated induced abortions. This paper aims to explore service providers’ perceptions of the current situation regarding family planning and abortion service needs, provision, utilization, and the feasibility and acceptability of high quality PAFP in the future. Qualitative methods, including in-depth interviews and focus group discussions, were used with family planning policy makers, health managers, and service providers. Three provinces—Zhejiang, Hubei and Yunnan—were purposively selected, representing high, medium and relatively undeveloped areas of China. A total of fifty-three in-depth interviews and ten focus-group discussions were conducted and analysed thematically. Increased numbers of abortions among young, unmarried women were perceived as a major reason for high numbers of abortions. Participants attributed this to increasing socio-cultural acceptability of premarital sex, and simultaneously, lack of understanding or awareness of contraception among young people. The majority of service stakeholders acknowledged that free family planning services were neither targeted at, nor accessible to unmarried people. The extent of PAFP provision is variable and limited. However, service providers expressed willingness and enthusiasm towards providing PAFP services in the future. Three main considerations were expressed regarding the feasibility of developing and implementing PAFP services: policy support, human resources, and financial resources. The study indicated that key service stakeholders show demand for and perceive considerable opportunities to develop PAFP in China. However, changes are needed to enable the systematic development of high quality PAFP, including actively targeting young and unmarried people in service provision, obtaining policy support and increasing the investment of human and financial resources

Carotenoid Distribution in Living Cells of Haematococcus pluvialis (Chlorophyceae)

Haematococcus pluvialis is a freshwater unicellular green microalga belonging to the class Chlorophyceae and is of commercial interest for its ability to accumulate massive amounts of the red ketocarotenoid astaxanthin (3,3′-dihydroxy-β,β-carotene-4,4′-dione). Using confocal Raman microscopy and multivariate analysis, we demonstrate the ability to spectrally resolve resonance–enhanced Raman signatures associated with astaxanthin and β-carotene along with chlorophyll fluorescence. By mathematically isolating these spectral signatures, in turn, it is possible to locate these species independent of each other in living cells of H. pluvialis in various stages of the life cycle. Chlorophyll emission was found only in the chloroplast whereas astaxanthin was identified within globular and punctate regions of the cytoplasmic space. Moreover, we found evidence for β-carotene to be co-located with both the chloroplast and astaxanthin in the cytosol. These observations imply that β-carotene is a precursor for astaxanthin and the synthesis of astaxanthin occurs outside the chloroplast. Our work demonstrates the broad utility of confocal Raman microscopy to resolve spectral signatures of highly similar chromophores in living cells