Sturm Bounds for Siegel Modular Forms

We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g

Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms

We investigate Poincar\'e series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincar\'e series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincar\'e series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel-Poincar\'e series. We surprisingly discover that these Poincar\'e series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls

3D Raman mapping of the collagen fibril orientation in human osteonal lamellae

AbstractChemical composition and fibrillar organization are the major determinants of osteonal bone mechanics. However, prominent methodologies commonly applied to investigate mechanical properties of bone on the micro scale are usually not able to concurrently describe both factors. In this study, we used polarized Raman spectroscopy (PRS) to simultaneously analyze structural and chemical information of collagen fibrils in human osteonal bone in a single experiment. Specifically, the three-dimensional arrangement of collagen fibrils in osteonal lamellae was assessed. By analyzing the anisotropic intensity of the amide I Raman band of collagen as a function of the orientation of the incident laser polarization, different parameters related to the orientation of the collagen fibrils and the degree of alignment of the fibrils were derived. Based on the analysis of several osteons, two major fibrillar organization patterns were identified, one with a monotonic and another with a periodically changing twist direction. These results confirm earlier reported twisted and oscillating plywood arrangements, respectively. Furthermore, indicators of the degree of alignment suggested the presence of disordered collagen within the lamellar organization of the osteon. The results show the versatility of the analytical PRS approach and demonstrate its capability in providing not only compositional, but also 3D structural information in a complex hierarchically structured biological material. The concurrent assessment of chemical and structural features may contribute to a comprehensive characterization of the microstructure of bone and other collagen-based tissues

Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues

Recent experimental data revealed a stiffening of aged cortical bone tissue, which could not be explained by common multiscale elastic material models. We explain this data by incorporating the role of mineral fusion via a new hierarchical modeling approach exploiting the asymptotic (periodic) homogenization (AH) technique for three-dimensional linear elastic composites. We quantify for the first time the stiffening that is obtained by considering a fused mineral structure in a softer matrix in comparison with a composite having non-fused cubic mineral inclusions. We integrate the AH approach in the Eshelby-based hierarchical mineralized turkey leg tendon model (Tiburtius et al 2014 Biomech. Model. Mechanobiol. 13 1003–23), which can be considered as a base for musculoskeletal mineralized tissue modeling. We model the finest scale compartments, i.e. the extrafibrillar space and the mineralized collagen fibril, by replacing the self-consistent scheme with our AH approach. This way, we perform a parametric analysis at increasing mineral volume fraction, by varying the amount of mineral that is fusing in the axial and transverse tissue directions in both compartments. Our effective stiffness results are in good agreement with those reported for aged human radius and support the argument that the axial stiffening in aged bone tissue is caused by the formation of a continuous mineral foam. Moreover, the proposed theoretical and computational approach supports the design of biomimetic materials which require an overall composite stiffening without increasing the amount of the reinforcing material

Harmonic Maa{\ss}-Jacobi forms of degree 1 with higher rank indices

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.Comment: 28 page

Achieving impact from ecosystem assessment and valuation of urban greenspace: The case of i-Tree Eco in Great Britain

Numerous tools have been developed to assist environmental decision-making, but there has been little examination of whether these tools achieve this aim, particularly for urban environments. This study aimed to evaluate the use of the i-Tree Eco tool in Great Britain, an assessment tool developed to support urban forest management. The study employed a documentary review, an online survey, and interviews in six case study areas to examine five impacts (instrumental, conceptual, capacity-building, enduring connectivity, and culture/attitudes towards knowledge exchange) and to identify which factors inhibited or supported achievement of impact. It revealed that the i-Tree Eco projects had helped to increase knowledge of urban forests and awareness of the benefits they provide. While there was often broad use of i-Tree Eco findings in various internal reports, external forums, and discussions of wider policies and plans, direct changes relating to improved urban forest management, increased funding or new tree policies were less frequent. The barriers we identified which limited impact included a lack of project champions, policy drivers and resources, problems with knowledge transfer and exchange, organisational and staff change, and negative views of trees. Overall, i-Tree Eco, similar to other environmental decision-making tools, can help to improve the management of urban trees when planned as one step in a longer process of engagement with stakeholders and development of new management plans and policies. In this first published impact evaluation of multiple i-Tree Eco projects, we identified eight lessons to enhance the impact of future i-Tree Eco projects, transferable to other environmental decision-making tools

Almost holomorphic Poincar\ue9 series corresponding to products of harmonic Siegel–Maass forms

\ua9 2016, The Author(s). We investigate Poincar\ue9 series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincar\ue9 series are almost holomorphic as well. In general, this is not the case. The main point of this paper is the study of Siegel–Poincar\ue9 series of degree\ua02 attached to products of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel–Poincar\ue9 series. We surprisingly discover that these Poincar\ue9 series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls

volumetric characterisation and correlation to established classification systems

Objective and sensitive assessment of cartilage repair outcomes lacks suitable methods. This study investigated the feasibility of 3D ultrasound biomicroscopy (UBM) to quantify cartilage repair outcomes volumetrically and their correlation with established classification systems. 32 sheep underwent bilateral treatment of a focal cartilage defect. One or two years post- operatively the repair outcomes were assessed and scored macroscopically (Outerbridge, ICRS-CRA), by magnetic resonance imaging (MRI, MOCART), and histopathology (O'Driscoll, ICRS-I and ICRS-II). The UBM data were acquired after MRI and used to reconstruct the shape of the initial cartilage layer, enabling the estimation of the initial cartilage thickness and defect volume as well as volumetric parameters for defect filling, repair tissue, bone loss and bone overgrowth. The quantification of the repair outcomes revealed high variations in the initial thickness of the cartilage layer, indicating the need for cartilage thickness estimation before creating a defect. Furthermore, highly significant correlations were found for the defect filling estimated from UBM to the established classification systems. 3D visualisation of the repair regions showed highly variable morphology within single samples. This raises the question as to whether macroscopic, MRI and histopathological scoring provide sufficient reliability. The biases of the individual methods will be discussed within this context. UBM was shown to be a feasible tool to evaluate cartilage repair outcomes, whereby the most important objective parameter is the defect filling. Translation of UBM into arthroscopic or transcutaneous ultrasound examinations would allow non-destructive and objective follow-up of individual patients and better comparison between the results of clinical trials