Finite Generation Research Articles

Parameter estimation in branching processes with almost sure extinction

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is z≥1. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation n and let n→∞; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property (C-consistency). This leads us to define the concept of Q-consistency, and we prove that our estimators are Q-consistent and asymptotically normal. To investigate the circumstances in which a C-consistent estimator is preferable to a Q-consistent estimator, we then provide two C-consistent estimators for subcritical Galton–Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.

Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras, we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties, we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper, we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem, we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.

Cohomology of finite tensor categories: Duality and Drinfeld centers

We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.

Finite generation of the cohomology rings for the extension algebras of monomial algebras

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring [Formula: see text] is finitely generated [Formula: see text] is [Formula: see text] is, where [Formula: see text] is the dual extension of a monomial algebra A and [Formula: see text] is the opposite algebra of A.

Large Deviation Rates for the Continuous‐Time Supercritical Branching Processes with Immigration

Let {Y(t); t ≥ 0} be a supercritical continuous‐time branching process with immigration; our focus is on the large deviation rates of Y(t) and thus extending the results of the discrete‐time Galton–Watson process to the continuous‐time case. Firstly, we prove that Z(t) = e−mt[Y(t) − ((em(t + 1) − 1)/(em − 1))ea+m] is a submartingale and converges to a random variable Z. Then, we study the decay rates of P(|Z(t) − Z| > ε) as t⟶∞ and P(|(Y(t + v)/Y(t)) − emv| > ε|Z ≥ α) as t⟶∞ for α > 0 and ε > 0 under various moment conditions on {bk; k ≥ 0} and {aj; j ≥ 0}. We conclude that the rates are supergeometric under the assumption of finite moment generation functions.

Additive decompositions for rings of modular forms

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We apply this to characterize which rings of modular forms are Cohen--Macaulay and to prove finite generation results. These theorems are based on decomposition results about vector bundles on the compactified moduli stack of elliptic curves.

Biomechanical analysis of the cervical spine segment as a method for studying the functional and dynamic anatomy of the human neck

BackgroundTraditionally, dynamic and functional anatomy, in particular the dynamic anatomy of the neck, is studied on cadaveric material. However, the development of in vivo visualization technologies and in silico modeling has made it possible to expand these possibilities. Despite significant progress in the study of dynamic and functional anatomy of the neck by means of in silico methods, the issues of validating the developed models and taking into account the pronounced nonlinearity of soft tissues as well as local anisotropy remain open. The aim of this study was to develop a virtual dynamic anatomical model of the human neck and reproduce the dynamic processes in the cervical spine from this model using the finite element method. Materials and methodsReverse engineering was used to generate a dynamic anatomical model of the neck from CT data (both male, 24 and 22 years old). Two segments of the cervical spine (C3-C5, C2-T1) were isolated from the resulting model for finite element analysis. Finite element mesh generation and contact interactions were performed using the HyperMesh software (Altair Engineering Inc, Troy, Michigan, USA). The anisotropic hyperelastic Holzapfel-Gasser-Ogden model was used to describe the material behavior of the fibrous rings of the disc. Material modeling and finite element analysis were performed using Abaqus CAE 6.14 software (Simulia, Johnston, Rhode Island, USA). ResultsA technique for creating a virtual dynamic anatomical model of the neck was elaborated and implemented. The model includes 79 major anatomical structures of the neck segmented from radiological data. A finite element analysis of the cervical spine was performed. The results of finite element analysis of the C3-C5 segment under axial load were compared with in vitro data. The proposed model shows nonlinear deformation of the disc under static loading; the model predicted displacement values agree well with the experimental ones. The displacement of the С3-С5 central vertebra with an axial load of 800 N reaches a value of 0.65 mm. For the segment C2-T1, data on intradiscal pressure, stress plots and displacements during flexion were obtained. The maximum stress value of 10.036 MPa is observed in the C3-C4 disc. ConclusionSimulation results using the proposed methodology are in good agreement with experimental data. The generated biomechanical models allow describing dynamic phenomena in the cervical spine and obtaining a wide range of quantitative properties of anatomical objects, which are otherwise inaccessible to classical methods for studying dynamic and functional anatomy.

Analysis and Testing of the Dynamic Characteristics of Rotor Worktable

The finite element modelling, mesh generation and modal analysis of the rotor worktable are carried out by combining the analysis technology of finite element software in this paper. Then, the software analysis results and the actual experimental results are compared and analyzed, so as to get the causes of error, which can provide good basic data for the use of the rotor table in the future that could better meet the needs of scientific research and teaching.

Ground view factor computation model for bifacial photovoltaic collector field: uniform and non-uniform surfaces

The photovoltaic collectors in a field are subject to three types of solar radiation: direct, diffuse, and reflected irradiance. The reflected irradiance received by solar photovoltaic (PV) depends on the view factor from solar PV to the ground. This view factor component is dominant for bifacial PV due to additional reflected irradiance gain, which can be achieved from the module’s rear side.This paper proposes and verifies a finite element method based view factor computation model, which can handle both uniform and non-uniform ground surfaces. The unique contribution of this work is that it introduces a geometric progression based finite element mesh generation process that forms the quasi-uniform grid. The generated grid values are fitted into the computation model to calculate the view factor from bifacial photovoltaics to the ground, known as the ground view factor (GVF). The proposed computation model can achieve an accuracy of 99%. To keep accuracy at this level, the smallest element size of the coarse mesh should be within 0.1%–0.4% of surface width or length. Moreover, the geometric progression ratio of the fine and coarse mesh should be in the range of 1.001–1.002 and 1.01–1.04, respectively. The model is analysed under six different PV field variables: multiple reflective ground surfaces, the height of PV, tilted ground surface, PV position in the ground, length and width of the ground, and PV string length. For the different string sizes considered here, the view factor model’s computation time varies from 180 s to 257 min for the iteration size of 7.67 billion to 765 billion. The view factor computation model will contribute to analyse reflected irradiance at the rear side of bifacial PV, which is essential to predict the energy generation accurately. The proposed model is also beneficial for urban planning and addressing heat gain of the building-integrated PV system and energy usage.

On finite generation of the Johnson filtrations

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally proved for the second terms by Ershov and He.

Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases.We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,…,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety (Remark 2.7).

New approaches to finite generation of cohomology rings

In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules.In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A, we show that the finite generation condition on A-modules can be replaced by a condition on any affine commutative A-module algebra R under the assumption that R is integral over its invariant subring RA. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A′ over a finite field, where A can be viewed as a deformation of A′, and prove that if the finite generation condition holds for A′, then the same condition holds for A.

On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

A high-fidelity crystal-plasticity finite element methodology for low-cycle fatigue using automatic electron backscatter diffraction scan conversion: Application to hot-rolled cobalt–chromium alloy

The common approach to crystal-plasticity finite element modeling for load-bearing prediction of metallic structures involves the simulation of simplified grain morphology and substructure detail. This paper details a methodology for predicting the structure–property effect of as-manufactured microstructure, including true grain morphology and orientation, on cyclic plasticity, and fatigue crack initiation in biomedical-grade CoCr alloy. The methodology generates high-fidelity crystal-plasticity finite element models, by directly converting measured electron backscatter diffraction metal microstructure grain maps into finite element microstructural models, and thus captures essential grain definition for improved microstructure–property analyses. This electron backscatter diffraction-based method for crystal-plasticity finite element model generation is shown to give approximately 10% improved agreement for fatigue life prediction, compared with the more commonly used Voronoi tessellation method. However, the added microstructural detail available in electron backscatter diffraction–crystal-plasticity finite element did not significantly alter the bulk stress–strain response prediction, compared to Voronoi tessellation–crystal-plasticity finite element. The new electron backscatter diffraction-based method within a strain-gradient crystal-plasticity finite element model is also applied to predict measured grain size effects for cyclic plasticity and fatigue crack initiation, and shows the concentration of geometrically necessary dislocations around true grain boundaries, with smaller grain samples exhibiting higher overall geometrically necessary dislocations concentrations. In addition, minimum model sizes for Voronoi tessellation–crystal-plasticity finite element and electron backscatter diffraction–crystal-plasticity finite element models are proposed for cyclic hysteresis and fatigue crack initiation prediction.

CRIMSON: An open-source software framework for cardiovascular integrated modelling and simulation.

In this work, we describe the CRIMSON (CardiovasculaR Integrated Modelling and SimulatiON) software environment. CRIMSON provides a powerful, customizable and user-friendly system for performing three-dimensional and reduced-order computational haemodynamics studies via a pipeline which involves: 1) segmenting vascular structures from medical images; 2) constructing analytic arterial and venous geometric models; 3) performing finite element mesh generation; 4) designing, and 5) applying boundary conditions; 6) running incompressible Navier-Stokes simulations of blood flow with fluid-structure interaction capabilities; and 7) post-processing and visualizing the results, including velocity, pressure and wall shear stress fields. A key aim of CRIMSON is to create a software environment that makes powerful computational haemodynamics tools accessible to a wide audience, including clinicians and students, both within our research laboratories and throughout the community. The overall philosophy is to leverage best-in-class open source standards for medical image processing, parallel flow computation, geometric solid modelling, data assimilation, and mesh generation. It is actively used by researchers in Europe, North and South America, Asia, and Australia. It has been applied to numerous clinical problems; we illustrate applications of CRIMSON to real-world problems using examples ranging from pre-operative surgical planning to medical device design optimization.

3DGShield: A new 3D gamma ray shielding code for arbitrary source and shield geometry based on point kernel technique

3DGShield, A new 3D gamma ray shielding code, based on the method of point kernel technique, has been developed for arbitrary regular & irregular shaped source and shield geometries. The 3D geometry modeling and its mesh generation are performed using Gmsh, a free open source finite element mesh generation software. The output of the Gmsh software is in Gmsh MSH (.msh) file format which will be used as an input of geometry description of our code. The salient features of 3DGShield are (i) ability to handle any type of complex source geometries including regular source shapes like point, line, disc, box, cylinder, sphere, hexagonal cylinder, cone etc. and irregular source shapes like pieces of broken glass, piece of brick etc., (ii) handling discrete as well as distributed photon energy sources, (iii) modeling any kind of regular and irregular complex multilayered shield geometries, (iv) having provision to export the output results into ParaView, an open source data analysis and visualization software, for quick visualization and analyzing of the results. Buildup factor is the correction factor which takes account the contribution of scattered photon. Various appropriate multilayer build-up factor formulas have been incorporated in our code. Out of these, the most popular methods are Blizard's method, Goldstein method of an effective atomic number and Broders's method. Verification and validation of the code have been done by a number of benchmark problems available in literature. The results are presented in this paper which demonstrate the accuracy and efficiency of the code in complex geometries. This code is written in Python-2.7 language and compatible both in Windows and Linux operating system.

Finite generation of cohomology for Drinfeld doubles of finite group schemes

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual to \({{\,\mathrm{rep}\,}}(G)\) are also of finite type (i.e. have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions. This paper completes earlier work of Friedlander and the author.

Modules over some group rings, having d-generator property

For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.

A Finite Element Model of the C1-C4 Human Spine Segment

A three-dimensional FE analysis of four adjacent cervical vertebrae (C1-C4) has been developed. The model is composed of vertebrae and skull but disc ligaments were not considered. The aim of this research is to address the new advancements in the finite element mesh generation of the human cervical spine. The procedure for creating a finite element model based on available biomedical closed volumes obtained from CT scans is introduced.

Automatic Finite Element Mesh Generation on 3D Surfaces Based on STEP

Computer Aided Engineering (CAE) have a vital value in die and mould manufacturing. For the purpose of enhancing the modelling ability of the existing CAE analysis system and improving the mesh generation quality and working efficiency, a system for automatic finite element mesh generation on 3D surfaces is designed and implemented. Applying the neutral mechanism for product information sharing provided by STEP, product model import and reconstruction process is first implemented. And then product model information extraction module is developed aims at preparing geometrical and topological information for finite element mesh generation. An improved strategy for automatic triangular mesh generation over 3D surface is adopted to realize the mesh generation of the system. Implemented results indicate that the presented system helps to improve the mesh quality and at the same time increase the process efficiency and reduce the time cost.