Finite Generation Research Articles

Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings

Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings

Finite generation for Hochschild cohomology of Gorenstein monomial algebras

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall–Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann–Holloway–Snashall–Solberg–Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

Some necessary and sufficient condition for finite generation of symbolic Rees rings

Consider the blow-up Y of a weighted projective plane at a point in the open orbit over a field of characteristic 0. We assume that there exists a curve C on Y such that C2<0 and C.E=1, where E is the exceptional curve.In this paper we give a (very simple) necessary and sufficient condition for finite generation of the Cox ring of Y (Theorem 1.2). It is an affirmative answer to a conjecture due to He and Kurano-Nishida.

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras.

Simulation of structure and power generation for Self-Compacting concrete hollow slab solar pavement with micro photovoltaic array

To explore new solar pavements, a self-compacting concrete hollow slab solar pavement based on a micro photovoltaic array was proposed. The hollow slab solar pavement is composed of three layers: a surface transparent protection slab, a middle micro photovoltaic array, and a bottom concrete base slab. Using ANSYS software and PVsyst software, through the 3D finite element numerical simulation and power generation simulation of the pavement structure, based on orthogonal experimental design and single-factor sensitivity analysis, the structure size of the hollow slab and the layout mode of the micro photovoltaic array were optimized. Then, the optimal hollow slab model was determined. The results show that the optimal size of the hollow slab is 1000 × 1000 × 250 mm; the tilt, azimuth, pitch and edge distance of the solar cells in the best layout mode of the micro photovoltaic array are 18°, 0°, 45 mm and 65 mm, respectively; the power generation capacity in the first year is 140.697 kWh. The life cycle economic analysis indicates that the increased cost can be recovered through the power generation income after 10.2 years, and 1 km of two-way four-lane pavement could reduce by 2.15 × 107 kg of CO2 emissions, with good economic, social, and environmental benefits.

Finite generation for valuations computing stability thresholds and applications to K-stability

We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.

Finite Generation of Cox Rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

2D Digital Reconstruction of Asphalt Concrete Microstructure for Numerical Modeling Purposes.

In this paper, we deal with the issue of asphalt concrete microstructure recognition for further numerical analysis. An efficient reconstruction of the underlying microstructure makes the composite analysis more reliable. We propose for this purpose a methodology based on the image processing and focus on a two-dimensional case (it can be easily used as a part of the 3D geometry reconstruction, however). Initially obtained geometry of the inclusions is further simplified to reduce the cost of the finite element mesh generation. Three straightforward geometry simplification algorithms are used to perform this process in a controlled way. Subsequently, we present the solutions of two problems, i.e., heat flow and elasticity (plane strain), in order to illustrate the effectiveness of the whole elaborated methodology. The numerical results were obtained using the finite element method. Consequently, an error analysis is demonstrated in order to refer the overkill mesh solutions to the ones presented in this study. The main finding of this paper is the efficient methodology dedicated to a digital reconstruction of the asphalt concrete microstructure by the image processing. It can be also extended to other materials exhibiting similar microstructure.

ReMo3D – an open-source Python package for 2D and 3D simulation of normal and lateral resistivity logs

An open-source Python package is presented, ReMo3D, which allows the generation of synthetic normal and lateral resistivity logs for 2D and 3D models. The package is built around a finite element mesh generator Gmsh and a high-performance multiphysics finite element software Netgen/NGSolve and supports distributed-memory parallel computation. The examples included in the paper show that the developed software can accurately simulate the measurement process and produce detailed synthetic normal and lateral resistivity logs. In addition, basic information about normal and lateral tools such as tool configurations, measurement principles, nomenclature and a brief history of utilization is included in the paper.

Rejuvenation improves the quality of Trichogramma brassicae Bezdenko (Hymenoptera: Trichogrammatidae) reared for many generations on Sitotroga cerealella (Olivier) (Lepidoptera: Gelechiidae)

BackgroundDomestication usually has negative effects on insect performance, especially when they are reared continuously for many generations. Rejuvenation can reduce the negative effects of domestication in the parasitoid, Trichogramma brassicae Bezdenko (Hymenoptera: Trichogrammatidae) reared for 30 generations (G) on Sitotroga cerealella (Olivier) (Lepidoptera: Gelechiidae). Life table and parasitism parameters in the primary colony (G31–G45) were compared with the respective colony rejuvenated with field-collected individuals.ResultsThere were significant differences in life history traits between domesticated and rejuvenated populations. The highest (38.4 eggs/female) and lowest (13.3 eggs/female) fecundity was observed in rejuvenated G31 and domesticated G45, respectively. The highest values of the life table and parasitism parameters were observed in rejuvenated G31 without significant difference with G32, G33 and G34. The relationship between finite parasitism rate (ω) and generation number in both rejuvenated and domesticated populations were fitted to the cubic regression model, indicating the parasitism potential of T. brassicae increased significantly after adding feral individuals but decreased remarkably over the generations.ConclusionsTherefore, it seems that adding 10% feral individuals re-established the reproductive performance of the wasps at least for four generations, and it needs to be repeated routinely or it needs to be made by adding a higher rate of feral individuals. However, this issue should be investigated by more studies in which different rates of feral individuals are added to the primary colony.

Tight bounds for the dihedral angle sums of a pyramid

We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval (3π, 5π). Moreover, for any number in (3π, 5π) there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound 4π is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation and analysis.

Some relational structures with polynomial growth and their associated algebras II. Finite generation.

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.

Finite Generation of Lie Derived Powers of Skew Lie Algebras

Let [Formula: see text] be a finitely generated associative algebra over a field of characteristic different from 2. Herstein asked when the Lie algebra [Formula: see text] is finitely generated. Recently, it was shown that for a finitely generated nil algebra [Formula: see text] all derived powers of [Formula: see text] are finitely generated Lie algebras. Let [Formula: see text] be the Lie algebra of skew-symmetric elements of an associative algebra with involution. We consider all derived powers of the Lie algebra [Formula: see text] and prove that for any finitely generated associative nil algebra with an involution, all derived powers of [Formula: see text] are finitely generated Lie algebras.

Essential finite generation of valuation rings in characteristic zero algebraic function fields

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega}$ of $\omega$ is essentially finitely generated over the valuation ring $V_{\nu}$ of $\nu$ if and only if the initial index $\epsilon(\omega|\nu)$ is equal to the ramification index $e(\omega|\nu)$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.

Invariant Chains in Algebra and Discrete Geometry

We relate finite generation of cones, monoids, and ideals in increasing chains (the local situation) to equivariant finite generation of the corresponding limit objects (the global situation). For cones and monoids there is no analog of Noetherianity as in the case of ideals and we demonstrate this in examples. As a remedy, we find local-global correspondences for finite generation. These results are derived from a more general framework that relates finite generation under closure operations to equivariant finite generation under general families of maps. We also give a new proof that non-saturated Inc-invariant chains of ideals stabilize, closing a gap in the literature.

Boundaries of relative factor graphs and subgroup classification for automorphisms of free products

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $\mathrm{FF}$ and the $\mathcal{Z}$-factor graph $\mathcal{Z}\mathrm{F}$, as spaces of equivalence classes of arational trees (respectively relatively free arational trees). We also identify the loxodromic isometries of $\mathrm{FF}$ with the fully irreducible elements of $Out(G,\mathcal{F})$, and loxodromic isometries of $\mathcal{Z}\mathrm{F}$ with the fully irreducible atoroidal outer automorphisms.

Homological invariants of the arrow removal operation

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.

Comparison and Verification of Dynamic Simulations and Experiments for a Modified Spur Gear Pair

In this study, static and dynamic analyses of a spur gear pair with linear tip relief and longitudinal crowning were performed. Dynamic experiments were conducted to compare and verify the dynamic analysis results. On the basis of the theory of gearing, the mathematical model of a modified spur gear pair was established. In addition, a finite element mesh generation program was developed according to the mathematical model. Finite element analysis was applied to calculate the contact stress, contact patterns, and mesh stiffness of the gear pair. Moreover, a dynamic analysis was performed to predict the dynamic transmission error (DTE) at various speeds and loads. Dynamic experiments were performed on the spur gear pair by using a gear dynamic tester. Accelerometers were mounted on the gear pair to record the vibrations during gear meshing. The experimental DTE was calculated and compared with the simulated DTE to verify the prediction of the vibration performance of the spur gear pair under different rotational conditions.

J estimation based on regression machine learning applied to circumferential surface clad pipes with V groove weld

A regression machine learning approach is applied to propose a new equation for J-integral estimation for circumferential surface clad pipelines. The training data is provided based on a large number of simulations accomplished using a parametric finite element model generator script. The analysis matrix includes several crack depth ratios a/t, crack aspect ratios c/a, pipe diameter ratios Dext/t, V-groove angle, and material strain hardening exponent, n. The clad layer thickness is considered fixed with 3 mm. A model generator based on the Python programming language is used to generate an extensive data set of models. Training and testing sets are formed for five features and one target to develop a machine learning model. This paper aims mainly to promote fracture assessments by presenting an easy-to-use regression model encompassing several geometrical and material parameters. This is for finding an initial conservative J-integral estimation for a fully plastic description of fracture behavior under considerable plastic strain to provide a numerical evaluation of the crack driving force. The validation of the proposed model is ensured by comparing the results herein with those from different approaches.

On the homotopy of closed manifolds and finite CW-complexes

We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when X X is a finite CW-complex of dimension n n and π 1 ( X ) \pi _1(X) is virtually a Poincaré duality group of dimension ≥ n − 1 \geq n-1 , then π i ( X ) \pi _i(X) is not finitely generated for some i i unless X X is homotopy equivalent to the Eilenberg–MacLane space K ( π 1 ( X ) , 1 ) K(\pi _1(X),1) ; when M M is an n n -dimensional closed manifold and π 1 ( M ) \pi _1(M) is virtually a Poincaré duality group of dimension ≥ n − 1 \ge n-1 , then for some i ≤ [ n / 2 ] i\leq [n/2] , π i ( M ) \pi _i(M) is not finitely generated, unless M M itself is an aspherical manifold. These generalize theorems of M. Damian [Trans. Amer. Math. Soc. 361 (2009), pp. 1791–1809] from polycyclic groups to any virtually Poincaré duality groups.