Finite Type Research Articles

A characterization of ramification groups via jet algebras

AbstractWe present a functorial method to define ramification groups, identifying them as inertia groups of an induced action on composite jet algebras. This framework lays the foundation for defining higher ramification groups for actions involving group schemes. To achieve this, we introduce Taylor maps within the category of commutative unitary rings at prime ideals of an $$R$$ R -algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.

Pinchuk scaling method on domains with non-compact automorphism groups

In this paper, we characterize weakly pseudoconvex domains of finite type in [Formula: see text] in terms of the boundary behavior of automorphism orbits by using the scaling method.

Shifted Yangians and Polynomial $R$-Matrices

We study the category \mathcal{O}^{\mathrm{sh}} of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or cocyclic. This implies the existence and uniqueness of morphisms, the R -matrices, for such tensor products. We prove the R -matrices are polynomial in the spectral parameter, and we establish functional relations for the R -matrices. As applications, we prove the Jordan–Hölder property in the category \mathcal{O}^{\mathrm{sh}} . We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian.

Geometrical Penrose tilings are characterized by their 1-atlas

Penrose rhombus tilings are tilings of the plane by two decorated rhombi such that the decorations match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by three different methods: two based on the substitutive structure of the Penrose tilings and the last on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, i.e., they form a tiling space of finite type. Though considered as folklore, no complete proof of this result has been published, to our knowledge.

Stability of $$\imath $$canonical Bases of Locally Finite Type

Stability of $$\imath $$canonical Bases of Locally Finite Type

Minimal and proximal examples of -stable and -approachable shift spaces

Abstract We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $\bar {d}$ metric ( $\bar {d}$ -approachable shift spaces). The class of $\bar {d}$ -approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar {d}$ -approachability, together with a closely connected notion of $\bar {d}$ -shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys.43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $\bar {d}$ -approachability and $\bar {d}$ -shadowing. Here, we study further properties and connections between $\bar {d}$ -shadowing and $\bar {d}$ -approachability. We prove that $\bar {d}$ -shadowing implies $\bar {d}$ -stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar {d}$ -shadowing property the Hausdorff pseudodistance ${\bar d}^{\mathrm {H}}$ between shift spaces induced by $\bar {d}$ is the same as the Hausdorff distance between their simplices of invariant measures with respect to the Hausdorff distance induced by Ornstein’s metric $\bar {d}$ between measures. We prove that without $\bar {d}$ -shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the $\bar {d}$ -shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that $\bar {d}$ -shadowing indeed generalizes the specification property.

Equilibrium measures for two-sided shift spaces via dimension theory

Abstract Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

Topological rigidity of maps in positive characteristic and anabelian geometry

We study pairs of non-constant maps between two integral schemes of finite type over two (possibly different) fields of positive characteristic. When the target is quasi-affine, Tamagawa showed that the two maps are equal up to a power of Frobenius if and only if they induce the same homomorphism on their étale fundamental groups. We extend Tamagawa’s result by adding a purely topological criterion for maps to agree up to a power of Frobenius.

Categorical resolutions of filtered schemes

We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the fact that every variety (over a field of characteristic zero) can be resolved by a finite sequence of blow-ups along smooth centres. We merely require the existence of (projective) resolutions. To accomplish this we put the \mathcal{A} -spaces of Kuznetsov and Lunts in a different light, viewing them instead as schemes endowed with finite filtrations. The categorical resolution is then constructed by gluing together differential graded categories obtained from a hypercube of finite length filtered schemes.

Some classification of affine homothetical surfaces of finite type in 𝕀

Abstract A Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian Δ. In this paper, we classify two types of affine homothetical surfaces of finite type in isotropic 3-space 𝕀 3 {\mathbb{I}^{3}} under the condition Δ ⁢ r i = λ i ⁢ r i {\Delta r_{i}=\lambda_{i}r_{i}} , where ( r i ) {(r_{i})} is the i-component function of the position vector r ( i = 1 , 2 , 3 {i=1,2,3} ), λ i ∈ ℝ {\lambda_{i}\in\mathbb{R}} and Δ denotes the Laplace operator.

On a hemivariational inequality with non-monotone operator

We investigate the existence of generalized variational solutions to abstract hemivariational inequality driven by a coercive and locally Lipschitz operator with no further monotonicity conditions imposed on it. A finite dimensional Ritz type approximation scheme is derived. Applications to competitive systems driven by the generalized ( p , q ) - Laplacian with convection are given.

Nonabelian descent types

We present the notion of nonabelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and nonabelian Galois 2-cohomology due to Springer and Borovoi. The necessity of descent types arises in the context of the descent theory where no torsors are given a priori, for example, when we wish to study the arithmetic properties such as the Brauer–Manin obstruction to the Hasse principle on homogeneous spaces without rational points. This new definition also unifies the types by Colliot-Thélène–Sansuc, the extended types by Harari–Skorobogatov, and the finite descent types by Harpaz–Wittenberg.

On Gorenstein algebras of finite Cohen-Macaulay type: Dimer tree algebras and their skew group algebras

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra A is a 2-cluster category of Dynkin type A. Here we show that, if A has an admissible action by the group G with two elements, then the stable Cohen-Macaulay category of the skew group algebra AG is a 2-cluster category of Dynkin type D. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible G-action on the path algebra of type A the resulting skew group algebra is of type D. Moreover, we provide a geometric model of the syzygy category of AG in terms of a punctured polygon P with a checkerboard pattern in its interior, such that the 2-arcs in P correspond to indecomposable syzygies in AG and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type A and D respectively. We also provide examples of types E6,E7, and E8.

Accuracy and efficiency of finite element head models: The role of finite element formulation and material laws.

Traumatic brain injury is a significant problem worldwide. In the United States of America, around 1.7 million cases are documented annually, displaying the need for a deeper understanding of the effects on the human brain. The tests required for this assessment are very complex. Tests on cadavers may raise serious ethical questions, and in vivo crash tests are not viable. In this context, there is a great need to developing finite element head models (FEHM) to study the biomechanics of the tissues when submitted to a certain impact or acceleration/deceleration scenario. An excellent compromise between accuracy and CPU efficiency is always desirable for a FEHM, For this reason, this work focuses on the improvement of an existing head model, including the study of the behavior of the brain using distinct finite element types. The finite element type and formulation is of utmost importance for the general accuracy and efficiency of the models. Several validations were performed, comparing the simulation results against experimental data. The simulations with hexahedral elements, under specific conditions, obtained more accurate results with a lower computational cost. Using hexahedrals, a comparison was also performed using two material characterizations with more than 10 years apart, using the latest finite element head model validation experiment. Overall, the newer material model displays a less stiff response, although its implementation must always depend on the overall purpose of the model it is being applied to.

On n-hereditary algebras and n-slice algebras

In this paper we show that acyclic n-slice algebras are exactly acyclic n-hereditary algebras whose (n+1)-preprojective algebras are (q+1,n+1)-Koszul. We also list the equivalent triangulated categories arising from the algebra constructions related to an n-slice algebra. We show that higher slice algebras of finite type appear in pairs and they share the Auslander-Reiten quiver for their higher-preprojective modules.

On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

We show that the underlying complex manifold of a complete non-compact two-dimen­sional shrinking gradient Kähler–Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature \operatorname{R}_{g} whose soliton vector field X has an integral curve along which \operatorname{R}_{g}\not\to0 is biholomorphic to either \mathbb{C}\times\mathbb{P}^{1} or to the blowup of this manifold at one point, and that the soliton metric g is toric. We also identify the corresponding soliton vector field X in each case. Given these possibilities, we then prove a strong form of the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.

Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type

Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type

Ng\^o support theorem and polarizability of quasi-projective commutative group schemes

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ng\^o. This extends the applicability of Ng\^o's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

Chain conditions on non-parallel submodules

In this paper, we investigate modules with ascending and descending chain conditions on non-parallel submodules. We call these modules np-Noetherian and np-Artinian respectively, and give structure theorems for them. It is proved that any np-Artinian module is either atomic or finitely embedded. Also, we give a sufficient condition for np-Noetherian (resp., np-Artinian) modules to be Noetherian (resp., Artinian). We study ascending (resp., descending) chain condition up to isomorphism on non-parallel submodules as npi-Noetherian (resp., npi-Artinian) modules and characterize these modules. It is shown that any npi-Noetherian module has finite type dimension. Next, we investigate some properties of semiprime right np-Artinian (resp., npi-Artinian) rings. In particular, it is proved that if $ R $ semiprime ring such that $ J(R) $ is not atomic, then $ R $ is right np-Artinian if and only if it is semisimple. Further, it is shown that if $ R $ is a semiprime right npi-Artinian ring, then either $ Z(R) $ is atomic or $ R $ is right non-singular. Finally, we investigate when np-Artinian (resp., np-Noetherian) rings and ne-Artinian (resp., ne-Noetherian) rings coincide.

A Comparison Theorem For The Pro-étale Fundamental Group

Abstract Let $X$ be a connected scheme locally of finite type over ${\mathbb{C}}$, and let $X^{\textrm{an}}$ be its associated analytic space. In this paper, we define a comparison map from the topological fundamental group of $X^{\textrm{an}}$ to the pro-étale fundamental group of $X$.