Finite Type Research Articles

Noninjectivity of the cycle class map in continuous -adic cohomology

AbstractJannsen asked whether the rational cycle class map in continuous$\ell $-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by Schreieder, the integral version of Jannsen’s question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. Our examples also have consequences for the coniveau filtration on Chow groups and the transcendental Abel-Jacobi map constructed by Schreieder.

Parametrization by horizontal constraints in the study of algorithmic properties of $ \mathbb{Z}^2 $-subshifts of finite type

The non-emptiness, called the Domino Problem, and the characterization of the possible entropies of $ \mathbb{Z}^2 $-subshifts of finite type are standard problems of symbolic dynamics. In this article we study these questions with horizontal constraints fixed beforehand as a parameter. We determine for which horizontal constraints the Domino Problem is undecidable and when all right-recursively enumerable numbers can be obtained as entropy, with two approaches: either the additional local rules added to the horizontal constraints can be of any shape, or they can only be vertical rules.

On constraints for knots to admit chirally cosmetic surgeries and their calculations

We discuss various constraints for knots in $S^{3}$ to admit chirally cosmetic surgeries, derived from invariants of 3-manifolds, such as, the quantum $SO(3)$-invariant, the rank of the Heegaard Floer homology, and finite type invariants. We apply them to show that a large portion (roughly 75$\%$) of knots which are neither amphicheiral nor $(2,p)$-torus knots with less than or equal to 10 crossings admits no chirally cosmetic surgeries.

FINITE ELEMENT MODEL FOR ANALYSIS OF CHARACTERISTICS OF SHROUDED ROTOR BLADE VIBRATIONS

The paper presents the approaches to FE modelling of blade airfoil, contact between the shrouds and operational damage. The regularities are established concerning the influence of the finite element type, finite element mesh and model of contact interaction on the spectrum of natural frequencies of blade assemblies. The use of the developed computational models is substantiated to determine the forced vibration characteristics of the selected objects of investigation. Based on the performed numerical experiments it was substantiated of finite element model selection for analysis of characteristics of shrouded rotor blade vibrations.

Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

Dynamics of Sofic Shifts

In this paper, we provide a characterization for the subshifts of finite type (SFT) in terms of Cellular automata (CA). In addition, we prove that 1. The following are equivalent for a non-singleton subshift of finite type XF. a) XF is transitive and Per(XF), the set of periodic points of XF, is cofinite b) XF is weak mixing c) XF is mixing. 2. For non-singleton sofic shifts, only the statements (a) and (b) are equivalent.

The horofunction boundary of a Gromov hyperbolic space

We highlight a condition, the approaching geodesics property, on a proper geodesic Gromov hyperbolic metric space, which implies that the horofunction compactification is topologically equivalent to the Gromov compactification. It is known that this equivalence does not hold in general. We prove using rescaling techniques that the approaching geodesics property is satisfied by bounded strongly pseudoconvex domains of $$\mathbb {C}^q$$ endowed with the Kobayashi metric. We also show that bounded convex domains of $$\mathbb {C}^q$$ with boundary of finite type in the sense of D’Angelo satisfy a weaker property, which still implies the equivalence of the two said compactifications. As a consequence we prove that on those domains big and small horospheres as defined by Abate in (Math Z 198(2):225–238, 1988) coincide. Finally we generalize the classical Julia’s lemma, giving applications to the dynamics of non-expanding maps.

Canonical tessellations of decorated hyperbolic surfaces

A decoration of a hyperbolic surface of finite type is a choice of circle, horocycle or hypercycle about each cone-point, cusp or flare of the surface, respectively. In this article we show that a decoration induces a unique canonical tessellation and dual decomposition of the underlying surface. They are analogues of the weighted Delaunay tessellation and Voronoi decomposition in the Euclidean plane. We develop a characterisation in terms of the hyperbolic geometric equivalents of Delaunay’s empty-discs and Laguerre’s tangent-distance, also known as power-distance. Furthermore, the relation between the tessellations and convex hulls in Minkowski space is presented, generalising the Epstein–Penner convex hull construction. This relation allows us to extend Weeks’ flip algorithm to the case of decorated finite type hyperbolic surfaces. Finally, we give a simple description of the configuration space of decorations and show that any fixed hyperbolic surface only admits a finite number of combinatorially different canonical tessellations.

Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.

Characterization of Pseudoconvex Domains of Finite Type with Locally Diagonalizable Levi Form by Their Automorphism Groups

Characterization of Pseudoconvex Domains of Finite Type with Locally Diagonalizable Levi Form by Their Automorphism Groups

NORMAL AND IRREDUCIBLE ADIC SPACES, THE OPENNESS OF FINITE MORPHISMS, AND A STEIN FACTORIZATION

AbstractWe transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces.

The parity of Lusztig's restriction functor and Green's formula

Our investigation in the present paper is based on three important results. (1) In [14], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula.

Characteristic measures for language stable subshifts

We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class of shifts, which we call language stable subshifts, and show that these shifts have characteristic measures. This is a large class that is generic in several senses and contains numerous positive entropy examples.

Extra-large type Artin groups are hierarchically hyperbolic

We show that Artin groups of extra-large type, and more generally Artin groups of large and hyperbolic type, are hierarchically hyperbolic. This implies in particular that these groups have finite asymptotic dimension and uniform exponential growth. We prove these results by using a combinatorial approach to hierarchical hyperbolicity, via the action of these groups on a new complex that is quasi-isometric both to the coned-off Deligne complex introduced by Martin–Przytycki and to a generalisation due to Morris-Wright of the graph of irreducible parabolic subgroups of finite type introduced by Cumplido–Gebhardt–González-Meneses–Wiest.

Ranks of fringe operators on finite Rudin type invariant subspaces

Let M be a finite Rudin type invariant subspace of the Hardy space over the bidisk with variables z,w. In this paper we determine the rank of the fringe operator Fz on M⊖wM.

BIG COHEN–MACAULAY TEST IDEALS IN EQUAL CHARACTERISTIC ZERO VIA ULTRAPRODUCTS

AbstractUtilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $\mathcal {B}(R)$ over a local domain R essentially of finite type over $\mathbb {C}$ . We show that if R is normal and $\Delta $ is an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier, then the BCM test ideal $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ coincides with the multiplier ideal $\mathcal {J}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ , where $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$ -adic completions of R and $\mathcal {B}(R)$ , respectively, and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$ . As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.

Tensor diagrams and cluster combinatorics at punctures

Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SLk-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k=2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the language of tensor diagrams, extending work of Fomin and the second author, and we formulate skein-algebraic tools for calculating in these cluster algebras. We analyze the finite mutation type examples in detail.

Asymptotic pressure on some self-similar trees

The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix, respecting nearest neighbor adjacency restrictions. To the set of these configurations one can associate a pressure, which is defined as the limit, when it exists, of averages of the logarithm of the partition function over certain finite subgraphs. We prove that for shifts of finite type on generalized Fibonacci trees and many primitive interaction matrices, the limit exists and is given by an infinite series. We also show that the limit of any cluster points of the pressure on finite subtrees as the number of generators grows without bound, which we call the asymptotic pressure, equals the logarithm of the maximum row sum of the interaction matrix.

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models.

Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics-a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.

Effective intrinsic ergodicity for countable state Markov shifts

For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.