Finite Type Research Articles

Discrete representations of orbit structures of flows for topological data analysis

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequences of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of two-dimensional (2D) incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse–Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem.

On dynamical finiteness properties of algebraic group shifts

Let G be a group and let V be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts Σ ⊂ VG which generalize both the class of algebraic sofic subshifts of VG and the class of closed group subshifts over finite group alphabets. When G is a polycyclic-by-finite group, we show that VG satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known results of Kitchens and Schmidt to cover the case of many non-compact group alphabets.

Riesz Projection and Essential S-spectrum in Quaternionic Setting

This paper is devoted to the investigation of the Weyl and the essential S-spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the S-eigenvalue of finite type is both introduced and studied. In particular, we have shown that the Weyl and the essential S-spectra do not contain eigenvalues of finite type. We have also described the boundary of the Weyl S-spectrum and the particular case of the spectral theorem of the essential S-spectrum.

Upper topology and its relation with the projective modules

In this article, first we obtain a number of new results on projective modules and on the upper topology of an ordinal number. Then it is shown that the rank map of a locally finite type projective module is continuous with respect to the upper topology (by contrast, it is well known that this map is not necessarily continuous with respect to the discrete topology).

Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups

AbstractIn this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G, if X is a G-SFT with positive topological entropy $h(X)> 0$ , then the entropies of the SFT subsystems of X are dense in the interval $[0, h(X)]$ . In fact, we prove a ‘relative’ version of the same result: if X is a G-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$ , then the entropies of the SFTs Z for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$ . We also establish analogous results for sofic G-shifts.

Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent

We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains.

Finite Domination Type for Monoid Presentations

In [5], Squier, Otto and Kobayashi explored a homotopical property for monoids called finite derivation type (FDT) and proved that FDT is a necessary condition that a finitely presented monoid must satisfy if it is to have a finite canonical presentation. In the latter development in [2], Kobayashi proved that the property <img src=image/13428295_01.gif> is equivalent with what is called in [2] finite domination type. It was indicated in the end of [2] that there are <img src=image/13428295_01.gif> monoids which are not even finitely generated, and as a consequence are not of FDT. It was this indication that inspired us to look for the possibility of defining a property of monoids which encapsulates both, FDT and finite domination type. This is realized in the current paper by extending the notion of finite domination from monoids to rewriting systems, and to achieve this, we are based on the approach of Isbell in [1], who defined the notion of the dominion of a subcategory <img src=image/13428295_02.gif> of a category <img src=image/13428295_03.gif> and characterized that dominion in terms of zigzags in <img src=image/13428295_03.gif> over <img src=image/13428295_02.gif>. The reason we followed this approach is that to every rewriting system <img src=image/13428295_04.gif> which gives a monoid <img src=image/13428295_06.gif>, there is always a category <img src=image/13428295_05.gif> associated to it which contains three types of information at the same time: (i) all the possible ways in which the elements of <img src=image/13428295_06.gif> are written in terms of words with letters from <img src=image/13428295_07.gif>, (ii) all the possible ways one can transform a word with letters from <img src=image/13428295_07.gif> into another one representing the same element of <img src=image/13428295_06.gif> by using rewriting rules from <img src=image/13428295_08.gif>. Each of such way gives is in fact a path in the reduction graph of <img src=image/13428295_04.gif>. The last information (iii) encoded in <img src=image/13428295_05.gif> is that <img src=image/13428295_05.gif> contains all the possible ways that two parallel paths of the reduction graph are linked to each other by a series of compositions of whiskerings of other parallel paths. This category <img src=image/13428295_05.gif> turns out to have the advantage that it can "measure" the extent to which a set <img src=image/13428295_09.gif> of parallel paths is sufficient to express any pair of parallel paths by composing whiskers from <img src=image/13428295_09.gif>. The gadget used to measure this, is the Isbell dominion of the whisker category <img src=image/13428295_10.gif> generated by <img src=image/13428295_09.gif> over <img src=image/13428295_05.gif>. We then define the monoid <img src=image/13428295_06.gif> given by <img src=image/13428295_04.gif> to be of finite domination type (FDOT) if both <img src=image/13428295_07.gif> and <img src=image/13428295_08.gif> are finite and there is a finite set <img src=image/13428295_09.gif> of morphisms such that <img src=image/13428295_11.gif> is exactly <img src=image/13428295_05.gif>. The first main result of our paper is that likewise FDT, FDOT is an invariant of the monoid presentation, and the second one is that that FDT implies FDOT, while remains open whether the converse is true or not. The importance of FDOT stands in the fact that not only it generalizes FDT, but the way it is defined has a lot in common with <img src=image/13428295_01.gif>, giving thus hope that FDOT is the right tool to put FDT and <img src=image/13428295_01.gif> into the same framework.

Tilting modules arising from knot invariants

We construct tilting modules over Jacobian algebras arising from knots. To a two-bridge knot L [ a 1 , … , a n ] , we associate a quiver Q with potential and its Jacobian algebra A . We construct a family of canonical indecomposable A -modules M ( i ) , each supported on a different specific subquiver Q ( i ) of Q . Each of the M ( i ) is expected to parametrize the Jones polynomial of the knot. We study the direct sum M = ⊕ i M ( i ) of these indecomposables inside the module category of A as well as in the cluster category. In this paper we consider the special case where the two-bridge knot is given by two parameters a 1 , a 2 . We show that the module M is rigid and τ -rigid, and we construct a completion of M to a tilting (and τ -tilting) A -module T . We show that the endomorphism algebra End A T of T is isomorphic to A , and that the mapping T ↦ A [ 1 ] induces a cluster automorphism of the cluster algebra A ( Q ) . This automorphism is of order two. Moreover, we give a mutation sequence that realizes the cluster automorphism. In particular, we show that the quiver Q is mutation equivalent to an acyclic quiver of type T p , q , r (a tree with three branches). This quiver is of finite type if ( a 1 , a 2 ) = ( a 1 , 2 ) , ( 1 , a 2 ) , or ( 2 , 3 ) , it is tame for ( a 1 , a 2 ) = ( 2 , 4 ) or ( 3 , 3 ) , and wild otherwise.

Bergman–Szegő kernel asymptotics in weakly pseudoconvex finite type cases

Abstract We construct a pointwise Boutet de Monvel–Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three-dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in ℂ 2 {\mathbb{C}^{2}} , in agreement with D’Angelo’s example. Finally, our results generalize a three-dimensional CR embedding theorem of Lempert.

Diagram automorphisms and canonical bases for quantum affine algebras, II

Let Uq− be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on Uq− induced from a diagram automorphism. Let U_q− be the quantum algebra obtained from σ, and B˜ (resp. B_˜) the canonical signed basis of Uq− (resp. U_q−). Assume that Uq− is simply-laced of finite or affine type. In our previous papers [10], [11], we have proved by an elementary method, that there exists a natural bijection B˜σ≃B_˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases.

The boundary at infinity of the curve complex and the relative Teichmüller space

In this paper we study the boundary at infinity of the curve complex \mathcal{C}(S) of a surface S of finite type and the relative Teichmüller space \mathcal{T}_{\mathrm{el}}(S) obtained from the Teichmüller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) are quasi-isometric, and Masur–Minsky have shown that \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) are hyperbolic in the sense of Gromov. We show that the boundary at infinity of \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) is the space of topological equivalence classes of minimal foliations on S .

Homology of étale groupoids a graded approach

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε-th component with ε the identity of the grade group. For Z-graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.

Weak damped wave equations defined by a class of self-similar measures with overlaps

We study weak damped wave equations defined by fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. First, we prove the weak well-posedness of these wave equations and some regularity results. Second, by using some important information about the structure of the three measures above, we set up a framework for one-dimensional measures to discretize the wave equations. Then we use the finite element and central difference methods to obtain numerical approximations of the weak solutions for such equations. Finally, we show that the numerical solutions converge to the actual weak solution and obtain the rate of convergence.

Lattice points in stretched finite type domains

We study an optimal stretching problem, which is a variant lattice point problem, for convex domains in Rd (d≥2) with smooth boundary of finite type that are symmetric with respect to each coordinate hyperplane/axis. We prove that optimal domains which contain the most positive (or least nonnegative) lattice points are asymptotically balanced.

Conjugacy of local homeomorphisms via groupoids and C*-algebras

AbstractWe investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.

Technical Note—An Improved Analysis of LP-Based Control for Revenue Management

Bounded Regret for LP-Based Revenue-Management ProblemsIn “An Improved Analysis of LP-Based Control for Revenue Management,” Chen, Li, and Ye study a class of quantity-based network revenue-management problems. The authors consider a stochastic setting where all the orders are i.i.d. sampled and the customers are of finite type. They focus on the classic LP-based adaptive algorithm and consider regret as the performance measure. They found that when the underlying LP is nondegenerate, the algorithm achieves a problem-dependent regret upper bound that is independent of the horizon/number of time periods T; when the underlying LP is degenerate, the algorithm achieves a tight regret upper bound that scales on the order of T log(T) and matches the lower bound up to a logarithmic order.

Specialization for the pro-étale fundamental group

For a formal scheme$\mathfrak {X}$of finite type over a complete rank-one valuation ring, we construct a specialization morphism\[ \pi^{\mathrm{dJ}}_1(\mathfrak{X}_\eta) \to \pi^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}_1(\mathfrak{X}_k) \]from the de Jong fundamental group of the rigid generic fiber to the Bhatt–Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of$\mathfrak {X}$and normalizations of the irreducible components of$\mathfrak {X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent.

Homomorphisms of Algebraic Groups: Representability and Rigidity

Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) Hom(G,H) and the subfunctor of homomorphisms (of algebraic groups) Homgp(G,H). We show that Hom(G,H) is represented by a group scheme, locally of finite type, if the k-vector space O(G) is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that Homgp(G,H) is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open.

On images of subshifts under embeddings of symbolic varieties

AbstractWe show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule $\Sigma \subset A^G$ and every injective G-equivariant uniformly continuous module homomorphism $\tau \colon \! \Sigma \to B^G$ , a subshift $\Delta \subset \Sigma $ is of finite type, respectively sofic, if and only if so is the image $\tau (\Delta )$ . Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.

Piatetski-Shapiro primes in the intersection of multiple Beatty sequences

Suppose that α1,α2,β1,β2∈ℝ. Let α1,α2>1 be irrational and of finite type such that 1,α1−1,α2−1 are linearly independent over ℚ. Let c be a real number in the range 1<c<12∕11. In this paper, it is proved that there exist infinitely many primes in the intersection of the Beatty sequences ℬα1,β1=⌊α1n+β1⌋, ℬα2,β2=⌊α2n+β2⌋ and the Piatetski-Shapiro sequence 𝒩(c)=⌊nc⌋. Moreover, we also give a sketch of the proof of Piatetski-Shapiro primes in the intersection of multiple Beatty sequences.