Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

FIm-modules over Noetherian rings

In this paper we study representation theory of the category ${\mathrm{FI}}^m$ introduced in [6], [7] which is a product of copies of the category FI, and show that quite a few interesting representation theoretic and homological properties of FI can be generalized to ${\mathrm{FI}}^m$ in a natural way. In particular, we prove the representation stability property of finitely generated ${\mathrm{FI}}^m$-modules over fields of characteristic 0.     

A model-theoretic characterization of constant-depth arithmetic circuits

We study the class ${\mathrm{\#}\text{AC}}^0$ of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean circuit class ${\text{AC}}^0$, we remedy this situation and develop such a characterization of ${\mathrm{\#}\text{AC}}^0$. Our characterization can be interpreted as follows: Functions in ${\mathrm{\#}\text{AC}}^0$ are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of ${\text{TC}}^0$, the class of languages accepted by constant-depth polynomial-size majority circuits.     

The Hilbert series and a-invariant of circle invariants

Let V be a finite-dimensional representation of the complex circle ${\mathbb{C}}^{\times }$ determined by a weight vector $\mathit{a}\in {\mathbb{Z}}^n$. We study the Hilbert series ${\mathrm{Hilb}}_{\mathit{a}}(t)$ of the graded algebra $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ of polynomial ${\mathbb{C}}^{\times }$-invariants in terms of the weight vector a of the ${\mathbb{C}}^{\times }$-action. In particular, we give explicit formulas for ${\mathrm{Hilb}}_{\mathit{a}}(t)$ as well as the first four coefficients of the Laurent expansion of ${\mathrm{Hilb}}_{\mathit{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.     

On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the $L_{\infty }$ norm. We consider algorithms that use standard information ${\mathrm{\Lambda }}^{\mathrm{std}}$ consisting of function values or general linear information ${\mathrm{\Lambda }}^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for ${\mathrm{\Lambda }}^{\mathrm{std}}$ and ${\mathrm{\Lambda }}^{\mathrm{all}}$ under the absolute or normalized error criterion, and show that the power of ${\mathrm{\Lambda }}^{\mathrm{std}}$ is the same as the one of ${\mathrm{\Lambda }}^{\mathrm{all}}$ for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].     

Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety X we study a category of modules that admit compatible actions of both the algebra A of functions on X and the Lie algebra of vector fields on X. In particular, for the case when X is the sphere ${\mathbb{S}}^2$, we construct a set of simple modules that are finitely generated over A. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational ${\mathrm{GL}}_2$-modules.     

On finite-dimensional copointed Hopf algebras over dihedral groups

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions ${\mathbb{k}}^{{\mathbb{D}}_m}$ over a dihedral group ${\mathbb{D}}_m$, with $m=4a\ge 12$. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups.     

The classification of 3-dimensional noetherian cubic Calabi–Yau algebras

It is known that every 3-dimensional noetherian Calabi–Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S.P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi–Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi–Yau algebras. As an application, we show that if S is a 3-dimensional noetherian cubic Calabi–Yau algebra and σ is a graded algebra automorphism of S, then the homological determinant of σ can be calculated by the formula $\mathrm{hdet}\sigma ={(\det \sigma )}^2$ with one exception.     

The Z2-orbifold of the universal affine vertex algebra

Let $\mathfrak{g}$ be a simple, finite-dimensional complex Lie algebra, and let $V^k(\mathfrak{g})$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ at level k. The Cartan involution on $\mathfrak{g}$ lifts to an involution on $V^k(\mathfrak{g})$, and we denote by $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ the orbifold, or fixed-point subalgebra, under this involution. Our main result is an explicit minimal strong finite generating set for $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ for generic values of k.     

Well-posedness of the free boundary problem for incompressible elastodynamics

The free boundary problem for the three dimensional incompressible elastodynamics system is studied under the Rayleigh–Taylor sign condition. Both the columns of the elastic stress $\mathbf{F}{\mathbf{F}}^{?}?\mathrm{I}$ and the transpose of the deformation gradient ${\mathbf{F}}^{?}?\mathrm{I}$ are tangential to the boundary which moves with the velocity, and the pressure vanishes outside the flow domain. The linearized equation takes the form of wave equation in terms of the flow map in the Lagrangian coordinate, and the local-in-time existence of a unique smooth solution is proved using a geometric argument in the spirit of [19].     

The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity ${\mathrm{e}}^{\frac{2\pi \mathbf{i}}{k}}$ (and its symmetric entry is ${\mathrm{e}}^{?\frac{2\pi \mathbf{i}}{k}}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set $\{1,{\mathrm{e}}^{\frac{2\mathit{\pi }\mathbf{i}}{k}},{\mathrm{e}}^{?\frac{2\pi \mathbf{i}}{k}}\}$, which is not closed under multiplication when $k?4$. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank ($H_k$-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with $H_k$-rank 2 and 3, respectively.