\alpha -Admissibility for Ritt Operators

Let $T:X\rightarrow X$ be a power bounded operator on Banach space. An operator $C:X\rightarrow Y$ is called admissible for $T$ if it satisfies an estimate $\sum _k\Vert CT^k(x)\Vert ^2\,\le M^2\Vert x\Vert ^2$ . Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when $X$ is reflexive and $T$ is a Ritt operator satisfying a appropriate square function estimate, $C$ is admissible for $T$ if and only if it satisfies a uniform estimate $(1-\vert \omega \vert ^2)^{\frac{1}{2}}\Vert C(I-\omega T)^{-1}\Vert \,\le K\,$ for $\omega \in \mathbb{C }$ , $\vert \omega \vert <1$ . We extend this result to the more general setting of $\alpha $ -admissibility. Then we investigate a natural variant of admissibility involving $R$ -boundedness and provide examples to which our general results apply.     

Open Gromov–Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X , \omega )$ . We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$ . Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$ , every relative homology class $d \in H_2 (X , L ; \mathbb{Z })$ and adequate number of incidence conditions in $L$ or $X$ , the weighted number of $J$ -holomorphic discs with boundary on $L$ , homologous to $d$ , and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$ , provided that at least one incidence condition lies in $L$ . These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring $A$ .     

Positive kernel operators in L^{p(x)} spaces

A characterization of a weight $v$ governing the boundedness/compactness of the weighted kernel operator $K_v$ in variable exponent Lebesgue spaces $L^{p(\cdot )}$ is established under the log-Hölder continuity condition on exponents of spaces. The kernel operator involves, for example, weighted variable parameter fractional integral operators. The distance between $K_v$ and the class of compact integral operators acting from $L^{p(\cdot )}$ to $L^{q(\cdot )}$ (measure of non-compactness) is also estimated from above and below.     

On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators

Given a positive, irreducible and bounded $C_0$ -semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non-trivial compact or kernel operator. For a discrete semigroup, i.e. for powers of a single operator $T$ , we show that the point spectrum of some power $T^k$ intersects the unit circle at most in $1$ . As a consequence, we obtain a sufficient condition for strong convergence of the $C_0$ -semigroup and for a subsequence of the powers of $T$ , respectively.     

Groups with the same orders of maximal abelian subgroups as A_2(q)

In Li and Chen (Sib. Math. J. 53(2), 243–247, 2012), it is proved that the simple group $A_1(p^n)$ is uniquely determined by the set of orders of its maximal abelian subgroups. Let $q=p^{\alpha }$ be a prime power and $L=A_2(q)$ . In this paper, we prove that if $q$ is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as $L$ , is isomorphic to $L$ or an extension of $L$ by a subgroup of the outer automorphism group of $L$ .     

A Characterization of Compact Operators Via the Non-Connectedness of the Attractors of a Family of IFSs

In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space $X$ , $S$ and $T$ bounded linear operators from $X$ to $X$ such that $\Vert S\Vert , \Vert T\Vert <1$ and $w\in X$ , let us consider the IFS $\mathcal S _{w}=(X,f_{1},f_{2})$ , where $f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $f_{2}(x)=T(x)+w$ , for all $x\in X$ . On one hand we prove that if the operator $S$ is compact, then there exists a family $(K_{n})_{n\in \mathbb N }$ of compact subsets of $X$ such that $A_{\mathcal S _{w}}$ is not connected, for all $w\in X-\bigcup _{n\in \mathbb N } K_{n}$ . On the other hand we prove that if $H$ is an infinite dimensional Hilbert space, then a bounded linear operator $S:H\rightarrow H$ having the property that $\Vert S\Vert <1$ is compact provided that for every bounded linear operator $T:H\rightarrow H$ such that $\Vert T\Vert <1$ there exists a sequence $(K_{T,n})_{n}$ of compact subsets of $H$ such that $A_{\mathcal S _{w}}$ is not connected for all $w\in H-\bigcup _{n}K_{T,n}$ . Consequently, given an infinite dimensional Hilbert space $H$ , there exists a complete characterization of the compactness of an operator $S:H\rightarrow H$ by means of the non-connectedness of the attractors of a family of IFSs related to the given operator. Finally we present three examples illustrating our results.     

Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices

A fine mixed subdivision of a $(d-1)$ -simplex $T$ of size $n$ gives rise to a system of  ${d \atopwithdelims ()2}$ permutations of $[n]$ on the edges of $T$ , and to a collection of $n$ unit $(d-1)$ -simplices inside $T$ . Which systems of permutations and which collections of simplices arise in this way? The Spread Out Simplices Conjecture of Ardila and Billey proposes an answer to the second question. We propose and give evidence for an answer to the first question, the Acyclic System Conjecture. We prove that the system of permutations of $T$ determines the collection of simplices of $T$ . This establishes the Acyclic System Conjecture as a first step towards proving the Spread Out Simplices Conjecture. We use this approach to prove both conjectures for $n=3$ in arbitrary dimension.     

Counting nonsingular matrices with primitive row vectors

We give an asymptotic expression for the number of nonsingular integer $n\times n$ -matrices with primitive row vectors, determinant $k$ , and Euclidean matrix norm less than $T$ , as $T\rightarrow \infty $ . We also investigate the density of matrices with primitive rows in the space of matrices with determinant $k$ , and determine its asymptotics for large $k$ .     

Euclidean Steiner trees optimal with respect to swapping 4-point subtrees

The Steiner tree problem in Euclidean space $E^3$ asks for a minimum length network $T$ , called a Euclidean Steiner Minimum Tree (ESMT), spanning a given set of points. This problem is NP-hard and the hardness is inherently due to the number of feasible topologies (underlying graph structure of $T$ ) which increases exponentially as the number of given points increases. Planarity is a very strong condition that gives a big difference between the ESMT problem in the Euclidean plane $E^2$ and in Euclidean $d$ -space $E^d (d\ge 3)$ : the ESMT problem in the plane is practically solvable whereas the ESMT problem in $d$ -space is really intractable. The simplest tree rearrangement technique is to repeatedly replace a subtree spanning four nodes in $T$ with another subtree spanning the same four nodes. This technique is referred to as the Swapping 4-point Topology/ Tree technique in this paper. An indicator (or quasi-indicator) of $T$ plays a similar role in the optimization of the length $L(T)$ of $T$ in the discrete topology space (the underlying graph structure of $T$ ) to the derivative of a differentiable function which indicates a fastest direction of descent. $T$ will be called S4pT-optimal if it is optimal with respect to swapping 4-point subtrees. In this paper we first make a complete analysis of 4-point trees in Euclidean space exploring all possible types of 4-point trees and their properties. We review some known indicators of 4-point ESMTs in $E^2$ , and give some simple geometric proofs of these indicators. Then, we translate these indicators to $E^3$ , producing eight quasi-indicators in $E^3$ using computational experiments, the best quasi-indicator $\rho _\mathrm{osr}$ is sifted with an effectiveness for randomly generated 4-point sets as high as 98.62 %. Finally we show how $\rho _\mathrm{osr}$ is used to find an S4pT-optimal ESMT on 14 probability vectors in $4d$ -space with a detailed example.     

Optimal L^2, H^1 and L^\infty analysis of finite volume methods for the stationary Navier–Stokes equations with large data

Previous work on the stability and convergence analysis of numerical methods for the stationary Navier–Stokes equations was carried out under the uniqueness condition of the solution, which required that the data be small enough in certain norms. In this paper an optimal analysis for the finite volume methods is performed for the stationary Navier–Stokes equations, which relaxes the solution uniqueness condition and thus the data requirement. In particular, optimal order error estimates in the $H^1$ -norm for velocity and the $L^2$ -norm for pressure are obtained with large data, and a new residual technique for the stationary Navier–Stokes equations is introduced for the first time to obtain a convergence rate of optimal order in the $L^2$ -norm for the velocity. In addition, after proving a number of additional technical lemmas including weighted $L^2$ -norm estimates for regularized Green’s functions associated with the Stokes problem, optimal error estimates in the $L^\infty $ -norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor $O(|\log h|)$ for the stationary Naiver–Stokes equations.     

Regularity estimates in Hölder spaces for Schrödinger operators via a T1 theorem

We derive Hölder regularity estimates for operators associated with a time-independent Schrödinger operator of the form $-\Delta +V$ . The results are obtained by checking a certain condition on the function $T1$ . Our general method applies to get regularity estimates for maximal operators and square functions of the heat and Poisson semigroups, for Laplace transform type multipliers and also for Riesz transforms and negative powers $(-\Delta +V)^{-\gamma /2}$ , all of them in a unified way.     

A gaussian estimate for the heat kernel on differential forms and application to the Riesz transform

Let $(M,g)$ be a complete Riemannian manifold which satisfies a Sobolev inequality of dimension $n$ , and on which the volume growth is comparable to the one of ${\mathbb{R }}^n$ for big balls; if there is no non-zero $L^2$ harmonic 1-form, and the Ricci tensor is in $L^{\frac{n}{2}-\varepsilon }\cap L^\infty $ for an $\varepsilon >0$ , then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian acting on 1-forms. This allows us to prove that, under the same hypotheses, the Riesz transform $d\varDelta ^{-1/2}$ is bounded on $L^p$ for all $1<p<\infty $ . Then, in presence of non-zero $L^2$ harmonic 1-forms, we prove that the Riesz transform is still bounded on $L^p$ for all $1<p<n$ , when $n>3$ .     

On the Degenerate Crossing Number

The degenerate crossing number ${\text{ cr}^{*}}(G)$ of a graph $G$ is the minimum number of crossing points of edges in any drawing of $G$ as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph $G$ with $n$ vertices and $e \ge 4n$ edges ${\text{ cr}^{*}}(G)=\Omega \big (e^4 / n^4\big )$ . In this paper we completely resolve the main open question about degenerate crossing numbers and show that ${\text{ cr}^{*}}(G)=\Omega \big (e^3 / n^2 \big )$ , provided that $e \ge 4n$ . This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.     

Strongly embedded subspaces of p-convex Banach function spaces

Let $X(\mu )$ be a p-convex ( $1\le p<\infty $ ) order continuous Banach function space over a positive finite measure  $\mu $ . We characterize the subspaces of  $X(\mu )$ which can be found simultaneously in  $X(\mu )$ and a suitable $L^1(\eta )$ space, where $\eta $ is a positive finite measure related to the representation of  $X(\mu )$ as an $L^p(m)$ space of a vector measure  $m$ . We provide in this way new tools to analyze the strict singularity of the inclusion of  $X(\mu )$ in such an $L^1$ space. No rearrangement invariant type restrictions on  $X(\mu )$ are required.     

L^{\infty } estimation of tensor truncations

Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to $\ell ^{2}$ or $L^{2}$ norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require $\ell ^{\infty }$ or $L^{\infty }$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $\left\| \cdot \right\| _{\infty }$ by $\left\| \cdot \right\| _{2}$ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to $\left\| \cdot \right\| _{\infty }$ can be derived from the Gagliardo–Nirenberg inequality because of the special nature of the singular value decomposition truncation.     

New results about normal pairs of rings with zero-divisors

Let $R\subset S$ be a (unital) extension of (commutative) rings. It is proved in Theorem 1, that $(R, S)$ is a normal pair (i.e. $T$ is integrally closed in $S$ for each ring $T$ such that $R \subseteq T \subseteq S$ ) if and only if $R\subset S$ is a $P$ -extension and $R$ is integrally closed in $S$ . Theorem 2 states that for rings $R\subseteq T \subseteq S, R\subseteq S$ is a $P$ -extension if and only if $R\subseteq T$ and $T\subseteq S$ are $P$ -extensions. As a consequence, we prove that if $R\subseteq T \subseteq B$ are rings and if $\overline{R}_T$ (respectively, $\overline{R}_B$ ) is the integral closure of $R$ in $T$ (respectively, in $B$ ), then $(\overline{R}_T, T)$ is a normal pair if and only if $(\overline{R}_B, \overline{R}_BT)$ is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of arbitrary rings.     

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters \alpha =\beta =-1

In this paper, we consider the second-order differential expression $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $\alpha =\beta =-1$ in contrast to the classical case when $\alpha ,\beta >-1$ . For fixed $k\ge 0$ and appropriate values of the spectral parameter $\lambda ,$ the equation $\ell [y]=\lambda y$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions $\{P_{n}^{(-1,-1)} \}_{n=0}^{\infty }.$ These Jacobi polynomial solutions of degree $\ge 2$ form a complete orthogonal set in the Hilbert space $L^{2}((-1,1);(1-x^{2})^{-1})$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $\ge 0$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $T$ , generated by $\ell [\cdot ],$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $T$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.     

Norm Transitive Semigroups on Operator Algebra

Let $\mathcal A $ be a semigroup of bounded linear operators on the Banach algebra $B(X)$ for a separable Banach space $X$ . We show the transitivity of $\mathcal A $ with the operator norm topology, implies hypercyclicity with the strong operator topology (SOT) while the converse may not be true. As a consequence, SOT-transitive semigroup of left multiplication operators on $B(X)$ is characterized.