Characterization of (generalized) Lie derivations on {\mathcal{J}}-subspace lattice algebras by local action

Let ${\mathcal{L}}$ be a ${\mathcal{J}}$ -subspace lattice on a Banach space X over the real or complex field ${\mathbb{F}}$ with dim X ≥ 2 and Alg ${\mathcal{L}}$ be the associated ${\mathcal{J}}$ -subspace lattice algebra. For any scalar ${\xi \in \mathbb{F}}$ , there is a characterization of any linear map L : Alg ${\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}$ satisfying ${L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}$ for any ${A, B \in{\rm Alg} {\mathcal{L}}}$ with AB = 0 (rep. ${[A,B]_ \xi = AB - \xi BA = 0}$ ) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg ${\mathcal{L}}$ is obtained.     

Forbidden configurations and Steiner designs

Let ${\mathcal{F}}$ be a (0, 1) matrix. A (0, 1) matrix ${\mathcal{M}}$ is said to have ${\mathcal{F}}$ as a configuration if there is a submatrix of ${\mathcal{M}}$ which is a row and column permutation of ${\mathcal{F}}$ . We say that a matrix ${\mathcal{M}}$ is simple if it has no repeated columns. For a given ${v \in \mathbb{N}}$ , we shall denote by forb ${(v, \mathcal{F})}$ the maximum number of columns in a simple (0, 1) matrix with v rows for which ${\mathcal{F}}$ does not occur as a configuration. We say that a matrix ${\mathcal{M}}$ is maximal for ${\mathcal{F}}$ if ${\mathcal{M}}$ has forb ${(v, \mathcal{F})}$ columns. In this paper we show that for certain natural choices of ${\mathcal{F}}$ , forb ${(v, \mathcal{F})\leq\frac{\binom{v}{t}}{t+1}}$ . In particular this gives an extremal characterization for Steiner t-designs as maximal (0, 1) matrices in terms of certain forbidden configurations.     

Atomic intersection of σ-fields and some of its consequences

Let ${(\Omega, \mathcal{F}, P)}$ be a probability space. For each ${\mathcal{G}\subset\mathcal{F}}$ , define ${\overline{\mathcal{G}}}$ as the σ-field generated by ${\mathcal{G}}$ and those sets ${F\in \mathcal{F}}$ satisfying ${P(F)\in\{0,1\}}$ . Conditions for P to be atomic on ${\cap_{i=1}^k\overline{\mathcal{A}_i}}$ , with ${\mathcal{A }_1,\ldots,\mathcal{A}_k\subset\mathcal{F}}$ sub-σ-fields, are given. Conditions for P to be 0-1-valued on ${\cap_{i=1}^k \overline{\mathcal{A}_i}}$ are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.     

Axiomatizability by {{\forall}{\exists}!}-sentences

A ${\forall\exists!}$ -sentence is a sentence of the form ${\forall x_{1}\cdots x_{n}\exists!y_{1}\cdots y_{m}O(\overline{x},\overline{y})}$ , where O is a quantifier-free formula, and ${\exists!}$ stands for ??there exist unique??. We prove that if ${\mathcal{C}}$ is (up to isomorphism) a finite class of finite models then ${\mathcal{C}}$ is axiomatizable by a set of ${\forall\exists!}$ -sentences if and only if ${\mathcal{C}}$ is closed under isomorphic images, ${\mathcal{C}}$ has the intersection property, and ${\mathcal{C}}$ is closed under fixed-point submodels. This result is employed to characterize the subclasses of finitely generated discriminator varieties axiomatizable by sentences of the form ${\forall\exists!\bigwedge p=q}$ .     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Saturating Sperner Families

A family ${\mathcal{F} \subseteq 2^{[n]}}$ saturates the monotone decreasing property ${\mathcal{P}}$ if ${\mathcal{F}}$ satisfies ${\mathcal{P}}$ and one cannot add any set to ${\mathcal{F}}$ such that property ${\mathcal{P}}$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.     

The universal Glivenko–Cantelli property

Let ${\mathcal{F}}$ be a separable uniformly bounded family of measurable functions on a standard measurable space ${(X, \mathcal{X})}$ , and let ${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$ be the smallest number of ${\varepsilon}$ -brackets in L ¹(μ) needed to cover ${\mathcal{F}}$ . The following are equivalent:

1. ${\mathcal{F}}$ is a universal Glivenko–Cantelli class.
2. ${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$ for every ${\varepsilon > 0}$ and every probability measure μ.
3. ${\mathcal{F}}$ is totally bounded in L ¹(μ) for every probability measure μ.
4. ${\mathcal{F}}$ does not contain a Boolean σ-independent sequence.

It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

Existence, automatic continuity and invariant submodules of generalized derivations on modules

Let ${\mathcal{A}}$ be a ${\mathbb{C}}$ -algebra, δ be a derivation on ${\mathcal{A}}$ and ${\mathcal{M}}$ be a left ${\mathcal{A}}$ -module. A linear map ${\tau : \mathcal{M} \rightarrow \mathcal{M}}$ is called a generalized derivation relative to δ if ${\tau(am)=a\tau(m)+\delta(a)m\,(a \in \mathcal{A}, m \in \mathcal{M})}$ . In this article first we study the existence of generalized derivations. In particular we show that free modules and projective modules always have nontrivial generalized derivations relative to nonzero derivations of ${\mathcal{A}}$ . Then we investigate the invariance of prime submodules under generalized derivations. Specifically we show that every minimal prime submodule of ${\mathcal{M}}$ is invariant under every generalized derivation. Moreover we obtain analogs of Posner’s theorem for generalized derivations. In the case that ${\mathcal{A}}$ is a Banach algebra and ${\mathcal{M}}$ is a Banach left ${\mathcal{A}}$ -module, we study the existence of continuous generalized derivations and automatic continuity of generalized derivations.     

On Minimum Saturated Matrices

Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let ${\mathcal{F}}$ be a family of k-row matrices. A matrix M is called ${\mathcal{F}}$ -admissible if M contains no submatrix ${F \in \mathcal{F}}$ (as a row and column permutation of F). A matrix M without repeated columns is ${\mathcal{F}}$ -saturated if M is ${\mathcal{F}}$ -admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat( ${n, \mathcal{F}}$ ) which is the minimal number of columns of an ${\mathcal{F}}$ -saturated matrix with n rows. We establish the estimate sat ${(n, \mathcal{F})=O(n^{k-1})}$ for any family ${\mathcal{F}}$ of k-row matrices and also compute the sat-function for a few small forbidden matrices.     

Forbidden Induced Subgraphs for Perfect Matchings

Let ${\mathcal{F}}$ be a family of connected graphs. A graph G is said to be ${\mathcal{F}}$ -free if G is H-free for every graph H in ${\mathcal{F}}$ . We study the problem of characterizing the families of graphs ${\mathcal{F}}$ such that every large enough connected ${\mathcal{F}}$ -free graph of even order has a perfect matching. This problems was previously studied in Plummer and Saito (J Graph Theory 50(1):1–12, 2005), Fujita et al. (J Combin Theory Ser B 96(3):315–324, 2006) and Ota et al. (J Graph Theory, 67(3):250–259, 2011), where the authors were able to characterize such graph families ${\mathcal{F}}$ restricted to the cases ${|\mathcal{F}|\leq 1, |\mathcal{F}| \leq 2}$ and ${|\mathcal{F}| \leq 3}$ , respectively. In this paper, we complete the characterization of all the families that satisfy the above mentioned property. Additionally, we show the families that one gets when adding the condition ${|\mathcal{F}| \leq k}$ for some k ≥ 4.     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Jordan Modules and Jordan Ideals of Reflexive Algebras

Let ${\mathcal{L}}$ be a completely distributive subspace lattice on a Banach space and alg ${\mathcal{L}}$ the associated reflexive algebra. Suppose that the following $$\mbox{Condition A:}\dim(F/F\wedge F_-)\ne1\;\; \mbox{for all}\;\;F\in\mathcal{L}$$ holds; note that if ${\mathcal{L}}$ is an atomic Boolean subspace lattice, this condition means that every atom of ${\mathcal{L}}$ has dimension at least two. It is shown that every reflexive Jordan Alg ${\mathcal{L}}$ -module is an associative Alg ${\mathcal{L}}$ -module. We give an example which shows that if the Condition A is removed, then the conclusion is not necessarily true. Moreover, we prove that all reflexive Jordan ideals of Alg ${\mathcal{L}}$ are associative ideals in the case that no the Condition A is assumed. The same conclusions hold for weakly closed Jordan modules and weakly closed Jordan ideals if the rank one subalgebra of Alg ${\mathcal{L}}$ is weakly dense in Alg ${\mathcal{L}}$ .     

A Geometry for the Set of Split Operators

We study the set ${\mathcal{X}}$ of split operators acting in the Hilbert space ${\mathcal{H}}$ : $$\mathcal{X}=\{T\in \mathcal{B}(\mathcal{H}): N(T)\cap R(T)=\{0\} \ {\rm and} \ N(T)+R(T)=\mathcal{H}\}.$$ Inside ${\mathcal{X}}$ , we consider the set ${\mathcal{Y}}$ : $$\mathcal{Y}=\{T\in\mathcal{X}: N(T)\perp R(T)\}.$$ Several characterizations of these sets are given. For instance ${T\in\mathcal{X}}$ if and only if there exists an oblique projection ${Q}$ whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. TS = ST, TST = T and STS = S). Analogous characterizations are given for ${\mathcal{Y}}$ . Two natural maps are considered: $${\bf q}:\mathcal{X} \to \mathbb{Q}:=\{{\rm oblique \ projections \ in} \, \mathcal{H} \}, \ {\bf q}(T)=P_{R(T)//N(T)}$$ and $${\bf p}:\mathcal{Y} \to \mathbb{P}:=\{{\rm orthogonal \ projections \ in} \ \mathcal{H} \}, \ {\bf p}(T)=P_{R(T)}, $$ where ${P_{R(T)//N(T)}}$ denotes the projection onto R(T) with nullspace N(T), and P _R(T) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets ${\mathcal{X}_{c_k}\subset \mathcal{X}}$ of operators with rank ${k<\infty}$ , and ${\mathcal{X}_{F_k}\subset\mathcal{X}}$ of Fredholm operators with nullity ${k<\infty}$ . For the map p there are analogous results. We show that the interior of ${\mathcal{X}}$ is ${\mathcal{X}_{F_0}\cup\mathcal{X}_{F_1}}$ , and that ${\mathcal{X}_{c_k}}$ and ${\mathcal{X}_{F_k}}$ are arc-wise connected differentiable manifolds.     

On a conjecture of Rapoport and Zink

In their book, Rapoport and Zink constructed rigid analytic period spaces ${\mathcal {F}}^{wa}$ for Fontaine’s filtered isocrystals, and period morphisms from PEL moduli spaces of p-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism ${\mathcal {F}}^{a}\to {\mathcal {F}}^{wa}$ of rigid analytic spaces and of a universal local system of ? _p -vector spaces on  ${\mathcal {F}}^{a}$ . Such a local system would give rise to a tower of étale covering spaces $\breve {{\mathcal {E}}}_{{\widetilde {K}}}$ of ${\mathcal {F}}^{a}$ , equipped with a Hecke-action, and an action of the automorphism group J(? _p ) of the isocrystal with extra structure. For Hodge-Tate weights n?1 and n we construct in this article an intrinsic Berkovich open subspace ${\mathcal {F}}^{0}$ of ${\mathcal {F}}^{wa}$ and the universal local system on ${\mathcal {F}}^{0}$ . We show that only in exceptional cases ${\mathcal {F}}^{0}$ equals all of ${\mathcal {F}}^{wa}$ and when the Shimura group is $\operatorname {GL}_{n}$ we determine all these cases. We conjecture that the rigid-analytic space associated with ${\mathcal {F}}^{0}$ is the maximal possible ${\mathcal {F}}^{a}$ , and that ${\mathcal {F}}^{0}$ is connected. We give evidence for these conjectures. For those period spaces possessing PEL period morphisms, we show that ${\mathcal {F}}^{0}$ equals the image of the period morphism. Then our local system is the rational Tate module of the universal p-divisible group and carries a J(? _p )-linearization. We construct the tower $\breve {{\mathcal {E}}}_{{\widetilde {K}}}$ of étale covering spaces, and we show that it is canonically isomorphic in a Hecke and J(? _p )-equivariant way to the tower constructed by Rapoport and Zink using the universal p-divisible group.     

Sharing Set and Normal Families of Entire Functions

Let ${\mathcal{F}}$ be a family of holomorphic functions defined in a domain ${\mathcal{D}}$ , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each ${f \in \mathcal{F}}$ , all zeros of f(z) are of multiplicity at least k, and f and f ^(k) share the set S in ${\mathcal{D}}$ , then ${\mathcal{F}}$ is normal in ${\mathcal{D}}$ . As an application, we prove a uniqueness theorem.     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

Complemented modular lattices with involution and orthogonal geometry

With each orthogeometry (P, ⊥) we associate ${{\mathbb {L}}(P, \bot)}$ , a complemented modular lattice with involution (CMIL), consisting of all subspaces X and X ^⊥ such that dim X < ?₀, and we study its rôle in decompositions of (P, ⊥) as directed (resp., disjoint) union. We also establish a 1–1 correspondence between ?-varieties ${\mathcal {V}}$ of CMILs with ${\mathcal {V}}$ generated by its finite dimensional members and ‘quasivarieties’ ${\mathcal {G}}$ of orthogeometries: ${\mathcal {V}}$ consists of the CMILs representable within some geometry from ${\mathcal {G}}$ and ${\mathcal {G}}$ of the (P, ⊥) with ${{\mathbb {L}}(P, \bot) \in {\mathcal {V}}}$ . Here, ${\mathcal {V}}$ is recursively axiomatizable if and only if so is ${\mathcal {G}}$ . It follows that the equational theory of ${\mathcal {V}}$ is decidable provided that the equational theories of the ${\{{\mathbb {L}}(P, \bot)\, |\, (P, \bot) \in \mathcal {G}, {\rm{dim}} P = n\}}$ are uniformly decidable.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .