Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields

For the extended Dirichlet space \(\mathcal {F}_{e}\) of a general irreducible recurrent regular Dirichlet form \((\mathcal {E},\mathcal {F})\) on L ²(E;m), we consider the family \(\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}\) of centered Gaussian random variables defined on a probability space \(({\Omega }, \mathcal {B}, \mathbb {P})\) indexed by the elements of \(\mathcal {F}_{e}\) and possessing the Dirichlet form \(\mathcal {E}\) as its covariance. We formulate the Markov property of the Gaussian field \(\mathbb {G}(\mathcal {E})\) by associating with each set A ? E the sub-σ-field σ(A) of \(\mathcal {B}\) generated by X _u for every \(u\in \mathcal {F}_{e}\) whose spectrum s(u) is contained in A. Under a mild absolute continuity condition on the transition function of the Hunt process associated with \((\mathcal {E}, \mathcal {F})\), we prove the equivalence of the Markov property of \(\mathbb {G}(\mathcal {E})\) and the local property of \((\mathcal {E},\mathcal {F})\). One of the key ingredients in the proof is in that we construct potentials of finite signed measures of zero total mass and show that, for any Borel set B with m(B) >?0, any function \(u\in \mathcal {F}_{e}\) with s(u) ? B can be approximated by a sequence of potentials of measures supported by B.     

Supercyclicity in Spaces of Operators

A sufficient criterion for the map \({C_{A, B}(S) = ASB}\) to be supercyclic on certain algebras of operators on Banach spaces is given. If T is an operator satisfying the Supercyclicity Criterion on a Hilbert space H, then the linear map \({C_{T}(V) = TVT^*}\) is shown to be norm-supercyclic on the algebra \({\mathcal{K}(H)}\) of all compact operators, COT-supercyclic on the real subspace \({\mathcal{S}(H)}\) of all self-adjoint operators and weak*-supercyclic on \({\mathcal{L}(H)}\) of all bounded operators on H. Examples including operators of the form \({C_{B_w, F_\mu}}\) are provided, where B_w and \({F_\mu}\) are respectively backward and forward shifts on Banach sequence spaces.     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

Conflict free colorings of nonuniform systems of infinite sets

If \(\mathcal{H}\) is a system of infinite sets, |A∩B|<r for \({A\ne B\in\mathcal{H}}\) (r<ω) then \(\mathcal{H}\) has a conflict free coloring with ω colors, i.e., a function \(F\colon {\bigcup\mathcal{H}\to\omega}\) so that each \(A\in\mathcal{H}\) has a color i<ω with |F ^?1(i)∩A|=1.     

Resolutions of Hilbert Modules and Similarity

Let \(H^{2}_{m}\) be the Drury–Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function \((z, w) \in\mathbb{B}^{m} \times\mathbb{B}^{m} \rightarrow (1 - \sum_{i=1}^{m}z_{i} \bar{w}_{i})^{-1}\). We investigate for which multipliers \(\theta: \mathbb{B}^{m} \rightarrow \mathcal{L}(\mathcal{E}, \mathcal {E}_{*})\) with ran?M _θ closed, the quotient module \(\mathcal{H}_{\theta}\), given by

$\cdots\longrightarrow H^2_m \otimes\mathcal{E} \stackrel{M_{\theta }}{\longrightarrow}H^2_m \otimes\mathcal{E}_* \stackrel{\pi_{\theta}}{\longrightarrow}\mathcal{H}_{\theta}\longrightarrow0,$

is similar to \(H^{2}_{m} \otimes \mathcal {F}\) for some Hilbert space \(\mathcal{F}\). Here M _θ is the corresponding multiplication operator in \(\mathcal{L}(H^{2}_{m} \otimes\mathcal{E}, H^{2}_{m} \otimes\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) and \(\mathcal {H}_{\theta}\) is the quotient module \((H^{2}_{m} \otimes\mathcal{E}_{*})/ M_{\theta}(H^{2}_{m} \otimes\mathcal{E})\), and π _θ is the quotient map. We show that a necessary condition is the existence of a multiplier ψ in \(\mathcal{M}(\mathcal{E}_{*}, \mathcal{E})\) such that

$\theta\psi\theta= \theta.$

Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of \(H^{2}_{m} \otimes\mathcal{E}\) for a Hilbert space \(\mathcal {E}\), which is valid for the case of m=1. The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a finite resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator m-tuple (or algebra) is not necessarily commuting (or commutative). In this case the converse to the similarity result is always valid.

     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).     

Weak module amenability of triangular Banach algebras

Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra \(\mathcal{T} = \left[ {_B^{AM} } \right]\) and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When \(\mathfrak{A}\) is a Banach algebra and A and B are Banach \(\mathfrak{A}\)-module with compatible actions, and M is a commutative left Banach \(\mathfrak{A}\)-A-module and right Banach \(\mathfrak{A}\)-B-module, we show that A and B are weakly \(\mathfrak{A}\)-module amenable if and only if triangular Banach algebra T is weakly \(\mathfrak{T}\)-module amenable, where \(\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} \).     

Relative Projectivity and Transferability for Partial Lattices

A partial lattice P is ideal-projective, with respect to a class \(\mathcal {C}\) of lattices, if for every \(K\in \mathcal {C}\) and every homomorphism φ of partial lattices from P to the ideal lattice of K, there are arbitrarily large choice functions f:P→K for φ that are also homomorphisms of partial lattices. This extends the traditional concept of (sharp) transferability of a lattice with respect to \(\mathcal {C}\). We prove the following: (1) A finite lattice P, belonging to a variety \(\mathcal {V}\), is sharply transferable with respect to \(\mathcal {V}\) iff it is projective with respect to \(\mathcal {V}\) and weakly distributive lattice homomorphisms, iff it is ideal-projective with respect to \(\mathcal {V}\), (2) Every finite distributive lattice is sharply transferable with respect to the class \(\mathcal {R}_{\text {mod}}\) of all relatively complemented modular lattices, (3) The gluing D ₄ of two squares, the top of one being identified with the bottom of the other one, is sharply transferable with respect to a variety \(\mathcal {V}\) iff \(\mathcal {V}\) is contained in the variety \(\mathcal {M}_{\omega }\) generated by all lattices of length 2, (4) D ₄ is projective, but not ideal-projective, with respect to \(\mathcal {R}_{\text {mod}}\) , (5) D ₄ is transferable, but not sharply transferable, with respect to the variety \(\mathcal {M}\) of all modular lattices. This solves a 1978 problem of G. Grätzer, (6) We construct a modular lattice whose canonical embedding into its ideal lattice is not pure. This solves a 1974 problem of E. Nelson.     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

Lie n-derivatio ns o n $ {\mathcal {}}$-subspace lattice algebras

Let \(\mathcal {L}\) be a \(\mathcal {J}\)-subspace lattice on a Banach space X over the real or complex field \(\mathbb {F}\) with dimX ≥ 3 and let n ≥ 2 be an integer. Suppose that dimK ≠ 2 for every \(K\in \mathcal {J}{(\mathcal L)}\) and \(L: \text {Alg}\, \mathcal {L}\rightarrow \text {Alg}\,\mathcal {L}\) is a linear map. It is shown that L satisfies \({\sum }_{i=1}^{n}p_{n} (A_{1}, \ldots , A_{i-1}, L(A_{i}), A_{i+1}, \ldots , A_{n})=0\) whenever p _n (A ₁,A ₂,…,A _n ) = 0 for \(A_{1},A_{2},\ldots ,A_{n}\in \text {Alg}\,\mathcal {L}\) if and only if for each \(K\in \mathcal {J}(\mathcal {L})\), there exists a bounded linear operator \(T_{K}\in \mathcal {B}(K)\), a scalar λ _K and a linear functional \(h_{K}: \text {Alg}\,\mathcal {L}\rightarrow \mathbb {F}\) such that L(A)x = (T _K A ? A T _K + λ _K A + h _K (A)I)x for all x ∈ K and all \(A\in \text {Alg}\,\mathcal {L}\). Based on this result, a complete characterization of linear n-Lie derivations on \(\text {Alg}\,\mathcal {L}\) is obtained.     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

Resilience of ranks of higher inclusion matrices

Let \(n \ge r \ge s \ge 0\) be integers and \(\mathcal {F}\) a family of r-subsets of [n]. Let \(W_{r,s}^{\mathcal {F}}\) be the higher inclusion matrix of the subsets in \({{\mathcal {F}}}\) vs. the s-subsets of [n]. When \(\mathcal {F}\) consists of all r-subsets of [n], we shall simply write \(W_{r,s}\) in place of \(W_{r,s}^{\mathcal {F}}\). In this paper we prove that the rank of the higher inclusion matrix \(W_{r,s}\) over an arbitrary field K is resilient. That is, if the size of \(\mathcal {F}\) is “close” to \({n \atopwithdelims ()r}\) then \({{\mathrm{rank}}}_{K}( W_{r,s}^{\mathcal {F}}) = {{\mathrm{rank}}}_{K}(W_{r,s})\), where K is an arbitrary field. Furthermore, we prove that the rank (over a field K) of the higher inclusion matrix of r-subspaces vs. s-subspaces of an n-dimensional vector space over \({\mathbb {F}}_q\) is also resilient if \(\mathrm{char}(K)\) is coprime to q.     

Nonlinear *-Lie-type derivations on standard operator algebras

Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{A}\) be a standard operator algebra on \(\mathcal{H}\) which is closed under the adjoint operation. It is shown that each nonlinear *-Lie-type derivation δ on \(\mathcal{A}\) is a linear *-derivation. Moreover, δ is an inner *-derivation as well.     

Dual geometry of Cartan distribution

The paper is devoted to the study of intrinsic geometry of a Cartan distribution \(\mathcal{M}\) in projective space P_2m . We essentially use the hyperband distribution \(\mathcal{H}\) and P_2m associated with \(\mathcal{M}\). Using the duality theory, we construct, in the 4th differential neighborhood, a series of normalizations of \(\mathcal{M}\). We also consider dual affine connections \(\mathop \nabla \limits^1 \) and \(\mathop \nabla \limits^2 \) induced by the dual normalization of the Cartan distribution \(\mathcal{M}\).     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.     

Extremes of Gaussian random fields with regularly varying dependence structure

Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ ²(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t ₀. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈?, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.