Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

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.     

On the pullback of an arithmetic theta function

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series ${\widehat{\phi}_{\mathcal C}(\tau)}$ defined using arithmetic 0-cycles on the moduli space ${\mathcal C}$ of elliptic curves with CM by the ring of integers ${O_{\kappa}}$ of an imaginary quadratic field. The second such series ${\widehat{\phi}_{\mathcal M}(\tau)}$ has weight 3/2 and takes values in the arithmetic Chow group ${\widehat{{\rm CH}}^1(\mathcal{M})}$ of the arithmetic surface associated to an indefinite quaternion algebra ${B/\mathbb{Q}}$ . For an embedding ${O_\kappa \rightarrow O_B}$ , a maximal order in B, and a two sided O _B -ideal Λ, there is a morphism ${j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}$ and a pullback ${j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}$ . Our main result is an expression for the pullback ${j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}$ as a linear combination of products of ${\widehat{\phi}_{\mathcal C}(\tau)}$ ’s and classical weight ${\frac{1}{2}}$ theta series.     

Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions

If ${\mathcal{A}}$ is a family of continuous functions on a locally compact Hausdorff space X, a boundary for ${\mathcal{A}}$ is a subset ${B \subset X}$ such that every ${f \in \mathcal{A}}$ attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for ${\mathcal{A}}$ . We give conditions under which all boundaries for ${\mathcal{A}}$ consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When ${\mathcal{A}}$ has sufficient algebraic structure it is proven that this construction provides a unique boundary for ${\mathcal{A}}$ consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras.     

Inductive Moving Frames

An inductive implementation of the equivariant moving frame method is introduced for both finite-dimensional Lie group actions and infinite-dimensional Lie pseudo-groups. Given two Lie (pseudo-)groups ${\mathcal{G}}$ and ${\mathcal{H}}$ with ${\mathcal{G} \subset \mathcal{H}}$ , the inductive method streamlines the construction of a moving frame for ${\mathcal{H}}$ using the already constructed moving frame for ${\mathcal{G}}$ . As a by-product, a systematic procedure for expressing ${\mathcal{H}}$ -invariant quantities in terms of their ${\mathcal{G}}$ -invariant counterparts is obtained.     

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra ${\mathcal{A}}$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that ${\mathcal{A}}$ separates points in the sense that for each distinct pair ${x, y \in X}$ , there exists an ${f \in \mathcal{A}}$ such that ${f(x) \neq f(y)}$ . If ${\mathcal{A}}$ does not separate points, it is known that there exists an algebra ${\widehat{\mathcal{A}}}$ on a compact Hausdorff space ${(\widehat{X}, \widehat{\tau})}$ that does separate points such that the map ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume ${\mathcal{A}}$ separates points. The construction of ${{\widehat{\mathcal{A}}}}$ and ${(\widehat{X}, \widehat{\tau})}$ does not require that ${\mathcal{A}}$ has any algebraic structure nor that ${(X, \tau)}$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family ${\mathcal{A}}$ of bounded, complex-valued, continuous functions on any topological space ${(X, \tau)}$ . We also demonstrate that further structures may be preserved by the mapping ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.     

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 ${\mathcal{L}}$ .     

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.     

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 ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

On Extending {\mathcal{A}}-Modules Through the Coefficients

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free ${\mathcal{A}}$ -modules, where ${\mathcal{A}}$ is an ${\mathbb{R}}$ -algebra sheaf. Complexification of free ${\mathcal{A}}$ -modules, which is defined to be the process of obtaining new free ${\mathcal{A}}$ -modules by enlarging the ${\mathbb{R}}$ -algebra sheaf ${\mathcal{A}}$ to a ${\mathbb{C}}$ -algebra sheaf, denoted ${\mathcal{A}_\mathbb{C}}$ , is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- ${_{\mathbb{R}}\mathcal{A}}$ -sheaf of almost complex structures on the sheaf ${{_\mathbb{R}}\mathcal{A}^{2n}}$ , the underlying ${\mathbb{R}}$ -algebra sheaf of a ${\mathbb{C}}$ -algebra sheaf ${\mathcal{A}}$ , and on the other hand, on the complexification of the functor ${\mathcal{H}om_\mathcal {A}}$ , with ${\mathcal{A}}$ an ${\mathbb{R}}$ -algebra sheaf.     

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 ${\mathbb{P\ast P}}$ .     

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.     

Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

Let ${(\mathcal{M}, \tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the problem ${\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}$ , where ${\varepsilon > 0}$ is a small parameter and V is a positive, smooth function in ${\mathcal{M}}$ . Let ${\kappa \subset \mathcal{M}}$ be an (N ? 1)-dimensional smooth submanifold that divides ${\mathcal{M}}$ into two disjoint components ${\mathcal{M}_{\pm}}$ . We assume κ is stationary and non-degenerate relative to the weighted area functional ${\int_{\kappa}V^{\frac{1}{2}}}$ . For each integer m ≥ 2, we prove the existence of a sequence ${\varepsilon = \varepsilon_\ell \rightarrow 0}$ , and two opposite directional solutions with m-transition layers near κ, whose mutual distance is ${{\rm O}(\varepsilon | \log \varepsilon | )}$ . Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

Generalized Jordan left derivations on semiprime algebras

Let ${\mathcal{A}}$ be a semiprime algebra of characteristic not 2. Then any generalized Jordan left derivation on ${\mathcal{A}}$ is a generalized left derivation and is also a generalized derivation. This gives an affirmative answer to a question in Ashraf and Ali (Bull Korean Math Soc 45:253–261, 2008). Moreover, we prove that there are no nonzero generalized Jordan left derivations that take only nilpotent values on ${\mathcal{A}}$ .     

Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$      

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.