Local derivations of full matrix rings

Let \({\mathcal{R}}\) be a unital commutative ring and \({\mathcal{M}}\) be a 2-torsion free central \({\mathcal{R}}\) -bimodule. In this paper, for \({n \geqq 3}\), we show that every local derivation from M _n (\({\mathcal{R}}\)) into M _n (\({\mathcal{M}}\)) is a derivation.     

Qualitative properties of ideal convergent subsequences and rearrangements

We investigate the Baire category of \({\mathcal{I}}\)-convergent subsequences and rearrangements of a divergent sequence s = (s_n) of reals if \({\mathcal{I}}\) is an ideal on \({\mathbb{N}}\) having the Baire property. We also discuss the measure of the set of \({\mathcal{I}}\)-convergent subsequences for some classes of ideals on \({\mathbb{N}}\). Our results generalize theorems due to H. Miller and C. Orhan [16].     

M-$${\mathcal{}}$$-Injective (Flat) and Strongly $${\mathcal{B}}$$B-Injective (Flat) Modules

Let M be a left R-module, \({\mathcal{A}}\)be a family of some submodules of M and \({\mathcal{B}}\)be a family of some left R-modules. In this article, we introduce and characterize \({\mathcal{A}}\)-coherent, \({P\mathcal{A}}\), \({F\mathcal{A}}\), M-\({\mathcal{A}}\)-injective (flat) and strongly \({\mathcal{B}}\)-injective (flat) modules, which are generalizations of coherent, PS, FS, M-injective (flat) and strongly M-injective modules, respectively. We extend some known results to this general structure.     

Partially isometric immersions and free maps

In this paper we investigate the existence of “partially” isometric immersions. These are maps \({f:M\rightarrow \mathbb{R}^q}\) which, for a given Riemannian manifold M, are isometries on some sub-bundle \({\mathcal{H}\subset TM}\). The concept of free maps, which is essential in the Nash–Gromov theory of isometric immersions, is replaced here by that of \({\mathcal{H}}\) –free maps, i.e. maps whose restriction to \({\mathcal{H}}\) is free. We prove, under suitable conditions on the dimension q of the Euclidean space, that \({\mathcal{H}}\) –free maps are generic and we provide, for the smallest possible value of q, explicit expressions for \({\mathcal{H}}\) –free maps in the following three settings: 1–dimensional distributions in \({\mathbb{R}^2}\), Lagrangian distributions of completely integrable systems, Hamiltonian distributions of a particular kind of Poisson Bracket.     

A multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians

Let M be a closed and connected manifold, \(H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}\) a Tonelli 1-periodic Hamiltonian and \({\mathscr {L}}\subset T^*M\) a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if \({\mathscr {L}}\) is invariant by the time-one map of H, then \({\mathscr {L}}\) is a graph over M. An interesting consequence in the autonomous case is that in this case, \({\mathscr {L}}\) is invariant by all the time t maps of the Hamiltonian flow of H.     

A deterministic sparse FFT algorithm for vectors with small support

In this paper we consider the special case where a signal x\({\in }\,\mathbb {C}^{N}\) is known to vanish outside a support interval of length m < N. If the support length m of x or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute x from its discrete Fourier transform \(\widehat {\mathbf x}\,{\in }\,\mathbb {C}^{N}\). In case of exact Fourier measurements we require only \({\mathcal O}\)(m\(\log \)m) arithmetical operations. For noisy measurements, we propose a stable \({\mathcal O}\)(m\(\log \)N) algorithm.     

Spectral properties of continuous representations of topological groups

Let \({\mathcal{L}(X)}\) be the algebra of all bounded operators on a Banach space X. \({\theta:G\rightarrow \mathcal{L}(X)}\) denotes a strongly continuous representation of a topological abelian group G on X. Set \({\sigma^1(\theta(g)):=\{\lambda/|\lambda|,\lambda\in\sigma(\theta(g))\}}\), where σ(θ(g)) is the spectrum of θ(g) and \({\Sigma:=\{g\in G/\enskip\text{there is no} \enskip P\in \mathcal{P}/P\subseteq \sigma^1(\theta(g))\}}\), where \({\mathcal{P}}\) is the set of regular polygons of \({\mathbb{T}}\) (we call polygon in \({\mathbb{T}}\) the image by a rotation of a closed subgroup of \({\mathbb{T}}\), the unit circle of \({\mathbb{C}}\)). We prove here that if G is a locally compact and second countable abelian group, then θ is uniformly continuous if and only if Σ is non-meager.     

On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

Spherical t-designs are point sets \(X_{M}=\{x_{1},\ldots, x_{M}\} \subset {\mathbb{S}^{2}}\) which provide quadrature rules with equal weights for the sphere which are exact for polynomials up to degree t. In this paper we consider the problem of finding numerical spherical t-designs on the sphere \({\mathbb{S}^{2}}\) for high polynomial degree \({t \in {\mathbb{N}}}\) . That is, we compute numerically local minimizers of a certain quadrature error A _t (X _M ). The quadrature error A _t was also used for a variational characterization of spherical t-designs by Sloan and Womersley (J Approx Theory 159:308–318, 2009). For the minimization problem we regard several nonlinear optimization methods on manifolds, like Newton and conjugate gradient methods. We show that by means of the nonequispaced fast spherical Fourier transforms we perform gradient and Hessian evaluations in \({\mathcal {O}(t^{2}\log t + M \log^{2}(1/\varepsilon))}\) arithmetic operations, where \({\varepsilon >0 }\) is a prescribed accuracy. Using these methods we present numerical spherical t-designs for t ≤  1,000, even in the case \({M\approx \frac{1}{2} t^{2}}\) .     

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

On a Class of Weakly Coupled Systems of Elliptic Operators with Unbounded Coefficients

We consider a class of weakly coupled systems of elliptic operators \({\mathcal{A}}\) with unbounded coefficients defined in \({\mathbb{R}^N}\). We prove that a semigroup (T(t))t ≥ 0 of bounded linear operators can be associated with \({\mathcal{A}}\), in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T(t)f, when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator \({\mathcal{A}}\). Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L ^p -spaces over \({\mathbb{R}^N}\), related to the Lebesgue measure, when \({p \in [1,\infty)}\). We also provide sufficient conditions for this semigroup to be analytic when \({p \in [1,\infty)}\). Finally, we prove some L ^p ?L ^q -estimates.     

Naturally dualizable algebras omitting types 1 and 5 have a cube term

An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if \({\mathcal{V}(\mathbb{A})}\) is congruence distributive and \({\mathbb{A}}\) is dualizable, then \({\mathbb{A}}\) has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if \({\mathbb{A}}\) omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then \({\mathbb{A}}\) has a cube term.     

A Unified Framework for Linear Dimensionality Reduction in L1

For a family of interpolation norms \({\| \cdot \|_{1,2,s}}\) on \({\mathbb{R}^{n}}\), we provide a distribution over random matrices \({\Phi_s \in \mathbb{R}^{m \times n}}\) parametrized by sparsity level s such that for a fixed set X of K points in \({\mathbb{R}^{n}}\), if \({m \geq C s \log(K)}\) then with high probability, \({\frac{1}{2}\| \varvec{x} \|_{1,2,s} \leq \| \Phi_s (\varvec{x}) \|_1 \leq 2 \| \varvec{x} \|_{1,2,s}}\) for all \({\varvec{x} \in X}\). Several existing results in the literature roughly reduce to special cases of this result at different values of s: For s = n, \({\| \varvec{x} \|_{1,2,n}\equiv \| \varvec{x} \|_{1}}\) and we recover that dimension reducing linear maps can preserve the ?₁-norm up to a distortion proportional to the dimension reduction factor, which is known to be the best possible such result. For s = 1, \({\| \varvec{x} \|_{1,2,1}\equiv \| \varvec{x} \|_{2}}\), and we recover an ?₂/?₁ variant of the Johnson–Lindenstrauss Lemma for Gaussian random matrices. Finally, if \({\varvec{x}}\) is s- sparse, then \({\| \varvec{x} \|_{1,2,s} = \| \varvec{x} \|_1}\) and we recover that s-sparse vectors in \({\ell_1^n}\) embed into \({\ell_1^{\mathcal{O}(s \log(n))}}\) via sparse random matrix constructions.     

Joins and subdirect products of varieties

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and P?onka proved that independent subvarieties \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) of a variety \({\mathcal{V}}\) are disjoint and such that their join \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) (in the lattice of subvarieties of \({\mathcal{V}}\)) is their direct product \({\mathcal{V}_{1} \times \mathcal{V}_{2}}\) . Jónsson and Tsinakis provided a partial converse to this result: if \({\mathcal{V}}\) is congruence permutable and \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) are disjoint, then they are independent (and so \({\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}\)). We show that (i) if \({\mathcal{V}}\) is subtractive, then Jónsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if \({\mathcal{V}}\) satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) is the subdirect product of \({\mathcal{V}_{1}}\) and \({\mathcal{V}_2}\) ; (iii) the same holds if \({\mathcal{V}}\) is congruence 3-permutable.     

The commutator in equivalential algebras and Fregean varieties

A class \({\mathcal {K}}\) of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in \({\mathcal {K}}\) are uniquely determined by their 0-cosets and Θ _A (0, a) = Θ _A (0, b) implies a = b for all \({a, b \in {\bf A} \in \mathcal {K}}\) . The structure of Fregean varieties was investigated in a paper by P. Idziak, K. S?omczyńska, and A. Wroński. In particular, it was shown there that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e., algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. In this paper we give a full characterization of the commutator for equivalential algebras and solvable Fregean varieties. In particular, we show that in a solvable algebra from a Fregean variety, the commutator coincides with the commutator of its purely equivalential reduct. Moreover, an intrinsic characterization of the commutator in this setting is given.     

Mutually prime sequences of inner functions and the ranks of subspaces over the bidisk

Let \({\{\varphi_n(z)\}_{n\ge0}}\) be a sequence of inner functions satisfying that \({\zeta_n(z):=\varphi_n(z)/\varphi_{n+1}(z)\in H^\infty(z)}\) for every n ≥ 0 and \({\{\varphi_n(z)\}_{n\ge0}}\) have no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace \({\mathcal{M}}\) of \({H^2(\mathbb{D}^2)}\) . We write \({\mathcal{N}= H^2(\mathbb{D}^2)\ominus\mathcal{M}}\) . If \({\{\zeta_n(z)\}_{n\ge0}}\) ia a mutually prime sequence, then we shall prove that \({rank_{\{T^\ast_z,T^\ast_w\}} \mathcal{N}=1}\) and \({rank_{\{\mathcal{F}^\ast_z\}}(\mathcal{M}\ominus w\mathcal{M})=1}\) , where \({\mathcal{F}_z}\) is the fringe operator on \({\mathcal{M}\ominus w\mathcal{M}}\) .     

Weighted Poincaré inequality and heat kernel estimates for finite range jump processes

It is well-known that there is a deep interplay between analysis and probability theory. For example, for a Markovian infinitesimal generator \({\mathcal{L}}\) , the transition density function p(t, x, y) of the Markov process associated with \({\mathcal{L}}\) (if it exists) is the fundamental solution (or heat kernel) of \({\mathcal{L}}\) . A fundamental problem in analysis and in probability theory is to obtain sharp estimates of p(t, x, y). In this paper, we consider a class of non-local (integro-differential) operators \({\mathcal{L}}\) on \({\mathbb{R}^d}\) of the form

$\mathcal{L}u(x) = \lim\limits_{{\varepsilon \downarrow 0}} \int\limits_{\{y\in \mathbb {R}^d: \, |y-x| > \varepsilon\}} (u(y)-u(x)) J(x, y) dy,$

where \({\displaystyle J(x, y)= \frac{c (x, y)}{|x-y|^{d+\alpha}} {\bf 1}_{\{|x-y| \leq \kappa\}}}\) for some constant \({\kappa > 0}\) and a measurable symmetric function c(x, y) that is bounded between two positive constants. Associated with such a non-local operator \({\mathcal{L}}\) is an \({\mathbb{R}^d}\) -valued symmetric jump process of finite range with jumping kernel J(x, y). We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for them. Along the way, some new heat kernel estimates are obtained for more general finite range jump processes that were studied in (Barlow et al. in Trans Am Math Soc, 2008). One of our key tools is a new form of weighted Poincaré inequality of fractional order, which corresponds to the one established by Jerison in (Duke Math J 53(2):503–523, 1986) for differential operators. Using Meyer’s construction of adding new jumps, we also obtain various a priori estimates such as Hölder continuity estimates for parabolic functions of jump processes (not necessarily of finite range) where only a very mild integrability condition is assumed for large jumps. To establish these results, we employ methods from both probability theory and analysis extensively.

     

Character Connes-amenability of dual Banach algebras

The concept of left character Connes-amenability for a dual Banach algebra \({\mathcal {A}}\) is introduced. We obtain a cohomological characterization of left character Connes-amenability as well as the relation between left \(\varphi \)-Connes-amenability and existence of left \(\varphi \)-normal virtual diagonals for a \(\omega ^{*}\)-continuous character \(\varphi \). We prove that left character amenability of \({\mathcal {A}}\) is equivalent to left character Connes-amenability of \({\mathcal {A}}^{**}\) when \({\mathcal {A}}\) is Arens regular. Moreover for a locally compact group G, we show that M(G) is left character Connes-amenable. In addition by means of some examples we show that for the new notion, the corresponding class of dual Banach algebras is larger than Connes-amenable dual Banach algebras.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).