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.     

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q ₁(r) + q ₂(r) such that q ₁(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane.     

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.     

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 \({T - 2}\).     

Finite groups with small automizers of {\mathcal {F}}-subgroups

Let \({ \mathcal {F}}\) be a saturated formation and G a finite group such that \({N_{G} (H^{\mathcal {F}})/C_{G} (H^{\mathcal {F}})\cong Inn(H^{\mathcal {F}})}\) for every subgroup H of G. If the minimal non-\({ \mathcal {F}}\)-group is soluble, then \({G \in \mathcal {F}}\).     

Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.     

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.     

A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B′in separable complex Hilbert spaces H and H′, respectively. It is found that if the mapping f ∈ C~(1+α)at z_0∈ ?B with f(z_0) = w_0∈ ?B′, then the Fr′echet derivative operator Df(z_0) maps the tangent space Tz_0(?B~n) to Tw_0(?B′), the holomorphic tangent space T_(z_0)~(1,0)(?B~n) to T_(w_0)~(1,0)(?B′),respectively.     

Lifting Fixed Points of Completely Positive Semigroups

Let \({\{\phi_s\}_{s\in S}}\) be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra N. Assume there exists a semigroup \({\{\alpha_s\}_{s\in S}}\) of weak*-continuous *-endomorphisms of some larger von Neumann algebra \({M\supset N}\) and a projection \({p\in M}\) with N = pMp such that α _s (1 ? p) ≤ 1 ? p for every \({s\in S}\) and \({\phi_s(y)=p\alpha_s(y)p}\) for all \({y\in N}\). If \({\inf_{s \in S}\alpha_s(1-p)=0}\) then we show that the map \({E:M\to N}\) defined by E(x) = pxp for \({x\in M}\) induces a complete isometry between the fixed point spaces of \({\{\alpha_s\}_{s\in S}}\) and \({\{\phi_s\}_{s\in S}}\).     

<Emphasis Type="Italic">k</Emphasis>-Extreme Points in Symmetric Spaces of Measurable Operators

Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

On Multiple and Infinite Log-Concavity

Following Boros-Moll, a sequence (a _n ) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j =  0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality \({a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}\) for large enough r. On the other hand, a recent result of Brändén shows that ∞-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator \({\mathcal{L}}\) and are therefore ∞-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators \({\mathcal{L}}\) and \({\mathcal{L}^{2}}\) are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become ∞-log-concave if convoluted with themselves a finite number of times.     

On the existence of self-complementary and non-self-complementary strongly regular graphs with Paley parameters

For p an odd prime, let \({{\mathcal A}_{p}}\) be the complete classical affine association scheme whose associate classes correspond to parallel classes of lines in the classical affine plane AG(2, p). It is known that \({{\mathcal A}_{p}}\) is an amorphic association scheme. We investigate rank 3 fusion schemes of \({{\mathcal A}_{p}}\) whose basis graphs have the same parameters as the Paley graphs \({P(p^{2})}\). In contrast to the Paley graphs, the great majority of graphs we detect are non-self-complementary and non-Schurian. In particular, existence of non-self-complementary graphs with Paley parameters is established for \({p \ge 17}\), with an analogous existence result for non-Schurian such graphs when \({p \ge 11}\). We demonstrate that the number of self-complementary and non-self-complementary strongly regular graphs with Paley parameters grows rapidly as \({p \to \infty}\).     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

Similarity Classification and Properties of Some Extended Holomorphic Curves

For \({\Omega\subseteq \mathbb{C}}\) a connected open set, and \({{\mathcal U}}\) a unital Banach algebra (or a unital C*-algebra), let \({{\xi (U)}}\) and \({ P({\mathcal U})}\) denote the sets of all idempotents and projections in \({{\mathcal U}}\), respectively. If \({e:\Omega\rightarrow \xi ({\mathcal U})}\) (resp.\({P({\mathcal U}))}\) is a holomorphic \({{\mathcal U}}\)-valued map, then e is called an extended holomorphic curve on \({ \xi ({\mathcal U})}\) (resp. \({P({\mathcal U})}\)). In this article, we focus on discussing the similarity classification problem of extended holomorphic curves. First, we introduce the definition of the commutant of extended holomorphic curves. By using K ₀-group of the commutant of the extended holomorphic curve, we characterize the curve which has unique finite (SI) decomposition up to similarity. Subsequently, we also obtain a similarity classification theorem. Second, we also discuss the unitary equivalence problem of some curves with respect to inductive limit C*-algebras.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Remarks on Complex Symmetric Operators

An operator \({T\in{\mathcal{L}}({\mathcal{H}})}\) is said to be complex symmetric if there exists a conjugation C on \({{\mathcal H}}\) such that \({T= CT^{\ast}C}\). In this paper, we study the spectral radius algebras for complex symmetric operators. In particular, we prove that if A is a complex symmetric operator, then the spectral radius algebra \({{\mathcal B}_{A}}\) associated with A has a nontrivial invariant subspace under some conditions. Finally, we give some relations between \({P_{\tilde{A}}}\) and \({P_{\widetilde{A^{\ast}}}}\) (defined below) when A is complex symmetric.     

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

In this paper the authors study the boundedness for a large class of sublinear operators \({T_{\alpha}, \alpha \in [0,n)}\) generated by Calderón–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces \({M_{p,\varphi}}\) . As an application of the above result, the boundeness of the commutator of sublinear operators \({T_{b,\alpha}, \alpha \in [0,n)}\) on generalized Morrey spaces is also obtained. In the case \({b \in BMO}\) and T _b,α is a sublinear operator, we find the sufficient conditions on the pair \({(\varphi_1,\varphi_2)}\) which ensures the boundedness of the operators \({T_{b,\alpha}, \alpha \in [0,n)}\) from one generalized Morrey space \({M_{p,\varphi_1}}\) to another \({M_{q,\varphi_2}}\) with 1/p ? 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({(\varphi_1,\varphi_2)}\) , which do not assume any assumption on monotonicity of \({\varphi_1, \, \varphi_2}\) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.     

Schatten-Herz Operators,Berezin Transform and Mixed Norm Spaces

For p, q > 0 we study operators T on the Bergman space \({A_{2}(\mathbb{D)}}\) in the disk such that \({\left(\sum_{j}\Vert T\Delta_{j}\Vert_{p}^{q}\right)^{1/q}<\infty,}\) where the norms \({\Vert\cdot\Vert_{p}}\) are in the Schatten class S _p (A ₂), the projection \({\Delta_{j}f=\sum_{n\in I_{j}}a_{n}z^{n}}\) for \({f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}}\) and \({I_{j}=[2^{j}-1,2^{j+1} )\cap(\mathbb{N}\cup\{0\})}\) for \({j\in\mathbb{N}\cup\{0\}.}\) We consider the relation of this property with mixed norms of the Berezin transform of T and of the related function \({f_{T}(z)={\Vert}T(k_{z})\Vert}\) where k _z is the normalized Bergman kernel. These classes of operators denoted by S(p, q) are closely related when assumed to be positive with other sets of operators, like the class of positive operators on A ₂ for which \({\left(\sum_{j\geq0}(\sum_{n\in I_{j}}|\left\langle T^pe_{n},e_{n}\right\rangle |)^{q/p}\right)^{1/q}<\infty}\) , where \({\{e_{n}\}_{n\geq0}}\) is the canonical basis of A ₂; also we study the relation of Toeplitz operators in S(p, q) with the Schatten-Herz classes, where the decomposition is through dyadic annuli of the domain \({\mathbb{D}}\) .