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.     

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle \({\mathcal {E}}\) on X. Let \(\mathfrak {h}\) be the Lie algebra of H. Let \(\mathcal {S}(X,{\mathcal {E}})\) be the space of Schwartz sections of \({\mathcal {E}}\). We prove that \(\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})\) is a closed subspace of \(\mathcal {S}(X,{\mathcal {E}})\) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let \(\pi \) be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants \(V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}\) is an isomorphism if and only if any linear \(\mathfrak {h}\)-invariant functional on V is continuous in the topology induced from \(\pi \). The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.

     

Some Properties of Approximately Dual Frames in Hilbert Spaces

In this paper, a new characterization is obtained for approximately dual frames of a given frame. Among other things, it is proved that if the sequence \({\Psi=(\psi_n)_n}\) is sufficiently close to the frame \({\Phi=(\varphi_n)_n}\), then \({\Psi}\) is a frame for \({\mathcal{H}}\) and approximately dual frames \({\Phi^{ad}=(\varphi^{ad}_n)_n}\) and \({\Psi^{ad}=(\psi^{ad}_n)_n}\) can be found which are close to each other and \({T_\Phi U_{\Phi^{ad}}=T_\Psi U_{\Psi^{ad}}}\), where T_X and U_X denote the synthesis and analysis operators of the frame X, respectively. Finally, the results are applied to Gabor systems to obtain some practical examples.     

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.     

Fixed points indices and period-doubling cascades

Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, \({F : \mathbb{R}\times\mathfrak{M} \rightarrow \mathfrak{M},}\) where \({\mathfrak{M}}\) is a locally compact manifold without boundary, typically \({\mathbb{R}^N}\). In particular, we investigate F(μ, ·) for \({\mu \in J = [\mu_{1}, \mu_{2}]}\), when F(μ ₁, ·) has only finitely many periodic orbits while F(μ ₂, ·) has exponential growth of the number of periodic orbits as a function of the period. For generic F, under additional hypotheses, we use a fixed point index argument to show that there are infinitely many “regular” periodic orbits at μ ₂. Furthermore, all but finitely many of these regular orbits at μ ₂ are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ ₂ lies in a connected component C(ρ) of regular orbits in \({J \times \mathfrak{M}}\); different regular orbits typically are contained in different components, and each component contains a period-doubling cascade. These components are one-manifolds of orbits, meaning that we can reasonably say that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ ₂) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in \({J \times \mathfrak{M}}\).As corollaries of our main results, we give several examples, we prove that the map in each example has infinitely many cascades, and we count the cascades.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

Stringy K-theory and the Chern character

We construct two new G-equivariant rings: \(\mathcal{K}(X,G)\), called the stringy K-theory of the G-variety X, and \(\mathcal{H}(X,G)\), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack \(\mathcal{X}\), we also construct a new ring \(\mathsf{K}_{\mathrm{orb}}(\mathcal{X})\) called the full orbifold K-theory of \(\mathcal{X}\). We show that for a global quotient \(\mathcal{X} = [X/G]\), the ring of G-invariants \(K_{\mathrm{orb}}(\mathcal{X})\) of \(\mathcal{K}(X,G)\) is a subalgebra of \(\mathsf{K}_{\mathrm{orb}}([X/G])\) and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading.We prove that there is a ring isomorphism \(\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)\), which we call the stringy Chern character. We also show that there is a ring homomorphism \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\), which we call the orbifold Chern character, which induces an isomorphism \(Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\) when restricted to the sub-algebra \(K_{\mathrm{orb}}(\mathcal{X})\). Here \(H_{\mathrm{orb}}^\bullet(\mathcal{X})\) is the Chen–Ruan orbifold cohomology. We further show that \(\mathcal{C}\mathbf{h}\) and \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}\) preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.We further prove that \(\mathcal{H}(X,G)\) is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.     

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 \({r\to 1-}\).     

Generalized commutativity of lattice-ordered groups II

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n.     

A Note on Derivations on the Algebra of Operators in Hilbert <Emphasis Type="Italic">C</Emphasis>*-Modules

Let \({\mathfrak{M}}\) be a Hilbert C*-module on a C*-algebra \({\mathfrak{A}}\) and let \({End_\mathfrak{A}(\mathfrak{M})}\) be the algebra of all operators on \({\mathfrak{M}}\). In this paper, first the continuity of \({\mathfrak{A}}\)-module homomorphism derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) is investigated. We give some sufficient conditions on which every derivation on \({End_\mathfrak{A}(\mathfrak{M})}\) is inner. Next, we study approximately innerness of derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) for a σ-unital C*-algebra \({\mathfrak{A}}\) and full Hilbert \({\mathfrak{A}}\)-module \({\mathfrak{M}}\). Finally, we show that every bounded linear mapping on \({End_\mathfrak{A}(\mathfrak{M})}\) which behave like a derivation when acting on pairs of elements with unit product, is a Jordan derivation.     

Homomorphisms from a finite group into wreath products

Let G be a finite group, A a finite abelian group. Each homomorphism \({\varphi:G\rightarrow A\wr S_n}\) induces a homomorphism \({\overline{\varphi}:G\rightarrow A}\) in a natural way. We show that as \({\varphi}\) is chosen randomly, then the distribution of \({\overline{\varphi}}\) is close to uniform. As application we prove a conjecture of T. Müller on the number of homomorphisms from a finite group into Weyl groups of type D _n .     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

Uniform continuity in {\omega_\mu}-metric spaces and uc {\omega_\mu}-metric extendability

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the \({\omega_\mu}\)-metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

Disintegration of positive isometric group representations on $$\varvec{\mathrm {L}^{p}}$$-spaces

Let G be a Polish locally compact group acting on a Polish space \({{X}}\) with a G-invariant probability measure \(\mu \). We factorize the integral with respect to \(\mu \) in terms of the integrals with respect to the ergodic measures on X, and show that \(\mathrm {L}^{p}({{X}},\mu )\) (\(1\le p<\infty \)) is G-equivariantly isometrically lattice isomorphic to an \({\mathrm {L}^p}\)-direct integral of the spaces \(\mathrm {L}^{p}({{X}},\lambda )\), where \(\lambda \) ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of \(\mathrm {L}^{p}({{X}},\mu )\) as an \({\mathrm {L}^p}\)-direct integral of order indecomposable representations. If \(({{X}}^\prime ,\mu ^\prime )\) is a probability space, and, for some \(1\le q<\infty \), G acts in a strongly continuous manner on \(\mathrm {L}^{q}({{X}}^\prime ,\mu ^\prime )\) as isometric lattice automorphisms that leave the constants fixed, then G acts on \(\mathrm {L}^{p}({{X}}^{\prime },\mu ^{\prime })\) in a similar fashion for all \(1\le p<\infty \). Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If \(({{X}}^\prime ,\mu ^\prime )\) is separable, the representation of G on \(\mathrm {L}^p(X^\prime ,\mu ^\prime )\) can then be disintegrated into order indecomposable representations. The notions of \({\mathrm {L}^p}\)-direct integrals of Banach spaces and representations that are developed extend those in the literature.     

Equivariant Euler–Poincaré characteristic in sheaf cohomology

Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports \({\Phi}\). Fix a number field F with ring of integers R. We study the class \({\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}\) in the character group of G over F for any flat G-sheaf \({\mathcal{E}}\) of R-modules over X. Under natural cohomological finiteness conditions we give a formula for \({\chi}\) with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups.     

Limit theorems for multidimensional renewal sets

Given a sequence \({\mathcal{U} =\{U_n: n \in \omega\}}\) of non-empty open subsets of a space X, a set \({\{x_n : n \in \omega\}}\) is a selection of \({\mathcal{U}}\) if \({x_n \in U_n}\) for every \({n \in \omega}\). We show that a space X is uncountable if and only if every sequence of non-empty open subsets of C _p (X) has a closed discrete selection. The same statement is not true for \({C_p(X,[0,1])}\) so we study when the above selection property (which we call discrete selectivity) holds in \({C_p(X,[0,1])}\). We prove, among other things, that \({C_p(X, [0,1])}\) is discretely selective if X is an uncountable Lindelöf \({\Sigma}\)-space. We also give a characterization, in terms of the topology of X, of discrete selectivity of \({C_p(X,[0,1])}\) if X is an \({\omega}\)-monolithic space of countable tightness.     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.