The Berry–Esseen bounds of the weighted estimator in a nonparametric regression model

Consider the following nonparametric model: \(Y_{ni}=g(x_{ni})+ \varepsilon _{ni},1\le i\le n,\) where \(x_{ni}\in {\mathbb {A}}\) are the nonrandom design points and \({\mathbb {A}}\) is a compact set of \({\mathbb {R}}^{m}\) for some \(m\ge 1\), \(g(\cdot )\) is a real valued function defined on \({\mathbb {A}}\), and \(\varepsilon _{n1},\ldots ,\varepsilon _{nn}\) are \(\rho ^{-}\)-mixing random errors with zero mean and finite variance. We obtain the Berry–Esseen bounds of the weighted estimator of \(g(\cdot )\). The rate can achieve nearly \(O(n^{-1/4})\) when the moment condition is appropriate. Moreover, we carry out some simulations to verify the validity of our results.

     

Inference on a distribution function from ranked set samples

Consider independent observations \((X_i,R_i)\) with random or fixed ranks \(R_i\), while conditional on \(R_i\), the random variable \(X_i\) has the same distribution as the \(R_i\)-th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.

     

On Neyman–Pearson minimax detection of Poisson process intensity

The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\), the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({\mathbf{p}},\mathcal{Q})\), is considered. What is the maximal set \(\mathcal{Q}\), which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance? In the asymptotic case (\(T\rightarrow \infty \)) that maximal set \(\mathcal{Q}\) is described.

     

Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let \(p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]\) be a variable exponent function satisfying the globally log-Hölder continuous condition, \(q\in (0,\infty ]\) and A be a general expansive matrix on \({\mathbb {R}}^n\). Let \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) by means of the Littlewood–Paley g-function or the Littlewood–Paley \(g_\lambda ^*\)-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space \(L^{p(\cdot ),q}({\mathbb {R}}^n)\). Moreover, the finite atomic characterization of \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) to \(L^{p(\cdot ),q}({\mathbb {R}}^n)\) and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

     

The Dual Kaczmarz Algorithm

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector \(x\) in a (separable) Hilbert space from the inner-products \(\{\langle x, \phi _{n} \rangle \}\). The Kaczmarz algorithm defines a sequence of approximations from the sequence \(\{\langle x, \phi _{n} \rangle \}\); these approximations only converge to \(x\) when \(\{\phi _{n}\}\) is effective. We dualize the Kaczmarz algorithm so that \(x\) can be obtained from \(\{\langle x, \phi _{n} \rangle \}\) by using a second sequence \(\{\psi _{n}\}\) in the reconstruction. This allows for the recovery of \(x\) even when the sequence \(\{\phi _{n}\}\) is not effective; in particular, our dualization yields a reconstruction when the sequence \(\{\phi _{n}\}\) is almost effective. We also obtain some partial results characterizing when the sequence of approximations from \(\{\langle x, \phi _{n} \rangle \}\) using \(\{\psi _{n}\}\) converges to \(x\), in which case \(\{(\phi _{n}, \psi _{n})\}\) is called an effective pair.

     

On the Coincidence of the Classes of Finite Groups $ E_{\pi_{x}} $ and $ D_{\pi_{x}} $

Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.

     

Bisymplectic Grassmannians of planes

The bisymplectic Grassmannian \({{\,\mathrm{I_2Gr}\,}}(k,V)\) parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano projective variety which admits an action of a torus with a finite number of fixed points. In this work, we study its equivariant cohomology with complex coefficients when \(k=2\); the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyperplane class by any Schubert class. Moreover, we study in detail the case of \({{\,\mathrm{I_2Gr}\,}}(2, {\mathbb {C}}^6)\), which is a quasi-homogeneous variety, we analyse its deformations, and we give a presentation of its cohomology.

     

$$L^2_f$$ L f 2 -harmonic 1-forms on smooth metric measure spaces with positive $$\lambda _1(\Delta _f)$$ λ 1 ( Δ f )

In this paper, we study vanishing and splitting results on a complete smooth metric measure space \((M^n,g,\mathrm {e}^{-f}\mathrm {d}v)\) with various negative m-Bakry-Émery Ricci curvature lower bounds in terms of the first eigenvalue \(\lambda _1(\Delta _f)\) of the weighted Laplacian \(\Delta _f\), i.e., \(\mathrm {Ric}_{m,n}\ge -a\lambda _1(\Delta _f)-b\) for \(0<a\le \dfrac{m}{m-1}, b\ge 0\). In particular, we consider three main cases for different a and b with or without conditions on \(\lambda _1(\Delta _f)\). These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung, and Vieira.

     

Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation \({\partial _t}u - \epsilon \partial _x^2u + {\cal H}\partial _x^2u + u{u_x} = 0\), where \({\cal H}\) denotes the Hilbert transform operator. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space \({\tilde H^\sigma }(\mathbb{R})\,\,(\sigma \geqslant 0)\), which is a subspace of L²(ℝ). It is worth noting that the low-frequency part of \({\tilde H^\sigma }(\mathbb{R})\) is scaling critical, and thus the small data is necessary. The high-frequency part of \({\tilde H^\sigma }(\mathbb{R})\) is equal to the Sobolev space H^σ (ℝ) (σ ⩾ 0) and reduces to L²(ℝ). Furthermore, we also obtain its inviscid limit behavior in \({\tilde H^\sigma }(\mathbb{R})\) (σ ⩾ 0).

     

A Shape Optimization Problem on Planar Sets with Prescribed Topology

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form \(P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}\), and the class of admissible domains consists of two-dimensional open sets \(\Omega \) satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when \(q<1/2\) an optimal relaxed domain exists. When \(q>1/2\), the problem is ill-posed, and for \(q=1/2\), the explicit value of the infimum is provided in the cases \(k=0\) and \(k=1\).

     

Finite Groups with Generalized Subnormal Schmidt Subgroups

Given a prime \( p \) and a partition \( \sigma=\{\{p\},\{p\}^{\prime}\} \) of the set of all primes, we describe the structure of the nonnilpotent finite groups whose every Schmidt subgroup is \( \sigma \)-subnormal.

     

Density lower bound estimate for local minimizer of free interface problem with volume constraint

We prove a density lower bound for some functionals involving bulk and interfacial energies. The bulk energies are convex functions with p-power growth not subjected to any further structure conditions. The interface \(\partial E\) is the boundary of a set \(E\subset \Omega \) such that \(|E|=d\) is prescribed. Then we get \(\mathcal {H}^{n-1}((\partial E{\setminus }\partial E^*)\cup \Omega )=0\).

     

Spatially homogeneous copulas

We consider spatially homogeneous copulas, i.e. copulas whose corresponding measure is invariant under a special transformations of \([0,1]^2\), and we study their main properties with a view to possible use in stochastic models. Specifically, we express any spatially homogeneous copula in terms of a probability measure on [0, 1) via the Markov kernel representation. Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly singular properties. Finally, a generalization of spatially homogeneous copulas to the so-called (m, n)-spatially homogeneous copulas is studied and a characterization of this new family of copulas in terms of the Markov \(*\)-product is established.

     

Zero sequences,factorization and sampling measures for weighted Bergman spaces

The zero sets of the Bergman space \(A^p_\omega \) induced by either a radial weight \(\omega \) admitting a certain doubling property or a non-radial Bekollé-Bonami type weight are characterized in the spirit of Luecking’s results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz’s factorization result from 1977 for functions in \(A^p_\omega \). The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space \(A^p_\omega \) induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.

     

On the 2-rainbow independent domination numbers of some graphs

By suitably adjusting the tropical algebra technique we compute the rainbow independent domination numbers of several infinite families of graphs including Cartesian products \(C_n \Box P_m\) and \(C_n \Box C_m\) for all n and \(m\le 5\), and generalized Petersen graphs P(n, 2) for \(n \ge 3\).

     

Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form

$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$

for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.

     

On the difference spaces of almost convergent and strongly almost convergent double sequences

In this paper, we study the difference spaces \({\mathcal {F}}(\varDelta )\), \({\mathcal {F}}_0(\varDelta )\), \({\mathcal {[F]}}(\varDelta )\) and \({\mathcal {[F]}}_0(\varDelta )\) of double sequences obtained as the domain of four-dimensional backward difference matrix \(\varDelta \) in the spaces \({\mathcal {F}}\), \({\mathcal {F}}_{0}\), \({\mathcal {[F]}}\) and \({\mathcal {[F]}}_{0}\) of almost convergent, almost null, strongly almost convergent and strongly almost null double sequences; respectively. We examine general topological properties of those spaces and give some inclusion theorems. Furthermore, we deal with their dual spaces.

     

Some properties and applications of non-trivial divisor functions

The jth divisor function \(d_j\), which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function\(c_j\), which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor function\(c_j^{(r)}\), for \(r\ge 0\), whose definition is motivated by the sum-over divisors recurrence for \(d_j\). We give an overview of properties of \(d_j\), \(c_j\) and \(c_j^{(r)}\), specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers.