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.

     

Gabor Frames in $${\ell }^2({\mathbb {Z}})$$ ℓ 2 ( Z ) and Linear Dependence

We prove that an overcomplete Gabor frame in \({\ell }^2({\mathbb {Z}})\) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in \({\ell }^2({\mathbb {Z}})\) with modulation parameter 1 / M and translation parameter N for some \(M,N\in {\mathbb {N}},\) and generated by a finite sequence g in \({\ell }^2({\mathbb {Z}})\) with K nonzero entries.

     

Eigenvalue inequalities for the clamped plate problem of $${\mathfrak{L}}_\nu ^2$$ operator

\({{\mathfrak{L}}_{II}}\) operator is introduced by Xin (2015), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend \({{\mathfrak{L}}_{II}}\) operator to a more general elliptic differential operator \({{\mathfrak{L}}_\nu}\), and investigate the clamped plate problem of bi-\({{\mathfrak{L}}_\nu}\) operator, which is denoted by \({\mathfrak{L}}_\nu ^2\) on the complete Riemannian manifolds. A general formula of eigenvalues for the \({\mathfrak{L}}_\nu ^2\) operator is established. Applying this formula, we estimate the eigenvalues on the Riemannian manifolds. As some further applications, we establish some eigenvalue inequalities for this operator on the translating solitons with respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit spheres and projective spaces. In particular, for the case of translating solitons, all of the eigenvalue inequalities are universal.

     

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 Menon-type identity with Dirichlet characters in residually finite Dedekind domains

This paper studies Menon–Sury’s identity in a general case, i.e., the Menon–Sury’s identity involving Dirichlet characters in residually finite Dedekind domains. By using the filtration of the ring \({\mathfrak {D}}/{\mathfrak {n}}\) and its unit group \(U({\mathfrak {D}}/{\mathfrak {n}})\), we explicitly compute the following two summations:

$$\begin{aligned} \sum _{\begin{array}{c} a\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a-1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi (a) \end{aligned}$$

and

$$\begin{aligned} \sum _{\begin{array}{c} a_{1},\ldots , a_{s}\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a_{1}-1,\ldots , a_{s}-1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi _{1}(a_1) \cdots \chi _{s}(a_s), \end{aligned}$$

where \({\mathfrak {D}}\) is a residually finite Dedekind domain and \({\mathfrak {n}}\) is a nonzero ideal of \({\mathfrak {D}}\), \(N({\mathfrak {n}})\) is the cardinality of quotient ring \({\mathfrak {D}}/{\mathfrak {n}}\), \(\chi _{i}~(1\le i\le s)\) are Dirichlet characters mod \({\mathfrak {n}}\) with conductor \({\mathfrak {d}}_i\).

     

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.

     

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.

     

On the Fourier coefficients of certain Hilbert modular forms

We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).

     

3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$

It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.     

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.

     

Non-separable Lattices,Gabor Orthonormal Bases and Tilings

Let \(K\subset {\mathbb {R}}^d\) be a bounded set with positive Lebesgue measure. Let \(\Lambda =M({\mathbb {Z}}^{2d})\) be a lattice in \({\mathbb {R}}^{2d}\) with density dens\((\Lambda )=1\). It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis for \(L^2({\mathbb {R}}^d)\) if and only if K tiles both by \(A({\mathbb {Z}}^d)\) and \(B^{-t}({\mathbb {Z}}^d)\). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis, then K can be written as a finite union of fundamental domains of \(A({{\mathbb {Z}}}^d)\) and at the same time, as a finite union of fundamental domains of \(B^{-t}({{\mathbb {Z}}}^d)\). If \(A^tB\) is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domain is also possible. We also provide a constructive way for forming a Gabor window function for a given upper triangular lattice. Our study is related to a Fuglede’s type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

     

Magnetic Trajectories in an Almost Contact Metric Manifold $${\mathbb{R}^{2N+1}}$$

In this paper we classify magnetic trajectories γ in \({{\mathbb{R}}^{2N+1}}\) endowed with a canonical quasi-Sasakian structure, corresponding to a magnetic field proportional to the fundamental 2-form. We prove that they are helices of order 5 and we show that there exists a totally geodesic \({{\mathbb{R}}^5}\) in \({\mathbb{R}^{2N+1}}\) such that γ lies in \({{\mathbb{R}}^5}\). Moreover, the quasi-Sasakian structure of \({{\mathbb{R}}^5}\) is that induced from the ambient manifold.     

A Sharp Weak-Type $$(\infty ,\infty )$$ Inequality for the Hilbert Transform

The paper is devoted to sharp weak type \((\infty ,\infty )\) estimates for \({\mathcal {H}}^{\mathbb {T}}\) and \({\mathcal {H}}^{\mathbb {R}}\), the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {T}}f\right\| _{W({\mathbb {T}})}\le \Vert f\Vert _{L^\infty ({\mathbb {T}})} \end{aligned}$$

and

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {R}}f\right\| _{W({\mathbb {R}})}\le \Vert f\Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$

where \(W({\mathbb {T}})\) and \(W({\mathbb {R}})\) stand for the weak-\(L^\infty \) spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant \(1\) on the right is shown to be the best possible.

     

New complete embedded minimal surfaces in $${{\mathbb {H} ^2\times \mathbb {R}}}$$

We construct three kinds of complete embedded minimal surfaces in \({\mathbb {H}^2\times \mathbb {R}}\) . The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in \({\mathbb {R}^3}\) . The third one is a finite total curvature surface which is conformal to \({\mathbb {S}^2\setminus\{p_1,\ldots,p_k\}, k\geq3.}\)     

Ground state solutions for a class of fractional Hamiltonian systems

In this paper we study the following fractional Hamiltonian systems

$$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))- L(t).x(t)+\nabla W(t,x(t))=0, \\ x\in H^{\alpha }(\mathbb {R}, \mathbb {R}^{N}), \end{array} \right. \end{aligned}$$

where \(\alpha \in \left( {1\over {2}}, 1\right] ,\ t\in \mathbb {R}, x\in \mathbb {R}^N,\ _{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are the left and right Liouville–Weyl fractional derivatives of order \(\alpha \) on the whole axis \(\mathbb {R}\) respectively, \(L:\mathbb {R}\longrightarrow \mathbb {R}^{2N}\) and \(W: \mathbb {R}\times \mathbb {R}^{N}\longrightarrow \mathbb {R}\) are suitable functions. One ground state solution is obtained by applying the monotonicity trick of Jeanjean and the concentration-compactness principle in the case where the matrix L(t) is positive definite and \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition.

     

Integer Decomposition Property of Free Sums of Convex Polytopes

Let \({\mathcal{P} \subset \mathbb{R}^{d}}\) and \({\mathcal{Q} \subset \mathbb{R}^{e}}\) be integral convex polytopes of dimension d and e which contain the origin of \({\mathbb{R}^{d}}\) and \({\mathbb{R}^{e}}\), respectively. We say that an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^{d}}\) possesses the integer decomposition property if, for each \({n\geq1}\) and for each \({\gamma \in n\mathcal{P}\cap\mathbb{Z}^{d}}\), there exist \({\gamma^{(1)}, . . . , \gamma^{(n)}}\) belonging to \({\mathcal{P}\cap\mathbb{Z}^{d}}\) such that \({\gamma = \gamma^{(1)} +. . .+\gamma^{(n)}}\). In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of \({\mathcal{P}}\) and \({\mathcal{Q}}\) to possess the integer decomposition property will be presented.     

Weighted $${{L^p}}$$-Liouville theorems for hypoelliptic partial differential operators on Lie groups

We prove weighted \({L^p}\)-Liouville theorems for a class of second-order hypoelliptic partial differential operators \({\mathcal{L}}\) on Lie groups \({\mathbb{G}}\) whose underlying manifold is \({n}\)-dimensional space. We show that a natural weight is the right-invariant measure \(\check{H}\) of \({\mathbb{G}}\). We also prove Liouville-type theorems for \({C^{2}}\) subsolutions in \({L^{p}(\mathbb{G},\check{H})}\). We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator \({\mathcal{L}-\partial_{t}}\).     

The Whitney Extension Theorem for $$C^1$$, Horizontal Curves in the Heisenberg Group

For a real-valued function defined on a compact set \(K \subset {\mathbb {R}}^m\), the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a \(C^k\) extension to \({\mathbb {R}}^m\). In this paper, we prove a version of the Whitney Extension Theorem in the case of \(C^1\), horizontal extensions for mappings defined on compact subsets of \({\mathbb {R}}\) taking values in the sub-Riemannian Heisenberg group \(\mathbb {H}^n\).     

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$

Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305–7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over \({\mathbb {Z}}_{p^{r}}\) from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) for any given set of parameters.