On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

$${G_\delta}$$ covers of compact spaces

We solve a long standing question due to Arhangel’skii by constructing a compact space which has a \({G_\delta}\) cover with no continuum-sized (\({G_\delta}\))-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every \({G_\delta}\) cover has a \({\mathfrak{c}}\)-sized subcollection with a \({G_\delta}\)-dense union and that in a Lindelöf space with a base of multiplicity continuum, every \({G_\delta}\) cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De la Vega’s celebrated theorem on the cardinality of homogeneous compacta of countable tightness.     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

$${\cal K}$$-Convexity1 in $$\Re^{n}$$

We generalize the concept of K-convexity to an n-dimensional Euclidean space. The resulting concept of -convexity is useful in addressing production and inventory problems where there are individual product setup costs and/or joint setup costs. We derive some basic properties of -convex functions. We conclude the paper with some suggestions for future research. Support from Columbia University and University of Texas at Dallas is gratefully acknowledged. Helpful comments from Qi Feng are appreciated.     

On two-weight $$\mathbb {Z}_{2^k}$$-codes

We determine the possible homogeneous weights of regular projective two-weight codes over \(\mathbb {Z}_{2^k}\) of length \(n>3\), with dual Krotov distance \(d^{\lozenge }\) at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When \(k=2\), we characterize the parameters of such codes as those of the inverse Gray images of \(\mathbb {Z}_4\)-linear Hadamard codes, which have been characterized by their types by several authors.     

Some Relations on the Double $$\mathcal {L}_{22}$$ -Integral Transform and Their Applications

Mediterranean Journal of Mathematics - A variant of the Jensen–Mercer operator inequality is proved for a superquadratic function and positive linear operators on a Hilbert space using a...     

Clifford Cauchy type integrals on Ahlfors-David regular surfaces in $$\mathbb{R}^{m + 1} $$

The main goal of this paper is centred around the study of the behavior of the Cauchy type integral and its corresponding singular version, both over nonsmooth domains in Euclidean space. This approach is based on a recently developed quaternionic Cauchy integrals theory [1, 5, 7] within the three-dimensional setting. The present work involves the extension of fundamental results of the already cited references showing that the Clifford singular integral operator has a proper invariant subspace of generalized H?lder continuous functions defined in a surface of the (m+1)-dimensional Euclidean space.     

A new characterization of Sobolev spaces on $${\mathbb{R}^{n}}$$

In this paper we present a new characterization of Sobolev spaces on . Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.     

Nonexistence of generalized bent functions from $$\mathbb {Z}_{2}^{n}$$ to $$\mathbb {Z}_{m}$$Zm

In this paper, several nonexistence results on generalized bent functions \(f:\mathbb {Z}_{2}^{n} \rightarrow \mathbb {Z}_{m}\) are presented by using the knowledge on cyclotomic number fields and their imaginary quadratic subfields.     

Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Any algebraic surface in which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in ruled in cubics. This work is within the framework of the national research project “Geometry on Algebraic Varieties” Cofin 2006 of MIUR.     

$$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral

The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μ_Ω, where w belongs to the Muckenhoupt weight class.     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.     

Geometric Separation in $$\mathbb {R}^3$$ R 3

Journal of Fourier Analysis and Applications - The geometric separation problem, initially posed by Donoho and Kutyniok (Commun Pure Appl Math 66:1–47, 2013), aims to separate a distribution...     

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.     

Borsuk’s Partition Problem in $$(\mathbb{R}^{n},\ell_{p})$$

Mathematical Notes - For a bounded set $$X$$ with diameter $$d_{C}(X)$$ in a finite-dimensional normed space with an origin-symmetric convex body $$C$$ as the unit ball, the Borsuk number of $$X$$...     

A note on compact Weingarten hypersurfaces embedded in $$\mathbb {R}^{n + 1}$$

We prove that the round sphere is the only compact Weingarten hypersurface embedded in the Euclidean space such that \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\). Here, \(H_r\) stands for the r-th mean curvature and H denotes the standard mean curvature of the hypersurface.     

A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.     

Lagrangian isotopy of tori in $${S^2\times S^2}$$ and $${{\mathbb{C}}P^2}$$CP2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.     

A note on the intersection property for flat boxes and boxicity in $${\mathbb{R}^{d}}$$

By extending the definition of boxicity, we extend a Hellytype result given by Danzer and Grünbaum on 2-piercings of families of boxes in d-dimensional Euclidean space by lowering the dimension of the boxes in the ambient space.