Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

On p-tuples of the Grassmann manifolds

We provide a matrix invariant for isometry classes of p-tuples of points in the Grassmann manifold ${G_{n}\left(\mathbb{K}^{d}\right) }$ ( ${\mathbb{K=\mathbb{R}}}$ or ${\mathbb{C}}$ ). This invariant fully characterizes the p-tuple. We use it to classify the regular p-tuples of ${G_{2}\left(\mathbb{R}^{d}\right) }$ , ${G_{3}\left( \mathbb{R}^{d}\right) }$ and ${G_{2}\left( \mathbb{C}^{d}\right) }$ .     

Characterization of (generalized) Lie derivations on {\mathcal{J}}-subspace lattice algebras by local action

Let ${\mathcal{L}}$ be a ${\mathcal{J}}$ -subspace lattice on a Banach space X over the real or complex field ${\mathbb{F}}$ with dim X ≥ 2 and Alg ${\mathcal{L}}$ be the associated ${\mathcal{J}}$ -subspace lattice algebra. For any scalar ${\xi \in \mathbb{F}}$ , there is a characterization of any linear map L : Alg ${\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}$ satisfying ${L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}$ for any ${A, B \in{\rm Alg} {\mathcal{L}}}$ with AB = 0 (rep. ${[A,B]_ \xi = AB - \xi BA = 0}$ ) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg ${\mathcal{L}}$ is obtained.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

On additive involutions and Hamel bases

We provide an example of a discontinuous involutory additive function ${a: \mathbb{R}\to \mathbb{R}}$ such that ${a(H) \setminus H \ne \emptyset}$ for every Hamel basis ${H \subset \mathbb{R}}$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from ${\mathbb{R}}$ to ${\mathbb{R}}$ with the Tychonoff topology induced by ${\mathbb{R}^{\mathbb{R}}}$ .     

Cohomology of Some Complex Laminations

In this paper we solve the ${\overline{\partial }}$ -problem along the leaves for two types of laminations: (i) Some open sets Ω of ${{\mathbb C}\times B}$ (where B is any differentiable manifold) endowed with the canonical foliation that is, the foliation whose leaves are the sections ${\Omega ^t=\{ z\in {\mathbb C}:(z,t)\in \Omega \}}$ . We construct a solution to the equation ${\overline{\partial }h=fd\overline z}$ for any function ${f:\Omega\longrightarrow {\mathbb C}}$ of class ${C^{s}\,(s\in \mathbb{N}\cup\{ \infty \}),\,C^\infty}$ along the leaves and satisfies some growth conditions near the singularities. (ii) A complex lamination by Riemann surfaces obtained by suspending a homeomorphism of a closed set of the Euclidean space ${\mathbb{C}\times \mathbb{R}}$ .     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

Biharmonic Hypersurfaces in a Conformally Flat Space

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and used to determine the conformally flat metric ${f^{-2}\delta_{ij}}$ on the Euclidean space ${\mathbb{R}^{m+1}}$ so that a minimal hypersurface ${M^{m}\longrightarrow (\mathbb{R}^{m+1}, \delta_{ij})}$ in a Euclidean space becomes a biharmonic hypersurface ${M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}\delta_{ij})}$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in Ou (Pac J Math 248(1):217–232, 2010) and Ou and Tang (Mich Math J 61:531–542, 2012) as special cases.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

On Extending {\mathcal{A}}-Modules Through the Coefficients

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free ${\mathcal{A}}$ -modules, where ${\mathcal{A}}$ is an ${\mathbb{R}}$ -algebra sheaf. Complexification of free ${\mathcal{A}}$ -modules, which is defined to be the process of obtaining new free ${\mathcal{A}}$ -modules by enlarging the ${\mathbb{R}}$ -algebra sheaf ${\mathcal{A}}$ to a ${\mathbb{C}}$ -algebra sheaf, denoted ${\mathcal{A}_\mathbb{C}}$ , is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- ${_{\mathbb{R}}\mathcal{A}}$ -sheaf of almost complex structures on the sheaf ${{_\mathbb{R}}\mathcal{A}^{2n}}$ , the underlying ${\mathbb{R}}$ -algebra sheaf of a ${\mathbb{C}}$ -algebra sheaf ${\mathcal{A}}$ , and on the other hand, on the complexification of the functor ${\mathcal{H}om_\mathcal {A}}$ , with ${\mathcal{A}}$ an ${\mathbb{R}}$ -algebra sheaf.     

Bent functions embedded into the recursive framework of {\mathbb{Z}} -bent functions

Suppose that n is even. Let ${\mathbb{F}_2}$ denote the two-element field and ${\mathbb{Z}}$ the set of integers. Bent functions can be defined as ± 1-valued functions on ${\mathbb{F}_2^n}$ with ± 1-valued Fourier transform. More generally we call a mapping f on ${\mathbb{F}_2^n}$ a ${\mathbb{Z}}$ -bent function if both f and its Fourier transform ${\widehat{f}}$ are integer-valued. ${\mathbb{Z}}$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and ${\widehat{f}}$ . It is shown how ${\mathbb{Z}}$ -bent functions of lower level can be built up recursively by gluing together ${\mathbb{Z}}$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of ${\mathbb{Z}}$ -bent functions and give some guidelines for further research.     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

Functional inequalities related to Mahler’s conjecture

Extending a result of Meyer and Reisner (Monatsh Math 125:219–227, 1998), we prove that if ${g: \mathbb{R}\to \mathbb{R}_+}$ is a function which is concave on its support, then for every m > 0 and every ${z\in\mathbb{R}}$ such that g(z) > 0, one has $$ \int\limits_{\mathbb{R}} g(x)^mdx\int\limits_{\mathbb{R}} (g^{*z}(y))^m dy\ge \frac{(m+2)^{m+2}}{(m+1)^{m+3}},$$ where for ${y\in \mathbb{R}}$ , ${g^{*z}(y)=\inf_x \frac{(1-(x-z)y)_+}{g(x)}}$ . It is shown how this inequality is related to a special case of Mahler’s conjecture (or inverse Santaló inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santaló inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218:1430–1452, 2008). Namely, if ${\phi:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}}$ is a convex function such that ${0 < \int e^{-\phi} < +\infty}$ then, for every ${z\in\mathbb{R}}$ such that ${\phi(z) < +\infty}$ , one has $$ \int\limits_{\mathbb{R}}e^{-\phi}\int\limits_{\mathbb{R}} e^{-\mathcal{L}^z\phi}\ge e,$$ where ${\mathcal {L}^z\phi}$ is the Legendre transform of ${\phi}$ with respect to z.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.     

A Uniqueness Theorem for Bounded Analytic Functions on the Polydisc

Given n, N ≥ 1 we construct a set of points ${\lambda_1,{\ldots},\lambda_{N^n}\in{\mathbb D}^n}$ such that for each rational inner function f on ${{\mathbb D}^n}$ of degree less than N the Pick problem on ${{\mathbb D}^n}$ with data ${\lambda_1,{\ldots},\lambda_{N^n}}$ and ${f(\lambda_1),{\ldots},f(\lambda_{N^n})}$ has a unique solution. In particular, we construct a 1-dimensional inner variety V and show that the points ${\lambda_1,{\ldots},\lambda_{N^n}}$ may be chosen almost arbitrarily in ${V\cap{\mathbb D}^n}$ . Our results state that f is uniquely determined in the Schur class of ${{\mathbb D}^n}$ by its values on ${\lambda_1,{\ldots},\lambda_{N^n}}$ .     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.