Two remarks on normality preserving Borel automorphisms of \boldsymbol{\mathbb{R}{\kern-8.3pt}\mathbb{R}}^{{\bf \textit{n}}}

Let T be a bijective map on ? ⁿ such that both T and T ^???1 are Borel measurable. For any θ?∈?? ⁿ and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ? ⁿ with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose $N(\boldsymbol{\theta}_j, I)T^{-1}$ is gaussian for 0?≤?j?≤?n, where I is the identity matrix and {θ _j ???θ ₀, 1?≤?j?≤?n } is a basis for ? ⁿ . Then T is an affine linear transformation; (2) Let $\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ 1?≤?j?≤?n where ε _j ?>???1 for every j and {u _j , 1?≤?j?≤?n } is a basis of unit vectors in ? ⁿ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u _j . Suppose N(0, I)T ^???1 and $N (\mathbf{0}, \Sigma_j)T^{-1},$ 1?≤?j?≤?n are gaussian. Then $T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}$ a.e. x, where s runs over the set of 2 ⁿ diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { E _s } is a collection of 2 ⁿ Borel subsets of ? ⁿ such that { E _s } and {V s U (E _s )} are partitions of ? ⁿ modulo Lebesgue-null sets and for every j, $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all s for which the Lebesgue measure of E _s is positive. The converse of this result also holds. Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Cluster Algebras of Type A _{\bf{2}}^{\bf {(1)}}

In this paper we study cluster algebras $\mathcal{A}$ of type $A_2^{(1)}$ . We solve the recurrence relations among the cluster variables (which form a T-system of type $A_2^{(1)}$ ). We solve the recurrence relations among the coefficients of $\mathcal{A}$ (which form a Y-system of type $A_2^{(1)}$ ). In $\mathcal{A}$ there is a natural notion of positivity. We find linear bases B of $\mathcal{A}$ such that positive linear combinations of elements of B coincide with the cone of positive elements. We call these bases atomic bases of $\mathcal{A}$ . These are the analogue of the “canonical bases” found by Sherman and Zelevinsky in type $A_{1}^{(1)}$ . Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of B are parameterized by ?³ via their g-vectors in every cluster. We prove that the denominator vector map in every acyclic seed of $\mathcal{A}$ restricts to a bijection between B and ?³. We find explicit recurrence relations to express every element of $\mathcal{A}$ as linear combinations of elements of B.     

Hyperflock determining line sets and totally regular parallelisms of PG {(3,\,{\mathbb{R}})}

By a totally regular parallelism of the real projective 3-space ${\Pi_3:={{\rm PG}}(3, \mathbb {R})}$ we mean a family T of regular spreads such that each line of Π ₃ is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ is on exactly one subquadric of H. Moreover, ${\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}$ is a hyperflock determining line set, i.e., a set ${\mathcal {Z}}$ of 0-secants of H ₅ such that each tangential hyperplane of H ₅ contains exactly one line of ${\mathcal {Z}}$ . We say that ${{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}$ is the dimension of T and that T is a d _T - parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If ${\mathcal{G}}$ is a hyperflock determining line set, then ${\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}$ is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Extended quasi-homogeneous polynomial system in {\mathbb{R}^{3}}

In this paper we define an extended quasi-homogeneous polynomial system d x/dt = Q = Q ₁ + Q ₂ + ... + Q _δ , where Q _i are some 3-dimensional quasi-homogeneous vectors with weight α and degree i, i = 1, . . . ,δ. Firstly we investigate the limit set of trajectory of this system. Secondly let Q _T be the projective vector field of Q. We show that if δ ≤ 3 and the number of closed orbits of Q _T is known, then an upper bound for the number of isolated closed orbits of the system is obtained. Moreover this upper bound is sharp for δ = 3. As an application, we show that a 3-dimensional polynomial system of degree 3 (resp. 5) admits 26 (resp. 112) isolated closed orbits. Finally, we prove that a 3-dimensional Lotka-Volterra system has no isolated closed orbits in the first octant if it is extended quasi-homogeneous.     

Finite determinacy and Whitney equisingularity of map germs from {\mathbb{C}^n} to {\mathbb{C}^{2n-1}}

We show that a holomorphic map germ ${f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}$ is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D ²(f)) = 2μ(D ²(f)/S ₂)+C(f)?1, where D ²(f) is the lifting of the double point curve in ${(\mathbb{C}^n\times \mathbb{C}^n,0)}$ , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f _t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f _t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.     

A minimum principle for a soap film problem in {\mathbb{R}^{2}}

In this note, we investigate the problem of a thin extensible film (a soap film), under the influence of gravity and surface tension, supported by the contour of a given strictly convex smooth domain Ω. Our main result is a minimum principle for an appropriate combination of u(x) and ${\left\vert \nabla u\left( \mathbf{x}\right) \right\vert }$ , that is, a kind of P-function in the sense of Payne (see the book of Sperb in Maximum Principles and Their Applications. Academic Press, New York, 1981), where u(x) is the solution of our problem. As an application of this minimum principle, we obtain some a priori estimates for the surface represented by the thin extensible film, in terms of the curvature of ${\partial \Omega}$ . The proofs make use of Hopf’s maximum principles, some topological arguments regarding the local behavior of analytic functions and some computations in normal coordinates with respect to the boundary ${\partial \Omega }$ .     

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 ${\mathcal{N}.}$      

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Finite groups with some NR-subgroups or {\mathcal{H}}-subgroups

Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever ${K \trianglelefteq H}$ , then ${K^G \cap H = K}$ , where K ^G is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of ${\mathcal{H}}$ -subgroups: a subgroup H of G is said to be an ${\mathcal{H}}$ -subgroup of G if ${H^g \cap N_G(H) \leq H}$ for all ${g \in G}$ . In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or ${\mathcal{H}}$ -subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.     

On real local isometric immersions of {\mathbb{C}Q^2_c} into {\mathbb{C} P^{3}}

We investigate real local isometric immersions of Kähler manifolds ${\mathbb{C}Q^2_c}$ of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of ${\mathbb{C}P^2}$ into ${\mathbb{C}P^3}$ has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of ${\mathbb{C}Q^2_c}$ into ${\mathbb{C}P^3}$ when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed ${\mathbb{C}Q^2_c}$ and the complex structure of the ambient space ${\mathbb{C}P^3}$ .     

Tridiagonal pairs and the quantum affine algebra {\boldmath U_q({\widehat {sl}}_2)}

Let ${\mathbb K}$ denote an algebraically closed field and let q denote a nonzero scalar in ${\mathbb K}$ that is not a root of unity. Let V denote a vector space over ${\mathbb K}$ with finite positive dimension and let A,A* denote a tridiagonal pair on V. Let θ₀, θ₁,…, θ _d (resp. θ*₀, θ*₁,…, θ* _d ) denote a standard ordering of the eigenvalues of A (resp. A*). We assume there exist nonzero scalars a, a* in ${\mathbb K}$ such that θ _i = aq ^2i?d and θ* _i = a*q ^d?2i for 0 ≤ i ≤ d. We display two irreducible ${\boldmath U_q({\widehat {sl}}_2)}$ -module structures on V and discuss how these are related to the actions of A and A*.     

Uniqueness of non-linear ground states for fractional Laplacians in {\mathbb{R}}

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ ( ? Δ ) s Q + Q ? Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1?2s) for ${s<\frac{1}{2}}$ s < 1 2 and 0 < α < ∞ for ${s\geq \frac{1}{2}}$ s ≥ 1 2 . Here (?Δ) ^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for ${s=\frac{1}{2}}$ s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation. As a technical key result in this paper, we show that the associated linearized operator L ₊ = (?Δ) ^s +1?(α+1)Q ^α is non-degenerate; i.e., its kernel satisfies ker L ₊ = span{Q′}. This result about L ₊ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.     

{\mathbb{G}_a^ M} degeneration of flag varieties

Let ${\mathcal{F}_\lambda}$ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module V _λ . We define a flat degeneration ${\mathcal{F}_\lambda^a}$ , which is a ${\mathbb{G}^M_a}$ variety. Moreover, there exists a larger group G ^a acting on ${\mathcal{F}_\lambda^a}$ , which is a degeneration of the group G. The group G ^a contains ${\mathbb{G}^M_a}$ as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde‘d into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of ${\mathcal{F}_\lambda}$ is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of ${\mathcal{F}_\lambda^a}$ is isomorphic to a direct sum of dual PBW-graded ${\mathfrak{g}}$ -modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.     

Selmer groups over {\mathbb {Z}}_p^d-extensions

Consider an abelian variety \(A\) defined over a global field \(K\) and let \(L/K\) be a \({\mathbb {Z}}_p^d\) -extension, unramified outside a finite set of places of \(K\) , with \({{\mathrm{Gal}}}(L/K)=\Gamma \) . Let \(\Lambda (\Gamma ):={\mathbb {Z}}_p[[\Gamma ]]\) denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the \(\Lambda (\Gamma )\) -module \(X_L\) , the dual \(p\) -primary Selmer group, varies when \(L/K\) is replaced by a strict intermediate \({\mathbb {Z}}_p^e\) -extension.     

There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-cyclic 1-perfect code

Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.