Affine Hypersurfaces Admitting a Pointwise SO(n − 1) Symmetry

In this article we obtain a classification of strictly locally convex affine hypersurfaces in ${\mathbb{A}^{n+1}}$ for which the geometrical structure is pointwise invariant under the group SO(n ? 1) represented by rotations around a fixed axis in the tangent space. This generalises the results obtained by Lu and Scharlach (Results Math 48:275–300, 2005) for n =  3.     

n-Barbilian Domains

The notion of Barbilian domain as introduced and studied by W. Leißner [14], [16] and others is refined here to n-Barbilian domain in a free module of rank n. This leads to results that bear on n-dimensional affine ring geometry. The case of infinite rank is also considered. AMS Classification: 51C05. Keywords: Free module, Barbilian domain, Hermite ring, affine ring geometry.     

Further remarks on the non-degeneracy condition

The structure of the set of positive solutions of the semilinear elliptic boundary value problem depends on a certain non-degeneracy condition, which was proved by K.J. Brown [2] and T. Ouyang and J. Shi [12], with a shorter proof given later by P. Korman [8]. In this note we present a more general result, communicated to us by L. Nirenberg [13]. We also discuss the extensions in cases when the domain D is in R ², and it is either symmetric or convex.     

Generalizations Of A “folk-Theorem” On Simple Functional Equations In A Single Variable

It is known (“mathematical folklore”) that, to every function defined on [1,2], there exists a solution of f(2x) = 2f(x) on ]0,∞[ of which the given function is a restriction to [1,2]. With a little care in the definition on [1,2], with still a lot of arbitrariness left, the resulting solution will be continuous, even C^∞ on ]0,∞[ (a behaviour markedly different from that of the Cauchy equation f(x + y) = f(x) + f(y), which has f(x) = cx as only continuous solution on ]0,∞[, even though, with y = x, it degenerates into the above equation). If 0 is added to the domain and we choose the “arbitrary function” bounded on [1,2[, then the solution will even be continuous (from the right) at 0. However, if f is supposed to be differentiable at 0 (from the right), then f(x) = cx is the only solution on [0,∞[. p In this paper we present similar and further results concerning general, Cⁿ (n ≤ ∞), analytic, locally monotonie or γ-th order convex solutions of the somewhat more general equation f(kx) = k^γf(x) (k ≠ 1 a positive, γ a real constant), which seems to be of importance in meterology. Some of the results are not quite what one expects.     

Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms

In Perez (Thesis, 2011), Perez proved some L ² inequalities for closed convex hypersurfaces immersed in the Euclidean space ? ⁿ⁺¹, and more generally for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is ? ⁿ⁺¹, the hyperbolic space ? ⁿ⁺¹, or the closed hemisphere \(\mathbb{S}_{+}^{n+1}\) . We also obtain a generalization of Perez’s theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

Ideal Divisibility Monoids

Inspired by the monograph of Larsen/McCarthy, [26], in [10] and [11] the author started a series of articles concerning abstract multiplicative ideal theory along the problem lines of [26]. In this paper we turn to multiplicative lattices having the left Priifer property, that is to m-lattices satisfying the implication a¹ + … + a_n ? B ? a₁ +… + a_n ¦? B or even the multiplication property A ? B ? A ¦B, respectively. Clearly, studying such structures includes studying substructures of d-semigroups.     

Combinatorial geometry of belt bodies

In this paper we consider a new class of convex bodies which was introduced in [11]. This is the class of belt bodies, and it is a natural generalization of the class of zonoids (see the surveys [18, 28, 24]). While the class of zonoids is not dense in the family of all centrally symmetric, convex bodies, the class of belt bodies is dense in the set of all convex bodies. Nevertheless, we shall extend solutions of combinatorial problems for zonoids (cf. [2, 12]) to the class of belt bodies. Therefore, we first introduce the set of belt bodies by using zonoids as starting point. (To make the paper self-contained, a few parts of the approach from [11] are given repeatedly.) Second, complete solutions of three well-known (and generally unsolved) problems from the combinatorial geometry of convex bodies are given for the class of belt bodies. The first of these, connected with the names of I. Gohberg and H. Hadwiger, is the problem of covering a convex body with smaller homothetic copies, or the equivalent illumination problem. The second is the Szökefalvi-Nagy problem, which asks for the determination of the convex bodies whose families of translates have a given Helly dimension. The third problem concerns special fixing systems, a notion which is due to L. Fejes Tóth. These solutions consist of improved and more general approaches to recently solved problems (as in the case of the Helly-dimensional classification of belt bodies) or new results (as those concerning minimal fixing systems, providing also an answer to a problem of B. Grünbaum which is not only restricted to belt bodies).     

The composition of solutions of the multidimensional translation equation

Under certain conditions, the general solution of the multidimensional translation equation was constructed locally in L. Berg [4] and globally in J. Aczél, L. Berg and Z. Moszner [2]. By composition of two of these solutions there arise new functional equations, which are solved here locally using generalized inverses, cf. A. Ben-Israel and T.N.E. Greville [5]. The results are illustrated by the linear case.     

Further results on factorization of meromorphic solutions of partial differential equations

This paper is a continuation of Hu-Yang [2]. Here we extend Malmquist type theorem ofalgebraic differential equations of Steinmetz [3] and Tu [4] to higher order partial differential equations. The results also generalize Theorems 4.2 and 4.3 in [2].     

Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q ⁺(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).     

Hypercyclic weighted translations generated by non-torsion elements

We characterize hypercyclic weighted translation operators generated by non-torsion elements of a locally compact group, extending the recent results in Chen and Chu [6], Chen and Chu [7].     

The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}

Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).     

Maximization of lower semi-continuous convex functional on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems

We prove, for a proper lower semi-continuous convex functional ? on a locally convex space E and a bounded subset G of E, a formula for sup ?(G) which is symmetric to the Lagrange multiplier theorem for convex minimization, obtained in [7], with the difference that for sup ?(G) Lagrange multiplier functionals need not exist. When ? is also continuous we give some necessary conditions for g₀ ∈ G to satisfy ?(g₀) = sup ?(G). Also, we give some applications to deviations and farthest points. Finally, we show the connections with the “hyperplane theorems” of our previous paper [8].     

The Total Absolute Curvature of Nonclosed Curves in S 2, (II)

Extending the results in [1], we determine the shape of the curve in S ² which minimize the total absolute curvature in the set of nonclosed curves with fixed endpoints, end-directions and length. We deal with a more general class of curves than in [1].     

A new existence proof for Steiner quadruple systems

A Steiner quadruple system of order v is an ordered pair ${(X, \mathcal{B})}$ , where X is a set of cardinality v, and ${\mathcal{B}}$ is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. Such designs exist if and only if ${v \equiv 2,4\, (\bmod\, 6)}$ . The first and second proofs of this result were given by Hanani in 1960 and in 1963, respectively. All the existing proofs are rather cumbersome, even though simplified proofs have been given by Lenz in 1985 and by Hartman in 1994. To study Steiner quadruple systems, Hanani introduced the concept of H-designs in 1963. The purpose of this paper is to provide an alternative existence proof for Steiner quadruple systems via H-designs of type 2 ⁿ . In 1990, Mills showed that for n > 3, n ≠ 5, an H-design of type g ⁿ exists if and only if ng is even and g(n ? 1)(n ? 2) is divisible by 3, where the main context is the complicated existence proof for H-designs of type 2 ⁿ . However, Mill’s proof was based on the existence result of Steiner quadruple systems. In this paper, by using the theory of candelabra systems and H-frames, we give a new existence proof for H-designs of type 2 ⁿ independent of the existence result of Steiner quadruple systems. As a consequence, we also provide a new existence proof for Steiner quadruple systems.