Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

Minimal helix surfaces in <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℝ

An immersed surface M in N ⁿ ×ℝ is a helix if its tangent planes make constant angle with ∂ _t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N ⁿ ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

<Emphasis Type="Italic">C</Emphasis><Superscript>1+<Emphasis Type="Italic">α</Emphasis></Superscript>-Regularity for Two-Dimensional Almost-Minimal Sets in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ³ are locally C ^1+α -equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that if X is the cone over an arc of small Lipschitz graph in the unit sphere, but X is not contained in a disk, we can use the graph of a harmonic function to deform X and substantially diminish its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in ℝ ⁿ , but in this setting our final regularity result on E may depend on the list of minimal cones obtained as blow-up limits of E at a point.     

Complete hypersurfaces with constant laguerre scalar curvature in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let x: M ^n?1 → R ⁿ be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of R ⁿ are Laguerre form C and Laguerre tensor L. In this paper, n > 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R ⁿ , denote the trace-free Laguerre tensor by ?\(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\) · Id. If \(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if \({\sup _M}\left\| {\widetilde L} \right\| = \frac{{\sqrt {\left( {n - 1} \right)\left( {n - 2} \right)} R}}{{\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}},\), M is Laguerre equivalent to the hypersurface ?x: H ¹ × S ^n?2 → R ⁿ .     

The index of semielliptic operators in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

The paper investigates the index of some linear, differential, semielliptic operators with variable coefficients of a special form in ? ⁿ . In particular, additional conditions on the symbol are found that render the index finite. The operators are considered in the weighted Sobolev spaces.     

The hot spots conjecture on a class of domains in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with <Emphasis Type="Italic">n</Emphasis> ⩾ 3

In this paper, we define a class of domains in R ⁿ . Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> Estimates of solution for <Emphasis Type="Italic">m</Emphasis>-Laplacian parabolic equation with a nonlocal term

In this paper, we consider the global existence, uniqueness and L ^∞ estimates of weak solutions to quasilinear parabolic equation of m-Laplacian type u _t − div(|∇u| ^m−2∇u) = u|u| ^β−1 ∫_Ω |u| ^α dx in Ω × (0,∞) with zero Dirichlet boundary condition in tdΩ. Further, we obtain the L ^∞ estimate of the solution u(t) and ∇u(t) for t > 0 with the initial data u ₀ ∈ L ^q (Ω) (q > 1), and the case α + β < m − 1.     

On a space of smooth functions on a convex unbounded set in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> admitting holomorphic extension in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ? ⁿ . Due to the conditions on M each function of this space admits a holomorphic extension in ? ⁿ . In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L ²-bounds in a domain of holomorphy in ? ⁿ are obtained with the aid of L. Hörmander’s method of L ²-bounds for the \(\bar \partial\) operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.     

Asymptotic purity for very general hypersurfaces of ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> × ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> of bidegree (<Emphasis Type="Italic">k,k</Emphasis>)

For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙ ⁿ × ℙ ⁿ . In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.     

Submanifolds with constant Möbius scalar curvature in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 Let M ^m be a m-dimensional submanifold in the n-dimensional unit sphere S ⁿ without umbilic point. Two basic invariants of M ^m under the M?bius transformation group of S ⁿ are a 1-form Φ called M?bius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let M ^m be a m-dimensional (m≥3) submanifold with vanishing M?bius form and with constant M?bius scalar curvature R in S ⁿ , denote the trace-free Blaschke tensor by . If , then either ||?||≡0 and M ^m is M?bius equivalent to a minimal submanifold with constant scalar curvature in S ⁿ ; or and M ^m is M?bius equivalent to in for some c≥0 and . Received: 15 May 2002 / Revised version: 3 February 2003 Published online: 19 May 2003 RID="*" ID="*" Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. RID="†" ID="†" Partially supported by the Alexander Humboldt von Stiftung and Zhongdian grant of NSFC. Mathematics Subject Classification (2000): Primary 53A30; Secondary 53B25     

On 2 – (<Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>, 2<Emphasis Type="Italic">n</Emphasis>, 2<Emphasis Type="Italic">n</Emphasis>–1) designs with three intersection numbers

The simple incidence structure , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane of order n > 4, is a 2 – (n ²,2n,2n–1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ≥ 5 is an odd integer. Supported by M.I.U.R., Università di Palermo.     

Free involutions on <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Topological free involutions on S ¹ × S ⁿ are classified up to conjugation. We prove that this is the same as classifying quotient manifolds up to homeomorphism. There are exactly four possible homotopy types of such quotients, and surgery theory is used to classify all manifolds within each homotopy type.     

On Sobolev solutions of poisson equations in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with a parameter

Smoothness with respect to a parameter is established under mild assumptions on the regularity of coefficients for Sobolev solutions of the Poisson equations in the whole ℝ ^d in the “ergodic case.” An assertion of this kind serves as one of the key tools in diffusion approximation and some other limit theorems. Bibliography: 12 titles.     

Solvability of a Class of Nonlinear Pseudo-Differential Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In the present work the existence of continuous and bounded solutions for a class of nonlinear pseudo-differential equations on ? ⁿ is proved. The monotonicity, asymptotic behavior and other properties for obtained solutions are also presented. Mentioned class of equations arises in p-adic string theory.     

On Solvability of Regular Hypoelliptic Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper the unique solvability of regular hypoelliptic equations in multianisotropic weighted functional spaces is proved by means of special integral representation of functions through a regular operator. The existence of the solutions is proved by constructing approximate solutions using multianisotropic integral operators.     

On Hamiltonian-Minimal Isotropic Homogeneous Tori in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We study isotropic homogeneous tori in ?ⁿ and ?Pⁿ. We find necessary and sufficient conditions for their Hamiltonian minimality.     

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.     

Kinetic Spanners in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

We present a new (1+ε)-spanner for sets of n points in ℝ ^d . Our spanner has size O(n/ε ^d−1) and maximum degree O(log  ^d n). The main advantage of our spanner is that it can be maintained efficiently as the points move: Assuming that the trajectories of the points can be described by bounded-degree polynomials, the number of topological changes to the spanner is O(n ²/ε ^d−1), and using a supporting data structure of size O(nlog  ^d n), we can handle events in time O(log  ^d+1 n). Moreover, the spanner can be updated in time O(log n) if the flight plan of a point changes. This is the first kinetic spanner for points in ℝ ^d whose performance does not depend on the spread of the point set.     

Existence,uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For a bounded domain Ω ⊂ ℝ ⁿ , n ⩾ 3, we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system − Δu + u · ∇u + ∇p = f, div u = k, u _|aΩ = g with u ∈ L ^q , q ⩾ n, and very general data classes for f, k, g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse & Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.