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.     

The colored Hadwiger transversal theorem in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Hadwiger’s transversal theorem gives necessary and suffcient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let F be a family of convex sets in ? ^d in bijection with a set P of points in ? ^d?1. Assume that there is a coloring of F with suffciently many colors such that any colorful Radon partition of points in P corresponds to a colorful Radon partition of sets in F. Then some monochromatic subfamily of F has a hyperplane transversal.     

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.     

<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.     

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

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.      

The Projection Median of a Set of Points in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

The projection median of a finite set of points in ℝ² was introduced by Durocher and Kirkpatrick [Computational Geometry: Theory and Applications, Vol. 42 (5), 364–375, 2009]. They proved that the projection median in ℝ² provides a better approximation of the two-dimensional Euclidean median than the center of mass or the rectilinear median, while maintaining a fixed degree of stability. In this paper we study the projection median of a set of points in ℝ ^d for d≥2. Using results from geometric measure theory we show that the d-dimensional projection median provides a (d/π)B(d/2,1/2)-approximation to the d-dimensional Euclidean median, where B(α,β) denotes the Beta function. We also show that the stability of the d-dimensional projection median is at least \frac1(d/p)B(d/2, 1/2)\frac{1}{(d/\pi)B(d/2, 1/2)}, and its breakdown point is 1/2. Based on the stability bound and the breakdown point, we compare the d-dimensional projection median with the rectilinear median and the center of mass, as a candidate for approximating the d-dimensional Euclidean median. For the special case of d=3, our results imply that the three-dimensional projection median is a (3/2)-approximation of the three-dimensional Euclidean median, which settles a conjecture posed by Durocher.     

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 sets of directions determined by subsets of ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Given E ? ? ^d , d ≥ 2, define

$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$

to be the set of directions determined by E. We prove that if the Hausdorff dimension of E is greater than d ? 1, then σ(D(E)) > 0, where σ denotes the surface measure on S ^d?1. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on D. This result is sharp, since the conclusion fails to hold if E is a (d ? 1)-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir ([22, 23]) on directions determined by finite subsets of ? ^d . We also discuss the case when the Hausdorff dimension of E is precisely d ? 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of E equals d ? 1 and E is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set P ? ? ^d , d ≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ? #P distinct directions.

     

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.     

On the complexity of the family of convex sets in ℝ<Superscript> <Emphasis Type="Italic">d</Emphasis> </Superscript>

Estimates of quantities characterizing the complexity of the family of convex subsets of the d-dimensional cube [1, n]^d as n→∞ are given. The geometric properties of spaces with norm generated by the generalized majorant of partial sums are studied.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

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.     

Bases of exponents in weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">−π, π</Emphasis>)

The system of exponents $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} is considered. A sufficient condition for a Riesz-property basis in the weighted space L ^p (−π, π) is obtained.     

A Jenkins-Serrin theorem in <Emphasis Type="Italic">M</Emphasis><Superscript>2</Superscript> × ℝ

In this paper we study minimal surfaces in M × ℝ, where M is a complete surface. Our main result is a Jenkins-Serrin type theorem which establishes necessary and sufficient conditions for the existence of certain minimal vertical graphs in M × ℝ. We also prove that there exists a unique solution of the Plateau’s problem in M × ℝ whoseboundaryisaNitschegraphandweconstructaScherk-typesurfaceinthisspace. Thanks to CNPq Agency for financial support.     

Pointwise multipliers and decomposition theorems in <Emphasis Type="Italic">F</Emphasis><Superscript><Emphasis Type="Italic">∞,q</Emphasis></Superscript><Subscript><Emphasis Type="Italic">s</Emphasis></Subscript>

In this paper we give a characterization of the pointwise multipliers of the holomorphic Triebel-Lizorkin spaces F^{, q}_s, and we apply these results to prove a corona decomposition theorem for these spaces.Mathematics Subject Classification (1991):32A37Both authors partially supported by DGICYT Grant BMF2002-04072-C02-01 and CIRIT Grant 2001-SGR-00172     

Free completely <Emphasis Type="Italic">J</Emphasis><Superscript>(<Emphasis Type="Italic">ℓ</Emphasis>)</Superscript>-simple Semigroups

A semigroup is called completely J^(ι)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J^(ι)-simple semigroups form a quasivarity. Moreover, the construction of free completely J^(ι)-simple semigroups is given. It is found that a free completely J^(ι)-simple semigroup is just a free completely J ^*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

The <Emphasis Type="Italic">ψ</Emphasis>-associate <Emphasis Type="Italic">X</Emphasis><Superscript>#</Superscript> of a flat <Emphasis Type="Italic">X</Emphasis> in PG(<Emphasis Type="Italic">n</Emphasis>, 2)

For a given hypersurface ψ in PG(n, 2), with equation Q(x) = 0, where Q is a polynomial of (reduced) degree d > 1, a definition is given of the ψ-associate X ^# of a flat X in PG(n, 2). The definition involves the fully polarized form of the polynomial Q; in the cubic case d = 3 it reads: X ^# = {z ∈ PG(n, 2) | T(x, y, z) = 0 for all x, y ∈ X}, where T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing Q. Some general results, valid for any degree d and projective dimension n, are first expounded. Thereafter several choices of ψ are visited, but for each choice just a few aspects are highlighted. Despite the partial nature of the survey quite a variety of behaviours of the ψ-associate are uncovered. Many of the choices of ψ which are considered are of cubic hypersurfaces in PG(5, 2). If ψ is the Segre variety it is shown that the 48 planes external to fall into eight pairs of ordered triplets {(P ₁, R ₁, S ₁), (P ₂, R ₂, S ₂)} such that and . Further those lines L of PG(5, 2) which are singular, satisfying that is L ^# = PG(5.2), are in this case shown to form a complete spread of 21 lines. Another result of note arises in the case where ψ is the underlying 35-set of a non-maximal partial spread Σ₅ of five planes in PG(5, 2), where it is shown that one plane is singled out by the property that every line is singular.