<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

Addendum to “Complete rotation hypersurfaces with <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> constant in space forms”*

Here we study complete rotation hypersurfaces with constant k-th mean curvature H_k in even and 2 < k < n. We prove the existence of a constant such that there are no such hypersurfaces for . We have only one compact hypersurface of this kind with . For each there is a corresponding family of complete immersed rotation hypersurfaces, each family containing two isoparametric hypersurfaces. For H_k ≥ 0, there is also such a family, now containing only one isoparametric hypersurface. Finally, we prove the existence of compact hypersurfaces with arbitrarily large H_k , neither isometric to a sphere nor to a product of spheres. *Bull. Braz. Math. Soc. 30 (2), 1999, 139–161. **Partially supported by FUNCAP, Brazil. ***Partially supported by CNPq, Brazil and DGAPA-UNAM, México.     

Extrinsic radius pinching in space forms of nonnegative sectional curvature

We first give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to these estimates. We show that under a suitable pinching condition, M is diffeomorphic and almost isometric to an n-dimensional sphere.      

Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

For a normal variation of a hypersurface M ⁿ in a space form Q _c ⁿ⁺¹ by a normal vector field fN, R. Reilly proved:

The Polynomial Degree of the Grassmannian {\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}

For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, r≤ N, if and only if ψ intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when ψ is the Grassmannian , to show that for n <8 the polynomial degree of is .     

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

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.      

The Product Formula in Unitary Deformation K-theory

For finitely generated groups G and H, we prove that there is a weak equivalence G H (G × H) of ku-algebra spectra, where denotes the “unitary deformation K-theory” functor. Additionally, we give spectral sequences for computing the homotopy groups of G and HG in terms of connective K-theory and homology of spaces of G-representations.     

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

On <Emphasis Type="Italic">Γ</Emphasis>-convergence of quadratic functionals in the plane

In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f _j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t ²log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.      

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

Strong Enumeration Reducibilities

We investigate strong versions of enumeration reducibility, the most important one being s-reducibility. We prove that every countable distributive lattice is embeddable into the local structure of the s-degrees. However, is not distributive. We show that on sets s-reducibility coincides with its finite branch version; the same holds of e-reducibility. We prove some density results for . In particular is upwards dense. Among the results about reducibilities that are stronger than s-reducibility, we show that the structure of the bs-degrees is dense. Many of these results on s-reducibility yield interesting corollaries for Q-reducibility as well.     

Exact Distribution of the Local Score for Markovian Sequences

Let be a sequence of letters taken in a finite alphabet Θ. Let be a scoring function and the corresponding score sequence where X _i = s(A _i ). The local score is defined as follows: . We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov model on the score sequence , and then on the letter sequence . The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous results using the independent model.     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

<Emphasis Type="Italic">q</Emphasis>-Besselian Frames in Banach Spaces

In this paper, we introduce the concepts of q-Besselian frame and （p, σ）-near Riesz basis in a Banach space, where a is a finite subset of positive integers and 1/p＋1/q = 1 with p 〉 1, q 〉 1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and （p, σ）-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X^＊ that if {xn}n=1^∞ C X is a q-Besselian frame for X, then there exists a p-Besselian frame {y＆＊}n=1^∞ belong to X^＊ for X^＊ such that x = ∑n=1^∞ yn^＊（x）xn for all x ∈ X, and x^＊ =∑n=1^∞ x^＊（xn）yn^＊ for all x^＊ ∈ X^＊. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.