Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(？) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(？)RPn(？) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (？) RPn(？) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.     

On the families of sets without the Baire property generated by the Vitali sets

Let A be the family of all meager sets of the real line ℝ, V be the family of all Vitali sets of ℝ, V ₁ be the family of all finite unions of elements of V and V ₂ = {(C \ A ₁) ∪ A ₂: C ∈ V ₁; A ₁, A ₂ ∈ A}. We show that the families V, V ₁, V ₂ are invariant under translations of ℝ, and V ₁, V ₂ are abelian semigroups with the respect to the operation of union of sets. Moreover, V ⊂ V ₁ ⊂ V ₂ and V ₂ consists of zero-dimensional sets without the Baire property. Then we extend the results above to the Euclidean spaces ℝ ⁿ , n ≥ 2, and their products with the finite powers of the Sorgenfrey line.     

Equipartition of a continuous mass distribution

Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ², and let ℓ = {ℓ₁, ..., ℓ₅ ⊂ ℝ²} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an affine image of ℓ subdivides m into five parts with any prescribed ratios. (2) For each finite continuous mass distribution m in ℝⁿ, there exist n mutually orthogonal hyperplanes any two of which quarter m. (3) Let m and m′ be two finite continuous mass distributions in ℝRⁿ with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106.     

A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains

We say that a domain U ⊂ ℝ ⁿ is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain V ⊂ ℝ ⁿ such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of U, is isometric to U in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination of a domain by the relative metric of its Hausdorff boundary.     

Properties of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in ℝ ^m×n to be the gradient range of a C ¹-smooth function υ: Ω ⊂ ℝ ⁿ → ℝ ^m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ ^m×n are hyperplanes.     

Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

The present paper studies the following constrained vector optimization problem: min  _C f(x), g(x)∈−K, h(x)=0, where f:ℝ ⁿ →ℝ ^m , g:ℝ ⁿ →ℝ ^p and h:ℝ ⁿ →ℝ ^q are locally Lipschitz functions and C⊂ℝ ^m , K⊂ℝ ^p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x ⁰ to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.     

Valuations on arithmetic surfaces

In this paper, we give the definition of the height of a valuation and the definition of the big field Cp,G, where p is a prime and GR is an additive subgroup containing 1. We conclude that Cp,G is a field and Cp,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ≤n∈ Z, let Vm,n be an R-vector space of dimension n-m + 1, whose coordinates are indexed from m to n. We generalize the definition of Cp,G, where p i...     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn

Let S ⊂ ℝⁿ be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x₁ (M₁) ∪ … ∪ x_r (M_r) where each M_j is connected, x_j: M_j → V_j is a classical minimal immersion into an affine subspace V_j of ℝⁿ, and the subspaces V₁,…, V_r are pairwise orthogonal. Here we prove that if M_j is orientable, then x_j (M_j) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola.     

On groups acting by cohomogeneity one on the euclidean space ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

The paper studies closed Lie subgroups G ⊂ Iso(ℝ ⁿ ) acting by cohomogeneity one on ℝ ⁿ and prove that when there is no singular orbit, then there is a simply connected, solvable and closed Lie subgroup F ⊂ G which acts by cohomogeneity one on ℝ ⁿ and the two actions are orbit equivalent.     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Area,coarea, and approximation in <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript>

Let Ω⊂ℝ ⁿ be an arbitrary open set. We characterize the space W ^1,1 _loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space W ^1,1(ℝ ⁿ ).     

Nilfactors of ℝ m and configurations in sets of positive upper density in ℝ m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. LetE be a measurable subset of ℝ ^m , with . LetV = {0,v ₁,...,v _k} ⊂ ℝ^m. We show that fort large enough, we can find an isometric copy oftV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FuKaW] showing a similar property form=k=2.     

Estimating the diameter of the space of plane convex figures with respect to an affine-invariant metric

A convex figure K ⊂ ℝ² is a compact convex set with nonempty interior, and αK is a homothetic image of K with coefficient α ∈ ℝ. It is proved that for any two convex figures K₁, K₂ ⊂ ℝ² there is an affine transformation T of the plane such that K₁ ⊂ T(K₂) ⊂ 2.7K₁. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 58–66.     

A Generalization of Bochner’s Extension Theorem to Rough Tubes

This paper generalizes Bochner’s extension theorem to tubes X+iℝ ^m where the set X⊂ℝ ^m is not necessarily a manifold.     

Spectral synthesis in the sobolev space generated by an integral metric

Let l ∈ ℕ, A ⊂ ℝⁿ . The main goal of this paper is to describe (in inner terms) the closure of the set {f ∈ W ₁ ^l : f=0 in a neighborhood of the set A} with respect to the norm of the space W ₁ ^l (ℝⁿ). Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 92–111.     

Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W _loc ^m,p (ℝ ⁿ ) or W ^m,p (Q) for some cube Q ⊂ ℝ ⁿ if and only if the formal (m − 1)-Taylor remainder R ^m−1 f(x,y) of f satisfies the pointwise inequality |R ^m−1 f(x,y)| ≤ |x − y| ^m [a(x) + a(y)] for some a ε L ^p (Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ ⁿ . Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005.     

Spaces of functions of fractional smoothness on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce spaces of functions of fractional smoothness s > 0. We prove embedding theorems relating these spaces to the Sobolev spaces W _p ^m (G) and Lebesgue spaces L _p (G).