Estimates of the number of singular points of a complex hypersurface and related questions

It is well known that the number of isolated singular points of a hypersurface of degree d in ℂP^m does not exceed the Arnol’d number A_m(d), which is defined in combinatorial terms. In the paper it is proved that if b _m−1 ^± (d) are the inertia indices of the intersection form of a nonsingular hypersurface of degree d in ℂP^m, then the inequality A_m(d)<min{b _m−1 ⁺ (d), b _m−1 ⁻ (d)} holds if and only if (m−5)(d−2)≥18 and (m,d)≠(7,12). The table of the Arnol’d numbers for 3≤m≤14, 3≤d≤17 and for 3≤m≤14, d=18, 19 is given. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 180–190. Translated by O. A. Ivanov and N. Yu. Netsvetev.     

Some Properties of Solutions to a Second Order Elliptic Equation with Principal Part of Divergence Form with Potential Concentrated on a Hypersurface

A number of classical results of the theory of second order elliptic equations of divergence form in ℝ ^d, d ⩾ 2, is generalized to problems with singular potential concentrated on a hypersurface of codimension 1. Such results are required, for example, for describing photon crystals. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 25–37.     

Singular homologically area minimizing surfaces of codimension one in Riemannian manifolds

For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (N ^{n +1},g), with non-trivial n dimensional integer homology, such that for some Γ∈H _n (N,Z), the hypersurface (n-current) M, which minimizes area among all hypersurfaces representing Γ, has singularities. The singular set of M consists of two isolated points, and the tangent cone at these points can be prescribed as any strictly stable, strictly minimizing, regular cone. To my knowledge these are the first examples of codimension one homological minimizers with singularities. Oblatum: 3-I-1997 & 13-II-1998 / Published online: 18 September 1998     

CR Runge Sets on Hypersurface Graphs

This work contains an improvement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if ω, an open subset of a real hypersurface in ℂ ⁿ , can be graphed over a convex subset in ℝ²ⁿ⁻¹, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂ ⁿ in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension. R. Dwilewicz is partially supported by the Polish Science Foundation (KBN) grant N201 019 32/805.     

Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.     

Factoriality of Certain Threefolds Complete Intersection in P 5 with Ordinary Double Points

Abstract

Let V ? P ⁵ be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t ² ? t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial.     

Two analogs of the Rudin-Carleson theorem

In this article, a new asymptotic estimate for embeddings of singular measures into H^1,∞ is proved and an application to multidimensional analogs of the theorem of Rudin-Carleson is given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 200–209. Translated by S. V. Kislyakov.     

Global solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables

The Cauchy-Dirichlet problem for quasilinear parabolic systems of second-order equations is considered in the case of two spatial variables. Under the condition that the corresponding elliptic operator has variational structure, the global in time solvability is established. The solution is smooth almost everywhere and the number of singular points is finite. Sufficient conditions that guarantee the absence of singular points are given. Bibliography: 23 titles. Translated fromProblemy Matematicheskogo Analiza No. 16, 1997, pp. 3–40.     

The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

Two questions on the exterior geometry of the plücker embeddings of the Grassmann manifolds

The Plücker model of the Grassmann manifold G _p,p+q ⁺ is considered. The structure of intersections of G _p,p+q ⁺ with tangent spaces of G _p,p+q ⁺ regarded as subspaces of the ambient exterior algebra is described. An explicit formula for the second fundamental form of G _2,4 ⁺ as of a hypersurface in the five-dimensional sphere is given. The level sets of the normal curvature functions for this hypersurface are studied. Bibliography: 3 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 5–12. Translated by N. Yu. Netsvetaev.     

Certain examples of birationally rigid varieties pencil of double quadrics

We prove that double covers ofP ¹×Q, where is a nondegenerate quadric, and the branch divisor cut out by a hypersurface of the type (2λ, 2k−2) are birationally superrigid. In particular, they have only one structure of a Fano fibration. Therefore, they are nonrational. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya, Tematicheskie Obzory. Vol. 56. Algebraic Geometry-9, 1998.     

Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞

Let Ω be a bounded circular domain in ℂ ^N , let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 9–91. Translated by B. Bekker.     

The number of components of complements to level surfaces of partially harmonic polynomials

In this paperk-harmonic polynomials in ℝ ⁿ , i.e. polynomials satisfying the Laplace equation with respect tok variables: ∂²/∂x ₁ ² +...+∂²/∂x _k ² F=0 are considered; here 1≤k≤n andn≥2. For a polynomialF (of degreem) of this type, it is proved that the number of components of the complements of its level sets does not exceed 2m ⁿ⁻¹+O(m ⁿ⁻²). Under the assumptions that the singular set of the level surface is compact or that the leading homogeneous part of thek-harmonic polynomialF is nondegenerate, sharper estimates are also established. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 831–835, December, 1997 Translated by S. S. Anisov     

On Multiple Singular Integrals along Polynomial Curves with Rough Kernels

This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.     

Invariant subspaces of a lightguide with periodic boundary

A waveguide operator is defined. It is proved that its spectrum coincides with the spectrum of a lightguide. The classification of singular points of the continuous spectrum is given. Invariant subspaces of the waveguide operator are distinguished that are related to an interval of the continuous spectrum without singular points. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 51–62.     

Existence, uniqueness and stability of C m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ℝ,G∈C ^m (J ⁿ⁺¹, ℝ) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

Singular parts of pluriharmonic measures

A measure μ defined on the complex sphere S is called pluriharmonic if its Poisson integral is a pluriharmonic function (in the unit ball of ℂⁿ). A probability measure ρ is called representing if ∫_Sfdp=f(0) for all f in the ball algebra. It is shown that singular parts of pluriharmonic measures and representing measures are mutually singular. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 54–58.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.