On the fundamental homology classes of a real algebraic variety

It is proved that there is only one relation between the homology classes determined by the real points of a special real algebraic variety. This relation is equal to the sum of all the homology classes. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 216–219, August, 1999.     

Albanese homomorphism of the Chow group of 0-cycles of a real Algebraic variety

We study a certain homomorphism of the Chow group of 0-cycles of degree zero of a real algebraic variety into the group of real points of the Albanese variety; this homomorphism is obtained from the Albanese mapping for the corresponding variety. The kernel of this homomorphism is calculated and estimates for the kernel of the mapping of the torsion groups are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 76–83, January, 1999.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 403–106, September, 1998.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m ³/36 + O(m ²) saddle points.     

Null-quadrics of codimension 4 in ℂ7

In this paper we study the automorphisms of nondegenerate quadrics of type (4, 3). In particular, we classify the so-called null-quadrics (we prove that there are exactly two null-quadrics up to equivalence), and find their automorphism groups. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 657–665, November, 1997. Translated by S. S. Anisov     

Topological complexity and real roots of polynomials

The topological complexity of an algorithm is the number of its branchings. In the paper we prove that the minimal topological complexity of an algorithm that approximately computes a root of a real polynomial of degreed equalsd/2 for evend, is greater than or equal to 1 for oddd>–3, and equals 1 ford=3 or 5.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 670–680, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00846a and by the INTAS under grant No. 4373.     

On the measure of algebraic independence of values of derivatives of ap-adic modular function

The measure of the relative transcendence for the values of the derivatives of ap-adic Weyl-Lutz modular functionJ(q) at algebraic points is estimated. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 431–440, March, 1997. Translated by O. V. Sipacheva     

Kaplansky Density and Kadison Transitivity Theorems for Irreducible Representations of Real C＊-Algebras

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C＊-algebra on a real Hilbert space. Specifically, if a C＊-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold.     

The impact of 2-neighborhoods on graph structure

Connected regular graphs of diameter greater than 2 with 2-neighborhoods isomorphic to a certain graph Δ of diameter 2 are studied. Regular graphs of diameter 2 with 2-neighborhoods isomorphic to the distance-regular graph Δ of diameter 3 are also studied. It is prowed that 2-locally Schrikhande graphs do not exist. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 892–897, December, 1997. Translated by S. S. Anisov     

Differential geometry “in the large” of plane algebraic curves and integral formulas for invariants of singularities

We generalize the Plücker formula for the number of inflection points of a complex projective curve and derive a formula for the number of sextatic points of such a curve. We also obtain an upper estimate for the number of vertices of a real algebraic curve. The proof uses a new result related with integration on the Euler characteristic. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 255–268. Translated by N. Yu. Netsvetaev.     

On a certain canonical form

We amplify the well-known result due to Dlab and Ringel on the reduction of a real rectangular matrix to canonical form by formally complex transformations of rows and columns. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 514–519, April, 2000.     

Rigid isotopy classification of real algebraic curves of bidegree (3,3) on quadrics

A rigid isotopy of nonsingular real algebraic curves on a quadric is a path in the space of such curves of a given bidegree. We obtain the rigid isotopy classification of nonsingular real algebraic curves of bidegree (3, 3) on a hyperboloid and on an ellipsoid. We also study of the space of real algebraic curves of bidegree (3, 3) with a single node or cusp. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 810–815, December, 1999.     

Multidimensional analogs of direct and converse Jackson and Bernshtein theorems and their generalizations

We obtain an instance of the multidimensional analogs of the classical direct and converse Jackson and Bernshtein-Vallée-Poussin theorems (a version stronger than those obtained by S. M. Nikol’skii) and significant generalizations of these theorems to polynomial approximations of functions of real variables from Nikol’skii and Besov spaces on bounded domains with Lipschitzian boundary. The direct theorem concerning the Nikol’skii spaces was previously obtained by Yu. A. Brudnyi and generalized to joint polynomial approximations of functions and their derivatives by B. N. Konovalov. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 608–615, April, 2000.     

Numerical Schottky Uniformizations

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.Partially supported by Projects Fondecyt 1030252 1030373 and UTFSM 12.03.21     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

Functions of two variables continuous along straigt lines

For a functionf(x, y), the setsJ _a of all its discontinuity points with a jump ofa or more (that is, such that the oscillation of the function in the neighborhood of any point fromJ _a is not smaller thana) are studied. Two cases are considered: (1)f is continuous along any straight line; (2)f is continuous along lines parallel to thex- andy-axes. In the first case, conditions that must be met by the setJ _a are given. In the second case, it is shown that a (closed) setF can be the setJ _a for a certain function if and only if the projections ofF on the coordinate axes nowhere dense. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 306–311, August, 1997. Translated by V. N. Dubrovsky     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

A representation of one-dimensional attractors ofA-diffeomorphisms by hyperbolic homeomorphisms

We give a representation for the restrictions ofA-diffeomorphisms of closed orientable surfaces of genus > 1 from a homotopy class containing a pseudo-Anosov diffeomorphism to all one-dimensional attractors that do not contain special pairs of boundary periodic points. The representation is given by the restriction of a hyperbolic homeomorphism to an invariant zero-dimensional set formed by the intersection of two transversal geodesic laminations. It is shown how this result can be generalized to the representation of the restrictions ofA-diffeomorphisms defined on a closed surface of any genus to arbitrary one-dimensional attractors. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 76–87, July, 1997. Translated by V. E. Nazaikinskii