Vector fields on the real flag manifolds ℝ<Emphasis Type="Italic">F</Emphasis>(1, 1, <Emphasis Type="Italic">n</Emphasis> − 2)

We obtain the span of the real flag manifolds ℝF(1, 1, n−2), n ≥ 3, for the cases n ≡ 2 (mod 4), n ≡ 4 (mod 8) and n ≡ 8 (mod 16) and use the results to deduce that certain Stiefel-Whitney classes of the manifold are zero.      

On a certain class of dynamical systems on the torus with rational rotation number

For system (1), the authors solve the problem of the existence of trajectories given by the linear equation y ≡ μx + y ₀ (mod 2π), μ ≥ 0, y ₀ ∈ ℝ, and in the case where such trajectories exist, they perform a complete qualitative study of the system on the two-dimensional torus under certain conditions. Translated From Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics And Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ² be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1. This work was partially completed while the author was at the Institut Fourier, Grenoble, France.     

Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

Projective graph theory and configurations of lines

The spaces of disjoint configurations of k-dimensional subspaces in ℝP ^2k+1 (for example, lines in ℝP ³) are studied. These spaces are modeled by various simplicial schemes, and the homology groups of the latter are computed in certain cases. We use the fact that every configuration can be assigned a so-called projective graph, which is a class of graphs with respect to a certain equivalence relation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 309–322. Translated by N. Yu. Netsvetaev.     

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

P 3-factorization of complete multipartite graphs

In this note it is shown that a necessary and sufficient condition for the existence of a P3-factorizatlon of complete multipartite graph λK, is (1) m≥3, (2) mn≡0(mod 3) and (3)λ(m-1)n≡0(mod 4).     

On perfect and multiply perfect numbers

Summary Denote by P(x) the number of integers n≤x satisfying σ(n)≡0 (mod n), and by P ₂ (x) the number of integers n ≤ x satisfying σ(n)=2n. The author proves that P(x)<x ^3/4+ɛ and P ₂ (x)<x ^(1−c)/2 for a certain c>0.     

The universal partition theorem for oriented matroids

We consider realization spaces of a family of oriented matroids of rank three as point configurations in the affine plane. The fundamental problem arises as to which way these realization spaces partition their embedding space. The Universal Partition Theorem roughly states that such a partition can be as complicated as any partition of ℝ ⁿ into elementary semialgebraic sets induced by an arbitrary finite set of polynomials in ℤ[X]. We present the first proof of the Universal Partition Theorem. In particular, it includes the first complete proof of the so-called Universality Theorem. This work was supported by the Deutsche Forschungsgemeinschaft, Graduiertenkolleg “Analyse und Konstruktion in der Mathematik”.     

Lattice of subalgebras of the ring of continuous functions and Hewitt spaces

The latticeA(X) of all possible subalgebras of the ring of all continuous ℝ-valued functions defined on an ℝ-separated spaceX is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line ℝ. The main achievement of the paper is the proof of the fact that any Hewitt spaceX is determined by the latticeA(X). An original technique of minimal and maximal subalgebras is applied. It is shown that the latticeA(X) is regular if and only ifX contains at most two points. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 687–693, November, 1997. Translated by A. I. Shtern     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

Frame Self-orthogonal Mendelsohn Triple Systems

A Mendelsohn triple system of order ν, MTS(ν) for short, is a pair (X, B) where X is a ν-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered pairs (a, b), (b, c) and (c, a). An MTS(ν) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order ν. An MTS(ν) is called frame self-orthogonal, FSOMTS for short, if its associated semisymmetric Latin square is frame self-orthogonal. It is known that an FSOMTS(1 ⁿ ) exists for all n≡1 (mod 3) except n=10 and for all n≥15, n≡0 (mod 3) with possible exception that n=18. In this paper, it is shown that (i) an FSOMTS(2 ⁿ ) exists if and only if n≡0,1 (mod 3) and n>5 with possible exceptions n∈{9, 27, 33, 39}; (ii) an FSOMTS(3 ⁿ ) exists if and only if n≥4, with possible exceptions that n∈{6, 14, 18, 19}. *Research supported by NSFC 10371002 *Partially supported by National Science Foundation under Grant CCR-0098093     

On quadratic congruences

It is proved that under appropriate assumptions, the solutions f of quadratic congruences Q(f) ≡ 0 (mod n), n ≤ x, with (f|n) = ±1, are distributed asymptotically equally as x → ∞. Bibliography: 5 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 195–200.     

Further Results on the Existence of Splitting BIBDs and Application to Authentication Codes

Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λ v(v−1)≡0 (mod (c ² u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes.     

Fundamental rectangles of admissible lattices

Let Λ be a unimodular lattice in ℝ², μ a homogeneous minimum of Λ; let P(a,b)⊂ℝ² be a rectangle with vertices at the points (a,0), ...(0,b), P(a, b)+X its image under the translation by a vector X ∈ ℝ². We prove that there exists a sequence of positive numbers v₁<v₂<...<v_k<... with , such that for u>μ the rectangle P(u, v_k)+X contains T=S(P)+R points of Λ, where |R|<5; here S(P) is the area of the rectangle. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 82–89. Translated by O. A. Ivanov.     

Approximation by harmonic functions in theC m -Norm and harmonicC m -capacity of compact sets in ℝ n

We study the function Λ ^m (X), 0<m<1, of compact setsX in ℝ ⁿ , n≥2, defined as the distance in the spaceC ^m (X)≡lip ^m(X) from the function |x|² to the subspaceH ^m (X) which is the closure inC ^m (X) of the class of functions harmonic in the neighborhood ofX (each function in its own neighborhood). We prove the equivalence of the conditions Λ ^m (X)=0 andC ^m (X)=H ^m (X). We derive an estimate from above that depends only on the geometrical properties of the setX (on its volume). Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 372–382, September, 1997. Translated by I. P. Zvyagin