Normal numbers and selection rules

Given anormal number x=0,x ₁ x ₂ ··· to base 2 and aselection rule S ?{0, 1}*=∪ _n=0 ^/t8 {0, 1} ⁿ , we define a subsequencex,=0, \(\chi _{t_1 } \chi _{t_2 } \) ·· where {t ₁<t ₂<···}={i; x ₁ x ₂···x _i?1 εS}.x _s is called aproper subsequence ofx if lim_i/∞ ti/S is said topreserve normality if for any normal numberx such thatx _s is a proper subsequence ofx, x _s is also a normal number. We prove that ifS/～ _s is a finite set, where ～ _s is an equivalence relation on {0, 1}* such that ξ ～ _s η if and only if {ζ; ξζ εS}={ζ; ηζ εS}, thenS preserves normality. This is a generalization of the known result in finite automata case, where {0, 1}*/～ _s is a finite set (Agafonov [1]).     

Normal numbers of piecewise-linear manifolds

A definition is given of the normal class of the immersion of a closed piecewise-linear manifold in a piecewise-linear manifold. It is shown that this number is zero for the immersion of an orientable manifold in euclidean space of any dimension. A complete investigation is carried out of normal classes of piecewise-smooth immersions of nonorientable manifolds in a euclidean space with dimension twice that of the manifolds.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 575–583, May, 1971.     

A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

     

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_η over K(T), Silverman's specialization theorem implies rank(E_η(K(T)))?rank(E_t(K)) for all but finitely many t∈P¹(K). If this inequality is strict for all but finitely many t, the elliptic curve E_η is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family E_c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ^×) such that, under the parity conjecture, each E_c,d has elevated rank.     

Normal Abelian subgroups and Euler characteristic

We generalize Rosset's theorem which states that the Euler characteristic of a group G of type FL_C vanishes if G contains a torsion free normal subgroup. In our case the subgroup is allowed to have torsion (but must also have elements of infinite order). Under similar conditions on the regular covering of a finite CW-complex X, it is shown that the Euler characteristic of X is 0; this includes the special case where X is nilpotent.     

Estimates of characteristic numbers of real algebraic varieties

We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than d, and for the size of the sum of Betti numbers with Z/2 coefficients for the real form of complex manifolds of complex degree less than d.     

On localization of the characteristic numbers of real matrices

In this paper we give domains for the distribution on the complex plane of the characteristic numbers of a set of real matrices, basing our work on results obtained by F. I. Karpelevich for matrices with nonnegative elements.Translated from Matematicheskie Zametki, Vol. 15, No. 5, pp. 765–768, May, 1974.In conclusion the author expresses his thanks to R. A. Poluéktov and G. S. Épel'man for useful discussions of his work.     

Recursions for characteristic numbers of genus one plane curves

Characteristic numbers of families of maps of nodal curves toP ² are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.     

On vanishing of characteristic numbers in Poincaré complexes

Let be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map is non-trivial and is a field. We will prove: if is a Poincaré complex oriented in -coefficient, all the characteristic numbers of in -coefficient vanish. Similarly, if and is a -Poincaré complex, then all the mod Wu numbers vanish. We will also show that the existence of a non-trivial derivation on with some suitable conditions implies vanishing of mod Wu numbers.

     

Explicit form of characteristic numbers of a periodic problem

A periodic problem for a linear differential equation of the second order is reduced to a periodic problem for a differential equation of the first order, but with deviation argument. We indicate the cases when the characteristic numbers are determined explicitly. This paper is the continuation of investigations commenced in “Differents. Uravneniya,” 44 (4) (2008).     

Indices, characteristic numbers and essential commutants of Toeplitz operators

For an essentially normal operatorT, it is shown that there exists a unilateral shift of multiplicitym inC ^* (T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic asC ^* -algebras. Finally, we construct a naturalC ^* -algebra ε + ε_* on the Bergman spaceL _a ² (B _n ), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators. Supported by NSFC and Laboratory of Mathematics for Nonlinear Science at Fudan University.     

Edge-partitions of graphs of nonnegative characteristic and their game coloring numbers

Let G be a graph of nonnegative characteristic and let g(G) and Δ(G) be its girth and maximum degree, respectively. We show that G has an edge-partition into a forest and a subgraph H so that (1) Δ(H)?1 if g(G)?11; (2) Δ(H)?2 if g(G)?7; (3) Δ(H)?4 if either g(G)?5 or G does not contain 4-cycles and 5-cycles; (4) Δ(H)?6 if G does not contain 4-cycles. These results are applied to find the following upper bounds for the game coloring number col_g(G) of G: (1) col_g(G)?5 if g(G)?11; (2) col_g(G)?6 if g(G)?7; (3) col_g(G)?8 if either g(G)?5 or G contains no 4-cycles and 5-cycles; (4) col_g(G)?10 if G does not contain 4-cycles.     

Circle action, lower bound of fixed points and characteristic numbers

Given an S ¹-manifold with isolated fixed points, some recent papers are concerned with the relationship between the least number of fixed points and the characteristic numbers of this manifold, and their proofs have some similar features. The main purpose of this note is, by using the language of equivariant cohomology, to present a unified method to deal with such problems, of which the related known results are direct corollaries.