We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature $K$, we give a description of the totally geodesic unit vector fields for $K=0$ and $K=1$ and prove a non-existence result for $K\ne 0,1$. We also found a family ${\xi }_{\omega }$ of vector fields on the hyperbolic 2-plane $L^2$ of curvature $-c^2$ which generate foliations on $T_1{L}^2$ with leaves of constant intrinsic...

The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

We give a short proof of the Jacobian criterion of formal smoothness using the Lichtenbaum-Schlessinger cotangent complex.

Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences with given joint 2-adic complexity.

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences...

Let $\overline{X}$ be a germ of normal surface with local ring $\overline{R}$ covering a germ of regular surface $X$ with local ring $R$ of characteristic $p\>0$. Given an extension of valuation rings $W/V$ birationally dominating $\overline{R}/R$, we study the existence of a new such pair of local rings ${\overline{R}}^{\text{'}}/R^{\text{'}}$ birationally dominating $\overline{R}/R$, such that $R^{\text{'}}$ is regular and ${\overline{R}}^{\text{'}}$ has only toric singularities. This is achieved when $W/V$ is defectless or when $[W:V]$ is equal to $p$

Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $𝔤=\mathrm{Lie}\left(G\right)$ and let $U$ be its enveloping algebra. Let $𝔥$ be a Cartan subalgebra of $𝔤$. For $\mu \in {𝔥}^{*}$, let $J_{\mu }$ be the corresponding minimal primitive ideal, let $U_{\mu }=U/J_{\mu }$, and let ${𝒯}_{U_{\mu }}:K_0(U_{m}u)\to ℂ$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the $ℂ$-algebras $U_{\mu }$. When $\mu$ is regular, Hodges has shown that $K_0(U_{\mu })\cong K_0\left(X\right)$. In this case $K_0(U_{\mu })$ is generated by the classes corresponding to...