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Vol 84, No 5 (2020)
- Year: 2020
- Articles: 9
- URL: https://journals.rcsi.science/1607-0046/issue/view/7546
Articles
Uniqueness theorems for one-dimensional and double Franklin series
Abstract
The paper contains two main results. First we describe one-dimensional Franklin series converging everywhereexcept possibly on a finite set to an everywhere-finite integrable function. Second we establish a class of subsetsof $[0, 1]^2$ with the following property. If a double Franklin series converges everywhere except on this set to an everywhere-finite integrable function, then it is the Fourier–Franklin series of this function. In particular, all countablesets are in this class.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):3-19
3-19
On the topology of non-compact simply connected homogeneous manifolds
Abstract
We study the covariant bundles (Mostow bundles) for simply connected homogeneous manifolds, establishtheir relation to homogeneous bundles and consider classes of homogeneous manifolds for which theMostow bundle is trivial (resp. non-trivial). We also give a classification of non-compact simply connectedhomogeneous manifolds of dimension not exceeding seven.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):20-39
20-39
Diffeomorphisms of 2-manifolds with one-dimensional spaciously situated basic sets
Abstract
We consider orientation-preserving $A$-diffeomorphismsof orientable surfaces of genus greater than onewith a one-dimensional spaciously situated perfect attractor.We show that the topological classificationof restrictions of diffeomorphisms to such basic sets can be reducedto that of pseudo-Anosovhomeomorphisms with a distinguished set of saddles. In particular, we prove a result announced by Zhirov and Plykin, which gives a topological classification of the $A$-diffeomorphisms of the surfaces under discussion under the additional assumption that the non-wandering set consists of a one-dimensional spaciously situated attractor and zero-dimensional sources.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):40-97
40-97
Subdivision schemes on the dyadic half-line
Abstract
We consider subdivision schemes, which are used for the approximation of functions and generation of curves on the dyadic half-line.In the classical case of functions on the real line, the theory of subdivision schemes is widely known becauseof its applications in constructive approximation theory and signal processing as well as for generatingfractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-lineendowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role ofexponentials is played by Walsh functions.We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectralproperties of matrices and in terms of the smoothness of solutions of the corresponding refinement equation. We also investigate the problem of convergence of subdivision schemes with non-negative coefficients. We obtain an explicit criterion for theconvergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-lineand prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):98-118
98-118
Blow-up and global solubility in the classical sense of the Cauchy problem for a formally hyperbolic equation with a non-coercive source
Abstract
We consider an abstract Cauchy problem with non-linear operator coefficients and prove the existence of a uniquenon-extendable classical solution. Under certain sufficient close-to-necessary conditions, we obtainfinite-time blow-up conditions and upper and lower bounds for the blow-up time. Moreover, under certain sufficientclose-to-necessary conditions, we obtain a result on the existence of a global-in-time solutionindependently of the size of the initial functions.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):119-150
119-150
Distribution of prime numbers and the discrete spectrum of the Laplace operator
Abstract
We obtain a class of explicit formulae each of which gives an expression for the remainder term in the asymptotic equation for the Chebyshev function in terms of the spectrum of the Laplace operator on the fundamental domain of the modular group.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):151-168
151-168
169-196
Isotopes of alternative algebras of characteristic different from $3$
Abstract
We study homotopes of alternative algebras over an algebraically closed field of characteristic different from $3$. We prove an analogue of Albert's theorem on isotopes of associative algebras: in the class of finite-dimensional unital alternative algebras every isotopy is an isomorphism. We also prove that every $(a,b)$-homotope of a unital alternative algebra preserves the identities of the original algebra. We also obtain results on the structure of isotopes of various simple algebras, in particular, Cayley–Dixon algebras.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):197-210
197-210
On the standard conjecture for a $3$-dimensional variety fibred by curves with a non-injective Kodaira–Spencer map
Abstract
We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of theJacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira–Spencer maphas rank $1$ on some non-empty open subset of the surface. When the generic fibre of the structure morphism is ofgenus $2$, the condition on the endomorphisms of the Jacobian may be omitted.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(5):211-232
211-232