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Тек жазылушылар үшін
Том 87, № 2 (2023)
Articles
A solution to the multidimensional additive homological equation
Аннотация
We prove that, for a finite-dimensional real normed space $V$,every bounded mean zero function $f\in L_\infty([0,1];V)$can be written in the form $f=g\circ T-g$ for some $g\in L_\infty([0,1];V)$and some ergodic invertible measure preserving transformation $T$of $[0,1]$.Our method moreover allows us to choose $g$, for any given $\varepsilon>0$,to be such that $\|g\|_\infty\leq (S_V+\varepsilon)\|f\|_\infty$,where $S_V$ is the Steinitz constant corresponding to $V$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):3-55
3-55
On the transference principle and Nesterenko's linear independence criterion
Аннотация
We consider the problem of simultaneous approximation of real numbers $\theta_1,…,\theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+\theta_1x_1+…+\theta_nx_n$ atinteger points. In this setting we analyse two transference inequalitiesobtained by Schmidt and Summerer. We present a rather simple geometricobservation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German'sinequalities for uniform exponents, Schmidt and Summerer's inequalities implythe inequalities by Bugeaud and Laurent and “one half” of the inequalitiesby Marnat and Moshchevitin. Moreover, we show that our main constructionprovides a rather simple proof of Nesterenko's linear independencecriterion.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):56-68
56-68
Green energy of discrete signed measure on concentric circles
Аннотация
We show that the difference between the Green energy of a discrete signed measurerelative to a circular annulus concentrated at some points on concentric circles and the energyof the signed measure at symmetric pointsis non-decreasing during the expansion of the annulus. As a corollary, generalizations ofthe classicalPolya–Schur inequality for complex numbers are obtained.Some open problems are formulated.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):69-88
69-88
Multivariate tile $\mathrm{B}$-splines
Аннотация
Tile $\mathrm{B}$-splines in $\mathbb R^d$ are defined as autoconvolutionsof indicators of tiles,which are special self-similar compact sets whose integer translates tilethe space $\mathbb R^d$. These functions are not piecewise-polynomial,however, being directgeneralizations of the classical $\mathrm{B}$-splines, they enjoy many oftheir properties and have some advantages. In particular, exactvalues of the Hölder exponents of tile $\mathrm{B}$-splinesare evaluated and are shown, in some cases, to exceed those of the classical $\mathrm{B}$-splines.Orthonormal systems of wavelets based on tile B-splines are constructed,and estimates of their exponential decay are obtained.Efficiency in applications of tile $\mathrm{B}$-splines is demonstrated on an example of subdivision schemesof surfaces. This efficiency is achieved due to high regularity, fast convergence, and small numberof coefficients in the corresponding refinement equation.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):89-132
89-132
A new class of fractional differential hemivariational inequalitieswith application to an incompressible Navier–Stokes system coupled witha fractional diffusion equation
Аннотация
This paper is devoted to the study of a new and complicated dynamical system,called a fractional differential hemivariational inequality, which consistsof a quasilinear evolution equation involving the fractional Caputo derivativeoperator and a coupled generalized parabolic hemivariational inequality.Undercertain general assumptions, existence and regularity of a mild solution tothe dynamical system are established by employing a surjectivity result forweakly–weakly upper semicontinuous multivalued mappings, and a feedbackiterative technique together with a temporally semi-discrete approach throughthe backward Euler difference scheme with quasi-uniform time-steps. Toillustrate the applicability of the abstract results, we consider a nonstationaryand incompressible Navier–Stokes system supplemented by a fractionalreaction–diffusion equation, which is studied as a fractional hemivariationalinequality.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):133-167
133-167
Hardy type inequalities for one weight function and their applications
Аннотация
New one-dimensional Hardy-type inequalities for a weight function of the form $x^\alpha(2-x)^\beta$ for positive and negative values of the parameters $\alpha$ and $\beta$ are put forward.In some cases, the constants in the resulting one-dimensional inequalities are sharp. We use one-dimensional inequalities with additional terms to establish multivariate inequalities with weight functions depending on the mean distance function or the distance function from the boundary of a domain. Spatial inequalities are proved in arbitrary domains, in Davies-regular domains, in domains satisfying the cone condition, in $\lambda$-close to convex domains,and in convex domains. The constant in the additional term in the spatial inequalities depends on the volume orthe diameter of the domain. As a consequence of these multivariate inequalities,estimates for the first eigenvalue of the Laplacian under the Dirichlet boundary conditions in various classes of domains are established. We also use one-dimensional inequalities to obtain new classes of meromorphic univalent functions in simply connected domains. Namely,Nehari–Pokornii type sufficient conditions for univalence are obtained.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):168-195
168-195
The nonarithmeticity of the predicate logicof primitive recursive realizability
Аннотация
The notion of primitive recursive realizability was introducedby S. Salehi as a kind of semantics for the language of basic arithmeticusing primitive recursive functions. It is of interest to studythe corresponding predicate logic. D. A. Viter proved that the predicatelogic of primitive recursive realizability by Salehi is not arithmetical.The technically complex proof combines the methods used by the authorof this article in the study of predicate logics of constructive arithmetictheories and the results of M. Ardeshir on the translation ofthe intuitionistic predicate logic into the basic predicate logic.The purpose of this article is to present another proof of Viter's resultby directly transferring the methods used earlier in provingthe nonarithmeticity of the predicate logic of recursive realizability.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):196-228
196-228
Corrections
Аннотация
Corrections to the papers:V. A. Krasnov, “The real Plücker–Klein map”, Izv. RAN. Ser. Mat., 86:3 (2022), 47–104; I. A. Panin, “An extended form of the Grothendieck–Serre conjecture”, Izv. RAN. Ser. Mat., 86:4 (2022), 175–191; S. G. Tankeev, “On the standard conjecture for compactifications of Néron models of 4-dimensional Abelian varieties”, Izv. RAN. Ser. Mat., 86:4 (2022), 192–232.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(2):229-236
229-236