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Vol 87, No 1 (2023)

Year: 2023
Articles: 6
URL: https://ogarev-online.ru/1607-0046/issue/view/8355

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Articles

Framed motivic $\Gamma$-spaces

Garkusha G.A., Panin I.A., Østvær P.A.

Abstract

We combine several mini miracles to achieve an elementary understandingof infinite loop spaces and very effective spectra in the algebro-geometricsetting of motivic homotopy theory. Our approach combines $\Gamma$-spacesand Voevodsky's framed correspondences into the concept of framed motivic$\Gamma$-spaces; these are continuous or enriched functors of two variablesthat take values in framed motivic spaces. We craft proofs of our mainresults by imposing further axioms on framed motivic $\Gamma$-spacessuch as a Segal condition for simplicial Nisnevich sheaves, cancellation,$\mathbb{A}^1$- and $\sigma$-invariance, Nisnevich excision,Suslin contractibility, and grouplikeness.This adds to the discussion in the literature on coexisting pointsof view on the $\mathbb{A}^1$-homotopy theory of algebraic varieties.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(1):3-32

3-32

(Rus)

(JATS XML)

Multiple positive solutions for a Schrödinger–Poisson system with critical and supercritical growths

Lei J., Suo H.

Abstract

In this paper, we are concerned with the following Schrödinger–Poisson system$$\begin{cases}-\Delta u+u+\lambda\phi u= Q(x)|u|^{4}u+\mu\dfrac{|x|^\beta}{1+|x|^\beta}|u|^{q-2}u&in \mathbb{R}^3, -\Delta \phi=u^{2} &in \mathbb{R}^3, \end{cases}$$where $0< \beta<3$, $60$ are real parameters. By the variationalmethod and the Nehari method, we obtain that the system has $k$ positivesolutions.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(1):33-48

33-48

(Eng)

(JATS XML)

Exact formulas in some boundary crossing problemsfor integer-valued random walks

Lotov V.I.

Abstract

For a wide class of integer-valued random walks,we obtain exact expressions for the distributionof the first excess over leveland the corresponding renewal function as well as for the distribution of the trajectory supremumif it is finite.We discuss possibilities of obtaining explicit expressionsfor pre-stationary and stationary distributionsof a random walk with switchings at the strip boundaries.The research is based on the factorization representationsfor the double moment generating functions of the distributions under study.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(1):49-64

49-64

(Rus)

(JATS XML)

“Far-field interaction” of concentrated masses in two-dimensional Neumann and Dirichlet problems

Nazarov S.A.

Abstract

We study the eigenvalues of the Neumann and Dirichlet boundary-value problems in a two-dimensionaldomain containing several small, of diameter $O(\varepsilon)$, inclusions of large “density”$O(\varepsilon^{-\gamma})$, $\gamma\geq2$, that is, the “mass” $O(\varepsilon^{2-\gamma})$ of each of them is comparable ($\gamma=2$) or much bigger ($\gamma>2$) than that of the embracingmaterial. We construct a model of such spectral problems on concentrated masses which (the model)provides an asymptotic expansions of the eigenvalues with remainders of power-law smallnessorder $O(\varepsilon^{\vartheta})$ as $\varepsilon\to+0$ and $\vartheta\in(0,1)$. Besides,the correction terms are real analytic functions of the parameter $|{\ln \varepsilon}|^{-1}$. A “far-field interaction”of the inclusions is observed at the levels $|{\ln \varepsilon}|^{-1}$or $|{\ln \varepsilon}|^{-2}$. The results are obtained with the help of the machineryof weighted spaces with detached asymptotics and also by using weighted estimates of solutions to limit problemsin a bounded punctured domain and in the intact plane.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(1):65-118

65-118

(Rus)

(JATS XML)

On Romanoff's theorem

Radomskii A.O.

Abstract

Some results related to Romanoff's theorem are obtained.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(1):119-160

119-160

(Rus)

(JATS XML)

Deterministic and random attractors for a wave equation with sign changing damping

Chang Q., Li D., Sun C., Zelik S.V.

Abstract

The paper gives a detailed study of long-time dynamics generated byweakly damped wave equations in bounded 3D domains where the dampingcoefficient depends explicitly on time and may change sign. It is shown thatin the case, where the non-linearity is superlinear, the considered equationremains dissipative if the weighted mean value of the dissipation rateremains positive and that the conditions of this type are not sufficient inthe linear case. Two principally different cases are considered. In thecase when this mean is uniform (which corresponds to deterministicdissipation rate), it is shown that the considered system possesses smoothuniform attractors as well as non-autonomous exponential attractors. In thecase where the mean is not uniform (which corresponds to the randomdissipation rate, for instance, when this dissipation rate is generated bythe Bernoulli process), the tempered random attractor is constructed. Incontrast to the usual situation, this random attractor is expected to haveinfinite Hausdorff and fractal dimensions. The simplified model exampledemonstrating infinite-dimensionality of the random attractor is alsopresented.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(1):161-210

161-210

(Rus)

(JATS XML)

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