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Volume 216, Nº 12 (2025)
Unitary transform diagonalizing the Confluent Hypergeometric kernel
Resumo
We consider the image of the operator inducing the determinantal point process with the confluent hypergeometric kernel. The space is described as the image of $L_2[0,1]$ under a unitary transform, which generalizes the Fourier transform. For the derived transform we prove a counterpart of the Paley-Wiener theorem. We use the theorem to prove that the corresponding analogue of the Wiener-Hopf operator is a unitary equivalent of the usual Wiener-Hopf operator, which implies that it shares the same factorization properties and Widom's trace formula. Finally, using the introduced transform we give explicit formulae for the hierarchical decomposition of the image of the operator induced by the confluent hypergeometric kernel.
Matematicheskii Sbornik. 2025;216(12):3-24
3-24
Functions of density with respect to a model function of growth
Resumo
The properties of general density functions with respect to a model function of growth $M$ and related semiadditive functions are discussed. The concept of function of slow growth with respect to the model function of growth $M$ is introduced; it is shown that the function $L(r)= M^{-\rho}(r)v()r)$ has a slow growth with respect to $M$ . The concept of $\rho$ -semiadditive function with respect to $M$ is also introduced and the main properties of such functions are established. Density functions are studied; a criterion of the continuity of the density $N_M(\alpha)$ and lower density $\underline N_M(\alpha)$ of a function $f$ is obtained. A uniformity theorem is proved. The main properties of $\rho$ -additive and -semiadditive functions with respect to the model function $M$ are presented. POn of the central results is a theorem that can be viewed as an extension of Polya's theorem on the existence of minimal and maximal densities to a wider class of functions, whose growth is bounded by an arbitrary model function of growth $M$ . Examples of function $f$ and their density functions are presented.
Matematicheskii Sbornik. 2025;216(12):25-56
25-56
On 2-categories of extensions
Resumo
The paper is essentially an illustration to the general technique of homotopical enhancements developed recently in [6]. Taking a derived category of an abelian category we consider its full subcategory generated by complexes of length 2. It has a natural refinement to a 2-category, which we call the ‘2-category of extensions’. However, it is not possible to construct this refinement by only using a triangulated structure. In this short note, first we construct a 2-category of enhancements by hand, using the techniques of abelian categories, and then we show how it can quite easily and naturally be recovered in the framework of the enhanced formalism of [6].
Matematicheskii Sbornik. 2025;216(12):57-78
57-78
Explicit formulae for extremals in sub-Lorentzian and Finsler problems on 2D and 3D Lie groups
Resumo
Questions relating to the search of geodesics in a series of left-invariant problems with sub-Lorentzian and Finsler structure are under consideration. Explicit formulae for extremals are found in terms of trigonometric functions of convex trigonometry. In sub-Lorentzian problems the machinery of the new trigonometric functions $\sinh_\Omega$ and $\cosh_\Omega$ , generalizing $\sinh$ and $\cosh$ to the case of an unbounded convex set $\Omega\subset\mathbb R^2$ , is particularly useful.
Matematicheskii Sbornik. 2025;216(12):79-124
79-124
Inversion of the Abel–Prym map in presence of an additional involution
Resumo
Unlike Abel map of the symmetric power of a Riemann surface onto its Jacobian, the Abel–Prym map generically can not be reversed by means of conventional technique related to the Jacobi inversion problem, and of its main ingredient, namely the Riemann vanishing theorem. It happens because the corresponding analog of the Riemann vanishing theorem gives twice as many points as the dimension of the Prym variety. However, if the Riemann surface has a second involution commuting with the one defining the Prym variety and satisfying a certain additional condition, an analog of the Jacobi inversion can be defined, and expressed in terms of the Prym theta function. We formulate these conditions and refer to the pairs of involutions satisfying them as to pairs of the first type. We formulate necessary conditions for the pair of involutions to be a pair of the first type, and give a series of examples of curves with such pairs of involutions, mainly spectral curves of Hitchin systems, and also a spectral curve of the Kovalewski system.
Matematicheskii Sbornik. 2025;216(12):125-144
125-144