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Том 242, № 4 (2019)
- Год: 2019
- Статей: 8
- URL: https://bakhtiniada.ru/1072-3374/issue/view/15030
Article
In Memory of Oleg Mstislavovich Fomenko (1936–2017)
469-469
Eisenstein Formula and Dirichlet Correspondence
Аннотация
In the paper, an exact formula for the number of integral points in the system of ellipses related, according to Dirichlet, to an arbitrary imaginary quadratic field is provided. The relation of this formula to an arithmetic Riemann–Roch theorem is discussed. Previously, only nine formulas of such a type have been known. They correspond to the imaginary quadratic fields with the trivial class group.
470-486
The Karyon Algorithm for Expansion in Multidimensional Continued Fractions
Аннотация
The paper presents a universal karyon algorithm, applicable to an arbitrary collection of reals α = (α1, . . . , αd), which is a modification of the simplex-karyon algorithm. The main distinction is that instead of a simplex sequence, an infinite sequence T = T0,T1, . . . ,Tn, . . . of d-dimensional parallelohedra Tn appear. Every parallelohedron Tn is obtained from the previous one Tn−1 by differentiation,\( {\mathbf{T}}_n={\mathbf{T}}_{n-1}^{\sigma n} \). The parallelohedra Tn are the karyons of some induced toric tilings. A certain algorithm (ϱ-strategy) for choosing infinite sequences σ|={σ1, σ2, …, σn, …} of differentiations σn is specified. This algorithm ensures the convergence ϱ(Tn) −→ 0 as n → +∞, where ϱ(Tn) denotes the radius of the parallelohedron Tn in the metric ϱ chosen as the objective function. It is proved that the parallelohedra Tn have the minimality property, i.e., the karyon approximation algorithm is the best one with respect to the karyon Tn-norms. Also an estimate for the rate of approximation of real numbers α = (α1, . . . , αd) by multidimensional continued fractions is derived.
487-508
Unimodularity of Induced Toric Tilings
Аннотация
Induced tilings \( \mathcal{T}=\mathcal{T}\left|{}_{\mathrm{Kr}}\right. \) of the d-dimensional torus ????d generated by an embedded karyon Kr are considered. The operations of differentiation \( \sigma :\mathcal{T}\to {\mathcal{T}}^{\sigma } \) are defined; as a result, the induced tilings \( {\mathcal{T}}^{\sigma }=\mathcal{T}\left|{}_{{\mathrm{Kr}}^{\sigma }}\right. \) of the same torus ????d, generated by the derivative karyon Krσ, are obtained. In terms of karyons Kr, the differentiations σ reduce to a combination of geometric transformations of the space ℝd. It is proved that if the karyon Kr is unimodular, then it generates an induced tiling \( \mathcal{T}=\mathcal{T}\left|{}_{\mathrm{Kr}}\right. \), and the derivative karyons Krσ are unimodular as well, whence the corresponding derivative tilings \( {\mathcal{T}}^{\sigma }=\mathcal{T}\left|{}_{{\mathrm{Kr}}^{\sigma }}\right. \) exist. Using unimodular karyons, one can construct an infinite family of induced tilings \( \mathcal{T}=\mathcal{T} \) (α, Kr*), depending on a shift vector α of the torus ????d and an initial karyon Kr*. Two algorithms for constructing such unimodular karyons Kr* are presented.
509-530
Unimodular Invariance of Karyon Expansions of Algebraic Numbers in Multidimensional Continued Fractions
Аннотация
By the method of differentiation of induced toric tilings, periodic expansions for algebraic irrationalities in multidimensional continued fractions are found. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that are expanded in purely periodic continued fractions. Karyon expansions have several invariants: recurrence relations for the numerators and denominators of the convergents of continued fractions and the rate of multidimensional approximation of irrationalities by rational numbers.
531-559
560-567
Kummer’s Tower and Big Zeta Functions
Аннотация
The paper discusses the statement of the problem of constructing a big zeta function. This problem is related to an arithmetic Hurwitz formula. Two candidates for the part of the big zeta are suggested. Representations and ramification structures related to Kummer’s tower are studied.
568-574
Number of Nonzero Cubic Sums
Аннотация
The exponential sums \( {S}_q\left(a,m\right)=\sum \limits_{l=1}^q\exp \left(2\pi i\left({al}^3+ ml\right){q}^{-1}\right) \) are considered. For every positive integer q, closed-form expressions for the number of nonzero sums occurring among Sq(a, 0), . . . , Sq(a, q − 1) are found.
575-585