Conformally convex functions and conformally flat metrics of nonnegative curvature

Kurkina, Mariya Viktorovna; Rodionov, E. D.; Slavskii, V. V.

doi:10.1134/s1064562415030096

Cited by 3 publications

(6 citation statements)

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“…: ff  и ** f есть выпуклая оболочка (или замыкание) функции f . Преобразование Лежандра применяется в физике, в теории дифференциальных уравнений, в теории выпуклых множеств [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Из новых применений преобразования Лежандра можно отметить применение в цифровой обработке сигналов [1][2][3].…”

Section: пусть функцияunclassified

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Kurkina¹,

Semenov²,

Slavsky³

et al. 2021

YuSU Bull

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Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra. By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

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Section: пусть функцияunclassified

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Kurkina¹,

Semenov²,

Slavsky³

et al. 2021

YuSU Bull

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show abstract

“…Преобразование Лежандра применяется в физике, в теории дифференциальных уравнений, в теории выпуклых множеств [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Из новых применений преобразования Лежандра можно отметить применения в цифровой обработке сигналов [1][2][3].…”

Section: Sobolevskyunclassified

“…здесь f (x) положительная конформно-выпуклая функция на сфере [6], то есть функция для которой конформно-плоская метрика ds…”

Section: Sobolevskyunclassified

Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

Куркина¹,

Semenov²,

Slavsky³

et al. 2020

Известия АлтГУ

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In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

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“…одномерная секционная кривизна конформно-плоской метрики (см. [7,8]). Здесь d 2 f /dξ 2 вторая производная функции в точке x ∈ R n вдоль единичного вектора ξ, ∇f градиент функции f в R n .…”

Section: коэффициент квазиконформности отображениеunclassified

“…Поэтому имеет смысл формулу (4) называть преобразованием Лежандра конформно-плоской метрики (см. [6][7][8]).…”

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Однородные Функции На Гильбертовом Пространстве И Квазиконформные Преобразования Сферы

Куркина¹,

Kurkina

Slavskii³

2020

Итоги науки и техники. Серия «Современная математика и ее прило

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Conformally convex functions and conformally flat metrics of nonnegative curvature

Cited by 3 publications

References 0 publications

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

Однородные Функции На Гильбертовом Пространстве И Квазиконформные Преобразования Сферы

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