title: ÐÐ¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ° Ð¸ ÑÐµÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°Ñ Ð¾Ð±ÑÐ°Ð±Ð¾ÑÐºÐ¸ Ð´Ð°Ð½Ð½ÑÑ Ð² ÑÐ²Ð¸Ð½Ð½Ð¾Ð¹ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸ÐºÐµ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Formulation and solution of data processing problems in twin arithmetic
Creators: name: ÐÐ°Ð²Ð¾ÑÐ¾Ð½ÐºÐ¾Ð²Ð°, ÐÐ»Ð¸Ð½Ð°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2022
Subjects: ÐÐ½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð· (Ð¼Ð°Ñ.)
ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÑÐ²Ð¸Ð½Ð½Ð°Ñ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸ÐºÐ°
ÑÐ²Ð¸Ð½
ÑÐ¾Ð»ÑÑÑÐ¹ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»
ÑÐ¾Ð»ÑÑÑÐ¹ Ð±ÑÑÑ
ÑÐ¾Ð»ÑÑÐ°Ñ ÑÑÐ½ÐºÑÐ¸Ñ
ÑÐ¾Ð»ÑÑÐ¾Ðµ Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð°
Ð·Ð°Ð´Ð°ÑÐ° Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ ÑÐ¾Ð»ÑÑÐ¾Ð³Ð¾ Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð°
Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ðµ ÑÐ²Ð¸Ð½Ð½ÑÑ Ð¿ÑÐ¾Ð´Ð¾Ð»Ð¶ÐµÐ½Ð¸Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¹
twin arithmetic
twin
thick interval
thick timber
thick function
thick set
the problem of inverting a thick set
inversion of twin continuations of functions

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«ÐÐ¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ° Ð¸ ÑÐµÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°Ñ Ð¾Ð±ÑÐ°Ð±Ð¾ÑÐºÐ¸ Ð´Ð°Ð½Ð½ÑÑ Ð² ÑÐ²Ð¸Ð½Ð½Ð¾Ð¹ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸ÐºÐµÂ». ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ð¹ ÑÐ²Ð¸Ð½Ð¾Ð² (Â«ÑÐ¾Ð»ÑÑÑÑÂ» Ð¸Ð½ÑÐµÑÐ²Ð°Ð»Ð¾Ð²), Ð¿Ð¾Ð´ÑÐ¾Ð±Ð½Ð¾Ð¼Ñ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ Â«ÑÐ¾Ð»ÑÑÑÑÂ» Ð¸Ð½ÑÐµÑÐ²Ð°Ð»Ð¾Ð² Ð² ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²ÐºÐµ ÑÑÐ°Ð½ÑÑÐ·ÑÐºÐ¸Ñ ÑÑÑÐ½ÑÑ, Ð° ÑÐ°ÐºÐ¶Ðµ ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ Â«ÑÐ¾Ð»ÑÑÐ¾Ð³Ð¾Â» Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð° (Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ðµ ÑÐ²Ð¸Ð½Ð½ÑÑ Ð¿ÑÐ¾Ð´Ð¾Ð»Ð¶ÐµÐ½Ð¸Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¹). ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ:Â ÐÐ·ÑÑÐµÐ½Ð¸Ðµ ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²ÐºÐ¸ ÑÐ²Ð¸Ð½Ð¾Ð² Ð² Ð½Ð¾ÑÐ°ÑÐ¸Ð¸ Ð. Ð. ÐÐµÑÑÐµÑÐ¾Ð²Ð°; ÐÐ·ÑÑÐµÐ½Ð¸Ðµ ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²ÐºÐ¸ Â«ÑÐ¾Ð»ÑÑÑÑÂ» Ð¸Ð½ÑÐµÑÐ²Ð°Ð»Ð¾Ð² Ð² Ð½Ð¾ÑÐ°ÑÐ¸Ð¸ ÑÑÐ°Ð½ÑÑÐ·ÑÐºÐ¸Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÐºÐ¾Ð² (Luc Jaulin); ÐÐ¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ° Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ Â«ÑÐ¾Ð»ÑÑÐ¾Ð³Ð¾Â» Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð° (Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ ÑÐ²Ð¸Ð½Ð½ÑÑ Ð¿ÑÐ¾Ð´Ð¾Ð»Ð¶ÐµÐ½Ð¸Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¹); ÐÐ·ÑÑÐµÐ½Ð¸Ðµ Ð¸ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ Â«ÑÐ¾Ð»ÑÑÐ¾Ð³Ð¾Â» Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð° thickSIVIA; ÐÐ½Ð°Ð»Ð¸Ð· ÑÐ²Ð¾Ð¹ÑÑÐ² Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° thickSIVIA; ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¿ÑÐ¸Ð¼ÐµÑÐ¾Ð² ÑÐ°Ð±Ð¾ÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° thickSIVIA. Ð ÑÐ¾Ð´Ðµ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ»Ð¸ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ñ Ð½ÐµÐ¾Ð±ÑÐ¾Ð´Ð¸Ð¼ÑÐµ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸, Ð° ÑÐ°ÐºÐ¶Ðµ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼ Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð½ÑÑ Ð·Ð°Ð´Ð°Ñ. Ð Ð¿ÐµÑÐ²ÑÑ Ð¾ÑÐµÑÐµÐ´Ñ Ð±ÑÐ»Ð° Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð° Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ ÑÐ²Ð¸Ð½Ð¾Ð² Ð² ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²ÐºÐµ Ð. Ð. ÐÐµÑÑÐµÑÐ¾Ð²Ð°, Ð° ÑÐ°ÐºÐ¶Ðµ Ð±ÑÐ»Ð° ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ð° Ð´Ð°Ð½Ð½Ð°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ. ÐÐ¾ Ð²ÑÐ¾ÑÑÑ Ð¾ÑÐµÑÐµÐ´Ñ Ð±ÑÐ»Ð° Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð° Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ Â«ÑÐ¾Ð»ÑÑÑÑÂ» Ð¸Ð½ÑÐµÑÐ²Ð°Ð»Ð¾Ð² (ÑÐ²Ð¸Ð½Ð¾Ð²) Ð² ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²ÐºÐµ ÑÑÐ°Ð½ÑÑÐ·ÑÐºÐ¸Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÐºÐ¾Ð² (Luc Jaulin). Ð ÑÑÐµÑÑÐµÐ¼ ÑÑÐ°Ð¿Ðµ Ð±ÑÐ»Ð° ÑÐ¾ÑÐ¼Ð°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ð° Ð·Ð°Ð´Ð°ÑÐ° Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ Â«ÑÐ¾Ð»ÑÑÐ¾Ð³Ð¾Â» Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð° (Ð¾Ð±ÑÐ°ÑÐµÐ½Ð¸Ñ ÑÐ²Ð¸Ð½Ð½ÑÑ Ð¿ÑÐ¾Ð´Ð¾Ð»Ð¶ÐµÐ½Ð¸Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¹), ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼ thickSIVIA Ð¸ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¾ ÑÐµÑÑÐ¾Ð²ÑÑ ÑÐ»ÑÑÐ°ÐµÐ² Ð´Ð»Ñ Ð¸Ð·ÑÑÐµÐ½Ð¸Ñ ÑÐ°Ð±Ð¾ÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð°. ÐÑÐ» ÑÐ´ÐµÐ»Ð°Ð½ Ð²ÑÐ²Ð¾Ð´, ÑÑÐ¾ ÑÑÑÐµÑÑÐ²ÑÑÑ Ð·Ð°Ð´Ð°ÑÐ¸, Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ ÐºÐ¾ÑÐ¾ÑÑÑ Ð½ÐµÐ¾Ð¿ÑÐ°Ð²Ð´Ð°Ð½Ð½Ð¾ Ð²ÑÐ±Ð¸ÑÐ°ÑÑ ÐºÐ»Ð°ÑÑÐ¸ÑÐµÑÐºÑÑ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÑ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸ÐºÑ. Ð ÐµÑÐµÐ½Ð¸Ðµ ÑÐ°ÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ Ð² Ð¿Ð¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐµ Â«ÑÐ¾Ð»ÑÑÑÑÂ» Ð¸Ð½ÑÐµÑÐ²Ð°Ð»Ð¾Ð² (ÑÐ²Ð¸Ð½Ð¾Ð²) Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÐµÑ Ð·Ð½Ð°ÑÐ¸ÑÐµÐ»ÑÐ½Ð¾ ÑÑÐºÐ¾Ð½Ð¾Ð¼Ð¸ÑÑ Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½Ð¾Ðµ Ð²ÑÐµÐ¼Ñ. Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð½Ð°Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ Ð¸ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½ÑÐ¹ ÐºÐ¾Ð´ Ð±ÑÐ´ÑÑ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°ÑÑÑÑ Ð´Ð»Ñ Ð´Ð°Ð»ÑÐ½ÐµÐ¹ÑÐµÐ³Ð¾ Ð¸Ð·ÑÑÐµÐ½Ð¸Ñ Â«ÑÐ¾Ð»ÑÑÑÑÂ» Ð¸Ð½ÑÐµÑÐ²Ð°Ð»Ð¾Ð² Ð¸ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð¿Ð¾ÑÑÐ°Ð½Ð¾Ð²Ð¾Ðº Ð·Ð°Ð´Ð°Ñ. ÐÑÐ¾Ð±ÐµÐ½Ð½Ð¾ Ð¿ÐµÑÑÐ¿ÐµÐºÑÐ¸Ð²Ð½ÑÐ¼ Ð²ÑÐ³Ð»ÑÐ´Ð¸Ñ Ð² Ð¿ÐµÑÑÐ¿ÐµÐºÑÐ¸Ð²Ðµ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ ÑÐ²Ð¸Ð½Ð½ÑÑ Ð°ÑÐ¸ÑÐ¼ÐµÑÐ¸Ðº Ð¸ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð´Ð»Ñ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð´Ð°Ð½Ð½ÑÑ Ñ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½Ð¾Ð¹ Ð½ÐµÐ¾Ð¿ÑÐµÐ´ÐµÐ»ÑÐ½Ð½Ð¾ÑÑÑÑ Ð´Ð»Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿Ð¾Ð´ÑÐ¾Ð´Ð° ÐÐµÑÑÐµÑÐ¾Ð²Ð° Ð¿Ð¾Ð»ÑÑÐµÐ½Ð¸Ñ Ð¾Ð´Ð½Ð¾Ð²ÑÐµÐ¼ÐµÐ½Ð½Ð¾ Ð²Ð½ÑÑÑÐµÐ½Ð½Ð¸Ñ Ð¸ Ð²Ð½ÐµÑÐ½Ð¸Ñ Ð¾ÑÐµÐ½Ð¾Ðº Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ² ÑÐµÑÐµÐ½Ð¸Ð¹.
description: The subject of the graduate qualification work is Â«Formulation and solution of data processing problems in twin arithmeticÂ». This given work is devoted to studying various definitions of twins (thick intervals), a detailed studying the definition of thick intervals in the formulation of French mathematics, and solving thick set inversion problem (inversions of twin continuations of functions). The research set the following goals: Studying the twin formulation in V. M. Nesterovâs notation; Studying the thick intervals (twins) formulation in Luc Jaulin notation; Statement of the thick set inversion problem (inversions of twin continuations of functions); Study and implementation of the thick set inversion algorithm thickSIVIA; Analysis of the properties of the thickSIVIA algorithm; Studying examples of of the thickSIVIA. In the course of the work, the necessary mathematical models were developed, as well as an algorithm for solving the tasks. First of all, a mathematical model of twins was constructed in the formulation of V. M. Nesterov, and this model was also implemented. In the second place, a mathematical model of thick intervals (twin) was constructed in the formulation of French mathematics (Luc Jaulin). In the third stage, the thick set inversion problem was formalized, the thickSIVIA algorithm was implemented, and several test cases were considered to study the work of the algorithm. It was concluded that there are problems for which it is not justified to choose classical interval arithmetic. Solving such problems in the formulation of thick intervals (twin) can significantly save computational time. The developed mathematical model and program code will be used for further study of thick intervals and various problem statements.Especially promising in the future is the use of twin arithmetic and methods for analyzing data with interval uncertainty to implement the Nesterov approach of obtaining both internal and external estimates of decision sets.