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Гончаров С.С., Молоков А.В., Романовский А.С. Нильпотентные группы конечной алгоритмической размерности. Сиб.мат. журнал // 1989.-№1(30),-С.82-88.
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УДК 513
q-ЕСЕПТЕУДЕГІ ХАРДИ ТЕҢСІЗДІГІ
Бақтияр С.Б., Шаймардан С.
Л.Н.Гумилев атындағы Еуразия ұлттық университеті, Астана
Ғылыми жетекші – Абылаева А.М., Темірханова А.М.
Есептеу техникасының қарқынды дамуына байланысты, матаматикада көптеген жаңа бағыттар пайда болды. Соның бірі q-есептеу деп аталады. q әрпі quantum сөзінің алғашқы әрпін білдіреді. Бұл бағыт бойынша алғашқы зерттеулер ХХ ғасырдың басында Ф.Г. Джексон (F.H. Jackson) [1], Р.Д. Кармайкл (R.D. Carmichael) [2] және т.б. математиктердің жұмыстарында қарастырылған. Тек 80 жылдардан ғана бастап бұл бағыт қарқынды зерттеліп, q-комбинаторика, q-арифметика, q-вариациялық есептеулер, q-интегралдық және q-дифференциалдық есептеулер тармақтары пайда болды.
1908 жылы Ф.Г.Джексон [1] ![](data:image/png;base64,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) жағдайы үшін туынды ұғымының q-аналогын енгізіп, келесі түрде анықтады:
Бұл формулада q-ді 1-ге ұмтылдырған жағдайда ![](data:image/png;base64,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) ![](data:image/png;base64,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) кәдімгі туындыны беретінін көруге болады:
мұндағы, ![](data:image/png;base64,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) ![](data:image/png;base64,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) .
Функцияның қосындысының, көбейтіндісінің және қатынасының q-туындысы, Лейбниц формуласы келесі түрде анықталған:
,
,
мұндағы, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAAUCAYAAADWQYA8AAAACXBIWXMAAA6/AAAOkwEBNDiKAAAHaElEQVR4nO2Ye1BU9xXHP7ssyz5ggX0JuxABUUBXRBRRgyAxIj4KSTTx0RZarY2xE006VpuJSbRO06kx1cy0TWsasWriA4rxUWOMryhRozhqghEjiLaiyMNFhV0WZLf3XipxA6YiOs0f+c7cub89v/M759zv/d1zzm8Vbrfbw/eQoLg9OHukkH9+5eGXORM7KNWU7uK9ohpypk5Gr1V0mH/Q2PXuQmRJcxkdb/KSe9wOClf+nV7Z00gIDWyXl+3fTXV4DMN7hXfLr/Rkjf8+zjsFXzF78Xxk7noOfXYei9GX8ms+PJbcF1NsBsNL3+KvBUW8lDuyWw7vBUEGE61GjUTKh5eMKF21pE16jnGJVnQBGnw83ptZptMSoPDttl+JjKN79xA3aixRWjm1FZXsKlhBg3kQ8ht+xA/pi1EGCSPGsW3xSs5kpRIXLO+247uhue4cR69pyVSpCI3qweYbkYwO88Xtckrz4VYl+46XE60Np/R8Jf7GcPQ9rGjV/t32rRB3QtHJ02S88AtJYIjsh9kYSlZGMu/v+gz7xUsYI8JQGnozOLye3UfOEjc2rtuO7wbRj025GbuzidYKB+N+8DgNhTMpd/QjfVg05aeqGTAhhaLCDZS6HNQ1JjI+6jqNUcMI0XePEIWnqoSSKj0/NKolgUjOoCGTiegZSkrEadQCMbcR16cvH5RUwEMkQ4Sf0kHe0nX8ZsUChgq7kvlrJXn1qa3kF5cxL/MqO11Wngopo1hnI+kxywPxq7hefYVWPws6TdvW98iDGDIqURpnT57hpawyBtC056KXTExqTmfHgqTRau87qGHPLBKujnLzgCxWr81C5rjA2/lzWKDsw6L3rGIU9+3rTijs9ir8rEbUsjbBiRMnKC4u9lKaOXOmdPczmWiu+cJrrrXZQ4tbhvuWS/otV/h5zdvtdgoKCh5IsLchcypInDKRpg2bWPnmCk730OFRNwgzqnu2ERMTQ2pqKs7r1fxs7I/YcaYfCqVCCU232pUiIiIICAi4qxGlr3fWLv2kgDV7Tnsr6W289uscxL0RHBxMenr6PQfZFaSlZ9/3Wn99iHRXBfjzRsFqFjsUKHRBZlx1ZYg7XStrC168OsPNqzXITXovmW1MLkvHfLvj6Ojo+w76YUPeWk9e3gahhIejCLD2xL91LzX1boz/LZl1lf9CFmTp0GDVlV8guNeoLjtsrDnO8iWfMPm1XPybWwkNNbfZK/uU1duPkZ3zPNF6H681jsZGKe+c+CiPT8+H8NPnxvK/slDDzRoaWtRSVXHUXiBvTR5DJs8nyeq98rZtEbWX7NQIZVsd3hOFOyiW5Eg3FZW1Qv/QFuTuVctxp80ie1BY+yIxUX7xZSWjf5LUZTLK9uxgwJxfYXGfZsnCHTz71jx6KHwwRD/K2AQ7pWcriB729e4RfR16dyORc2dwsrSCJ2bkIBMeACGW6zcdBAqNV2eoOrmFvx0w8+oLwgvztzAlK4OPj5aQ9GSyl97nG/6BZuozxGtUuGlhxqwX6W/RCH2GXENCbBSHhJI5ztZGRtDAwXCjmOy0p1hY+CVpj3jw1JRxrDqSRf2EfOLpWvY2mnVc9KjRKFWEjEzi6OtTWeXMoHD5bFp9FWjl3nlIjAmLhUDh2GQwGYTm7yqvz5uOPDaFt9cfYuuefIaGyLhSVd+mL1MQGByEzmwhPtHIrBE9SfjtEabFufGhE+LCjBhaWxATrikygdtNv/QdJD+Ry6kVKzl82cbwkBYOvr+Wz5tC+f2mImJEIoQ3tSN/G9lzZhDaRSLcjlp2lt8iPqaBc0ePcHBjPta06Wx+ebwUpjFAyUfHz5Bsk5O/6gN8o5OZmNmfMJOZWxcPUtJkISXQzISsJEojn2WRfwsqeyUOSx/Meg/N+KJRKXF5nBzeuJlVeyuY/pcSsoeE4KhooLSihEb6U19SxPZ9JfQekUmELgS1XN0hVokMucZIzs+nsX7nVvpMGIrt6cVM0ZSyZtM2qSqcPbAF+eBJ0tmgqxBtJ+mauOj0QaWxsnDJ79ix+0P2n60X7GmpEtp/S0gM+zduo/dIG4d3X4NMqK+8QuDICYTefAWnO5PmW31JtRk4uPcyV1xaBorGVdr2k6avsxFt34msy9Gy7I+rSe49B/UNwYY2DP+Wy6zfe46IqADsV4V8UXcF4hPQfyPW9gypMUaQO8mES3Aw5clIQZLcXiUeScoithtNVM9BSeT94U88/8aL0vln4EDpUaQEuv1UBRMmJrP52DUSXftRxk7BT6Zm6LTx0ucYYdKyYU0RubOnSmekqJfeuSvpj0/KlMbLlo2grr6Cddv3Ca37HOETP8+JGgctjaX0eTqboWP6dWrDq1wo7mD6TnSnmxQRGJ3B8j9ndJCLCfSVVx+VxjrXKPZv+ZheIwxeOunTFnA/XYohKJK5Ly+Vxh63nh+n3ODIgUYshs6Tr4iH/+fEPaJX4jBUylauarp/FP8mxIScPHokarUWnUKoSnTeVH5nyBBhtaXQ9ax0bxAJGZCa9q063yky/t/4Dz+Ncl02WQ9iAAAAAElFTkSuQmCC) ; ; ![](data:image/png;base64,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) ![](data:image/png;base64,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)
Сонымен қатар, Ф.Г. Джексон -интеграл ұғымын енгізіп, келесі түрде анықтаған:
мұндағы,
Функционалдық кеңістіктерде интегралдық, матрицалық операторлардың шенелгенділігі сызықты операторлар теориясында ең негізгі мәселелердің бірі болып табылады. Осы бағытта Харди теңсіздігі [3] айрықша орын алып, қазіргі уақытта дамып, зерттелу үстінде. Бұл жұмыстың мақсаты: q-есептеудегі Харди теңсіздігінің аналогын алып, оның орындалу шарттарын анықтау.
![](data:image/png;base64,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) ![](data:image/png;base64,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) –арқылы [0,1]-де анықталған және нормасы: ![](data:image/png;base64,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) ![](data:image/png;base64,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) = ![](data:image/png;base64,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) болатын f функциялар кеңістігін белгілейік. -![](data:image/png;base64,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) [0,1]-де анықталған теріс емес функциялар. Келесі түрдегі Харди теңсіздігінің q-аналогын қарастырайық:
, ![](data:image/png;base64,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) (1)
q→1 болғанда (1) теңсіздігі кәдімгі Харди теңсіздігін [3] береді.
Келесі белгілеулерді енгізейік:
(2)
![](data:image/png;base64,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) (3)
Теорема 1. болсын. Онда (1) – теңсіздіктің орындалуы үшін А![](data:image/png;base64,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) болуы қажетті және жеткілікті, сонымен қатар, А≈С, мұндағы, С шамасы (1) теңсіздігін қанағаттандыратын ең кіші оң сан.
Теорема 2. ![](data:image/png;base64,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) болсын. Онда (1.1) – теңсіздіктің орындалуы үшін В![](data:image/png;base64,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)
болуы қажетті және жеткілікті сонымен қатар, В≈С, мұндағы, С шамасы (1) теңсіздігін қанағаттандыратын ең кіші оң сан.
Достарыңызбен бөлісу: |