В статье изучается проблема подбора оптимального числа итераций алгоритма Гровера при выполнении квантового поиска в неупорядоченном пространстве состояний. В центре анализа находится зависимость вероятности успешного измерения от размера поискового пространства, количества отмеченных элементов и числа применений оператора Гровера. Увеличение числа итераций, как ни странно, не всегда ведёт к лучшему результату. После выхода в область максимальной вероятности дополнительные повторения уже снижают шанс успешного измерения. Для расчётов используются математические выражения, которые связывают , , угол , вероятность и приближённое значение . Полученные результаты могут быть полезны при учебном моделировании квантовых алгоритмов, а также на раннем этапе разработки программных модулей для анализа квантового поиска.

Mazkur maqolada yosh velosportchilarning maxsus jismoniy tayyorgarligini takomillashtirishda tezlik va chidamlilik sifatlarini rivojlantirishning samarali mashq tizimlari tahlil qilingan. Velosportda yuqori natijalarga erishish uchun sportchilarning funksional imkoniyatlarini oshirish, aerob va anaerob quvvatini rivojlantirish hamda mashg‘ulot yuklamalarini ilmiy asosda rejalashtirish muhim ahamiyat kasb etadi. Tadqiqot davomida intervalli mashqlar, takroriy usul va uzluksiz yuklama mashqlarining yosh velosportchilar organizmiga ta’siri o‘rganildi. Olingan natijalar tezlik va chidamlilik sifatlarini kompleks rivojlantirishga qaratilgan mashg‘ulotlar sport natijalarining sezilarli yaxshilanishiga xizmat qilishini ko‘rsatdi.

Ushbu maqolada raqamli iqtisodiyotning zamonaviy iqtisodiy tizimdagi o‘rni, uning tarkibiy elementlari hamda kambag‘allikni kamaytirishdagi ijtimoiy-iqtisodiy ahamiyati keng tahlil qilinadi. Raqamli iqtisodiyot bugungi kunda global iqtisodiy rivojlanishning eng muhim yo‘nalishlaridan biri bo‘lib, u axborot-kommunikatsiya texnologiyalari (AKT), internet tarmoqlari, sun’iy intellekt, katta ma’lumotlar (big data), elektron platformalar va mobil texnologiyalar asosida shakllangan yangi iqtisodiy muhitni ifodalaydi. Ushbu tizim an’anaviy iqtisodiyotdan farqli ravishda tezkorlik, shaffoflik va yuqori samaradorlik bilan ajralib turadi.

The increasing integration of technology into educational environments has created new opportunities for improving language instruction. Among various digital resources, podcasts have become an effective tool for developing listening comprehension, vocabulary acquisition, pronunciation, and communicative competence. Their flexibility and accessibility allow learners to engage with educational content beyond classroom boundaries. However, the successful use of podcasts depends on the application of appropriate methodological principles that guide the teaching process. This article examines the pedagogical foundations of podcast-based instruction, analyzes key methodological principles, and discusses their contribution to language learning effectiveness. The study also highlights practical recommendations for integrating podcasts into foreign language education.

The usefulness of code-switching as a teaching strategy in bilingual foreign language instruction is examined in this article. In order to improve comprehension, make difficult topics easier to understand, and foster a positive learning environment, it examines how teachers deliberately switch between the target language and the students' mother tongue. The usage of code-switching by students at primary, secondary, and high school levels is examined in this study. Additionally, it emphasizes the benefits and possible disadvantages of this method for learning a language. According to the research, code-switching can improve overall learning results, lessen anxiety, and increase students' comprehension when applied properly. However, the development of communicative competence in the target language may be hampered by an over-reliance on the original tongue. Therefore, the article recommends a balanced and purposeful use of code-switching in bilingual classrooms.