On integer linear combinations of terms of rational cycles for the generalized 3x+1 problem

In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear combinations are integers. The discussed results are demonstrated on some examples. In some particular cases the obtained results can be used to explain some patterns of digits in $p$-adic representations of the terms of the rational cycles

How to Intercept a High-Speed Rocket with a Pair of Compasses and a Straightedge?

In this paper a nonlinear differential equation arising from an elementary geometry problem is discussed. This geometry problem was inspired by one of the proofs of the first remarkable limit discussed in a typical first semester undergraduate Calculus course. It is known that the involved differential equation can be reduced to Abel’s differential equation of the first kind. In this paper the problem was solved using an approximate geometric method which constructs a piecewise linear solution approximation for the curve. The compass tool of GeoGebra was extensively used for these constructions. At the end of the paper, some generalizations are discussed. A new transformation of curves, named “Interception”, is introduced and its approximate construction using GeoGebra is described. Some possible applications include geometry, calculus, ordinary differential equations, and military interceptions

The basis property in Lp of the boundary value problem rationally dependent on the eigenparameter

We consider a Sturm–Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in Lp of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L2 we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness in Lp we use F. Riesz’s theorem

The Basis Property Of Sturm–Liouville Problems With Boundary Conditions Depending Quadratically On The Eigenparameter

We study basisness of root functions of Sturm–Liouville problems with a boundary condition depending quadratically on the spectral parameter. We determine the explicit form of the biorthogonal system. Using this we prove that the system of root functions, with arbitrary two functions removed, form a minimal system in L2, except some cases where this system is neither complete nor minimal. For the basisness in L2 we prove that the part of the root space is quadratically close to systems of sines and cosines. We also consider these basis properties in the context of general Lp

General Boundary Value Problem for the Third Order Linear Differential Equation of Composite Type

The boundary value problem is considered for the linear two-dimensional integro-di®erential loaded third order equation of composite type with nonlocal terms in the boundary conditions. The principal part of the equation is a derivative of the two-dimensional Laplace equation with respect to the variable x₂. Taking into account the ill-posedness of boundary value problems for hyperbolic di®erential equations, the principal part of the boundary conditions is chosen in a special form dictated by the obtained necessary conditions.Рассмотрена граничная задача для линейного двумерного интегро-дифференциального нагруженного уравнения композитного типа третьего порядка с нелокальными граничными условиями. Основная часть уравнения - производная по переменной от двумерного уравнения Лапласа относительно переменной x₂. Учитывая некорректность граничной задачи для гиперболических уравнений, главная часть граничных условий выбрана в специальной форме, продиктованной полученными необходимыми условиями

ABOUT THE SOCIALIZATION OF LABOR AND INCREASE OF SAFETY

A number of aspects of socialization of labor and safety in domestic conditions are revealed in the article, which conceptual bases are based on scientific, standard and empirical levels. The social principles-imperatives and their role in increase of socialization of labor and safety and also neoliberal, conservative and social and democratic social models of the state are considered. The dual character of labor socialization is established as a condition and a source of financing of the public social expenditures and as the instrument of regulation of the social and economic safe relations

ABOUT THE SOCIALIZATION OF LABOR AND INCREASE OF SAFETY

A number of aspects of socialization of labor and safety in domestic conditions are revealed in the article, which conceptual bases are based on scientific, standard and empirical levels. The social principles-imperatives and their role in increase of socialization of labor and safety and also neoliberal, conservative and social and democratic social models of the state are considered. The dual character of labor socialization is established as a condition and a source of financing of the public social expenditures and as the instrument of regulation of the social and economic safe relations

Морфобиологические особенности и продуктивность артишока в условиях Южного Узбекистана

Relevanc and methods. The article presents the research results on the introduction of artichoke in Uzbekistan. In the process of research, some issues of artichoke seed production in the conditions of dry subtropics of Uzbekistan were also studied. Studies were carried out on the Krasavets cultivar. When identifying the most effective method for forming a seed bush, the comparison of growing seed plants without formation and 4 methods of formation was compared: 1 stalk on a plant and 3 inflorescences on a stalk, 2 stems on a plant and three inflorescences on each, 3 stems on a plant and three inflorescences on each, 4 stems on the plant and three inflorescences on each. Results.It was revealed that artichoke plants grow and develop, form full-fledged seeds during the growing season. The manifestation of morphobiological and economically valuable traits largely depends on the age of the plants. A significant reduction in the growing season, an increase in the height of plants and the number of stems, an increase in the average mass and number of inflorescences and, ultimately, yield occurs in the second year of plant life. It was also revealed that the potential productivity of artichoke plants in Uzbekistan is very high. The yield of the Krasavets cultivar in the second year of life is 55.5 t/ha, and that of the Green Globe cultivar is 39.2 t/ha. This is significantly higher than the global average. The highest seed productivity was noted in the variant without the formation of a seed plant: 734.2 g/plant or 15789 seeds/plant. However, in such indicators as the mass and number of seeds from one inflorescence, the mass of 1000 seeds of a plant without formation is significantly inferior to other options. Based on the results of experiment 3, 4, 5, the options can be considered appropriate to use in artichoke seed production. Актуальность, материал и методика. В статье излагаются результаты исследований по интродукции артишока в условиях Узбекистана. Были изучены также некоторые вопросы семеноводства артишока в условиях сухих субтропиков Узбекистана. Исследования проводили на сорте Красавец. При выявлении наиболее эффективного способа формирования семенного куста сравнивали выращивание семенных растений без формирования и 4 способа формирования: 1 стебель на растении и 3 соцветия на стебле, 2 стебля на растении и по три соцветия на каждом, 3 стебля на растении и по три соцветия на каждом, 4 стебля на растении и по три соцветия на каждом.Результаты.Выявлено, что растения артишока растут и развиваются, образуя полноценные семена в течение вегетационного периода. Проявление морфобиологических и хозяйственно-ценных признаков в значительной степени зависит от возраста растений. Существенное сокращение вегетационного периода, увеличение высоты растений и количество стеблей, увеличение средней массы и количество соцветий и в конечном итоге урожайности происходит на втором году жизни растений. Также выявлено, что потенциальная продуктивность растений артишока в условиях Узбекистана очень высокая. Урожайность сорта Красавец на втором году жизни составляет 55,5 т/га, у сорта Green Globe – 39,2 т/га. Это значительно выше общемировых. Самая высокая семенная продуктивность отмечена в варианте без формирования семенного растения: 734,2 г/растения или 15789 семян/растении. Однако по таким показателям, как масса и количество семян с одного соцветия, масса 1000 семян, растения без формирования существенно уступают другим вариантам. Исходя из результатов опыта, варианты 3, 4, 5 можно считать целесообразным для использования в семеноводстве артишока.