Balanced factorisations

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.Comment: 7 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.ht

О ДОКАЗАТЕЛЬСТВЕ Н. П. РОМАНОВА ЕГО АДДИТИВНОЙ ТЕОРЕМЫ И ЕЕ АНАЛОГАХ

In paper we describe the way N. P. Romanoff proved his additive theorem and sufficient conditions to obtain its analogues for sets with similar distribution and arithmetic. Also the example of set with similar distribution but with different arithmetic is given. We prove that the Romanoff theorem’s analogue for this set is incorrect.В работе описывается техника, придуманная Н. П. Романовым для доказательства его теоремы о том, что нижняя асимптотическая плотность суммы множества простых и множества степеней фиксированного натурального числа положительна, которая также позволяет заменить в этой теореме второе множество другим — с похожими распределением и арифметикой. Описываются условия на второе множество, достаточные для получения аналога теоремы, и приводится пример множества с похожим распределением, но с другой арифметикой, для которого эти достаточные условия не выполняются. Доказывается, что для указанного множества аналог теоремы Романова неверен

ОБ ОЦЕНКЕ СРЕДНЕГО РАЦИОНАЛЬНЫХ ТРИГОНОМЕТРИЧЕСКИХ СУММ СПЕЦИАЛЬНОГО ВИДА И ЕЕ ПРИЛОЖЕНИЯХ

In this paper we obtain some arithmetic properties of generalized Fibonacci sequence and consider their applications.  В работе изучены некоторые свойства распределения членов обобщенной последовательности Фибоначчи по бесквадратному модулю и получены следствия из этих свойств. 

SENSING of the structure of the radio wave refractivity in the troposphere by a network of satellite navigation system receivers in the city of Kazan

We present the results of an experimental study of the structure of the radio-wave refractivity and its dynamics by a network of seven ground-based GPS and GLONASS receivers in Kazan, Russia. It is shown that the remote sensing results agree well with the data of weather stations, radio sounding, and reanalysis. The standard deviation of the refractivity value, which are obtained using the sensing results from the radio sounding data amounts to 2% of the average value at altitudes of up to 500 m. It is found that the refractivity structure has overnight variations, as well as and mesoscale spatial and temporal variability. ©2011 Springer Science+Business Media, Inc

Flower palate ultrastructure of the carnivorous plant Genlisea hispidula Stapf with remarks on the structure and function of the palate in the subgenus Genlisea (Lentibulariaceae)

In the genus Genlisea as well as in its sister genus Utricularia, the palate probably plays a key role in providing the colour, mechanical and olfactory stimuli to attract insect pollinators and to guide them to the generative structures and the nectary spur. However, information about the micro-morphology of the palate of Genlisea is scarce. This study aims to examine the structure of the palate in Genlisea hispidula in detail as well as the palate from other five species from the subgenus Genlisea. In particular, its aim is to ascertain whether these palates function as an area for the osmophores in the flower or whether they produce nectar. We showed that the palate in all of the species that were examined was the glandular type and that it had capitate, glandular trichomes, which had a similar general architecture across the species that were examined. No nectar secretion was observed on the palates. The ultrastructure of the palate trichomes showed that the palate glandular trichomes most probably function as scent glands that produce an olfactory stimulus for flower pollinators

SENSING of the structure of the radio wave refractivity in the troposphere by a network of satellite navigation system receivers in the city of Kazan

We present the results of an experimental study of the structure of the radio-wave refractivity and its dynamics by a network of seven ground-based GPS and GLONASS receivers in Kazan, Russia. It is shown that the remote sensing results agree well with the data of weather stations, radio sounding, and reanalysis. The standard deviation of the refractivity value, which are obtained using the sensing results from the radio sounding data amounts to 2% of the average value at altitudes of up to 500 m. It is found that the refractivity structure has overnight variations, as well as and mesoscale spatial and temporal variability. ©2011 Springer Science+Business Media, Inc