On the hierarchy of partially invariant submodels of differential equations

It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given

Imaging Molecules from Within: Ultra-fast, {\AA}ngstr\"om Scale Structure Determination of Molecules via Photoelectron Holography using Free Electron Lasers

A new scheme based on (i) upcoming brilliant X-ray Free Electron Laser (FEL) sources, (ii) novel energy and angular dispersive, large-area electron imagers and (iii) the well-known photoelectron holography is elaborated that provides time-dependent three-dimensional structure determination of small to medium sized molecules with {\AA}ngstr\"om spatial and femtosecond time resolution. Inducing molecular dynamics, wave-packet motion, dissociation, passage through conical intersections or isomerization by a pump pulse this motion is visualized by the X-ray FEL probe pulse launching keV photoelectrons within few femtoseconds from specific and well-defined sites, deep core levels of individual atoms, inside the molecule. On their way out the photoelectrons are diffracted generating a hologram on the detector that encodes the molecular structure at the instant of photoionization, thus providing 'femtosecond snapshot images of the molecule from within'. Detailed calculations in various approximations of increasing sophistication are presented and three-dimensional retrieval of the spatial structure of the molecule with {\AA}ngstr\"om spatial resolution is demonstrated. Due to the large photo-absorption cross sections the method extends X-ray diffraction based, time-dependent structure investigations envisioned at FELs to new classes of samples that are not accessible by any other method. Among them are dilute samples in the gas phase such as aligned, oriented or conformer selected molecules, ultra-cold ensembles and/or molecular or cluster objects containing mainly light atoms that do not scatter X-rays efficiently.Comment: 18 pages, 11 figure

Фрагментація пакетів в радіоканалах передачі даних

Introduction. Effective data transmission speed in the radio data channels network is determined by the parameters of the exchange report on the data link layer and physical levels and the wave emission channel characteristics. For guaranteed delivery of packets the technology with the supervision of packages receiving validity by consumer is used. Errors availabilities in packets demand retransmission, which leads to a decrease the effective data transmission speed. Fragmentation of packets reduces the probability of an error in the packet, the delay of the fragment retransmission, which improves the effective transmission speed. Problem statement. The maximum packet length for data link layer and physical levels is determined according to the conditions of the effective network functioning in general, consequently the packet length with low values of bit error in the channel can be significantly less than optimal, and with large values of bit error the possibility to select the division fragments of the maximum length into equal parts exists, under the terms of improving effective transmission speed. The aim of this work is to build analytical model for points determination on the scale of probability of bit errors, where the length of packet fragments is necessary to be changed in order to better efficiency of data transmission speed. Theoretical results. Based on the analysis of data transmission report with a stoppage and expectations the analytical model of effective data transmission speed in the radio data channel with bit errors and random distribution is received. There is equation to calculate the bit errors probability at the points of the fragments length changing. The obtained analytical solutions of the equations allow calculating the value of the bit errors probability for any parameters of data transmission cycle. Also, analytical expressions for calculating the optimal fragments length with a given probability of bit errors and probability of bit errors for a given fragment length are obtained. Conclusion. Proposed analytical model and analytical calculations can be used on data link layer and physical levels of the channels with packet data transmission according to the ARQ SAW algorithm to select parameters of fragmentation depending on the bit errors in the channel.В статье предложена аналитическая модель фрагментации пакетов в радиоканале передачи данных. Так как максимальная длина пакета для канального и физического уровней определена из условий эффективной работы сети в целом, то длина пакета при малых значения битовой ошибки в канале может быть значительно меньше оптимальной и не может быть изменена, а при больших значения битовой ошибки предлагается выбирать длину фрагментов делением максимальной длины на равные части,при условии улучшения эффективной скорости передачи. Разработана аналитическая модель позволяет для выбранных параметров цикла передачи данных протокола ARQ с Stay And Wait, рассчитать значение вероятности битовых ошибок при которых необходимо изменить параметры фрагментации для получения лучшей эффективной скорости передачи.В статі запропонована аналітична модель фрагментації пакетів в радіоканалі передачі даних. Так як максимальна довжина пакету для канального та фізичного рівнів визначена з умов ефективної роботи мережі в цілому, то довжина пакету при малих значення бітової похибки в каналі може бути значно меншою від оптимальної, а при великих значення бітової похибки пропонується вибирати довжину фрагментів діленням максимальної довжини на рівні частини, за умови поліпшення ефективної швидкості передачі. Розроблена аналітична модель дозволяє, для вибраних параметрів циклу передачі даних протоколу ARQ з Stay And Wait, розрахувати значення ймовірності бітових помилок при яких необхідно змінити параметри фрагментації для отримання кращої ефективної швидкості передачі