Inverse Problem for the Sobolev Type Equation of Higher Order

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study.

High fidelity epigenetic inheritance: Information theoretic model predicts k-threshold filling of histone modifications post replication

Beyond the genetic code, there is another layer of information encoded as chemical modifications on histone proteins positioned along the DNA. Maintaining these modifications is crucial for survival and identity of cells. How the information encoded in the histone marks gets inherited, given that only half the parental nucleosomes are transferred to each daughter chromatin, is a puzzle. Mapping DNA replication and reconstruction of modifications to equivalent problems in communication of information, we ask how well enzymes can recover the parental modifications, if they were ideal computing machines. Studying a parameter regime where realistic enzymes can function, our analysis predicts that, pragmatically, enzymes may implement a threshold-k filling algorithm which fills unmodified regions of length at most k. This algorithm, motivated from communication theory, is derived from the maximum a` posteriori probability (MAP) decoding which identifies the most probable modification sequence based on available observations. Simulations using our method pro- duce modification patterns similar to what has been observed in recent experiments. We also show that our results can be naturally extended to explain inheritance of spatially distinct antagonistic modifications.

Impulse noise treatment in magnetotelluric inversion

The problem of model recovering in the presence of impulse noise on the data is considered for the magnetotelluric (MT) inverse problem. The application of total variation regularization along with L1-norm penalized data fitting (TVL1) is the usual approach for the impulse noise treatment in image recovery. This combination works poorly when a high level of impulse noise is present on the data. A nonconvex operator named smoothly clipped absolute deviation (TVSCAD) was recently applied to the image recovery problem. This operator is solved using a sequence of TVL1 equivalent problems, providing a significant improvement over TVL1. In practice, TVSCAD requires the selection of several parameters, a task that can be very difficult to attain. A more simple approach to the presence of impulse noise in data is presented here. A nonconvex function is also considered in the data fitness operator, along with the total variation regularization operator. The nonconvex operator is solved by following a half-quadratic procedure of minimization. Results are presented for synthetic and also for field data, assessing the proposed algorithm’s capacity in model recovering under the influence of impulse noise on data for the MT problem.

Search time optimization for random pulse sources with given accuracy

Исследуются вопросы построения быстродействующих алгоритмов обнаружения и локализации точечных источников, имеющих случайное распределение и обнаруживающих себя в случайные моменты времени генерацией мгновенных дельта-импульсов. Поиск осуществляется системой, включающей одно либо несколько приемных устройств, и выполняется с соблюдением требований по точности локализации. Предполагается, что все приемные устройства имеют произвольно перестраиваемые во времени окна обзора. Оптимальной считается процедура, которая в статистическом плане (т.е. по ансамблю реализаций) минимизирует среднее время локализации. Показано, что даже при сравнительно невысоких требованиях к точности локализации оптимальная процедура состоит из нескольких этапов, каждый из которых заканчивается в момент регистрации очередного импульса. Вполне допускается ситуация, когда в процессе оптимального поиска часть генерируемых источником импульсов может быть пропущена приемной системой. В работе рассчитаны и систематизированы параметры оптимального поиска в зависимости от количества приемных устройств и требуемой точности локализации. Для случая предельно высоких требований к точности локализации рассчитаны параметры асимптотически оптимальных поисковых алгоритмов. Показана возможность использования полученных результатов в многомерном случае. Purpose. The main goal of the research is to develop time-optimal algorithms for the localization of point sources that have a random spatial distribution and indicate themselves by generating instantaneous delta pulses at random time points. Methods. In many practically important problems requiring the highest reduction in the average time of localization of signal objects, the complexity of constructing optimal search algorithms forces researchers to resort to various kinds of simplifications or to the use of methods of numerical and simulation modelling. The mathematical apparatus used in the article belongs to probabilistic-statistical and non-linear programming methods. In a number of sections of the study (in particular, when constructing optimal control algorithms for multi-receiving search engines), traditional methods of discrete analysis and applied programming were used. Results. The solution of the variational problem is found, which minimizes the average localization time in the class of one-stage search algorithms with a known distribution density and the simultaneous absence of a priori information about the intensity of a random pulse source. For random point sources with a priori known intensity of the instantaneous generation of pulses, physically realizable multistage search algorithms have been constructed that have a significant gain in speed over single-stage algorithms, especially with increased requirements for localization accuracy. For a uniform distribution of a random source, an optimal strategy of multi-stage search was calculated, depending on the required localization accuracy and the number of receivers used. Findings. A distinctive feature of the studies is their universality, since in mathematical terms, the discussed problems and algorithms for the time-optimal search of random point-pulse objects arise in many scientific and technical applications. In particular, such studies are needed when developing methods for intermittent failures troubleshooting in the theory of reliability, in mathematical communication theory and in problems of technical diagnostics. Scientifically equivalent problems appear in the problems of detection, localization and tracking of radiation targets for eliminating malfunctions that manifest themselves in the form of intermittent failures. Scientifically equivalent problems arise in the problems of detecting, localizing and tracking radiation source targets.

A Simple and Efficient Way to Save Energy in Multihop Wireless Networks with Flow Aggregation

Today, whether in industry, research, or civil applications, there are many incentives to reduce the energy footprint of automated systems. In multihop wireless networks, the main objective in that regard is usually to maximize the lifetime of the network by distributing the load over all nodes. In this paper, we improve a solution that aggregates flows to optimize the number of nodes that can be turned off. We introduce interference awareness in a routing metric designed to aggregate flows to avoid overloading the network and to preserve the quality of service required by the flows. This way, it becomes possible to integrate this metric into classical shortest path routing algorithms that do not consider interference. We also show that flow aggregation and overall energy consumption are equivalent problems.

Potential Effect of PISA Equivalent Questions Using the Context of Aceh Traditional Houses

The achievement of Indonesian students in PISA remains unsatisfactory, as evidenced by their scores that are below the baseline level set in PISA. One of the contributing factors is that Indonesian students are less trained in solving problems with characteristics such as in PISA items. Therefore, it is necessary to familiarize Indonesian students with PISA equivalent problems, but the availability of these questions is still limited, especially concerning reasoning ability. In addition, the use of context is also crucial, especially the local contexts, which can help students understand mathematical phenomena from the perspective of their life experiences. The purpose of this study was to develop the questions of PISA equivalent mathematical reasoning ability using the context of Aceh traditional houses. This research used formative evaluation type development research from Tessmer. The results of this study were 12 mathematical questions equivalent to PISA using the context of Aceh traditional house. The potential effect of the questions developed was analyzed based on the student's responses to the questionnaire, including two aspects: 1) students are interested and seriously working on the questions and 2) the students are interested in using the equivalent PISA questions in the future.

Conjugate (nil) clean rings and Köthe’s problem

Question 3 of [3] asks whether the matrix ring [Formula: see text] is nil clean, for any nil clean ring [Formula: see text]. It is shown that, positive answer to this question is equivalent to positive solution for Köthe’s problem in the class of algebras over the field [Formula: see text]. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.

When Instructional Guidance is Needed

Studying worked examples providing problem solutions to learners usually leads to better test performance than solving the equivalent problems without guidance, demonstrating the worked-example effect. The generation effect occurs when learners who generate answers without guidance learn better than those who read answers that provide guidance. The contradiction between these results can be hypothesised to be due to differences in the element interactivity of the learning tasks. Primary school students in Year 6 participated in the experiment, which investigated the hypothesis by using geometry materials. A disordinal interaction was obtained between levels of guidance and levels of element interactivity. Higher levels of guidance facilitated learning using high element interactivity information, while lower levels of guidance facilitated learning for low element interactivity information. Cognitive load theory was used to explain these contrasting results. From an educational perspective, it was suggested that when determining levels of guidance, a consideration of element interactivity is essential.

VARIATIONALLY EQUIVALENT PROBLEMS AND VARIATIONS OF NOETHER CURRENTS

We consider systems of local variational problems defining nonvanishing cohomology classes. Symmetry properties of the Euler–Lagrange expressions play a fundamental role since they introduce a cohomology class which adds up to Noether currents; they are related with invariance properties of the first variation, thus with the vanishing of a second variational derivative. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order that such a current be global.