On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments seem to suggest that the solutions of the nonlocal problems converge to the entropy admissible solution of the conservation law in the singular local limit. However, recent analytical results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii)~convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytical results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that Lax-Friedrichs type schemes may provide the wrong intuition and erroneously suggest that the solutions of the nonlocal problems converge to the entropy admissible solution of the conservation law in cases where this is ruled out by analytical results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide an intuition more in accordance with the analytical results.

Analytical Limit Load Procedure for The Axial Complex Shaped Defect in a Pressurized Pipe

The paper is devoted to elaboration of the analytical O-procedure for limit load analysis of complex shaped axial defect in a pressurized pipe. It is based on the classical lower bound theorem of the theory of plasticity, and consists in construction of the statically admissible solution, where distribution of stress satisfies to the equilibrium equations and strength conditions. O-procedure is an optimization process to get the most favorable stress distribution for providing the maximal pressure. It allows to explicitly account for the variable geometrical and physical parameters. Contrary to other approaches, the derived formula for rectangular defect is only a particular case of the general procedure application. Four different methods for the complex defects are compared. They are: first, ASME, A-, rectangular defect formula combined with RSTRENG, R-, procedure, i.e., A-R approach; second, PCORRC, P-, formula with R-procedure, P-R approach; third, Orynyak's, O-, formula with R-procedure, O-R approach; and fourth, our universal O-procedure. The verification begins for rectangular defects where both theoretical and experimental comparison is performed for A-, P-, and O- formulas. The difference between them is small, provided that all three employ the same characteristic of material, here the ultimate strength. Then theoretical comparison for A-R, P-R, O-R approaches and O-procedure is performed for the artificial complex defects, for two symmetrical rectangular defects, for triangular defect. Experimental comparison between four methods is made based on the well-known University of Waterloo full scale tests.

Nonexistence of entire positive solutions for conformal Hessian quotient inequalities

<p style='text-indent:20px;'>In this paper, we consider the nonexistence problem for conformal Hessian quotient inequalities in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We prove the nonexistence results of entire positive <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-admissible solution to a conformal Hessian quotient inequality, and entire <inline-formula><tex-math id="M3">\begin{document}$ (k, k') $\end{document}</tex-math></inline-formula>-admissible solution pair to a system of Hessian quotient inequalities, respectively. We use the contradiction method combining with the integration by parts, suitable choices of test functions, Taylor's expansion and Maclaurin's inequality for Hessian quotient operators.</p>

A New Optimal Ensemble Algorithm Based on SVDD Sampling for Imbalanced Data Classification

Nowadays, imbalanced data classification is a hot topic in data mining and recently, several valuable researches have been conducted to overcome certain difficulties in the field. Moreover, those approaches, which are based on ensemble classifiers, have achieved reasonable results. Despite the success of these works, there are still many unsolved issues such as disregarding the importance of samples in balancing, determination of proper number of classifiers and optimizing weights of base classifiers in voting stage of ensemble methods. This paper intends to find an admissible solution for these challenges. The solution suggested in this paper applies the support vector data descriptor (SVDD) for sampling both minority and majority classes. After determining the optimal number of base classifiers, the selected samples are utilized to adjust base classifiers. Finally, genetic algorithm optimization is used in order to find the optimum weights of each base classifier in the voting stage. The proposed method is compared with some existing algorithms. The results of experiments confirm its effectiveness.

Algorithm of Rational Planning and Resource Distribution in the Task of Preparing the Aircraft Group for Use

The work is devoted to solving the problem of justifying the rational composition of a team of specialists who provide preparing for a group of aircraft for a given time. To substantiate the optimal composition of the team, it is necessary to solve the problem of scheduling work on a group of aircraft with different composition of specialists. This, in turn, requires consideration of the huge number of options for streamlining work performed on each aircraft, and options for organizing the sequence of maintenance by one specialist of several aircraft. Finding solutions using combinatorial optimization requires an unacceptably high computational cost. The article proposes an approach for finding not the optimal, but some rational admissible solution, which is not much worse than the optimal one, but its definition does not require large computational resources. An algorithm for rational work scheduling based on discrete-event modeling is proposed. Planning is carried out sequentially in time. When planning the sequence of work, it was suggested first of all to put the work with the maximum duration possible. The developed algorithm is software implemented, which allowed to investigate some properties of the solutions obtained. Examples of calculating the schedule of work on a group of aircraft with a different composition of the team of specialists are given. The problem of justification of rational structure of the team is solved by rational planning algorithm works by sequentially increasing the number of specialists. An example of substantiating the rational composition of a team of specialists performing preparing of a group of eight aircraft, each of which performs five types of work, is given and analyzed in details. The high speed of the calculations for the rational planning of work by a given team allowed to consider all possible options for the team (tens of thousands of options) and substantiate such an option that the number of specialists in the team would be minimal, but they would ensure the preparation of aircraft for a given time. Low requirements for computing resources allow solving problems with a sufficiently large number of types of work performed on each aircraft of the group.

Mask-based approach to phasing of single-particle diffraction data. II. Likelihood-based selection criteria

A new type of mask-selection criterion is suggested for mask-based phasing. In this phasing approach, a large number of connected molecular masks are randomly generated. Structure-factor phases corresponding to a trial mask are accepted as an admissible solution of the phase problem if the mask satisfies some specified selection rules that are key to success. The admissible phase sets are aligned and averaged to give a preliminary solution of the phase problem. The new selection rule is based on the likelihood of the generated mask. It is defined as the probability of reproducing the observed structure-factor magnitudes by placing atoms randomly into the mask. While the result of the direct comparison of mask structure-factor magnitudes with observed ones using a correlation coefficient is highly dominated by a few very strong low-resolution reflections, a new method gives higher weight to relatively weak high-resolution reflections that allows them to be phased accurately. This mask-based phasing procedure with likelihood-based selection has been applied to simulated single-particle diffraction data of the photosystem II monomer. The phase set obtained resulted in a 16 Å resolution Fourier synthesis (more than 4000 reflections) with 98% correlation with the exact phase set and 69% correlation for about 2000 reflections in the highest resolution shell (20–16 Å). This work also addresses another essential problem of phasing methods, namely adequate estimation of the resolution achieved. A model-trapping analysis of the phase sets obtained by the mask-based phasing procedure suggests that the widely used `50% shell correlation' criterion may be too optimistic in some cases.

Quasi-completeness of the class of problems on the "weight-minimax edge"

The tasks of multi-criteria optimization in the general formulation do not have a trivial solution, which gives rise to a multitude of approaches in determining the most “successful” solution from a certain set of solutions that satisfy the problem conditions. One of the ways of formal defining of the possible alternative solution set is to isolate the Pareto set, i.e. the set of unimprovable alternatives. The previously developed approach was applied for studying some classes of multi-criteria problems, the objective functions of which have certain properties, and its productivity was confirmed. The concept of complete problems was introduced, for which the equality of the sets of feasible solutions, the Pareto and the full set of alternatives was fulfilled. In previous works, the authors introduced the concept of quasi-completeness. In the article the class of two-criterion problems, for which the admissible solution for the first criterion has a constant number of edges, and objective function contains the criterion of weight and the criterion of the minimax edge, is distinguished. The problem on the graph of the general structure and the problem on the bichromatic graph, for which the feasible solutions have a constant number of edges, were selected as representatives of this class. A method for studying the properties and estimating the powers of an admissible set of solutions, a Pareto set and a complete set of alternatives for the problems of the selected class, has been formulated. A theorem on the quasi-completeness for the selected class problems is proved. There were obtained estimates for two representatives of this class: “about a spanning tree and a minimax edge”, “about a perfect matching on a bichromatic graph and a minimax edge”. Polynomial algorithms for solving the problems under study are proposed. Estimates of the computational complexity of these algorithms are given

Transport-collapse scheme for heterogeneous scalar conservation laws

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

Robust STAP for MIMO Radar Based on Direct Data Domain Approach

The detection performance of direct data domain (D3) space-time adaptive processing (STAP) will be extremely degraded when there are mismatches between the actual and the presumed signal steering vectors. In this paper, a robust D3 STAP method for multiple-input multiple-output (MIMO) radar is developed. The proposed method utilizes the worst-case performance optimization (WCPO) to prevent the target self-nulling effect. An upper bound for the norm of the signal steering vector error is given to ensure that the WCPO problem has an admissible solution. Meanwhile, to obtain better detection performance in the low signal-to-noise ratio (SNR) environment, the proposed method gives a modified objective function to minimize the array noise while mitigating the interferences. Simulation results demonstrate the validity of our proposed method.