Search the digital library catalog

Abstract: This doctoral dissertation is devoted to the studies of some qualitative properties of certain polynomial systems of ordinary differential equations. The main problems that are considered in this thesis are the problems of integrability and cyclicity. Some results on the classification of the global phase portraits of a family of cubic systems are presented as well. In the first chapter basic notions and results of the qualitative theory of systems of ODE's are introduced. Since one of important tools for our study of these problems is the commutative computational algebra, some main notions and properties of polynomial ideals and their varieties, including various algorithms related to them, are also presented in the introduction. In the second chapter methods for investigation of trajectories near degenerated singularities are presented. They are further used for the classification of global phase portraits of a family of cubic systems with the nilpotent center at the origin. In the third chapter the main problem of these thesis is studied, the problem of integrability. The problem of integrability which is connected to the problem of distinguishing between a center and a focus is studied for two different families of cubic polynomial systems of ODE's. With the computational algebra approach the necessary conditions for the existence of the first integral of these systems were obtained. For all but one condition was proven, using various approaches, the existence of the first integrals. The center problem for the real systems can be generalized to the complex systems. The origin of the system obtained after the complexification of the real system is the so-called 1:-1 resonant singular point, from which one additional generalization follows. This is the generalization to the p:-q resonant center. In the third chapter the :-3 resonant singular point of a quadratic family of complex systems is studied. The fourth chapter is devoted to the study of the problem of integrability of a three dimensional polynomial system with quadratic nonlinearities. The problem of existence of two independent first integrals and the existence of one first integral in the system was investigated. In the last chapter local bifurcations of limit cycles of a family of cubic systems are studied. Estimations for the number of limit cycles bifurcated from each components of the center variety are obtained.
Keywords: planar systems of ODE's, higher dimensional systems of ODE's, phase portrait, nilpotent center, limit cylces, Poincaré compactification, center problem, Bautin ideal, focus quantities, time-reversibility, integrability problem, Darboux method, linearizability, limit cycle, cyclicity
Published in DKUM: 19.07.2016; Views: 2201; Downloads: 284
Full text (12,26 MB)

Abstract: Z uporabo algebraičnega pristopa implementiramo konstantni algoritem za računanje dominantnega števila kartezičnih produktov poti in ciklov. Podamo formule za dominantna števila ▫$gamma(P_n Box C_k)$▫ (za ▫$k leq 11$▫, ▫$n in {mathbb N}$)▫ in dominantna števila ▫$gamma(C_n Box P_k)$▫ in ▫$gamma(C_n Box C_k)$▫ (za ▫$k leq 6$▫, ▫$n in {mathbb N}$▫).
Keywords: teorija grafov, kartezični produkt, grid, torus, dominacija, algebra poti, konstantni algoritem, graph theory, Cartesian product, grid graph, torus, graph domination, path algebra, constant time algorithm
Published in DKUM: 10.07.2015; Views: 1740; Downloads: 40
Link to full text

Abstract: The obnoxious center problem in a graph ▫$G$▫ asks for a location on an edge of the graph such that the minimum weighted distance from this point to a vertex of the graph is as large as possible. An algorithm is given which finds the obnoxious center on a weighted cactus graph in ▫$O(cn)$▫ time, where ▫$n$▫ is the number of vertices and ▫$c$▫ is the number of different vertex weights (called marks).
Keywords: matematika, operacijsko raziskovanje, teorija grafov, lokacijski problemi, problem centra, nezaželjeni centri, algoritmi z linearno časovno zahtevnostjo, mathematics, operations research, graph theory, location problems, center problem, obnoxious facilities, linear time algorithm
Published in DKUM: 10.07.2015; Views: 1343; Downloads: 97
Link to full text

Keywords: operating systems, embedded systems, real time, task scheduling, FPGA
Published in DKUM: 10.07.2015; Views: 1334; Downloads: 99
Link to full text

Abstract: This study addresses online decomposition of high-density surface electromyograms (EMG) in real-time. The proposed method is based on previouslypublished Convolution Kernel Compensation (CKC) technique and sharesthe same decomposition paradigm, i.e. compensation of motor unit action potentials and direct identification of motor unit (MU) discharges. In contrast to previously published version of CKC, which operates in batch mode and requires ~ 10 s of EMG signal, the real-time implementation begins with batch processing of ~ 3 s of the EMG signal in the initialization stage and continues on with iterative updating of the estimators of MU discharges as blocks of new EMG samples become available. Its detailed comparison to previously validated batch version of CKC and asymptotically Bayesian optimal Linear Minimum Mean Square Error (LMMSE) estimator demonstrates high agreementin identified MU discharges among all three techniques. In the case of synthetic surface EMG with 20 dB signal-to-noise ratio, MU discharges were identified with average sensitivity of 98 %. In the case of experimental EMG, real-time CKC fully converged after initial 5 s of EMG recordings and real-time and batch CKC agreed on 90 % of MU discharges, on average. The real-time CKC identified slightly fewer MUs than its batch version (experimental EMG, 4 MUs versus 5 MUs identified by batch CKC, on average), but required only 0.6 s of processing time on regular personal computer for each second of multichannel surface EMG.
Keywords: discharge pattern, high-density EMG, surface EMG, motor unit, real time decomposition
Published in DKUM: 25.05.2015; Views: 1639; Downloads: 0