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Estimating the number of limit cycles in polynomials systems
Maoan Han, Valery Romanovski, 2010, original scientific article

Abstract: We describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field. We apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system.
Keywords: mathematics, limit cycles, bifurcation
Published: 07.06.2012; Views: 684; Downloads: 62
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4.
Limit cycle bifurcations from a nilpotent focus or center of planar systems
Maoan Han, Valery Romanovski, 2012, original scientific article

Abstract: We study analytic properties of the Poincaré return map and generalized focal values of analytic planar systems with a nilpotent focus or center. We use the focal values and the map to study the number of limit cycles of this kind of systems and obtain some new results on the lower and upper bounds of the maximal number of limit cycles bifurcating from the nilpotent focus or center. The main results generalize the classical Hopf bifurcation theory and establish the new bifurcation theory for the nilpotent case.
Keywords: mathematic, limit cycles, bifurcation, center problem
Published: 10.07.2015; Views: 485; Downloads: 184
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5.
Center conditions and cyclicity for a family of cubic systems
Brigita Ferčec, Adam Mahdi, 2013, original scientific article

Abstract: Using methods of computational algebra we obtain an upper bound for the cyclicity of a family of cubic systems. To that end we overcome the problem of nonradicality of the associated Bautin ideal by moving from the ring of polynomials to a coordinate ring. Finally, we also determine the number of limit cycles bifurcating from each component of the center variety.
Keywords: matematika, cikličnost, limitni cikli, računalniška algebra, mathematics, cyclicity, limit cycles, computer algebra system
Published: 10.07.2015; Views: 336; Downloads: 16
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6.
An almost complete description of perfect codes in direct products of cycles
Sandi Klavžar, Simon Špacapan, Janez Žerovnik, 2006, original scientific article

Abstract: Naj bo ▫$G = times_{i=1}^nC_{ell_i}$▫ direktni produkt ciklov. Dokazano je, da za vsak ▫$r ge 1$▫ in za vsak ▫$n ge 2$▫ velja naslednje. Če je vsak ▫$ell_i$▫ večkratnik od ▫$r^n + (r+1)^n$▫, tedaj vsaka povezana komponenta grafa ▫$G$▫ vsebuje ▫$r$▫-popolno kodo. Po drugi strani je tudi dokazano, da če koda grafa ▫$G$▫ vsebuje izbrano točko in njene lokalno kanonične točke, tedaj je vsak ▫$ell_i$▫ večkratnik od ▫$r^n + (r+1)^n$▫. Nadalje je dokazano, da je ▫$r$▫-popolna koda ▫$(r ge 2)$▫ grafa ▫$G$▫ enolično določena z ▫$n$▫ točkami. Postavljena je domneva, da za ▫$r ge 2$▫ ne obstajajo nobene druge kode v $G$ razen tistih, ki so konstruirane v članku.
Keywords: matematika, teorija grafov, korekcijske kode, direktni produkt grafov, popolne kode, cikli, mathematics, graph theory, error-correcting codes, direct product of graphs, perfect codes, cycles
Published: 10.07.2015; Views: 401; Downloads: 63
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7.
Perfect codes in direct products of cycles - a complete characterization
Janez Žerovnik, 2008, original scientific article

Abstract: Let ▫$G = times^n_{i=1}C_{ell_i}$▫ be a direct product of cycles. It is known that for any ▫$r le 1$▫, and any ▫$n le 2▫$, each connected component of ▫$G$▫ contains a so-called canonical ▫$r$▫-perfect code provided that each ▫$ell_i$▫ is a multiple of ▫$r^n + (r+1)^n$▫. Here we prove that up to a reasonably defined equivalence, these are the only perfect codes that exist.
Keywords: matematika, teorija grafov, korekcijske kode, direktni produkt grafov, popolne kode, cikli, mathematics, graph theory, error-correcting codes, direct product of graphs, perfect codes, cycles
Published: 10.07.2015; Views: 575; Downloads: 53
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8.
NZ-flows in strong products of graphs
Wilfried Imrich, Iztok Peterin, Simon Špacapan, Cun-Quan Zhang, 2010, original scientific article

Abstract: Za krepki produkt ▫$G_1 boxtimes G_2$▫ grafov ▫$G_1$▫ in ▫$G_2$▫ dokažemo, da je ▫${mathbb{Z}}_3$▫-pretočno kontraktibilen natanko tedaj, ko ▫$G_1 boxtimes G_2$▫ ni izomorfen ▫$Tboxtimes K_2$▫ (kar poimenujemo ▫$K_4$▫-drevo), kjer je ▫$T$▫ drevo. Sledi, da za ▫$G_1 boxtimes G_2$▫ obstaja NZ 3-pretok, razen če je ▫$G_1 boxtimes G_2$▫ ▫$K_4$▫-drevo. Dokaz je konstruktiven in implicira polinomski algoritem, ki nam vrne NZ 3-pretok, če ▫$G_1 boxtimes G_2$▫ ni ▫$K_4$▫-drevo, oziroma NZ 4-pretok sicer.
Keywords: matematika, teorija grafov, celoštevilski pretoki, krepki produkt, poti, cikli, nikjer ničelni pretok, mathematics, graph theory, integer flows, strong product, paths, cycles
Published: 10.07.2015; Views: 360; Downloads: 60
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9.
Domination game critical graphs
Csilla Bujtás, Sandi Klavžar, Gašper Košmrlj, original scientific article

Abstract: The domination game is played on a graph ▫$G$▫ by two players who alternately take turns by choosing a vertex such that in each turn at least one previously undominated vertex is dominated. The game is over when each vertex becomes dominated. One of the players, namely Dominator, wants to finish the game as soon as possible, while the other one wants to delay the end. The number of turns when Dominator starts the game on ▫$G$▫ and both players play optimally is the graph invariant ▫$\gamma_g(G)$▫, named the game domination number. Here we study the ▫$\gamma_g$▫-critical graphs which are critical with respect to vertex predomination. Besides proving some general properties, we characterize ▫$\gamma_g$▫-critical graphs with ▫$\gamma_g =2$▫ and with ▫$\gamma_g =3$▫, moreover for each ▫$n$▫ we identify the (infinite) class of all ▫$\gamma_g$▫-critical ones among the ▫$n$▫th powers ▫$C_N^n$▫ of cycles. Along the way we determine ▫$\gamma_g(C_N^n)$▫ for all ▫$n$▫ and ▫$N$▫. Results of a computer search for ▫$\gamma_g$▫-critical trees are presented and several problems and research directions are also listed.
Keywords: domination game, domination game critical graphs, powers of cycles, trees
Published: 31.03.2017; Views: 487; Downloads: 222
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10.
Visible symptoms of the financial crisis
Matic Žargi, 2017, undergraduate thesis

Abstract: The Great Recession of 2008 had such a huge impact on the global economy, hence it should be predictable? With a theoretical analysis of Early Warning Systems (EWS) we came to the conclussion that they do in fact work. However, due to complex dimensions of the crisis, there was no particular set of EWS which could actually predict the impact of the crisis. This complexity stems overleveraging and various financial innovations which made the financial markets riskier in unobservable ways. In order to observe the symptoms of a serious crisis we need to look at the economy as a simple machine. Ray Dalios principle about the long and short term debt cycle gave us a different approach. It enables us to spot the symptoms simply by analysisng interest rates, base money creation and federal spending. Rajan (2005) has proved that thesis in his paper, where he showed that the financial instruments became so complex, that they became too risky. Our analysis has brought us to a conclussion that right now interest rates in the US are starting ot rise, the inflation is rising and GDP growth stagnating. With the application of our system we can see that right now we are in the middle of a business cycle. We can predict that a downturn in the capital markets is about to occur. However no system is able to predict the impact of the crisis.
Keywords: financial crisis, debt, leveraging, leverage, economic cycles
Published: 03.11.2017; Views: 357; Downloads: 34
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