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Abstract: The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three distinct vertices from S lie on a common geodesic; such sets are refereed to as gp-sets of G. The general position number of cylinders Pr ◻ Cs is deduced. It is proved that (Cr ◻ Cs)∈{6,7} whenever r ≥ s ≥ 3, s ≠ 4, and r ≥ 6. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in Pr ◻ Ps, where r,s ≥ 2, is also determined.
Keywords: general position problem, Cartesian product of graphs, paths and cycles, probabilistic constructions, exact enumeration
Published in DKUM: 27.08.2024; Views: 39; Downloads: 7
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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 in DKUM: 03.11.2017; Views: 1368; Downloads: 121
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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 in DKUM: 10.07.2015; Views: 1067; Downloads: 124
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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 in DKUM: 10.07.2015; Views: 24186; Downloads: 101
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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 in DKUM: 10.07.2015; Views: 1259; Downloads: 349
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