Abel's method on summation by parts and theta hypergeometric series

Abel's lemma on summation by parts is reformulated to investigate systematically terminating theta hypergeometric series. Most of the known identities are reviewed and several new transformation and summation formulae are established. The authors are convinced by the exhibited examples that the iterating machinery based on the modified Abel lemma is powerful and a natural choice for dealing with terminating theta hypergeometric series.     

On solutions to two-dimensional systems realizing the transition from an unstable equilibrium to a stable cycle

For a two-dimensional dynamical system at ?∞ < t < +∞, a process describing the transition from an arbitrary neighborhood of an unstable equilibrium to a stable limit cycle is studied. The system is reduced to the Poincaré normal form. An approximate solution is constructed as a polynomial of degree 2N containing only even degrees of the small parameter ε. The functional classes to which the coefficients of this polynomial belong are described. The function space containing the exact solution differing from the approximate one by O(ε^2N+1) is determined.     

一类具功能反应的食饵-捕食者系统的定性分析

运用微分方程定性与稳定性分析的方法研究了一类食饵-捕食者系统的定性行为,讨论了该系统的平衡点的性态,给出了该系统极限环不存在性存在性的充分条件以及至多存在一个极限环的充分条件,从而补充和推广了前人已有的结果.     

New i‐Perfect Cycle Decompositions via Vertex Colorings of Graphs

Generalizing a result by Buratti et al.[M. Buratti, F. Rania, and F. Zuanni, Some constructions for cyclic perfect cycle systems, Discrete Math 299 (2005), 33–48], we present a construction for i‐perfect k‐cycle decompositions of the complete m‐partite graph with parts of size k. These decompositions are sharply vertex‐transitive under the additive group of with R a suitable ring of order m. The construction works whenever a suitable i‐perfect map exists. We show that for determining the set of all triples for which such a map exists, it is crucial to calculate the chromatic numbers of some auxiliary graphs. We completely determine this set except for one special case where is the product of two distinct primes, is even, and . This result allows us to obtain a plethora of new i‐perfect k‐cycle decompositions of the complete graph of order (mod 2k) with k odd. In particular, if k is a prime, such a decomposition exists for any possible i provided that .     

Decomposing Complete Equipartite Multigraphs into Cycles of Variable Lengths: The Amalgamation‐detachment Approach

Using the technique of amalgamation‐detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1‐factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1‐factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .     

Complex dynamic behaviour of an economic cycle model

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark–Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Variability in different antimony,arsenic and chromium species in waters and bottom sediments of three water reservoirs in Upper Silesia (Poland): a comparative study

Migration of metals/metalloids and their inorganic forms from bottom sediments into waters and from waters into bottom sediments is a complex process. Understanding this mechanism is essential for recognising transformations that occur in water reservoir ecosystems. The following research was conducted in 2012–2014. The aim was to define the time and spatial correlations in the changes of total and ionic antimony, arsenic and chromium concentrations in waters and bottom sediments of three water reservoirs located in Upper Silesia (Poland), i.e. P?awniowice, Rybnik and Gocza?kowice. The ionic form contents were determined with high-performance liquid chromatography-inductively coupled plasma-mass spectrometry (HPLC-ICP-MS). The research was carried out to verify whether the bottom sediments deposited in the discussed reservoirs could have posed an environmental threat due to the possible release of toxic ionic forms (i.e. As(III), Sb(III) and Cr(VI)) into the pelagic zone. The comparison of the research results obtained for three reservoirs (being under different anthropopressure influence) enabled the researchers to describe the bioavailability of specific elements and their species. It also helped to discern the existing environmental threats for such water reservoirs and their users.