Narrow Results

Let $G=(V,E)$ be a graph and $k$ be an integer representing $k$ colors. There is a function $f$ from $V$ to the power set of $k$ colors satisfying every vertex $v\in V$ assigned $\varnothing$ under $f$ in its neighborhood has all the colors, then $f$ is called a $k$-rainbow dominating function ($k$RDF) on $G$. The weight of $f$ is the sum of the number of colors on each vertex all over the graph. The $k$-rainbow domination number of $G$ is the minimum weight of $k$RDFs on $G$, denoted by ${\gamma }_{rk}(G)$. The aim of this paper is to investigate the $k$-rainbow ($k=3,4$) domination number of the Cartesian product of paths $P_m\square P_n$ and the Cartesian product of paths and cycles $P_m\square C_n$. For $P_m\square P_n$, we determine the value ${\gamma }_{r3}(P_4\square P_n)=2n+2$ and present ${\gamma }_{r3}(P_m\square P_n)\le \frac{mn}{2}+2$ for $m\ge 5$. For $P_m\square C_n$, we determine the values of ${\gamma }_{r3}(P_m\square C_n)$ for $m=3,4$ or $n=3,4$ and ${\gamma }_{r4}(P_m\square C_n)$ for $m=3$ or $n=3$.

A new program created in C/C$++$ language generates automatically the analytic expression of grand potential and prints it in Latex2e format and in textual data. Another code created in Mathematica language can translate the textual data into a mathematical expression and help any interested to evaluate the thermodynamic quantities in analytic or numeric forms.

This paper is an analytical study of Boolean networks. The motivation is our desire to understand the large, complicated and interconnected pathways which comprise intracellular biochemical signal transduction networks. The simplest possible conceptual model that mimics signal transduction with sigmoidal kinetics is the n-node Boolean network each of whose elements or nodes has the value 0 (off) or 1 (on) at any given time T = 0, 1, 2, …. A Boolean network has 2ⁿ states all of which are either on periodic cycles (including fixed points) or transients leading to cycles. Thus one understands a Boolean network by determining the number and length of its cycles. The problem one must circumvent is the large number of states (2ⁿ) since the networks we are interested in have 100 or more elements. Thus we concentrate on developing size n methods rather than the impossible task of enumerating all 2ⁿ states. This is done as follows: the dynamics of the network can be described by n polynomial equations which describe the logical function which determines the interaction at each node. Iterating the equations one step at a time finds all fixed points, period two cycles, period three cycles, etc. This is a general method that can be used to determine the fixed points and moderately large periodic cycles of any size network, but it is not useful in finding the largest cycles in a large network. However, we also show that the network equations can often be reduced to scalar form, which makes the cycle structure much more transparent. The scalar equations method is a true "size n" method and several examples (including nontrivial biochemical systems) are examined.

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained.

We examine and assess deterministic chaos as an observable. First, we present the development of model ecological systems. We illustrate how to apply the Kolmogorov theorem to obtain limits on the parameters in the system, which assure the existence of either stable equilibrium point or stable limit cycle behavior in the phase space of two-dimensional (2D) dynamical systems. We also illustrate the method of deriving conditions using the linear stability analysis. We apply these procedures on some basic existing model ecological systems. Then, we propose four model ecological systems to study the dynamical chaos (chaos and intermittent chaos) and cycles. Dynamics of two predation and two competition models have been explored. The predation models have been designed by linking two predator–prey communities, which differ from one another in one essential way: the predator in the first is specialist and that in the second is generalist. The two competition models pertain to two distinct competition processes: interference and exploitative competition. The first competition model was designed by linking two predator–prey communities through inter-specific competition. The other competition model assumes that a cycling predator–prey community is successfully invaded by a predator with linear functional response and coexists with the community as a result of differences in the functional responses of the two predators. The main criterion behind the selection of these two model systems for the present study was that they represent diversity of ecological interactions in the real world in a manner which preserves mathematical tractability. For investigating the dynamic behavior of the model systems, the following tools are used: (i) calculation of the basin boundary structures, (ii) performing two-dimensional parameter scans using two of the parameters in the system as base variables, (iii) drawing the bifurcation diagrams, and (iv) performing time series analysis and drawing the phase space diagrams. The results of numerical simulation are used to distinguish between chaotic and cyclic behaviors of the systems.

The conclusion that we obtain from the first two model systems (predation models) is that it would be difficult to capture chaos in the wild because ecological systems appear to change their attractors in response to changes in the system parameters quite frequently. The detection of chaos in the real data does not seem to be a possibility as what is present in ecological systems is not robust chaos but short-term recurrent chaos. The first competition model (interference competition) shares this conclusion with those of predation ones. The model with exploitative competition suggests that deterministic chaos may be robust in certain systems, but it would not be observed as the constituent populations frequently execute excursions to extinction-sized densities. Thus, no matter how good the data characteristics and analysis techniques are, dynamical chaos may continue to elude ecologists. On the other hand, the models suggest that the observation of cyclical dynamics in nature is the most likely outcome.

The onset and bifurcation points of n-cycles of several polynomial maps are located through a characteristic equation connecting cyclic polynomials of the cycle points. The polynomials satisfied by the parameters of the logistic, Hénon, and cubic maps at the onset and bifurcation points are obtained for n up to 14, 9, and 9, respectively.

The behaviour of a food chain with three populations of the type prey-predator-superpredator is studied in the case where the growth rates of the three populations are highly diversified. The analysis shows that the system can have high-frequency oscillations, due to the interaction between prey and predator, which arise during a low-frequency cycle, due to the interaction between predator and superpredator. Using an averaging method, the dynamics of the superpredator is determined, during the high-frequency oscillations of the prey and predator.

It has been recently proved that the non-existence of certain type of cycles of pre-imputation, fundamental cycles, is equivalent to the balancedness of a TU-games (Cesco (2003)). In some cases, the class of fundamental cycles can be narrowed and still obtain a characterization theorem. In this paper we prove that existence of maximal U-cycles, which are related to a transfer scheme designed for computing a point in the core of a game, is condition necessary and sufficient for a TU-game be non-balanced, provided n - 1 and n-person are the only coalitions with non-zero value. These games are strongly related to games with only 1, n - 1 and n-person permissible coalitions (Maschler (1963)).

Motivated by the study of rationally connected fibrations, we study different notions of birationally simple fibrations. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. This paper is largely self-contained and we prove a number of basic properties of these fibrations. One application is to the classification of “rationalizations of singularities of cones.” We also consider consequences for the Chow groups of the generic fiber of a Chow constant fibration.

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by $\phi \left(G\right)$, is the maximal integer k such that G has a b-coloring with k colors. In this paper, the b-chromatic numbers of the coronas of cycles, star graphs and wheel graphs with different numbers of vertices, respectively, are obtained. Also the bounds for the b-chromatic number of corona of any two graphs is discussed.

A network is said to be conditionally faulty if its every vertex is incident to at least g fault-free neighbors, where $g\ge 1$. An n-dimensional folded hypercube $F{Q}_n$ is a well-known variation of an n-dimensional hypercube $Q_n$, which can be constructed from $Q_n$ by adding an edge to every pair of vertices with complementary addresses. In this paper, we define that a network is said to be g-conditionally faulty if its every vertex is incident to at least g fault-free neighbors, and let $F{F}_v$ (respectively, $F{F}_e$) denote the set of faulty vertices (respectively, faulty edges) in $F{Q}_n$. Then, we consider for the cycles embedding properties in $F{Q}_n-F{F}_v-F{F}_e$ with 4-conditionally faulty, as follows:

(1) For $n\ge 3$, $F{Q}_n-F{F}_v-F{F}_e$ contains a fault-free cycle of every even length from 4 to $2^n-2\left|F{F}_v\right|$, where $\left|F{F}_n\right|+\left|F{F}_e\right|\le 2n-5$;

(2) For even $n\ge 4$, $F{Q}_n-F{F}_v-F{F}_e$ contains a fault-free cycle of every odd length from $n+1$ to $2^n-2\left|F{F}_v\right|-1$, where $\left|F{F}_v\right|+\left|F{F}_e\right|\le 2n-5$.

The temporal fluctuations of the 1880–1994 seismicity recorded in Pannonian Basin were investigated by means of the Fano Factor and Allan Factor methods, which allows us to estimate the power spectrum for point processes. Up to our knowledge, this is the first scaling analysis performed on earthquake series spanning for almost one century. Our findings point out a presence of two temporal regimes in the analyzed seismicity, which is characterized by quasi-Poissonian behavior for low to intermediate timescales for up to about 1/2 years, and by scaling behavior for higher timescales with scaling exponent around 0.3–0.4, indicating a rather high time-clustering of the events. Furthermore the Allan Factor shows the presence of a periodicity at about 2.23 years for events with magnitude larger or equal to 3.5.

The monthly rainfall time series, spanning more than a century, recorded in several sites in the middle Argentina were analyzed. The power spetral density (PSD) method reveals the presence of annual and semi-annual cyclic fluctuations. The detrended fluctuation analysis (DFA) performed on the residual times series (after removing the periodicities) shows a scaling behavior, characterized by DFA scaling exponents ranging between 0.54 and 0.58. These findings could contribute to a better understanding of rainfall dynamics.

We study cycles of the discrete-time nonlinear dynamical system under parametric random disturbances. A dynamics of deviations of the random solutions from the deterministic cycle is analyzed with the help of first approximation linear systems. A closed system for second moments of these deviations is obtained and explicit formulas for the stable periodic solution are given. Constructive abilities of our mathematical technique are demonstrated for parametrical analysis of stochastically forced periodic regimes in the Ricker population model with Allee effect. A threshold noise intensity corresponding to the onset of stochastic extinction is estimated.

Stochastic dynamics of the FitzHugh–Nagumo (FHN) neuron model in the limit cycles zone is studied. For weak noise, random trajectories are concentrated in the small neighborhood of the unforced deterministic cycle. As the noise intensity increases, in the Canard-like cycles zone of the FHN model, a bundle of the stochastic trajectories begins to split into two parts. This phenomenon is investigated using probability density functions for the distribution of random trajectories. It is shown that the intensity of noise generating this splitting bifurcation significantly depends on the stochastic sensitivity of cycles. Using the stochastic sensitivity function (SSF) technique, we find a critical value of the parameter corresponding to the supersensitive cycle. For the neighborhood of this critical value, a comparative parametrical analysis of the phenomenon of the stochastic cycle splitting is performed. To predict the splitting bifurcation and estimate a threshold value of the noise intensity, we use a confidence domains method based on SSF. A phenomenon of the noise-induced chaotization is studied. We show that P-bifurcation of the splitting of stochastic cycles implies a D-bifurcation of a noise-induced chaotization.

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.

We attach a diagraph with generalized Quaternion group of order $4n$ by utilizing the power map $f:Q_{4n}\to Q_{4n}$ defined by $f(x)=x^k$ for all $x\in Q_{4n}$, where $k$ is a fixed natural number. We examine the structure of these power digraphs and establish numerous results encapsulating the existence of cycle vertices, derivation of different formulae concerning the number of cycles, length of cycles and most importantly in-degree of vertices. Moreover, we categorize the regular and semi-regular power digraphs.

In this paper, we study the generalized path ideals, which is a new class of path ideals of cycle graphs. These ideals naturally generalize the standard path ideals of cycles, as studied by Alilooee and Faridi [On the resolution of path ideals of cycles, Comm. Algebra43 (2015) 5413–5433]. We give some formulas to compute all the top degree graded Betti numbers of these path ideals of cycle graphs. As a consequence, we can give some formulas to compute their projective dimension and regularity.

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of $h$-wheel graphs, under certain condition, are normal.

In the past three decades, there has been a rise in neo-Schumpeterian approaches for understanding the role of innovation in technological development cycles. However, a literature review reveals two important factors are missing: first, an understanding of the role of the Global Network, defined as the connective network by which people, goods, and information move around the world, and second, a more holistic view of innovation cycles that is based on hierarchies of technologies and encompasses the influence of sociopolitical interactions and market integration worldwide. This paper presents a new conceptual framework for understanding the development of technology, taking these factors into account through the grouping of innovations into Global Development Cycles. With increasing globalization and radical technological development, the attempt to understand predictable, long-term pattern changes has never been more important if we are to effectively manage global development. The implications of this more comprehensive global perspective are discussed.