Narrow Results

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.

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.

A 2-distance coloring of $G$ is a function $\phi$: $V(G)\to \{1,2,\dots ,k\}$, such that for every two distinct vertices $u$, $v$ in $G$, $\phi (u)\ne \phi (v)$ if $0<d_G(u,v)\le 2$. The 2-distance chromatic number of $G$ is the least integer $k$ such that $G$ has a $k$-$2$-distance coloring, denoted by ${\chi }_2(G)$. Similarly, the list 2-distance chromatic number of $G$ is denoted by ${\chi }_2^l(G)$. In this paper, we proved that: (1) for every planar graph with $g(G)\ge 5$ and $\mathrm{\Delta }(G)\ge 12$, ${\chi }_2^l(G)\le \mathrm{\Delta }(G)+6$; (2) for every planar graph with $g(G)\ge 6$ and $\mathrm{\Delta }(G)\ge 9$, ${\chi }_2^l(G)\le \mathrm{\Delta }(G)+3$.

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.

We study the digraphs based on dihedral group $D_n$ by using the power mapping, i.e., the set of vertices of these digraphs is $D_n$ and the set of edges is $\{(a,b):a^k=b,\forall a,b\in D_n\}$. These are called the power digraphs and denoted by ${\gamma }_{D_n}(2n,k)$. The cycle and in-degree structure of these digraphs are completely examined. This investigation leads to the derivation of various formulae regarding the number of cycle vertices, the length of the cycles, the number of cycles of certain lengths and the in-degrees of all vertices. We also establish necessary and sufficient conditions for a vertex to be a cycle vertex. The analysis of distance between vertices culminates at different expressions in terms of $n$ and $k$ to determine the heights of vertices, components and the power digraph itself. Moreover, all regular and semi-regular power digraphs ${\gamma }_{D_n}(2n,k)$ are completely classified.

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.

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.

Given n men, n women, and n dogs, each man has an incomplete preference list of women, each woman has an incomplete preference list of dogs, and each dog has an incomplete preference list of men. We understand a family as a triple consisting of one man, one woman, and one dog such that the dog belongs to the preference list of the woman, who, in turn, belongs to the preference list of the man, while the latter belongs to the preference list of the dog. We understand a matching as a collection of nonintersecting families (some agents, possibly, remain single). A matching is said to be nonstable, if one can find a man, a woman, and a dog who do not live together currently but each of them would become “happier” if they do. Otherwise, the matching is said to be stable (a weakly stable matching). We give an example of this problem for $n=3$ where no stable matching exists. Moreover, we prove the absence of such an example for $n<3$. Such an example was known earlier only for $n=6$ [P. Biró and E. McDermid, Three-sided stable matchings with cyclic preferences, Algorithmica 58 (2010) 5–18]. The constructed examples also allow one to halve the size of the recently constructed analogous example for complete preference lists [C.-K. Lam and C.G. Plaxton, On the existence of three-dimensional stable matchings with cyclic preferences, in Algorithmic Game Theory, Lecture Notes in Computer Science, Vol. 11801 (Springer, 2019), pp. 329–342].

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 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.

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.

As the backdrop for contemporary international relations, globalization reflects the way economic and political power are distributed, and provides the grand context for China’s strategic planning. The history and logic of globalization have shown that underpinned by a system of nation-states, globalization proceeds according to an inescapable cyclical pattern. Globalization suffered major setbacks in the aftermath of the 2008 financial crisis and is likely to further lose steam amid an evolving Covid-19 pandemic. A low-ebb phase of globalization will present an increasingly complicated strategic environment featuring intensifying great power rivalry, regionalized supply chains, and growing technology competition. Beijing remains determined to integrate further into the world, but to adapt to a new strategic environment, will vigorously implement the newly unveiled dual circulation strategy. As China sees it, despite all the major setbacks, globalization is an irreversible mega-trend but it will be driven by a new underlying logic.