In this paper, we consider a modification of the well-known logistic family using a family of fuzzy numbers. The dynamics of this modified logistic map is studied by computing its topological entropy with a given accuracy. This computation allows us to characterize when the dynamics of the modified family is chaotic. Besides, some attractors that appear in bifurcation diagrams are explained. Finally, we will show that the dynamics induced by the logistic family on the fuzzy numbers need not be complicated at all.
For a revised model of Caldentey and Stacchetti (Econometrica, 2010) in continuous-time insider trading with a random deadline which allows market makers to observe some information on a risky asset, a closed form of its market equilibrium consisting of optimal insider trading intensity and market liquidity is obtained by maximum principle method. It shows that in the equilibrium, (i) as time goes by, the optimal insider trading intensity is exponentially increasing even up to infinity while both the market liquidity and the residual information are exponentially decreasing even down to zero; (ii) the more accurate information observed by market makers, the stronger optimal insider trading intensity is such that the total expect profit of the insider is decreasing even go to zero while both the market liquidity and the residual information are decreasing; (iii) the longer the mean of random time, the weaker the optimal insider trading intensity is while the more both the residual information and the expected profit are, but there is a threshold of trading time, half of the mean of the random time, such that if and only if after it the market liquidity is increasing with the mean of random time increasing.
In present work studied the new boundary value problem for semi linear (Power-type nonlinearities) system equations of mixed hyperbolic -elliptic Keldysh type in the multivariate dimension with the changing time direction. Considered problem and equation belongs to the modern level partial differential equations. Applying methods of functional analysis, topological methods, “ε” -regularizing. and continuation by the parameter at the same time with aid of a prior estimates, under assumptions conditions on coefficients of equations of system, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev’s space. In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations, moreover in this case constructed new weithged spaces, the identity between of strong and weak solutions is established.
In this paper, we study the global behavior of solutions to the spherically symmetric(it means that the problem is 2+2 framework)coupled Nonlinear-Einstein-Klein-Gordon (NLEKG) system in the presence of a negative cosmological constant. We prove the well posedness of the NLEKG system in the Schwarzschild-AdS spacetimes and that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which ϕ = 0 identically) are asymptotically stable. Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a SchwarzschildAdS spacetime. Bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.
Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the concepts of Eulerian and bipartite delta-matroids and then extend the result from embedded graphs to arbitrary binary delta-matroids. The dual of any bipartite embedded graph is Eulerian. We also extend the result from embedded graphs to the class of delta-matroids that arise as twists of binary matroids. Several related results are also obtained.
In this paper, we consider the statistical inferences in a partially linear model when the model error follows an autoregressive process. A two-step procedure is proposed for estimating the unknown parameters by taking into account of the special structure in error. Since the asymptotic matrix of the estimator for the parametric part has a complex structure, an empirical likelihood function is also developed. We derive the asymptotic properties of the related statistics under mild conditions. Some simulations, as well as a real data example, are conducted to illustrate the finite sample performance.
Joint analysis of multiple phenotypes can have better interpretation of complex diseases and increase statistical power to detect more significant single nucleotide polymorphisms (SNPs) compare to traditional single phenotype analysis in genome-wide association analysis. Principle component analysis (PCA), as a popular dimension reduction method, has been broadly used in the analysis of multiple phenotypes. Since PCA transforms the original phenotypes into principal components (PCs), it is natural to think that by analyzing these PCs, we can combine information across phenotypes. Existing PCA-based methods can be divided into two categories, either selecting one particular PC manually or combining information from all PCs. In this paper, we propose an adaptive principle component test (APCT) which selects and combines the PCs adaptively by using Cauchy combination method. Our proposed method can be seen as a generalization of traditional PCA based method since it contains two existing methods as special situation. Extensive simulation shows that our method is robust and can generate powerful result in various situations. The real data analysis of stock mice data also demonstrate that our proposed APCT can identify significant SNPs that are missed by traditional methods.
In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)-dimensional KdV equation are constructed by the Bell polynomials approach. The N-soliton solution is derived directly from the bilinear form. Especially, based on the two-soliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one- and two-periodic wave solution, which reveal the relations with the single and two-soliton solution, are presented in detail.
A bio-safe dengue control strategy is to use Wolbachia, which can induce incomplete cytoplasmic incompatibility (CI) and reduce the mating competitiveness of infected males. In this work, we formulate a delay differential equation model, including both the larval and adult stages of wild mosquitoes, to assess the impacts of CI intensity ξ and mating competitiveness θ of infected males on the suppression efficiency. Our analysis identifies a CI intensity threshold ξ^{*} below which a successful suppression is impossible. When ξ≥ξ^{*}, the wild population will be eliminated ultimately if the releasing level exceeds the release amount threshold R^{*} uniformly. The dependence of R^{*} on ξ and θ, and the impact of temperature on suppression are further exhibited through numerical examples. Our analyses indicate that a slight reduction of ξ is more devastating than significantly decrease of θ in the suppression efficiency. To suppress more than 95% wild mosquitoes during the peak season of dengue in Guangzhou, the optimal starting date for releasing is sensitive to ξ but almost independent of θ. One percent reduction of ξ from 1 requires at least one week earlier in the optimal releasing starting date from 7 weeks ahead of the peak season of dengue.
It is difficult to characterize graphs which contain no a 2-connected graph as a minor in graph theory. Let V_{8} be a graph constructed from an 8-cycle by connecting the antipodal vertices. There are thirteen 2-connected spanning subgraphs of V_{8}. In particular, one of them is obtained from the Petersen graph by deleting two vertices and it is also a hard problem to characterize Petersen-minor-free graphs. In this paper, we characterize internally 4-connected graphs which contain a 2-connected spanning subgraph of V_{8} as a forbidden minor.
Let $K_{1,k}$ be a star of order $k+1$ and $K_n\sqcup K_{1,k}$ the graph obtained from a complete graph $K_n$ and an additional vertex $v$ by joining $v$ to $k$ vertices of $K_n$. For graphs $G$ and $H$, the star-critical Ramsey number $r_*(G,H)$ is the minimum integer $k$ such that any red/blue edge-coloring of $K_{r-1}\sqcup K_{1,k}$ contains a red copy of $G$ or a blue copy of $H$, where $r$ is the classical Ramsey number $R(G,H)$. Let $C_m$ denote a cycle of order $m$ and $W_n$ a wheel of order $n+1$. Hook (2010) proved that $r_*(W_n,C_3)=n+3$ for $n\geq6$. In this paper, we show that $r_*(W_n,C_m)=n+3$ for $m$ odd, $m\geq5$ and $n\geq 3(m-1)/2+2$.
In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations $$ \left \{ \begin{array}{l} \text{det}\ D^2u_1=\lambda h_1(|x|)f_1(-u_2), \qquad \text{in} \ \ \Omega,\\ \text{det}\ D^2u_2=\lambda h_2(|x|)f_2(-u_1), \qquad \text{in} \ \ \Omega,\\ u_1=u_2=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on} \ \ \partial \Omega \end{array} \right. $$ for a certain range of $\lambda >0$, $h_i$ are weight functions, $f_i$ are continuous functions with possible singularity at $0$ and satisfy a combined $N$-superlinear growth at $\infty$, where $i\in \{1,2\}$, $\Omega$ is the unit ball in $\mathbb{R}^N$. We establish the existence of a nontrivial radial convex solution for small $\lambda$, multiplicity results of nontrivial radial convex solutions for certain ranges of $\lambda$, and nonexistence results of nontrivial radial solutions for the case $\lambda\gg 1$. The asymptotic behavior of nontrivial radial convex solutions is also considered.
We consider non-autonomous ordinary differential equations in two cases. One is the one-dimensional case that admits a condition of hyperbolicity, and the other one is the higher-dimensional case that admits an exponential dichotomy. For differential equations of this kind, we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.
The perfect matching polytope of a graph $G$ is the convex hull of the incidence vectors of all perfect matchings of $G$. A graph $G$ is PM-compact if the 1-skeleton graph of the prefect matching polytope of $G$ is complete. Equivalently, a matchable graph $G$ is PM-compact if and only if for each even cycle $C$ of $G$, $G-V(C)$ has at most one perfect matching. This paper considers the class of graphs from which deleting any two adjacent vertices or nonadjacent vertices, respectively, the resulting graph has a unique perfect matching. The PM-compact graphs in this class of graphs are presented.
A vertex-colored path $P$ is rainbow if its internal vertices have distinct colors; whereas $P$ is monochromatic if its internal vertices are colored the same. For a vertex-colored connected graph $G$, the rainbow vertex-connection number ${\rm rvc} (G)$ is the minimum number of colors used such that there is a rainbow path joining any two vertices of $G$; whereas the monochromatic vertex-connection number ${\rm mvc} (G)$ is the maximum number of colors used such that any two vertices of $G$ are connected by a monochromatic path. These two opposite concepts are the vertex-versions of rainbow connection number ${\rm rc} (G)$ and monochromatic connection number ${\rm mc} (G)$ respectively. The study on ${\rm rc} (G)$ and ${\rm mc} (G)$ of random graphs drew much attention, and there are few results on the rainbow and monochromatic vertex-connection numbers. In this paper, we consider these two vertex-connection numbers of random graphs and establish sharp threshold functions for them, respectively.