A fractional [a, b]-factor of a graph G is a function h from E(G) to [0, 1] satisfying a ≤ d_{G}^{h}(v) ≤ b for every vertex v of G, where d_{G}^{h}(v)=Σ_{e∈ E(v)}h(e) and E(v)={e=uv: u∈ V(G)}. A graph G is called fractional [a, b]-covered if G contains a fractional [a, b]-factor h with h(e)=1 for any edge e of G. A graph G is called fractional (a, b, k)-critical covered if G-Q is fractional [a, b]-covered for any Q⊂eqV(G) with Q=k. In this article, we demonstrate a neighborhood condition for a graph to be fractional (a, b, k)-critical covered. Furthermore, we claim that the result is sharp.
In this paper, a model of mumps transmission with quarantine measure is proposed and then the control reproduction number R_{c} of the model is obtained. This model admits a unique endemic equilibrium P^{*} if and only if R_{c} > 1, while the disease-free equilibrium P^{0} always exists. By using the technique of constructing Lyapunov functions and the generalized Lyapunov-LaSalle theorem, we first show that the equilibrium P^{0} is globally asymptotically stable (GAS) if R_{c} ≤ 1; second, we prove that the equilibrium P^{*} is GAS if R_{c} > 1. Our results reveal that mumps can be eliminated from the community for R_{c} ≤ 1 and it will be persistent for R_{c} > 1, and quarantine measure can also effectively control the mumps transmission.
Our work is concerned with the bifurcation of critical period for a quartic Kolmogorov system. By computing the periodic constants carefully, we show that point (1, 1) can be a weak center of fourth order, and the weak centers condition is given. Moreover, point (1, 1) can bifurcate 4 critical periods under a certain condition. In terms of multiple bifurcation of critical periodic problem for Kolmogorov model, studied results are less seen, our work is good and interesting.
This paper proposes a belief-dependent utility function(BDU). The BDU is based on the fact that the beliefs of acts have inconsistent impacts on utilities. The existence and uniqueness result of the BDU function is provided. It is then proved that the BDU function can provide risk attitude analysis and general comparative analysis as powerful as those presented by the classical expected utility theories.
Let F_{d} be the finite field with d elements. Extraordinary subgroups in F_{d}×F_{d} play an important role in the field of quantum information theory, especially for the study of mutually unbiased bases. Recently, Ghiu et al. introduced the concept of supersquare of order d which is related to extraordinary subgroups. They have given a method of construction of the mutually orthogonal supersquares, and determined all the complete sets of mutually orthogonal extraordinary supersquares of order 4. In this article, we present the construction of a complete set of mutually orthogonal extraordinary supersquares of order p^{n} where p is a prime. We also determine all the complete sets of mutually orthogonal extraordinary supersquares of order 9.
It is well known that a shaded link diagram corresponds to a signed plane multi-graph. In graph theory, line graph is an old and important concept originally introduced by H. Whitney in 1932. In this paper we define the line graph link to be a link which has a diagram whose corresponding signed plane graph is a signed line graph. The main purpose of the paper is to illustrate the structure of planar line graphs, which permits us to deal with its signed Tutte polynomial and the Jones polynomials of line graph links.
In this paper we study optimization problems involving convex nonlinear semidefinite programming (CSDP). Here we convert CSDP into eigenvalue problem by exact penalty function, and apply the U-Lagrangian theory to the function of the largest eigenvalues, with matrix-convex valued mappings. We give the first-and second-order derivatives of U-Lagrangian in the space of decision variables R^{m} when transversality condition holds. Moreover, an algorithm frame with superlinear convergence is presented. Finally, we give one application: bilinear matrix inequality (BMI) optimization; meanwhile, list their UV decomposition results.
Given a graph G=(V, E) and a positive integer k, a k-total-coloring of G is a mapping φ: V ∪ E → {1, 2, ..., k} such that no two adjacent or incident elements receive the same color. The central problem of the total-colorings is the Total Coloring Conjecture, which asserts that every graph of maximum degree Δ admits a (Δ+2)-total-coloring. More precisely, this conjecture has been verified for Δ ≤ 5, and it is still open when Δ =6, even for planar graphs. Let mad(G) denote the maximum average degree of the graph G. In this paper, we prove that every graph G with Δ(G)=6 and mad(G) < 23/5 admits an 8-total-coloring.
In this paper, using the fractional step Lax-Friedrichs difference scheme, we establish the stability of the entropy solution on flow function and relaxation function for a class of conservation law systems with a source term and a relaxation term.
The binomial autoregressive (BAR(1)) process is very useful to model the integer-valued time series data defined on a finite range. It is commonly observed that the autoregressive coefficient is assumed to be a constant. To make the BAR(1) model more practical, this paper introduces a new random coefficient binomial autoregressive model, which is driven by covariates. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators of the model parameters are derived, and the asymptotic properties are obtained. The performance of these estimators is compared via a simulation study. An application to a real data example is also provided. The results show that the proposed model and methods perform well for the simulations and application.
In this paper, we establish a large deviation principle for two-dimensional primitive equations driven by multiplicative Lévy noises. The proof is based on the weak convergence approach.
Let G be a multigraph. Suppose that e=u_{1}v_{1} and e'=u_{2}v_{2} are two edges of G. If e ≠ e', then G(e, e') is the graph obtained from G by replacing e=u_{1}v_{1} with a path u_{1}v_{e}v_{1} and by replacing e'=u_{2}v_{2} with a path u_{2}v_{e'}v_{2}, where v_{e}, v_{e'} are two new vertices not in V(G). If e = e', then G(e, e'), also denoted by G(e), is obtained from G by replacing e=u_{1}v_{1} with a path u_{1}v_{e}v_{1}. A graph G is strongly spanning trailable if for any e, e' ∈ E(G), G(e, e') has a spanning (v_{e}, v_{e'})-trail. The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties. A sequence d=(d_{1}, d_{2}, ..., d_{n}) is multigraphic if there is a multigraph G with degree sequence d, and such a graph G is called a realization of d. A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph, and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected. In this paper, we prove that a nonincreasing multigraphic sequence d=(d_{1}, d_{2}, ..., d_{n}) is strongly spanning trailable if and only if either n=1 and d_{1}=0 or n ≥ 2 and d_{n} ≥ 3. Applying this result, we prove that for a nonincreasing multigraphic sequence d=(d_{1}, d_{2}, ..., d_{n}), if n ≥ 2 and d_{n} ≥ 3, then d is line-hamiltonian-connected.
In this paper, a mean-variance hedging portfolio problem is considered for mean-field stochastic differential equations. The original problem can be reformulated as a nonhomogeneous linear-quadratic optimal control problem with mean-field type. By virtue of the classical completion of squares, the optimal control is obtained in the form of state feedback. We use the theoretical results to the mean-variance hedging portfolio problem and get the optimal portfolio strategy.
In this paper, we are concerned with the autonomous Choquard equation -Δu+u=(I_{α}*u^{α/N+1})u^{α/N-1}u+u^{2*-2u}+f(u) in R^{N}, where N ≥ 3, I_{α} denotes the Riesz potential of order α∈(0, N), the exponents α/N+1 and 2^{*}=2N/N-2 are critical with respect to the Hardy-Littlewood-Sobolev inequality and Sobolev embedding, respectively. Based on the variational methods, by using the minimax principles and the Pohožaev manifold method, we prove the existence of ground state solution under some suitable assumptions on the perturbation f.
Let r ≥ 3 be an integer such that r-2 is a prime power and let H be a connected graph on n vertices with average degree at least d and α(H) ≤ βn, where 0 < β < 1 is a constant. We prove that the size Ramsey number R(H; r)>nd/2(r-2)^{2}-C√n for all sufficiently large n, where C is a constant depending only on r, d and β. In particular, for integers k ≥ 1, and r ≥ 3 such that r-2 is a prime power, we have that there exists a constant C depending only on r and k such that R(P_{n}^{k}; r)> kn(r - 2)^{2}-C√n -(k^{2} + k)/2(r - 2)^{2} for all sufficiently large n, where P_{n}^{k} is the kth power of P_{n}.
In this paper, we consider a non-standard renewal risk model with dependent claim sizes, where an insurance company is allowed to invest his/her wealth in financial assets, leading to some stochastic investment log-returns described as a general adapted càdlàg process. Under the assumptions that the claim sizes are heavy-tailed and the stochastic log-return process on investments is bounded from below almost surely, we derive some asymptotic formulas for the finite-time ruin probability holding uniformly in any finite time horizon.
A classical result of Motzkin and Straus established the connection between the Lagrangian of a graph and its maximum cliques. Applying it, they gave a new proof of Turán's theorem. This aroused the interests in studying the connection between continuous optimization and extremal problems in combinatorics. In 2009, S. Rota Bulò and M. Pelillo extended the result of Motzkin-Straus to r-uniform hypergraphs. Recently, Johnston and Lu initiated the study of the Turán density of a non-uniform hypergraph. Polynomial optimization problems related to several types of non-uniform hypergraphs and its applications on Turán densities have also been studied. In this paper, we obtain a Motzkin-Straus type of results for all non-uniform hypergraphs. Applying it, we give an upper bound of the Turán density of a complete non-uniform hypergraph.