Abstract:
Using an asymptotically additive sequence of continuous functions as a restrictive condition, this paper studies the relations of several ergodic averages for asymptotically additive potentials. Basic properties of conditional maximum ergodic averages are studied. In particular, if the dynamical systems satisfy the specification property, the maximal growth rate of an asymptotically additive potential on the level set is equal to its conditional maximum ergodic averages and the maximal growth rates on the irregular set is its maximum ergodic averages. Finally, the applications for suspension flows are given in the end of the paper.

The limitation of accounting experimental teaching software
leads to the level of teaching falling behind that of practical applications.
In order to solve this problem, it is necessary to construct an accounting
comprehensive experimental platform. It is highly feasible to establish such a platform
based on SaaS mode. Furthermore, when developing this platform, some new requirements
are proposed to the functional design of the platform.

Abstract:
Polypyrrole (PPy) and its
derivatives containing different were
synthesized by the interfacial polymerization reaction using
ammonium per sulfate as oxidant, and were characterized by using
Fourier transform infrared, Ultraviolet/visible absorption, and Raman
spectroscopic techniques. The third-order nonlinear optical properties of all
samples were measured by using nanosecond Z-scan measurements at 532 nm. At the
identical mass concentration of 0.15 mg·mL^{-1}, all PPy derivatives
exhibited an obvious reverse saturable absorption performance,
while the saturable absorption response was observed for PPy. In addition,
significant differences in their nonlinear optical performances were observed,
highlighting the influence on optical nonlinearity of the aromatic segments of
conjugated polymers.

Abstract:
A novel molecule tetra-N-alkylation of cyclen (1,4,7,10-tetraazacyclododecane), 1,4,7,10-tetrakis(2-((4-methoxy)phenoxy)ethyl)-1,4,7,10-tetraazacyclododecane 1, was synthesized and structurally characterized by the single-crystal X-ray diffraction. The crystals were obtained from ethanol by slow evapora-tion at room temperature and the four hydroquinone groups of the benzene ring formed a π-electron-rich cavity by C-H···Br stacking interaction. The crystal belongs to the orthorhombic system, space group Pbcn with a = 17.3174(15), b = 12.9891(11), c = 19.3379(17) ？, α= β = γ = 90°, V = 4349.8(7) ？^{3}, Z = 4, Dc = 1.304 g/cm^{3}, C_{44}H_{61} BrN_{4}O_{8}, Mr = 853.88, F(000) = 1808, μ = 1.001 mm^{？1}, CuKa radiation (λ = 0.71073), R = 0.0434 and wR = 0.1091 for 5200 observed reflections with I > 2σ(I).

Abstract:
A generalized -expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics. 1. Introduction In recent years, with the development of symbolic computation packages like Maple and Mathematica, searching for solutions of nonlinear differential equations directly has become more and more attractive [1–7]. This is because of the availability of computers symbolic system, which allows us to perform some complicated and tedious algebraic calculation and help us find new exact solutions of nonlinear differential equations. In 2008, Wang et al. [8] introduced a new direct method called the -expansion method to look for travelling wave solutions of nonlinear evolution equations (NLEEs). The method is based on the homogeneous balance principle and linear ordinary differential equation (LODE) theory. It is assumed that the traveling wave solutions can be expressed by a polynomial in , and that satisfies a second-order LODE . The degree of the polynomial can be determined by the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given NPDEs. The coefficients of the polynomial can be obtained by solving a set of algebraic equations. Many literatures have shown that the -expansion method is very effective, and many nonlinear equations have been successfully solved. Later, the further developed methods named the generalized -expansion method [9], the modified -expansion method [10], the extended -expansion method [11], the improved -expansion method [12], and the -expansion method [13] have been proposed. As we know, when using the direct method, the choice of an appropriate auxiliary LODE is of great importance. In this paper, by introducing a new auxiliary LODE of different literature [8], we propose the generalized -expansion method, which can be used to obtain travelling wave solutions of NLEEs. In our contribution, we will seek exact solutions of the Zakharov equations [14]: which are one of the classical models on governing the dynamics of nonlinear waves and describing the interactions between high- and low-frequency waves, where is the perturbed number density of the ion (in the low-frequency response), is the slow variation amplitude of the electric field intensity, is the thermal transportation velocity of the

Abstract:
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization problem can be approximated to any level of accuracy via continuous optimization (especially, linear and nonlinear semidefinite programming) problems. One of the main results in this paper shows that if the feasible set of the problem has a minimum rank element with the least F-norm (i.e., Frobenius norm), then the solution of the approximation problem converges to the minimum rank solution of the original problem as the approximation parameter tends to zero. The tractability under certain conditions and convex relaxation of the approximation problem are also discussed. The methodology and results in this paper provide a new theoretical basis for the development of some efficient computational methods for solving rank minimization problems. An immediate application of this theory to the system of quadratic equations is presented in this paper. It turns out that the condition for such a system without a nonzero solution can be characterized by a rank minimization problem, and thus the proposed approximation theory can be used to establish some sufficient conditions for the system to possess only zero solution.

Abstract:
The uniqueness of sparsest solutions of underdetermined linear systems plays a fundamental role in the newly developed compressed sensing theory. Several new algebraic concepts, including the sub-mutual coherence, scaled mutual coherence, coherence rank, and sub-coherence rank, are introduced in this paper in order to develop new and improved sufficient conditions for the uniqueness of sparsest solutions. The coherence rank of a matrix with normalized columns is the maximum number of absolute entries in a row of its Gram matrix that are equal to the mutual coherence. The main result of this paper claims that when the coherence rank of a matrix is low, the mutual-coherence-based uniqueness conditions for the sparsest solution of a linear system can be improved. Furthermore, we prove that the Babel-function-based uniqueness can be also improved by the so-called sub-Babel function. Moreover, we show that the scaled-coherence-based uniqueness conditions can be developed, and that the right-hand-side vector $b$ of a linear system, the support overlap of solutions, the orthogonal matrix out of the singular value decomposition of a matrix, and the range property of a transposed matrix can be also integrated into the criteria for the uniqueness of the sparsest solution of an underdetermined linear system.

Abstract:
Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that l1-minimization is efficient for solving some classes of l0-minimization problems. From a mathematical point of view, however, the understanding of the relationship between l0- and l1-minimization remains incomplete. In this paper, we further discuss several theoretical questions associated with these two problems. For instance, how to completely characterize the uniqueness of least l1-norm nonnegative solutions to a linear system, and is there any alternative matrix property that is different from existing ones, and can fully characterize the uniform recovery of K-sparse nonnegative vectors? We prove that the fundamental strict complementarity theorem of linear programming can yield a necessary and sufficient condition for a linear system to have a unique least l1-norm nonnegative solution. This condition leads naturally to the so-called range space property (RSP) and the `full-column-rank' property, which altogether provide a broad understanding of the relationship between l0- and l1-minimization. Motivated by these results, we introduce the concept of the `RSP of order K' that turns out to be a full characterization of the uniform recovery of K-sparse nonnegative vectors. This concept also enables us to develop certain conditions for the non-uniform recovery of sparse nonnegative vectors via the so-called weak range space property.

Abstract:
In this paper, we define the topological pressure for sub-additive potentials via separated sets in random dynamical systems and we give a proof of the relativized variational principle for the topological pressure.

Abstract:
We obtain large deviation bounds for the measure of deviation sets associated to asymptotically additive and sub-additive potentials under some weak specification properties. In particular a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some examples in connection with the convergence of Lyapunov exponents are given.