Generalized criterion of solvent polarity as a tool for control of chromatographic process

The generalized (reductive) criterion of solvent polarity was obtained by the method of multiparametric optimization of the Snyder index P", Hildebrand parameter _T , permittivity _r , and solvatochromism parameter E _T (30). Possibilities of employing this criterion for estimation of the elution power of normal and reversed mobile phases used in high-performance liquid chromatography were considered.     

病态分析体系有偏估计的研究

运用广义岭估计和Ｌｉｕｋｅｊｉａｎ提出的有偏估计，对病态分析体系进行了数值模拟和实际光度测定，结果表明，广义岭估计显优于最小二乘估计，Ｌｉｕｋｅｊｉａｎ法有功效，可和为解析病态分析体系的化学计量学方法。     

New Results on Generalized $c$-Distance Without Continuity in Cone $b$-Metric Spaces over Banach Algebras

In this work, some new fixed point results for generalized Lipschitz mappings on generalized $c$-distance in cone $b$-metric spaces over Banach algebras are obtained, not acquiring the condition that the underlying cone should be normal or the mappings should be continuous. Furthermore, the existence and the uniqueness of the fixed point are proven for such mappings. These results greatly improve and generalize several well-known comparable results in the literature. Moreover, some examples and an application are given to support our new results.     

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

分块算子矩阵的加权EP

本文在加权广义Schur补的基础上, 引入并研究了Hilbert空间上分块算子矩阵的加权Moore-Penrose逆和加权EP. 进一步, 给出了加权EP算子在算子方程中的一个应用.     

Multiple separatrix crossing in multi-degree-of-freedom Hamiltonian flows

Summary We study separatrix crossing in near-integrablek-degree-of-freedom Hamiltonian flows, 2 <k < , whose unperturbed phase portraits contain separatrices inn degrees of freedom, 1 <n <k. Each of the unperturbed separatrices can be recast as a codimension-one separatrix in the 2k-dimensional phase space, and the collection of these separatrices takes on a variety of geometrical possibilities in the reduced representation of a Poincaré section on the energy surface. In general 0 l n of the separatrices will be available to the Poincaré section, and each separatrix may be completely isolated from all other separatrices or intersect transversely with one or more of the other available separatrices. For completely isolated separatrices, transitions across broken separatrices are described for each separatrix by the single-separatrix crossing theory of Wiggins, as modified by Beigie. For intersecting separatrices, a possible violation of a normal hyperbolicity condition complicates the analysis by preventing the use of a persistence and smoothness theory for compact normally hyperbolic invariant manifolds and their local stable and unstable manifolds. For certain classes of multi-degree-of-freedom flows, however, a local persistence and smoothness result is straightforward, and we study the global implications of such a local result. In particular, we find codimension-one partial barriers and turnstile boundaries associated with each partially destroyed separatrix. From the collection of partial barriers and turnstiles follows a rich phase space partitioning and transport formalism to describe the dynamics amongst the various degrees of freedom. A generalization of Wiggins' higher-dimensional Melnikov theory to codimension-one surfaces in the multi-separatrix case allows one to uncover invariant manifold geometry. In the context of this perturbative analysis and detailed numerical computations, we study invariant manifold geometry, phase space partitioning, and phase space transport, with particular attention payed to the role of a vanishing frequency in the limit approaching the intersection of the partially destroyed separatrices. The class of flows under consideration includes flows of basic physical relevance, such as those describing scattering phenomena. The analysis is illustrated in the context of a detailed study of a 3-degree-of-freedom scattering problem.     

Dynamic complexity of optimal paths and discount factors for strongly concave problems

We study the relationship between the dynamical complexity of optimal paths and the discount factor in general infinite-horizon discrete-time concave problems. Given a dynamic systemx _t+1=h(x _t ), defined on the state space, we find two discount factors 0 < ^{* **} < 1 having the following properties. For any fixed discount factor 0 < < ^*, the dynamic system is the solution to some concave problem. For any discount factor ^** < < 1, the dynamic system is not the solution to any strongly concave problem. We prove that the upper bound ^** is a decreasing function of the topological entropy of the dynamic system. Different upper bounds are also discussed.This research was partially supported by MURST, National Group on Nonlinear dynamics in Economics and Social Sciences. The author would like to thank two anonymous referees for helpful comments and suggestions.     

On invexity-type nonlinear programming problems

In this paper, we propose a new class of nonlinear programing, called SFJ-invex programming. The optimality characterization shows that a problem is SFJ-invex if and only if a Fritz John point together with its multiplier, is a Fritz John saddle point of the problem. Under any constraint qualification assumption, a problem is SFJ-invex if and only if a Kuhn-Tucker point together with its multiplier is a Kuhn-Tucker saddle point of the problem. Furthermore, a generalization of the SFJ-invex, class is developed; the applications to (h, )-convex programming, particularly geometric programming, and to generalized fractional programming provide a relaxation in constraint qualification for differentiable problems to get saddle-point type optimality criteria.The author wishes to thank the referee for helpful comments.     

Probabilistic approach to homoclinic chaos

Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i, – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived.     

An existence result for linear partial differential equations with coefficients in an algebra of generalized functions

We prove the existence of solutions for essentially all linear partial differential equations with -coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy's equation has solutions whenever its right-hand side is a classical -function.