Finite Dimensional Filters with Nonlinear Drift, XI: Explicit Solution of the Generalized Kolmogorov Equation in Brockett–Mitter Program

Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late 1970s the concept of the estimation algebra of a filtering system was introduced. Brockett, Clark, and Mitter proposed to use the Wei–Norman approach to solve the nonlinear filtering problem. In 1990, Tam, Wong, and Yau presented a rigorous proof of the Brocket–Mitter program which allows one to construct finite-dimensional recursive filters from finite–dimensional estimation algebras. Later Yau wrote down explicitly a system of ordinary differential equations and generalized Kolmogorov equation to which the robust Duncan–Mortenser– Zakai equation can be reduced. Thus there remains three fundamental problems in Brockett–Mitter program. The first is the problem of finding explicit solution to the generalized Kolmogorov equation. The second is the problem of finding real-time solution of a system of ODEs. The third is the Brockett's problem of classification of finite–dimensional estimation algebras. In this paper, we solve the first problem.     

Solutions for the quasilinear elliptic problems involving critical Hardy–Sobolev exponents

In this article, we study the quasilinear elliptic problem involving critical Hardy–Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign–changing solutions to the problem.     

DEA–DA for bankruptcy-based performance assessment: Misclassification analysis of Japanese construction industry

This study describes a practical use of Data Envelopment Analysis–Discriminant Analysis (DEA–DA) for bankruptcy-based performance assessment. DEA–DA is useful for classifying non-default and default firms based upon their financial performance. However, when we apply DEA–DA to a data set on corporate bankruptcy, we usually face three problems. First, there is a sample imbalance problem because the number of default firms is often limited. In contrast, we can easily obtain a large number of non-default firms. Second, there is a computational problem to deal with a large data set. We need to consider a computational strategy to reduce the dimension of a large data set. Finally, we need to consider data alignment because the location of default firms may exist within that of non-default firms. This study discusses a simultaneous occurrence of the three problems from the perspective of Japanese industrial policy on construction business. To handle the three problems, this study combines DEA–DA with principal component analysis to reduce the computational burden and then alters DEA–DA weights to address both the sample imbalance problem and the location problem. This study also discusses a combined use between DEA–DA and rank sum tests to examine statistically hypotheses related to bankruptcy assessment. As an important application, we apply the proposed approach to the Japanese construction industry and discuss why many Japanese construction firms are misclassified.     

Long time regularity of solutions of the Hele–Shaw problem

In this paper we study the long-time behavior of solutions of the one phase Hele–Shaw problem without surface tension. We show that after a finite time solutions of the Hele–Shaw problem become starshaped and Lipschitz continuous in space. Based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed.     

Integrated portfolio management with options

In this paper we examine the problem of managing portfolios consisting of both, stocks and options. For the simultaneous optimization of stock and option positions we base our analysis on the generally accepted mean–variance framework. First, we analyze the effects of options on the mean–variance efficient frontier if they are considered as separate investment alternatives. Due to the resulting asymmetric portfolio return distribution mean–variance analysis will be not sufficient to identify optimal optioned portfolios. Additional investor preferences which are expressed in terms of shortfall constraints allow a more detailed portfolio specification. Under a mean–variance and shortfall preference structure we then derive optioned portfolios with a maximum expected return. To circumvent the technical optimization problems arising from stochastic constraints we use an approximation of the return distribution and develop economically meaningful conditions under which the complex optimization problem can be transformed into a linear problem being comparably easy to solve. Empirical results based on both, empirical market data and Monte Carlo simulations, illustrate the portfolio optimization procedure with options.     

A mean–variance optimization problem for discounted Markov decision processes

In this paper, we consider a mean–variance optimization problem for Markov decision processes (MDPs) over the set of (deterministic stationary) policies. Different from the usual formulation in MDPs, we aim to obtain the mean–variance optimal policy that minimizes the variance over a set of all policies with a given expected reward. For continuous-time MDPs with the discounted criterion and finite-state and action spaces, we prove that the mean–variance optimization problem can be transformed to an equivalent discounted optimization problem using the conditional expectation and Markov properties. Then, we show that a mean–variance optimal policy and the efficient frontier can be obtained by policy iteration methods with a finite number of iterations. We also address related issues such as a mutual fund theorem and illustrate our results with an example.     

On the equivalence of quadratic optimization problems commonly used in portfolio theory

In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e., the Markowitz mean–variance problem as well as the problems based on the mean–variance utility function and the quadratic utility. Conditions are derived under which the solutions of these three optimization procedures coincide and are lying on the efficient frontier, the set of mean–variance optimal portfolios. It is shown that the solutions of the Markowitz optimization problem and the quadratic utility problem are not always mean–variance efficient.     

Zero–nonzero and real–nonreal sign determination

We consider first the zero–nonzero determination problem, which consists in determining the list of zero–nonzero conditions realized by a finite list of polynomials on a finite set Z⊂C^k$Z\subset {\mathrm{C}}^k$ with C an algebraic closed field. We describe an algorithm to solve the zero–nonzero determination problem and we perform its bit complexity analysis. This algorithm, which is in many ways an adaptation of the methods used to solve the more classical sign determination problem, presents also new ideas which can be used to improve sign determination. Then, we consider the real–nonreal sign determination problem, which deals with both the sign determination and the zero–nonzero determination problem. We describe an algorithm to solve the real–nonreal sign determination problem, we perform its bit complexity analysis and we discuss this problem in a parametric context.     

High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system

In this paper we derive the high-electric-field limit of the three-dimensional Vlasov–Maxwell–Fokker–Planck system. We use the relative entropy method which requires the smoothness of the solution of the limit problem. We obtain convergences of the electro-magnetic field, charge and current densities.     

Standing waves for the Chern–Simons–Schrödinger systems without (AR) condition

In this paper we study the nonlinear Chern–Simons–Schrödinger systems with an external potential. We show the existence, non-existence, and multiplicity of standing waves to this problem with asymptotically linear nonlinearities, which do not hold the Ambrosetti–Rabinowitz condition.     

On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight μ, we define a metrized polyhedral complex, called the directed tight span T_μ, and prove that the dual of the μ-weighted maximum multiflow problem reduces to a facility location problem on T_μ. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μ. By utilizing this duality, we establish the classifications of terminal weights admitting a combinatorial min–max relation (i) for every network and (ii) for every Eulerian network. Our result includes the Lomonosov–Frank theorem for directed free multiflows and Ibaraki–Karzanov–Nagamochi’s directed multiflow locking theorem as special cases.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

Monge–Ampère measures on pluripolar sets

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko?odziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.     

On multistep interval methods for solving the initial value problem

In this paper we shortly complete our previous considerations on interval versions of Adams multistep methods [M. Jankowska, A. Marciniak, Implicit interval multistep methods for solving the initial value problem, Comput. Meth. Sci. Technol. 8(1) (2002) 17–30; M. Jankowska, A. Marciniak, On explicit interval methods of Adams–Bashforth type, Comput. Meth. Sci. Technol. 8(2) (2002) 46–57; A. Marciniak, Implicit interval methods for solving the initial value problem, Numerical Algorithms 37 (2004) 241–251]. It appears that there exist two families of implicit interval methods of this kind. More considerations are dealt with two new kinds of interval multistep methods based on conventional well-known Nyström and Milne–Simpson methods. For these new interval methods we prove that the exact solution of the initial value problem belongs to the intervals obtained. Moreover, we present some estimations of the widths of interval solutions. Some conclusions bring this paper to the end.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

Oleinik type estimates for the Ostrovsky–Hunter equation

The Ostrovsky–Hunter equation provides a model of small-amplitude long waves in a rotating fluid of finite depth. This is a nonlinear evolution equation. In this study, we consider the well-posedness of the Cauchy problem associated with this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities with an Oleinik-type estimate and we prove the uniqueness via a nonlocal adjoint problem. This implies that a shock wave in an entropy weak solution to the Ostrovsky–Hunter equation is admissible only if it jumps down in value (similar to the inviscid Burgers' equation).     

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.     

Pseudoinvexity,optimality conditions and efficiency in multiobjective problems; duality

In this paper, we establish characterizations for efficient solutions to multiobjective programming problems, which generalize the characterization of established results for optimal solutions to scalar programming problems. So, we prove that in order for Kuhn–Tucker points to be efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we obtain characterizations for efficient solutions by using Fritz–John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results. We study the dual problem and establish weak, strong and converse duality results.     

Large-time behavior for the solution to the generalized Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space

In this paper, we consider the global existence as well as the optimal decay estimates of the Cauchy problem for the multi-dimensional Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space. And these results are obtained by Green?s function method, Fourier analysis method, energy estimates method combined with the time-frequency decomposition method.     

On the wave front solution of a competition–diffusion system in population dynamics

In this paper, we consider a competition–diffusion system of two equations [Zhou and Pao, Asymptotic behavior of a competition–diffusion system in population dynamics, Nonlinear Anal. 6 (11) (1982) 1163–1184]. The diffusion coefficients of the system are not equal. We prove existence of a wave front solution which connects two nonzero restpoints of the system. In the proof, we rely essentially on the results of Kolmogorov et al. [A study of diffusion with increase in the quantity of matter, and its application to a biological problem, Bull. Moscow State Univ. 17 (1937) 1–72]. We also estimate the wave speed.