A structural connection between linear and 0–1 integer linear formulations

The connection between linear and 0–1 integer linear formulations has attracted the attention of many researchers. The main reason triggering this interest has been an availability of efficient computer programs for solving pure linear problems including the transportation problem. Also the optimality of linear problems is easily verifiable through existing algorithms. However, there is no efficient general technique available to solve 0–1 integer linear problems or to verify their optimality. This paper shows that in the case of one of the easier 0–1 integer linear problems, namely a single assignment problem, such a relation between linear and 0–1 integer linear formulation can be built. The theory behind the proposed ‘bridge’ is based on the combination of the absolute point principle and shadow price theory. The main practical benefit of this work is in providing an algorithm to find a MFL (more-for-less) solution for the assignment problem. To the best of our knowledge, this is one of the first efforts to provide a ‘more-for-less’ result for a 0–1 integer linear problem.     

On Bayes linear unbiased estimation of estimable functions for the singular linear model

The unique Bayes linear unbiased estimator (Bayes LUE) of estimable functions is derived for the singular linear model. The superiority of Bayes LUE over ordinary best linear unbiased estimator is investigated under mean square error matrix (MSEM) criterion.     

Optimal ℋ∞-state feedback control for continuous-time linear systemsfeedback control for continuous-time linear systems

This paper proposes a convex programming method to achieve optimal -state feedback control for continuous-time linear systems. State space conditions, formulated in an appropriate parameter space, define a convex set containing all the stabilizing control gains that guarantee an upper bound on the -norm of the closed-loop transfer function. An optimization problem is then proposed, in order to minimize this upper bound over the previous convex set, furnishing the optimal -control gain as its optimal solution. A limiting bound for the optimum -norm can easily be calculated, and the proposed method will achieve minimum attenuation whenever a feasible state feedback controller exists. Generalizations to decentralized and output feedback control are also investigated. Numerical examples illustrate the theory.This research has been supported in part by grants from Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP and Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq—Brazil. The authors are grateful to the anonymous referees for their useful comments on this paper.     

On orthogonal linear ℓ1 approximation

Summary The problem is considered of orthogonal ₁ fitting of discrete data. Local best approximations are characterized and the question of the robustness of these solutions is considered. An algorithm for the problem is presented, along with numerical results of its application to some data sets.     

Ont ∞ quasi-similarity of linear systems

Summary Conti defined t similarity of systems (1)x=tA(t)xand (2)y=B(t)y, and showed that it is an equivalence relation which preserves uniform and strict stability. Here the definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relating A and B. The extended relation, t quasi-similarity, is not symmetric or transitive; however, it is shown that if (2) is t quasi-similar to (1)and (1)is uniformly, uniformly asymptotically, or strictly stable, then so is (2).Results are also given concerning linear asymptotic equilibrium of (2)in the case where (1)is stricly stable or has linear asymptotic equilibrium.     

Several splittings for non-Hermitian linear systems

For large sparse non-Hermitian positive definite system of linear equations,we present several variants of the Hermitian and skew-Hermitian splitting(HSS)about the coefficient matrix and establish correspondingly several HSS-based iterative schemes.Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations,and they may show advantages on problems that the HSS method is ineffiective.     

Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Itô stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.     

Hierarchical linear regression models for conditional quantiles

The quantile regression has several useful features and therefore is gradually developing into a comprehensive approach to the statistical analysis of linear and nonlinear response models, but it cannot deal effectively with the data with a hierarchical structure. In practice, the existence of such data hierarchies is neither accidental nor ignorable, it is a common phenomenon. To ignore this hierarchical data structure risks overlooking the importance of group effects, and may also render many of the traditional statistical analysis techniques used for studying data relationships invalid. On the other hand, the hierarchical models take a hierarchical data structure into account and have also many applications in statistics, ranging from overdispersion to constructing min-max estimators. However, the hierarchical models are virtually the mean regression, therefore, they cannot be used to characterize the entire conditional distribution of a dependent variable given high-dimensional covariates. Furthermore, the estimated coefficient vector (marginal effects) is sensitive to an outlier observation on the dependent variable. In this article, a new approach, which is based on the Gauss-Seidel iteration and taking a full advantage of the quantile regression and hierarchical models, is developed. On the theoretical front, we also consider the asymptotic properties of the new method, obtaining the simple conditions for an n1/2-convergence and an asymptotic normality. We also illustrate the use of the technique with the real educational data which is hierarchical and how the results can be explained.     

Computer programs for non‐linear programming

In this paper we consider the problem of maximizing a non‐linear or linear objective function subject to non‐linear and/or linear constraints. The approach used is an adaptive random search with some non‐random searches built‐in. The algorithm begins with a given point which is replaced by another point if the latter satisfies each of the constraints and results in a bigger functional value. The process of moving from one point to a better point is repeated many times. The value of each of the coordinates of the next point is determined by one of several ways; for example, a coordinate is sometimes forced to have the same value as the value of the corresponding coordinate of the current feasible point. In this algorithm, a candidate point receives no further computational considerations as soon as it is found to be unfeasible; this makes the algorithm general. Computer programs illustrating the details of the new algorithm are given and computational results of two numerical test problems from the literature are presented. Optimality was reached in each of these two problems.     

Ax–Schanuel for linear differential equations

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.     

Multitime-scale singularly perturbed linear stochastic systems ∗

By applying diagonalization transformation, generalized variation of constants formula and theory of differential inequalities,the mean square convergence of solution process of a shingularly perturbed linear stochastic differential system of Itô-type is investigated. Moreover, slow and fast modes decomposition provides an auxiliary decoupled system whose solution processes are incorporated in approximating the solution processes of the original system     

Local times of linear multifractional stable sheets

Let X^H(u)(u)={X^H(u)(u);u∈R^N+}be linear multifractional stable sheets with index functional H(u),where H(u)=(H1(u),…,HN(u))is a function with values in(0;1)N.Based on some assumptions of H(u),we obtain the existence of the local times of X^H(u)(u)and establish its joint continuity and the Holder regularity.These results generalize the corresponding results about fractional stable sheets to multifractional stable sheets.     

Correlation functions of gauged linear σ-model

This is the second paper in a series following Tian and Xu(2015), on the construction of a mathematical theory of the gauged linear σ-model(GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.     

Invariant tori for asymptotically linear impact oscillators

The existence of invariant tori and quasi-periodic solutions for asymptotically linear impact oscillators is proved by using the successor map and some generalized versions of the Moser's twist theorem.     

Multisequences with large linear and k-error linear complexity from a tower of Artin–Schreier extensions of function fields

In this paper, we construct multisequences with both large (joint) linear complexity and k-error (joint) linear complexity from a tower of Artin–Schreier extensions of function fields. Moreover, these sequences can be explicitly constructed.