Local search of orthogonal packings using the lower bounds

Consideration was given to the problem of packing the rectangular half-infinite strip. It was suggested to solve it using a one-point evolutionary algorithm with search of the best solution in the neighborhood of the local lower bound. Proposed was an algorithm to construct this neighborhood including the packings obtained by solving a special problem of one-dimensional cutting. Its solution was shown to be the local lower bound in the considered neighborhood. An improved global lower bound was proposed. The results of numerical modeling were presented. The record value obtained was compared with the global bounds.     

Nexus VII: Relationships Between Architecture and Mathematics

Sylvie Duvernoy reports on the seventh international, interdisciplinary Nexus conference for architecture and mathematics. Point Loma Nazarene University, San Diego, USA 23-25 June 2008     

Gaining insight through case-based explanation

Traditional explanation strategies in machine learning have been dominated by rule and decision tree based approaches. Case-based explanations represent an alternative approach which has inherent advantages in terms of transparency and user acceptability. Case-based explanations are based on a strategy of presenting similar past examples in support of and as justification for recommendations made. The traditional approach to such explanations, of simply supplying the nearest neighbour as an explanation, has been found to have shortcomings. Cases should be selected based on their utility in forming useful explanations. However, the relevance of the explanation case may not be clear to the end user as it is retrieved using domain knowledge which they themselves may not have. In this paper the focus is on a knowledge-light approach to case-based explanations that works by selecting cases based on explanation utility and offering insights into the effects of feature-value differences. In this paper we examine to two such a knowledge-light frameworks for case-based explanation. We look at explanation oriented retrieval (EOR) a strategy which explicitly models explanation utility and also at the knowledge-light explanation framework (KLEF) that uses local logistic regression to support case-based explanation.

Partial orders and Jordan normal form

Consideration is given to the relation between the structure of the acyclic binary relation and the adjacency matrix of its corresponding graph. In this case, the existing methods for studying the binary relations and their corresponding graphs in terms of the spectrum, that is, the set of eigenvalues, of the adjacency matrix are inapplicable because for the acyclic relations this matrix is nilpotent and its spectrum is identically zero. Therefore, a more refined characteristic of the matrix is required. The present paper considers the Jordan normal form (JNF) as such.     

Inverse NP Problems

The inverse problem relative to a verifier V of proofs of membership for a NP language is the problem of deciding, given a set π of proofs, whether or not there exists a string x having exactly π as its set of proofs. In this paper, we study the complexity of inverse problems. We develop a new notion of reduction which allows one to compare the complexity of inverse problems. Using this notion, we classify as coNP-complete the inverse problems for the “natural” verifiers of many NP-complete problems. We also show that the inverse complexity of a verifier for a language L cannot be predicted solely from the complexity of L, but rather, is highly dependent upon the choice of verifier used to accept L. In this context, a verifier with a Σ₂ ^p -complete inverse problem is exhibited, giving a new and natural example of a Σ₂ ^p -complete problem.      

Quasi-invariant control: Design and functional capabilities

The questions of physical realizability, stability, and control errors were considered by the example of a simplest mathematical model of the linear minimum phase system of quasi-invariant control. The problem of designing the quasi-invariant control was solved on this basis, thus providing the opportunity to extend control by designing a nonlinear system combining the quasi-invariant and classical control strategies.     

Invariance and astatism in systems not measuring perturbations

Control systems not being able to measure perturbations are considered. A necessary and sufficient condition for the astatism with respect to external perturbations to be possible is formulated basing on the comparison of absolute invariance and astatism. An unconventional method to ensure astatism implementing jet pulse equipment is analysed.     

Anisotropic magnetoresistive sensors of the magnetic field and current

Results are presented of the work in the domain of investigation and production of thin-filmed anisotropic magnetoresistive sensors (AMRSs) of the magnetic field and current on the basis of the single-layer or the double-layer metallic ferromagnetic nanostructure with a thickness of the magnetic film of 12–25 nm. The problems of design of the AMRS structures and their efficiency are considered with the investigation of the basic characteristics and technology of their production.     

Feedback equivalence and global stabilization of nonstationary discrte systems

A problem is considered for the global stabilization of nonstationary nonlinear discrete systems without outputs. Here, the key instant is the establishment of equivalence of an arbitrary discrete system and a lossless system with the use of the appropriate feedback. The construction of control laws is performed on the basis of new methods for investigating the stability of nonautonomous systems of difference equations, using the theory of limit systems and Lyapunov limit functions.     

Local and semiglobal stabilization in a cascade with delay

Consideration is given to the problem of stability and stabilization for a system of differential equations with delay of a “cascade” structure. Sufficient conditions for local asymptotic stability as well as “semiglobal” stabilization for a cascade with a linear subsystem are obtained.     

Systems with a switch and estimation of absolute stability region

Two methods are proposed for estimation of absolute stability region in the space of parameters. The first method uses nontrivial necessary conditions, which follow from stability of switching systems. The second method is a modification of the harmonic balance approach; it is used for approximate determination of existence of periodic modes that arise when stability is lost.     

Attracting ellipsoids and synthesis of oscillatory regimes

An approach to stabilization of nonlinear oscillations in multidimensional spaces is proposed on the basis of the V.I. Zubov’s stability theory for invariant sets. As a special case, the derived controls make it possible to excite self-oscillating regimes in specified state subspaces R ^2k ? R ²ⁿ with simultaneous oscillation damping on Cartesian products R ^2n?2k .     

Comparison of the quantile and guaranteeing approaches to system analysis

The worst value of the quantile of the distribution of the linear loss function which depends on the uncertain stochastic parameters was compared with the maximum value of this function. The stochastic uncertainty is modelled by distributions from the Barmish class.     

Exponential queuing system with negative customers and bunker for ousted customers

Consideration was given to the queuing system with Poisson flows of incoming positive and negative customers. For the positive customers, there is an infinite-capacity buffer. The arriving negative customer knocks out a positive customer queued in the buffer and moves it to an infinite-capacity buffer of ousted customers (bunker). If the buffer is empty, then the negative customer discharges the system without affecting it. After servicing the current customer, the server receives a customer from the buffer or, if the buffer is empty, the bunker. The customers arriving from both the buffer and bunker are distributed exponentially with the same parameter. Relations for calculation of the stationary distributions of the queues in the buffer and bunker were obtained.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.