SECOND-ORDER CONVERGENCE PROPERTIES OF TRUST-REGION METHODS USING INCOMPLETE CURVATURE INFORMATION, WITH AN APPLICATION TO MULTIGRID OPTIMIZATION

Convergence properties of trust-region methods for unconstrained nonconvex optimiza-tion is considered in the case where information on the objective function's local curvatureis incomplete,in the sense that it may be restricted to a fixed set of "test directions"and may not be available at every iteration.It is shown that convergence to local "weak"minimizers can still be obtained under some additional but algorithmically realistic condi-tions.These theoretical results are then applied to recursive multigrid trust-region meth-ods,which suggests a new class of algorithms with guaranteed second-order convergenceproperties.     

THE GAUSS MAP OF TIMELIKE SURFACES IN R_1~n

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Risk measurement in the presence of background risk

A distortion-type risk measure is constructed, which evaluates the risk of any uncertain position in the context of a portfolio that contains that position and a fixed background risk. The risk measure can also be used to assess the performance of individual risks within a portfolio, allowing for the portfolio’s re-balancing, an area where standard capital allocation methods fail. It is shown that the properties of the risk measure depart from those of coherent distortion measures. In particular, it is shown that the presence of background risk makes risk measurement sensitive to the scale and aggregation of risk. The case of risks following elliptical distributions is examined in more detail and precise characterisations of the risk measure’s aggregation properties are obtained.     

A natural modification of a random process and its application to stochastic functional series and Gaussian measures

It is known that a Gaussian stochastic process can be expanded in a functional series with random independent coefficients. In the case where the process is continuous in mean but there exists no modification of it with continuous simple functions, the series does not converge uniformly. In what cases does it converge pointwise? This question reduces to the well-studied problem of the boundedness of the sample functions. It is shown that the pointwise convergence of the expansion mentioned above is equivalent to the continuity of the sample functions of the process in a certain separable metric. Some other properties of Gaussian processes and measures are considered, ansd generalizations to the non-Gaussian case are given.     

Contact problems for bodies with complex coatings

The paper deals with various statements and mathematical models of contact and contact-wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.     

Influence of transfer,release, and absorption of energy on vortex properties of ideal fluids

The vortex properties of ideal fluid flows are examined. It is shown that the barotropy assumption in classical theorems of the conservation of vortex properties of a fluid can be replaced by the adiabaticity assumption. In that case, the connection between the formation and evolution of anomalous atmospheric phenomena and significant violations of adiabaticity in the movement of air masses can be traced.     

Risk-sensitive capacity control in revenue management

Both the static and the dynamic single-leg revenue management problem are studied from the perspective of a risk-averse decision maker. Structural results well-known from the risk-neutral case are extended to the risk-averse case on the basis of an exponential utility function. In particular, using the closure properties of log-convex functions, it is shown that an optimal booking policy can be characterized by protection levels, depending on the actual booking class and the remaining time. Moreover, monotonicity of the protection levels with respect to the booking class and the remaining time are proven.     

Derived and integrated sets of simple sets of polynomials in two complex variables

In the present paper, we illustrate the contrast, concerning the effectiveness properties of the derived sets of simple sets of polynomials, between the single- and the two-variable cases. Moreover, a positive result is established for the relationship between the Cannon functions of simple sets of polynomials in two complex variables and those of the derived sets. Finally, it is shown that concerning the effectiveness of the integrated sets, the result of the single-variable case can be extended, without undue difficulty, to two variables.     

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.     

Log-linear Poisson autoregression

We consider a log-linear model for time series of counts. This type of model provides a framework where both negative and positive association can be taken into account. In addition time dependent covariates are accommodated in a straightforward way. We study its probabilistic properties and maximum likelihood estimation. It is shown that a perturbed version of the process is geometrically ergodic, and, under some conditions, it approaches the non-perturbed version. In addition, it is proved that the maximum likelihood estimator of the vector of unknown parameters is asymptotically normal with a covariance matrix that can be consistently estimated. The results are based on minimal assumptions and can be extended to the case of log-linear regression with continuous exogenous variables. The theory is applied to aggregated financial transaction time series. In particular, we discover positive association between the number of transactions and the volatility process of a certain stock.     

Basic definition and properties of Bessel multipliers

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a special case the multipliers for Riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted. Finally the continuous dependence of a Bessel multiplier on the parameters (i.e., the involved sequences and the symbol in use) is verified, using a special measure of similarity of sequences.     

Atomic and Subatomic Decompositions in Anisotropic Function Spaces

This work deals with decompositions in anisotropic function spaces. Defining anisotropic atoms as smooth building blocks which are the counterpart of the atoms from the works of M. Frazier and B. Jawerth , it is shown that the study of anisotropic function spaces can be done with the help of some sequence spaces in a similar way as it is done in the isotropic case. It is also shown that the subatomic decomposition theorem for isotropic function spaces, recently proved by H. Triebel , can be extended to the anisotropic case if the mean smoothness parameter is sufficiently large.     

Information in Quantal Response Data and Random Censoring

In this paper we study interesting properties of Fisher and divergence type measures of information for quantal, complete and incomplete random censoring, and not censoring at all. It is shown that, while quantal random censoring is less expensive, it is less informative than complete random censoring. It is also shown that in experiments which are mixtures of quantal and complete random censoring, the information received from these experiments is a convex combination of quantal information and the information in complete random censoring. Finally, the "total information" property is studied, in which the information received by the uncensored experiment can be expressed as the sum of the information provided by random censoring and the loss of information due to censoring. The results for Fisher's measure of information are an extension of already known results to the multiparameter case. The investigation of the previous information properties for divergence type measures is a new element, as well as the comparison of byproducts of Fisher information matrices.     

On the Variational Approach to Systems of Quasilinear Conservation Laws

The paper contains results concerning the development of a new approach to the proof of existence theorems for generalized solutions to systems of quasilinear conservation laws. This approach is based on reducing the search for a generalized solution to analyzing extremal properties of a certain set of functionals and is referred to as a variational approach. The definition of a generalized solution can be naturally reformulated in terms of the existence of critical points for a set of functionals, which is convenient within the approach proposed. The variational representation of generalized solutions, which was earlier known for Hopf-type equations, is generalized to systems of quasilinear conservation laws. The extremal properties of the functionals corresponding to systems of conservation laws are described within the variational approach, and a strategy for proving the existence theorem is outlined. In conclusion, it is shown that the variational approach can be generalized to the two-dimensional case.     

Pitman closeness of record values to population quantiles

In this paper, we examine the Pitman closeness of record statistics to the population quantiles of a location-scale family of distributions and study its monotonicity properties. Even though in general it depends on the parent distribution, exact expressions are derived for the required probabilities in the case of Uniform (−1,1) and exponential distributions. For the population median, it is shown that the first upper record is the Pitman-closest among all upper record values. Moreover, for the population median, in the case of symmetric distributions, the Pitman closeness probabilities of records are shown to be distribution-free and explicit expressions are also derived for these probabilities.     

(Some of) the many uses of Eulerian graphs in graph theory (plus some applications)

In this survey type article, various connections between eulerian graphs and other graph properties such as being hamiltonian, nowhere-zero flows, the cycle-plus-triangles problem and problems derived from it, are demonstrated. It is also shown how compatible cycle decompositions can be used to construct loopless 4-regular graphs having precisely one hamiltonian cycle, or to prove the equivalence between the Chinese Postman Problem and the Minimum Cycle Covering Problem in the planar bridgeless case.     

Visualization of network structure by the application of hypernodes

In the literature several authors describe methods to construct simplified models of networks. These methods are motivated by the need to gain insight into the main properties of medium sized or large networks. The present paper contributes to this research by setting focus on weighted networks, the geographical component of networks and introducing a class of functions to model how the weights propagate from one level of abstraction to the next. Hierarchies of network models can be constructed from reordering of the adjacency matrix of the network; this is how “hypernodes” are derived in the present paper. The hypernode algorithm is explored and it is shown how it can be formulated to handle weighted networks. Weighted networks enable handling the uncertainty or the strength of the components which make up the network. The hypernode algorithm can be run in an iterative manner so that a hierarchy of simplified models of the network can be derived. Some case studies demonstrate the hypernode algorithm. In the first case the algorithm is compared with a similar implementation described in the literature. In the second case an airline dataset is analysed. This study shows that when networks are embedded in the geographical space hypernodes may relate to clusters in the spatial domain. The selection of the visual variables to illustrate the strength of the edges and nodes in a weighted network is discussed.     

Estimation of Binary Markov Random Fields Using Markov chain Monte Carlo

This article compares three binary Markov random fields (MRFs) which are popular Bayesian priors for spatial smoothing. These are the Ising prior and two priors based on latent Gaussian MRFs. Concern is given to the selection of a suitable Markov chain Monte Carlo (MCMC) sampling scheme for each prior. The properties of the three priors and sampling schemes are investigated in the context of three empirical examples. The first is a simulated dataset, the second involves a confocal fluorescence microscopy dataset, while the third is based on the analysis of functional magnetic resonance imaging (fMRI) data. In the case of the Ising prior, single site and multi-site Swendsen-Wang sampling schemes are both considered. The single site scheme is shown to work consistently well, while it is shown that the Swendsen-Wang algorithm can have convergence problems. The sampling schemes for the priors are extended to generate the smoothing parameters, so that estimation becomes fully automatic. Although this works well, it is found that for highly contiguous images fixing smoothing parameters to very high values can improve results by injecting additional prior information concerning the level of contiguity in the image. The relative properties of the three binary MRFs are investigated, and it is shown how the Ising prior in particular defines sharp edges and encourages clustering. In addition, one of the latent Gaussian MRF priors is shown to be unable to distinguish between higher levels of smoothing. In the context of the fMRI example we also undertake a simulation study.     

The basic cyclic scheduling problem with deadlines

The purpose of this paper is to study the latest schedule existence, calculation and properties of a basic cyclic scheduling problem with deadlines. First it is shown that, in the general case, a latest schedule exists but may be difficult to compute. Then we focus on a special case we call the optimal cyclic production problem. We derive an upper bound for the number of maximal-path values needed to compute the latest starting times and show the K-periodic structure of the latest starting time sequences.