Quantifying Dynamical Predictability： the Pseudo-Ensemble Approach

The ensemble technique has been widely used in numerical weather prediction and extended-range forecasting. Current approaches to evaluate the predictability using the ensemble technique can be divided into two major groups. One is dynamical, including generating Lyapunov vectors, bred vectors, and singular vectors, sampling the fastest error-growing directions of the phase space, and examining the dependence of prediction efficiency on ensemble size. The other is statistical, including distributional analysis and quantifying prediction utility by the Shannon entropy and the relative entropy. Currently, with simple models, one could choose as many ensembles as possible, with each ensemble containing a large number of members. When the forecast models become increasingly complicated, however, one would only be able to afford a small number of ensembles, each with limited number of members, thus sacrificing estimation accuracy of the forecast errors. To uncover connections between different information theoretic approaches and between dynamical and statistical approaches, we propose an （ε, T）-entropy and scale-dependent Lyapunov exponent--based general theoretical framework to quantify information loss in ensemble forecasting. More importantly, to tremendously expedite computations, reduce data storage, and improve forecasting accuracy, we propose a technique for constructing a large number of ＂pseudo＂ ensembles from one single solution or scalar dataset. This pseudo-ensemble technique appears to be applicable under rather general conditions, one important situation being that observational data are available but the exact dynamical model is unknown.     

Noncommutative Dynamical Models with Quantum Symmetries

We define dynamical models on the q-Minkowski space algebra (which is a particular case of the Reflection Equation Algebra) as deformations (quantizations) of dynamical models with rotational symmetries, and we find their integrals. In particular, we introduce a q-analog of the Runge-Lenz vector and a q-analog of the dynamics in space-time with a spherically symmetric metric.     

A Mechanical Approach to the Dynamical Behavior of Non-Autonomous Van der Pol Equation

A mechanical reasoning method,based on Ritt-Wu’s method,for analytic equa-lities is developed to obtain a result on the Lyapunov characteristic exponent(LCE)ofthe following non-autonomous Van der Pol system     

Spiral Anchoring in Media with Multiple Inhomogeneities: A Dynamical System Approach

The spiral is one of nature’s more ubiquitous shapes: It can be seen in various media, from galactic geometry to cardiac tissue. Mathematically, spiral waves arise as solutions to reaction–diffusion partial differential equations (RDS). In the literature, various experimentally observed dynamical states and bifurcations of spiral waves have been explained using the underlying Euclidean symmetry of the RDS—see for example (Barkley in Phys. Rev. Lett. 68:2090–2093, 1992; Phys. Rev. Lett. 76:164–167, 1994; Sandstede et al. in C. R. Acad. Sci. 324:153–158, 1997; J. Differ. Equ. 141:122–149, 1997; J. Nonlinear Sci. 9:439–478, 1999), or additionally using the concept of forced Euclidean symmetry-breaking for situations where an inhomogeneity or anisotropy is present—see (LeBlanc in Nonlinearity 15:1179–1203, 2002; LeBlanc and Wulff in J. Nonlinear Sci. 10:569–601, 2000). In this paper, we further investigate the role of medium inhomogeneities on spiral wave dynamics by considering the effects of several localized sites of inhomogeneity. Using a model-independent approach based on n>1 simultaneous translational symmetry-breaking perturbations of the dynamics near rotating waves, we fully characterize the local anchoring behavior of the spiral wave in the n-dimensional parameter space of relative “amplitudes” of the individual perturbations. For the case n=2, we supplement the local anchoring results with a classification of the generic one-parameter bifurcation diagrams of anchored states which can be obtained by circling the origin of the two-dimensional amplitude parameter space. Numerical examples are given to illustrate our various results.     

Models of stochastic geometry — A survey

This paper discusses some models of stochastic geometry which are of potential interest for operations research. These are the Boolean model, a certain model for random compact sets and marked point processes. The Boolean model is a generalization of the well-known queueing systemM/G/. The random compact set model may be useful for modelling spatial spreading processes such as fires, cancers or holes in the Earth's surface. Marked point processes are used here as models of forests and used for a statistical study of the spatial distribution of damaged trees.Extended version of an Invited Lecture on the 16th Symposium for OR in Hamburg 1992.     

KKM—A Topological Approach For Trees

The Knaster–Kuratowski–Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this paper we introduce a version of the KKM theorem for trees and use it to prove several combinatorial theorems.A 2-tree hypergraph is a family of nonempty subsets of T R (where T and R are trees), each of which has a connected intersection with T and with R. A homogeneous 2-tree hypergraph is a family of subsets of T each of which is the union of two connected sets.For each such hypergraph H we denote by (H) the minimal cardinality of a set intersecting all sets in the hypergraph and by (H) the maximal number of disjoint sets in it.In this paper we prove that in a 2-tree hypergraph (H)2(H) and in a homogeneous 2-tree hypergraph (H)3(H). This improves the result of Alon [3], that (H)8(H) in both cases.Similar results are proved for d-tree hypergraphs and homogeneous d-tree hypergraphs, which are defined in a similar way. All the results improve the results of Alon [3] and generalize the results of Kaiser [1] for intervals.     

Oracle Inequality for Sparse Trace Regression Models with Exponential β-mixing Errors

In applications involving,e.g.,panel data,images,genomics microarrays,etc.,trace regression models are useful tools.To address the high-dimensional issue of these applications,it is common to assume some sparsity property.For the case of the parameter matrix being simultaneously low rank and elements-wise sparse,we estimate the parameter matrix through the least-squares approach with the composite penalty combining the nuclear norm and the l₁ norm.We extend the existing analysis of th...     

A Frisch-Newton Algorithm for Sparse Quantile Regression

Recent experience has shown that interior-point methods using a log barrier approach are far superior to classical simplex methods for computing solutions to large parametric quantile regression problems. In many large empirical applications, the design matrix has a very sparse structure. A typical example is the classical fixed-effect model for panel data where the parametric dimension of the model can be quite large, but the number of non-zero elements is quite small. Adopting recent developments in sparse linear algebra we introduce a modified version of the Prisch-Newton algorithm for quantile regression described in Portnoy and Koenker~([28]). The new algorithm substantially reduces the storage (memory) requirements and increases computational speed. The modified algorithm also facilitates the development of nonparametric quantile regression methods. The pseudo design matrices employed in nonparametric quantile regression smoothing are inherently sparse in both the fidelity and roughness penalty components. Exploiting the sparse structure of these problems opens up a whole range of new possibilities for multivariate smoothing on large data sets via ANOVA-type decomposition and partial linear models.     

A Kind of Nonconvex—valued Functional Differential Inclusions Problems

51.IntroductionLetXbeaBanachspaceandI=[O,7'](7'>O),L=[-a,Oj,J=[-a,7']bethreeinter-valsofR.WesetF:IXC[L;Xj-2"\diisaset-valuesmap.Weintroducethetimelagopera-tor7'.):CLJ;Xj-CLLiXj,definedby7'(t)x(s)=X(t s)foreveryseL-LetusassumethatA:Dom(A)-2"\diisaset-values))l-accretiveoperator.Inthispaper,w3discusstheini-tialvalueprob1emsasfoIlows:Sincethedevelopmentoftl1eControlTheoryandEconomics,theinitialva1ueproblem(1.1)havebeendiscussinginrecentyears,someimportantresultshavebeenobtainecl,for…     

Dynamical systems — Past and present

Plenary lecture at the International Congress of Mathematicians (Berlin 1998, August 18–27). Reprinted from Doc. Math. J., DMV, Extra Volume ICM I, 1998, pp. 381–402. ©J.Moser, 1998.     

A Dynamical model of the atmospheric turbulencefromtheunstablestratificationtothestablestratification

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Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach

There is ample evidence that in applications of self-exciting point-process models, the intensity of background events is often far from constant. If a constant background is imposed that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modeling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this article, we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam’s razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy, in cases where there is not sufficient empirical evidence to suggest that the background intensity function is more complex than those models. This approach is seldom, if ever, used for nonparametric function estimation in other settings, not least because in those cases more data are typically available. However, our “ontological parsimony” argument is appropriate in the context of self-exciting point-process models. Supplementary materials are available online.     

A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f1 （2x ＋ y） ＋ f2（2x - y） = f3（x ＋ y） ＋ f4（x - y） ＋ f5（x） without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 ： R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f（2x ＋ y） ＋ f（2x - y） = g（x ＋ y） ＋ g（x - y） ＋ h（x） and f（2x ＋ y） - f（2x - y） = g（x ＋ y） - g（x - y）. The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.     

On Knowing How I Feel About That—A Process-Reliabilist Approach

Human subjects seem to have a type of introspective access to their mental states that allows them to immediately judge the types and intensities of their occurrent emotions, as well as what those emotions are about or “directed at”. Such judgments manifest what I call “emotion-direction beliefs”, which, if reliably produced, may constitute emotion-direction knowledge. Many psychologists have argued that the “directed emotions” such beliefs represent have a componential structure, one that includes feelings of emotional responses and related but independent representations of what those feelings are about. I argue that such componentiality may help to explain how emotion-direction knowledge is achievable. I begin by developing a hybrid view of introspection that combines David Chalmers’ phenomenal realism with Alvin Goldman’s “partial redeployment” account of meta-belief content. I then provide a process-reliabilist account of introspectively gained emotion-direction knowledge that outlines the minimum conditions of reliably forming emotion-direction beliefs, and specifies several ways in which the warrant of such beliefs could be defeated by relevant counterfactual alternatives. The overall account suggests how distinct introspective processes might be epistemically synergistic.     

A THEORY FOR MARSHALLING RALLWAY CARS INTO A TRAIN——An Approach to a Multiple Objective Problem

About twefity years ago, on some marshalling yards in China, an empirical method, alled the Tabulation Method, was used to make feasible and effective plans for marshalling railway cars into a train, especially, into a local train. Theoretical studies of this method developed some mathematical models of the nature of of combinatorial optimization. The main part of this paper is a survey of a mathematical theory, called the Theory of Standard Numeration. When there is one loco (there are two locos) in operation on the marshalling yard (one on each side of the yard), this theory says that every feasible plan for making up a local train can be characterized by a finite sequence of binary(ternary) numbers which are called the characterlshc numbers of the cars. If the minimum value of the number of ordered subsequences in a standard partition of the initial train is n (which can easily be obtained by the Tabulation Method)and 2~(m-2)相似文献   

Stochastic Differentiation – A Generalized Approach

The space (D ^*) of Wiener distributions allows a natural Pettis-type stochastic calculus. For a certain class of generalized multiparameter processes X: R ^N (D ^*) we prove several differentiation rules (Itô formulas); these processes can be anticipating. We then apply these rules to some examples of square integrable Wiener functionals and look at the integral versions of the resulting formulas.     

Estimates for Schur Multipliers and Double Operator Integrals—A Wavelet Approach

Functional Analysis and Its Applications - We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in...