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The paper presents a novel model order reduction technique for large-scale linear parameter varying (LPV) systems. The approach is based on decoupling the original dynamics into smaller dimensional LPV subsystems that can be independently reduced by parameter varying reduction methods. The decomposition starts with the construction of a modal transformation that separates the modal subsystems. Hierarchical clustering is applied then to collect the dynamically similar modal subsystems into larger groups. The subsystems formed from the groups are then independently reduced. This approach substantially differs from most of the previously proposed LPV model reduction techniques, since it performs manipulations on the LPV model and not on a set of linear time-invariant (LTI) models defined at fixed scheduling parameter values. Therefore the model interpolation, which is the most challenging part of most reduction techniques, is avoided. The applicability of the developed algorithm is thoroughly investigated and demonstrated by numerical case studies., Comment: 26 pages, 11 figures, 1 table

Berwald metrics are particular Finsler metrics which still have linear Berwald connections. Their complete classification is established in an earlier work, [Sz1], of this author. The main tools in these classification are the Simons-Berger holonomy theorem and the Weyl-group theory. It turnes out that any Berwald metric is a perturbed-Cartesian product of Riemannian, Minkowski, and such non-Riemannian metrics which can be constructed on irreducible symmetric manifolds of $rank > 1$. The existence of these metrics are well established by the above theories. The present paper has several new features. First, the Finsler functions of Berwald manifolds are explicitly described by the Chevalley polynomials. New results are also the complete lists of reversible (d(x,y)=d(y,x)) resp. irreversible ($d(x,y)\not =d(y,x)$) Berwald metrics. The Cartan symmetric Finsler manifolds are also completely determined. The paper is concluded by proving that a Berwald metric is uniquely determined by the Minkowski metric induced on an arbitrarily fixed maximal totalgeodesic flat submanifold (Cartan flat). Moreover, any two Cartan flats are isometric., Comment: 28 pages, submitted, minor revisions were implemented in Section 5

The intertwining operator constructed in [Sz1,Sz2] does not appear in the right form. It is established there by using only the anticommutators. The correct operator must involve all endomorphisms, which are unified by the Z-Fourier transform. Although some of the correct elements of the previous constructions are kept, this idea is established by a new technique which yields the various isospectrality theorems stated in the papers on a much larger scale. The new results include new isospectrality examples living on sphere$X$ball- and sphere$X$sphere-type manifolds. Among them, there are such discrete isospectrality families where one of the members is homogeneous while the others are locally inhomogeneous (striking examples). Furthermore, a large class of new isospectrality families are constructed by $\sigma$ deformations., Comment: 43 pages. Completely new version. The constructions developed in the previous version are moved to a new paper

Emergent behaviors are in the focus of recent research interest. It is then of considerable importance to investigate what optimizations suit the learning and prediction of chaotic systems, the putative candidates for emergence. We have compared L1 and L2 regularizations on predicting chaotic time series using linear recurrent neural networks. The internal representation and the weights of the networks were optimized in a unifying framework. Computational tests on different problems indicate considerable advantages for the L1 regularization: It had considerably better learning time and better interpolating capabilities. We shall argue that optimization viewed as a maximum likelihood estimation justifies our results, because L1 regularization fits heavy-tailed distributions -- an apparently general feature of emergent systems -- better., Comment: 13 pages, 4 figures

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such three-manifolds., Comment: 35 pages, 2 figures

This article concludes the comprehensive study started in [Sz5], where the first non-trivial isospectral pairs of metrics are constructed on balls and spheres. These investigations incorporate 4 different cases since these balls and spheres are considered both on 2-step nilpotent Lie groups and on their solvable extensions. In [Sz5] the considerations are completely concluded in the ball-case and in the nilpotent-case. The other cases were mostly outlined. In this paper the isospectrality theorems are completely established on spheres. Also the important details required about the solvable extensions are concluded in this paper. A new so called anticommutator technique is developed for these constructions. This tool is completely different from the other methods applied in the field so far. It brought a wide range of new isospectrality examples. Those constructed on geodesic spheres of certain harmonic manifolds are particularly striking. One of these spheres is homogeneous (since it is the geodesic sphere of a 2-point homogeneous space) while the other spheres, although isospectral to the previous one, are not even locally homogeneous. This shows how little information is encoded about the geometry (for instance, about the isometries) in the spectrum of Laplacian acting on functions., Comment: The paper has been completely revised. Accepted for publication in Annals of Mathematics

The first isospectral pairs of metrics are constructed on balls and spheres. This long standing problem, concerning the existence of such pairs, has been solved by a new method called "Anticommutator Technique." Among the wide range of such pairs, the most striking examples are provided on (4k-1)-dimensional spheres, where k > 2. One of these metrics is homogeneous (since it is the metric on the geodesic sphere of a 2-point homogeneous space), while the other is locally inhomogeneous. These examples demonstrate the surprising fact that no information about the isometries is encoded in the spectrum of Laplacian acting on functions. In other words, "The group of isometries, even the local homogeneity property, is lost to the "Non-Audible" in the debate of "Audible versus Non-Audible Geometry".", Comment: 43 pages. After retrieving source, read README file or type tex whole to typeset (in Unix)