Duality in reconstruction systems

We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.     

Homological projective duality

We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.     

Strict Mittag–Leffler Conditions on Gorenstein Modules

In this article, we investigate the relations between Gorenstein projective modules and Gorenstein flat modules in terms of strict Mittag–Leffler condition. We give some conditions under which Gorenstein projectives are Gorenstein flat, and discuss when the direct limits of Gorenstein projective modules are Gorenstein flat. Moreover, we study the dual of Gorenstein injective modules with strict Mittag–Leffler condition.     

State reconstruction of a distributed parameter system with exact observability

We study the problem of state reconstruction for exactly observable systems. Here we present two approaches for state reconstruction, the Riesz basis approach and the dual Hilbert uniqueness method (HUM) based on an iteration algorithm. The Riesz basis approach gives an explicit representation of the initial state of the system, whereas the HUM dual method gives an efficient approximation of the initial state.     

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld–Mukai bundles on smooth projective surfaces X using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld–Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of X corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld–Mukai bundles. This gives semistable Lazarsfeld–Mukai bundles on Kawamata covers of the projective plane and of certain K3 surfaces.     

Strong duals of projective limits of (LB)-spaces

We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called strong P-type. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.     

Dualizable Shearlet Frames and Sparse Approximation

Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proved to form frames for \(L^2({\mathbb {R}}^2)\). For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be stated in closed form and is efficiently computed. We also show that dualizable shearlet frames still provide near optimal sparse approximations of anisotropic features.     

On delta spaces satisfying Pasch's axiom

We consider partial linear spaces all of whose lines contain at least three points and in which every pair of intersecting lines generates a subspace isomorphic to a projective or dual affine plane. In particular, we classify in this paper those spaces that contain a projective plane.     

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.     

On tangential deformations of homogeneous polynomials

The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is indeed trivial, and show that the answer is no in a general situation. We also give a characterization of tangentially smoothable hypersurfaces with isolated singularities. Our results have applications in the local study of variations of projective hypersurfaces, complementing the global versions given by J. Carlson and P. Griffiths, R. Donagi and the author, and in the study of isotrivial linear systems on the projective space, showing that a general divisor does not belong to an isotrivial linear system of positive dimension.     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

Affine and Projective Tree Metric Theorems

The tree metric theorem provides a combinatorial four-point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A related weaker four point condition characterizes dissimilarity maps derived from circular split systems known as Kalmanson metrics. The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics, and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.     

Singularities of Dual Varieties in Characteristic 3

We investigate singularities of a general plane section of the dual variety of a smooth projective variety, or more generally, the discriminant variety associated with a linear system of divisors on a smooth projective variety. We show that, in characteristic 3, singular points of E ₆-type take the place of ordinary cusps in characteristic 0.     

Classification of Griesmer codes and dual transform

In this work, we consider a classification of infinite families of linear codes which achieve the Griesmer bound, using the projective dual transform. We investigate the correspondence between families of linear codes with given properties via dual transform.     

An Isometric Representation of the Dual of {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)}

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.     

Quantum differential systems and construction of rational structures

We consider mirror symmetry (A-side vs B-side, namely singularity side) in the framework of quantum differential systems. We focuse on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau–Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani’s results), depending on numerous combinatorial data which are rearranged after the mirror transformation.     

Hitchin System on Singular Curves

We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of SL(2) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.     

On chaos control and synchronization of the commensurate fractional order Liu system

In this work, we study chaos control and synchronization of the commensurate fractional order Liu system. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate fractional order systems are discussed. The existence and uniqueness of solutions for a class of commensurate fractional order Liu systems are investigated. We also obtain the necessary condition for the existence of chaotic attractors in the commensurate fractional order Liu system. The effect of fractional order on chaos control of this system is revealed by showing that the commensurate fractional order Liu system is controllable just in the fractional order case when using a specific choice of controllers. Moreover, we achieve chaos synchronization between the commensurate fractional order Liu system and its integer order counterpart via function projective synchronization. Numerical simulations are used to verify the analytical results.     

On Bessel systems in a Banach space

We consider Bessel systems in a Banach space and study their operator and projective characteristics. Using an operator construction for the continuation of the system, we prove an analog of Schur’s theorem on continuable systems.     

Poisson Brackets after Jacobi and Plücker

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.