Two canonical passive state/signal shift realizations of passive discrete time behaviors

A discrete time invariant linear state/signal system Σ with a Hilbert state space and a Kren signal space has trajectories (x(),w()) that are solutions of the equation , where F is a bounded linear operator from into with a closed domain whose projection onto is all of . This system is passive if the graph of F is a maximal nonnegative subspace of the Kren space . The future behavior of a passive system Σ is the set of all signal components w() of trajectories (x(),w()) of Σ on with x(0)=0 and . This is always a maximal nonnegative shift-invariant subspace of the Kren space , i.e., the space endowed with the indefinite inner product inherited from . Subspaces of with this property are called passive future behaviors. In this work we study passive state/signal systems and passive behaviors (future, full, and past). In particular, we define and study the input and output maps of a passive state/signal system, and the past/future map of a passive behavior. We then turn to the inverse problem, and construct two passive state/signal realizations of a given passive future behavior , one of which is observable and backward conservative, and the other controllable and forward conservative. Both of these are canonical in the sense that they are uniquely determined by the given data , in contrast earlier realizations that depend not only on , but also on some arbitrarily chosen fundamental decomposition of the signal space . From our canonical realizations we are able to recover the two standard de Branges–Rovnyak input/state/output shift realizations of a given operator-valued Schur function in the unit disk.     

On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations–subrelations

It is shown that for the inclusion of factors corresponding to an inclusion of ergodic discrete measured equivalence relations , is normal in in the sense of Feldman–Sutherland–Zimmer [J. Feldman, C.E. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989) 239–269] if and only if A is generated by the normalizing groupoid of B. Moreover, we show that there exists the largest intermediate equivalence subrelation which contains as a normal subrelation. We further give a definition of “commensurability groupoid” as a generalization of normality. We show that the commensurability groupoid of B in A generates A if and only if the inclusion BA is discrete in the sense of Izumi–Longo–Popa [M. Izumi, R. Longo, S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998) 25–63]. We also show that there exists the largest equivalence subrelation such that the inclusion is discrete. It turns out that the intermediate equivalence subrelations and thus defined can be viewed as groupoid-theoretic counterparts of a normalizer subgroup and a commensurability subgroup in group theory.     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

Approximation of Sobolev classes by polynomials and ridge functions

Let be the usual Sobolev class of functions on the unit ball in , and be the subclass of all radial functions in . We show that for the classes and , the orders of best approximation by polynomials in coincide. We also obtain exact orders of best approximation in of the classes by ridge functions and, as an immediate consequence, we obtain the same orders in for the usual Sobolev classes .     

Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I

Let be the (2ν+1+l)-dimensional vector space over the finite field . In the paper we assume that is a finite field of characteristic 2, and the singular pseudo-symplectic groups of degree 2ν+1+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in and the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , and the characteristic polynomial of .     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

Formal extension and quotients of the invariant holonomic system

Let be a semisimple Lie algebra and a Cartan subalgebra of . Fix . Let be the invariant holonomic system (see [R. Hotta, M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984) 327–358]). First we investigate its formal extension . In the sequel we calculate the characteristic variety of some simple quotients of and its Fourier transform .     

On rank-one perturbations of normal operators

This paper is concerned with operators on Hilbert space of the form T=D+uv where D is a diagonalizable normal operator and uv is a rank-one operator. It is shown that if and the vectors u and v have Fourier coefficients and with respect to an orthonormal basis that diagonalizes D that satisfy , then T has a nontrivial hyperinvariant subspace. This partially answers an open question of at least 30 years duration.     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

On the sizes of Delaunay meshes

Let be a polyhedral domain occupying a convex volume. We prove that the size of a graded mesh of with bounded vertex degree is within a factor of the size of any Delaunay mesh of with bounded radius-edge ratio. The term depends on the geometry of and it is likely a small constant when the boundaries of are fine triangular meshes. There are several consequences. First, among all Delaunay meshes with bounded radius-edge ratio, those returned by Delaunay refinement algorithms have asymptotically optimal sizes. This is another advantage of meshing with Delaunay refinement algorithms. Second, if no input angle is acute, the minimum Delaunay mesh with bounded radius-edge ratio is not much smaller than any minimum mesh with aspect ratio bounded by a particular constant.     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces

By a recent result of M. De La Rosa and C. Read, there exist hypercyclic Banach space operators which do not satisfy the Hypercyclicity Criterion. In the present paper, we prove that such operators can be constructed on a large class of Banach spaces, including or .