Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Under the appropriate definition of sampling density D_ϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D_ϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B _1/2 . If the shift invariant space consists of polynomial splines, then we show that D_ϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B _1/2 .     

Characterizations for Besov spaces and applications. Part I

The main theorem of this paper gives a characterization for holomorphic Besov space B^p(D) over a large class of bounded domains D in , which states that there is a bounded linear operator so that PV_D=I on B^p(D), where P is the Bergman projection, and is the biholomorphic invariant measure with K(z,z) being Bergman kernel function for D. Moreover, some application for characterizing Schatter von Neumann p-class small Hankel operation is given as a direct consequence of this theorem.     

Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D. The research of this author is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

Limit points of eigenvalues of (di)graphs

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph D, the set of limit points of eigenvalues of iterated subdivision digraphs of D is the unit circle in the complex plane if and only if D has a directed cycle. 3. Every limit point of eigenvalues of a set D of digraphs (graphs) is a limit point of eigenvalues of a set of bipartite digraphs (graphs), where consists of the double covers of the members in D. 4. Every limit point of eigenvalues of a set D of digraphs is a limit point of eigenvalues of line digraphs of the digraphs in D. 5. If M is a limit point of the largest eigenvalues of graphs, then −M is a limit point of the smallest eigenvalues of graphs.     

Restricted Symmetric Permutations

Pattern-avoiding involutions, which have received much enumerative attention, are pattern-avoiding permutations which are invariant under the natural action of a certain subgroup of D ₈, the symmetry group of a square. Three other nontrivial subgroups of D ₈ also have invariant permutations under this action. For each of these subgroups, we enumerate the set of permutations which are invariant under the action of the subgroup and which also avoid a given set of forbidden patterns. The sets of forbidden patterns we consider include all subsets of S ₃. For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations. Received September 14, 2006     

Transgression of the index gerbe

 The holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection on B is a line bundle with connection over LB. Given a family of graded Dirac operators on B and some additional geometric data one can define the determinant line bundle with Quillen metric and Bismut-Freed connection. According to Witten, Bismut-Freed the holonomy of this determinant bundle can be expressed in terms of an adiabatic limit of eta invariants of an associated family of Dirac operators over LB. Recently, for a family of ungraded Dirac operators on B. Lott constructed an index gerbe with connection. In the present paper we show, in analogy to the holonomy formula for the determinant bundle, that the holonomy of the index gerbe coincides with an adiabatic limit of determinant bundles of the associated family of Dirac operators over LB. Received: 17 October 2001 / Revised version: 5 August 2002     

Über Bewertungen,welche unter einer reduktiven Gruppe invariant sind

Summary LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toK ^U . Then we study the set of invariant valuations having some fixed restrictionv ₀, toK ^B . Ifv ₀ is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupW ^X . Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofK ^B .

Unterstützt durch den Schweizerischen Nationalfonds zur Förderung der wissenschaftlichen Forschung.     

Constructing Covering Codes with Given Automorphisms

Let K(n,r) denote the minimum cardinality of a binary covering code of length n and covering radius r. Constructions of covering codes give upper bounds on K(n,r). It is here shown how computer searches for covering codes can be sped up by requiring that the code has a given (not necessarily full) automorphism group. Tabu search is used to find orbits of words that lead to a covering. In particular, a code D found which proves K(13,1) 704, a new record. A direct construction of D given, and its full automorphism group is shown to be the general linear group GL(3,3). It is proved that D is a perfect dominating set (each word not in D is covered by exactly one word in D) and is a counterexample to the recent Uniformity Conjecture of Weichsel.     

On the exact constant in the quantitative Steinitz theorem in the plane

We determine the maximal value ofr with the following property. If the convex hull of a set inR ² contains a unit circleB, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radiusr concentric withB. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribed aroundB. Imre Bárány was partially supported by Hungarian National Science Foundation Grant Nos. 1907 and 1909. Aladár Heppes was partially supported by Hungarian National Science Foundation Grant No. 2583.     

On the best interval quadrature formulae for classes of differentiable periodic functions

In this paper we solve the problem about optimal interval quadrature formula for the class W^rF of differentiable periodic functions with rearrangement invariant set F of their derivatives of order r. We prove that the formula with equal coefficients and n node intervals having equidistant midpoints is optimal for considering classes. To this end a sharp inequality for antiderivatives of rearrangements of averaged monosplines is proved.     

On Brauer’s <Emphasis Type="Italic">k</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)-problem for blocks with metacyclic defect groups of odd order

Let G be a finite group, and suppose that B is a p-block of G with defect group D. Let k(B) denote the number of ordinary irreducible characters in B. It was conjectured by Brauer that k(B) ≤ |D|. In this paper, we will prove Brauer’s inequality in the case that D is metacyclic and p is odd.     

On smoothness and invariance properties of the gauss-newton method

We consider systems of m nonlinear equations in m + p unknowns which have p-dimensional solution manifolds. It is well-known that the Gauss-Newton method converges locally and quadratically to regular points on this manifold. We investigate in detail the mapping which transfers the starting point to its limit on the manifold. This mapping is shown to be smooth of one order less than the given system. Moreover, we find that the Gauss-Newton method induces a foliation of the neighborhood of the manifold into smooth submanifolds. These submanifolds are of dimension m, they are invariant under the Gauss-Newton iteration, and they have orthogonal intersections with the solution manifold.     

Lie Triple Derivable Mappings on Rings

Let ? be a ring containing a nontrivial idempotent. In this article, under a mild condition on ?, we prove that if δ is a Lie triple derivable mapping from ? into ?, then there exists a Z _{A, B} (depending on A and B) in its centre 𝒵(?) such that δ(A + B) = δ(A) + δ(B) + Z _{A, B} . In particular, let ? be a prime ring of characteristic not 2 containing a nontrivial idempotent. It is shown that, under some mild conditions on ?, if δ is a Lie triple derivable mapping from ? into ?, then δ = D + τ, where D is an additive derivation from ? into its central closure T and τ is a mapping from ? into its extended centroid 𝒞 such that τ(A + B) = τ(A) + τ(B) + Z _{A, B} and τ([[A, B], C]) = 0 for all A, B, C ∈ ?.     

On the regularity of the axially symmetric solutions to the three‐dimensional MHD equations

We present the improved three‐dimensional axially symmetric incompressible magnetohydrodynamics (MHD) equations with nonzero swirl. We consider three kinds of smooth axially symmetric particular solutions to the MHD equations: (1) u^θ=0,B^r=B^z=0, (2) B^r=B^z=0, and (3) B^θ=0. In particular, we derive new regularity criteria for these three kinds of the three‐dimensional axially symmetric smooth solutions to the MHD equations. Our results also reveal some interesting dynamic behavior of the interaction by the angular vorticity field ω^θ and the angular current density field j^θ. Copyright © 2016 John Wiley & Sons, Ltd.     

Polynomial parallel volume, convexity and contact distributions of random sets

We characterize convexity of a random compact set X in ℝ^d via polynomial expected parallel volume of X. The parallel volume of a compact set A is a function of r≥0 and is defined here in two steps. First we form the parallel set at distance r with respect to a one- or two-dimensional gauge body B. Then we integrate the volume of this (relative) parallel set with respect to all rotations of B. We apply our results to characterize convexity of the typical grain of a Boolean model via first contact distributions.     

On Olsson’s conjecture for blocks with metacyclic defect groups of odd order

Let G be a finite group, and let B be a p-block of G with defect group D. Let k ₀(B) denote the number of ordinary irreducible characters of height 0 in B. In 1984 Olsson proposed a conjecture: k₀(B)\leqq |D:D￠|{k_{0}(B)\leqq |D:D'|}. In this paper, we will verify Olsson’s conjecture in the case that D is metacyclic and p is odd.     

Relative distance between two scalar fields. Application to mathematical modelling approximation

Inspired in the relative error between two quantities, we define a functional Δ_D that operates on two non‐negative scalar fields, which are integrable in a given open connected set D. We prove that Δ_D defines a metric, but not an ultrametric nor a translation invariant metric. We show how to apply Δ_D to evaluate the performance of analytical approximations of PDEs. The principal advantage of using Δ_D instead of other distances given in the literature is that the value given by Δ_D has a very easy interpretation: a value close to 0 means relatively near, but a value close to 1 means relatively infinitely far. Copyright © 2013 John Wiley & Sons, Ltd.