An essential spectra mapping theorem

Let T be a closed operator on a Hilbert Space H, such that α?p(T), the resolvent of T. Set A = (T ? αI)^?1. For μ ≠ 0, define λ such that (λ ? α)μ = 1. It is shown that λ ? essential spectra of T iff μ ? essential spectra of A for various definitions of the essential spectra. A number of immediate corollaries are then derived.     

Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Let ? _n [i] be the ring of Gaussian integers modulo n. We construct for ?n[i] a cubic mapping graph Γ(n) whose vertex set is all the elements of ?n[i] and for which there is a directed edge from a ∈ ?n[i] to b ∈ ?n[i] if b = a ³. This article investigates in detail the structure of Γ(n). We give suffcient and necessary conditions for the existence of cycles with length t. The number of t-cycles in Γ₁(n) is obtained and we also examine when a vertex lies on a t-cycle of Γ₂(n), where Γ₁(n) is induced by all the units of ?n[i] while Γ₂(n) is induced by all the zero-divisors of ?n[i]. In addition, formulas on the heights of components and vertices in Γ(n) are presented.     

On the period spectrum of a symplectic mapping

Let M be a compact Riemannian manifold. It has been known for a long time that the singularities of the wave trace, trace(cos √Δt), are located at the periods of the closed geodesics. Do these singularities also contain information about the geometry of M in the neighborhood of a closed geodesic? We prove that the Birkhoff canonical form of the Poincaré map can be determined from the singularities of the wave trace.     

On the domain dependence of the inf-sup and related constants via conformal mapping

In this paper we investigate the domain dependence of the inf-sup stability constant in the family of two-dimensional simply connected domains using its connection to the optimal constant figuring in Friedrichs? inequality for conjugate harmonic functions and the conformal mapping of the domain. A lower estimation of the inf-sup constant is also given in terms of the conformal mapping provided the boundary of the domain is smooth enough. We illustrate the results with several examples.     

Branch point area methods in conformal mapping

The classical estimate of Bieberbach that ?a ₂?≤2 for a given univalent function ?(z)=z+a ₂ z ²+… in the classS leads to the best possible pointwise estimates of the ratio ?"(z)/?'(z) for ?∈S, first obtained by K?be and Bieberbach. For the corresponding class Σ of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of ?"(z)/?'(z) for ψ∈Σ. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K?be-Bieberbach estimate, is firmly rooted in areabased methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.     

Constrained open mapping theorem with applications

We prove a generalization of Graves? Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a set-valued one having particular properties. The nonlinear mapping is restricted to a closed convex subset of a Banach space. We apply the results to derive necessary and sufficient conditions ensuring the existence of a differentiable selection for the inverse mapping. A slight generalization of a sufficient condition by J. Klamka on so-called constrained exact local controllability of nonlinear and semi-linear dynamic systems is also proved.     

Loops on spheres having a compact-free inner mapping group

We prove that any topological loop homeomorphic to a sphere or to a real projective space and having a compact-free Lie group as the inner mapping group is homeomorphic to the circle. Moreover, we classify the differentiable 1-dimensional compact loops explicitly using the theory of Fourier series. Authors’ addresses: ágota Figula, Mathematisches Institut der Universit?t Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany and Institute of Mathematics, University of Debrecen, P.O.B. 12, H-4010 Debrecen, Hungary; Karl Strambach, Mathematisches Institut der Universit?t Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany     

Bounded cohomology and non-uniform perfection of mapping class groups

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective. Oblatum 8-IX-2000 & 20-X-2000?Published online: 29 January 2001     

Finiteness of mapping degree sets for 3-manifolds

By constructing certain maps, this note completes the answer of the question: For which closed orientable 3-manifold N, is the set of mapping degrees D(M,N) finite for any closed orientable 3-manifold M?     

The Fueter mapping theorem in integral form and the ℱ‐functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n‐tuples of commuting operators (called ?‐functional calculus) based on a new notion of spectrum, called ?‐spectrum, for the n‐tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ?‐functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.     

Cones with a mapping cone symmetry in the finite-dimensional case

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio?kowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.     

Difference analogue of second main theorems for meromorphic mapping into algebraic variety

Analysis Mathematica - In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from ?m into an algebraic variety V intersecting a finite set of fixed...     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S~2×S~1 or RP~3#RP~3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni(2014) in an earlier paper.     

A nuclear construction of loops with small inner mapping groups

We construct an infinite class of left conjugacy closed loopsQ such that the inner mapping group InnQ is a nonabelian group of orderpq. The constructions are done in a broader context of loopsQ =N _λ N _ρ such thatN _λ =N _μ ?Q and InnQ can be embedded into the holomorph of a cyclic group.     

The combinatorial derivation and its inverse mapping

Let G be a group and P _G be the Boolean algebra of all subsets of G. A mapping Δ: P _G → P _G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P _X → P _X , A ? A ^d , where X is a topological space and A ^d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ?. For example, we show that if G is infinite and I is an ideal in P _G such that Δ(A) ∈ I and ?(A) ? I for each A ∈ I then I = P _G .     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

A remark on equivariance of a moment mapping

It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension $ \tilde G It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension × M → M. Original Russian Text ? I.V. Mikityuk, A.M. Stepin, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 30–33.     

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk $\mathbb{D}$ with φ( $\mathbb{D}$ ) ? $\mathbb{D}$ . We give new characterizations of the bounded and compact weighted composition operators W _ψ,? from the Hardy spaces H ^p , 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A _α ^p , α > ? 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal{D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W _ψ,? f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H ¹ as well as of compactness for H ^p , 1 ≤ p < ∞, and $\mathcal{D}$ purely in terms of the symbols ψ and φ.     

Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups

We prove that the mapping torus F_n \rtimes_f \Bbb Z F_n \rtimes_\phi {\Bbb Z} of a polynomially growing automorphism f: F_n ? F_n \phi : F_n \to F_n of finitely generated free group F_n satisfies the quadratic isoperimetric inequality.