M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

In this paper, we combine results on extensions of operators with recent results on the relation between the M ‐function and the spectrum, to examine the spectral behaviour of boundary value problems. M ‐functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M ‐function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

When Some Complement of a Submodule Is a Summand

A module M is said to satisfy the C ₁₁ condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C ₁ condition implies the C ₁₁ condition and that the class of C ₁₁-modules is closed under direct sums but not under direct summands. We show that if M = M ₁ ⊕ M ₂, where M has C ₁₁ and M ₁ is a fully invariant submodule of M, then both M ₁ and M ₂ are C ₁₁-modules. Moreover, the C ₁₁ condition is shown to be closed under formation of the ring of column finite matrices of size Γ, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C ₁₁ are essential extensions of C ₁₁-modules constructed from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C ₁₁-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.     

On Rings Having McCoy-Like Conditions

ABSTRACT

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M _R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ^?1], and M[x, x ^?1] over suitable skew extension rings.     

On the canonical extension of a differentiable manifold

In this paper, the differential geometry of second canonical extension² M of a differentiable manifoldM is studied. Some vector fields tangent to² M inTTM are determined. In addition we obtain that the second canonical extensions ofM and a totally geodesic submanifold inM are totally geodesic submanifolds inTTM and² M respectively.     

M -Hypoelliptic pseudo-differential operators on Lp (ℝ n )

Following Wong's point of view, we construct the minimal and maximal extension in L^p (? ⁿ ), 1 < p < ∞ for M-hypoelliptic pseudo-differential operators, which have been introduced and studied by Garello and Morando. We give some facts about the domain of minimal and maximal extensions of M-hypoelliptic pseudo-differential operators. For M-hypoelliptic pseudo-differential operators with constant coefficients, the spectrum and essential spectrum are computed.     

Central extensions of groups of sections

If K is a Lie group and q : P → M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra \mathfrakk{\mathfrak{k}} of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact 1-forms. In this article, we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components, we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by the specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context, we provide sufficient conditions for integrability in terms of data related only to the group K.     

Defining Sets and Critical Sets in (0,1)‐Matrices

If D is a partially filled‐in (0, 1)‐matrix with a unique completion to a (0, 1)‐matrix M (with prescribed row and column sums), we say that D is a defining set for M. If the removal of any entry of D destroys this property (i.e. at least two completions become possible), we say that D is a critical set for M. In this note, we show that the complement of a critical set for a (0, 1)‐matrix M is a defining set for M. We also study the possible sizes (number of filled‐in cells) of defining sets for square matrices M with uniform row and column sums, which are also frequency squares. In particular, we show that when the matrix is of even order 2m and the row and column sums are all equal to m, the smallest possible size of a critical set is precisely m². We give the exact structure of critical sets with this property. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 253–266, 2013     

Properties of Hadamard product of inverse M‐matrices

This paper concerns with the properties of Hadamard product of inverse M‐matrices. Structures of tridiagonal inverse M‐matrices and Hessenberg inverse M‐matrices are analysed. It is proved that the product A ○ A^T satisfies Willoughby's necessary conditions for being an inverse M‐matrix when A is an irreducible inverse M‐matrix. It is also proved that when A is either a Hessenberg inverse M‐matrix or a tridiagonal inverse M‐matrix then A ○ A^T is an inverse M‐matrix. Based on these results, the conjecture that A ○ A^T is an inverse M‐matrix when A is an inverse M‐matrix is made. Unfortunately, the conjecture is not true. Copyright © 2004 John Wiley Sons, Ltd.     

Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups

We investigate the simple modules for the sporadic simple Mathieu groups M₂₂, M₂₃ and M₂₄ as well as those of the automorphism group, the covering groups and the bicyclic extensions of M₂₂ in characteristics 2 and 3. We determine the vertices and sources as well as the Green correspondents of these simple modules. We also find two 3-blocks with elementary abelian defect groups of order 9 in these groups which are Morita equivalent to their Brauer correspondents.     

Orders of accumulation of entropy on manifolds

For a continuous self-map T of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold M and any countable ordinal α, we construct a continuous, surjective self-map ofM having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism.     

Nowhere zero 4‐flow in regular matroids

Jensen and Toft 8 conjectured that every 2‐edge‐connected graph without a K₅‐minor has a nowhere zero 4‐flow. Walton and Welsh 19 proved that if a coloopless regular matroid M does not have a minor in {M(K_3,3), M*(K₅)}, then M admits a nowhere zero 4‐flow. In this note, we prove that if a coloopless regular matroid M does not have a minor in {M(K₅), M*(K₅)}, then M admits a nowhere zero 4‐flow. Our result implies the Jensen and Toft conjecture. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

We study a stochastic model for the spread of two pathogen strains—termed type 1 and type 2—among a homogeneously mixing community consisting of a finite number of individuals. In the model, we assume partial cross‐immunity, exogenous streams of infection, and that the degree of severity of a newly infective individual depends on who this infective individual was infected by. The aim is to characterize the joint probability distribution of the numbers M₁ and M₂ of type‐1 and type‐2 infections suffered by a focal individual during an outbreak of the disease. We present iterative procedures for computing the probability mass function of (M₁,M₂) under the assumption that the initial state of the focal individual is known, and a numerical study of the model is performed to investigate the influence of certain key parameters on the spread of resistant bacteria in hospitals.     

Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

International trade and biological invasions: A queuing theoretic analysis of the prevention problem

We propose and develop a new framework for studying the problem of preventing biological invasions caused by ships transporting internationally traded goods between countries and continents. Specifically, we apply the methods of queuing theory to analyze the problem of preventing a biological invasion from a long run perspective. First, we characterize two simple regulatory regimes as two different kinds of queues. Second, we show how to pose a publically owned port manager’s decision problem as an optimization problem using queuing theoretic techniques. Third, we compare and contrast the optimality conditions emanating from our analysis of the M/M/I/U and the M/M/I/I inspection regimes. We conclude by discussing possible extensions to our basic models.     

EXTENSIONS OF CLEAN RINGS

It is shown that if e is an idempotent in a ring R such that both eRe and (1 ? e)R(1 ? e) are clean rings, then R is a clean ring. This implies that the matrix ring M _n (R) over a clean ring is clean, and it gives a quick proof that every semiperfect is clean. Other extensions of clean rings are studied, including group rings.     

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM _t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.     

Essentiality and Injectivity

Essentiality is an important notion closely related to injectivity. Depending on a class M\mathcal M of morphisms of a category A{\mathcal A}, three different types of essentiality are considered in literature. Each has its own benefits in regards with the behaviour of M\mathcal M-injectivity. In this paper we intend to study these different notions of essentiality and to investigate their relations to injectivity and among themselves. We will see, among other things, that although these essential extensions are not necessarily equivalent, they behave almost equivalently with regard to injectivity.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.