一类SIRS传染病模型

This paper considers an SIRS epidemic model that incorporates constant immigration rate, a general population-size dependent contact rate and proportional transfer rate from the infective class to susceptible class. A threshold parameter a is identified. If σ≤1, the disease-free equilibrium is globally stable. If σ＞1, a unique endemic equilibrium is locally asymptotically stable. For two important special cases of mass action incidence and standard incidence,global stability of the endemic equilibrium is proved provided the threshold is larger than unity. Some previous results are extended and improved.     

Normal families and uniqueness of entire functions and their derivatives

In this paper we use the theory of normal families to prove: Let a and b be two complex numbers such that b ≠ a, 0, and let f be a non-constant entire function. If f and f′ share the value a CM and if f = b whenever f′ = b, then f ≡ f′. This result improves a result due to Rubel and Yang. Received: 13 June 2005     

Remarks on η-Einstein unit tangent bundles

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is the space of constant sectional curvature 1 or 2. Authors’ addresses: Y. D. Chai, S. H. Chun, J. H. Park, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea; K. Sekigawa, Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

On the Bloch Heuristic Principle, normality, and totally ramified values

Guided by the Bloch Heuristic Principle, we give a characterization of entire functions of order ≦ 1 with two totally ramified values, a normality criterion for analytic functions in plane domains, and a Picard type theorem. The results are complemented by various examples for completeness. Received: 7 November 2005     

Toeplitz operators and arguments of analytic functions

For a Toeplitz operator T _φ , we study the interrelationship between smoothness properties of the symbol φ and those of the functions annihilated by T _φ . For instance, it follows from our results that if φ is a unimodular function on the circle lying in some Lipschitz or Zygmund space Λ^α with 0 < α < ∞, and if f is an H ^p -function (p ≥ 1) with T _φ f = 0, then f ∈ Λ^α and

On the asymptotic convergence of sequences of analytic functions

We consider sequences {f _n } of analytic self mappings of a domain and the associated sequence {Θ _n } of inner compositions given by . The case of interest in this paper concerns sequences {f _n } that converge assymptotically to a function f, in the sense that for any sequence of integers {n _k } with n ₁ < n ₂ < ... one has that locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω. Received: 16 December 2006     

A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements

We establish an asymptotic representation formula for the steady state current perturbations caused by internal corrosive boundary parts of small surface measure. Based on this formula we design a non-iterative method of MUSIC (multiple signal classification) type for localizing the corrosive parts from voltage-to-current observations. We perform numerical experiments to test the viability of the algorithm and the results clearly demonstrate that the algorithm works well even in the presence of relatively high noise ratios. H. Ammari is partially supported by the Brain Pool Korea Program at Seoul National University, H. Kang is partially supported by KOSEF grant R01-2006-000-10002-0, E. Kim is supported by BK21 Math. Division at Seoul National University, and M.S. Vogelius is partially supported by NSF grant DMS-0604999.     

Birkhoff-functions that are bounded on prescribed sets

We prove the existence of entire functions which are universal under translations and bounded on certain prescribed sets. It is also shown that the family of all these universal functions is a dense but not a G_δ-subset in the space of entire functions provided with a natural metric. Received: 24 November 2004; revised: 12 April 2005     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0. The first author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039. Authors’ addresses: T. Sugawa and T. Terada, Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan Current address: T. Sugawa, Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan     

Asymptotic compactness of global trajectories generated by the Navier–Stokes–Poisson equations of a compressible fluid

In this paper, we consider the global behavior of weak solutions of Navier–Stokes–Poisson equations in time in a bounded domain–arbitrary forces. After proving the existence of bounded absorbing sets, we also obtain the conclusion on asymptotic compactness of global trajectories generated by the Navier–Stokes–Poisson equations of a compressible fluid. Supported by NSF(No:10531020) of China and the Program of 985 Innovation Engineering on Information in Xiamen University (2004–2007).     

Adaption allows efficient integration of functions with unknown singularities

We study numerical integration for functions f with singularities. Nonadaptive methods are inefficient in this case, and we show that the problem can be efficiently solved by adaptive quadratures at cost similar to that for functions with no singularities. Consider first a class of functions whose derivatives of order up to r are continuous and uniformly bounded for any but one singular point. We propose adaptive quadratures Q*_n, each using at most n function values, whose worst case errors are proportional to n^−r. On the other hand, the worst case error of nonadaptive methods does not converge faster than n⁻¹. These worst case results do not extend to the case of functions with two or more singularities; however, adaption shows its power even for such functions in the asymptotic setting. That is, let F^∞_r be the class of r-smooth functions with arbitrary (but finite) number of singularities. Then a generalization of Q*_n yields adaptive quadratures Q**_n such that |I(f)−Q**_n(f)|=O(n^−r) for any f ∈ F^∞_r. In addition, we show that for any sequence of nonadaptive methods there are `many' functions in F^∞_r for which the errors converge no faster than n⁻¹. Results of numerical experiments are also presented. The authors were partially supported, respectively, by the State Committee for Scientific Research of Poland under Project 1 P03A 03928 and by the National Science Foundation under Grant CCR-0095709.     

Bloch’s theorem for mappings of bounded and finite distortion

We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural form. Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author was visiting at the University of Michigan. He wishes to thank the department for hospitality.     

Lower and upper regularizations of frame semicontinuous real functions

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), 271–281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem. The first and second named authors acknowledge financial support from the Ministry of Education and Science of Spain and FEDER under grant MTM2006-14925-C02-02. The first named author also acknowledges financial support from the University of the Basque Country under grant UPV05/101. The third named author acknowledges financial support from the Centre of Mathematics of the University of Coimbra/FCT.     

The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).     

Jump problem and removable singularities for monogenic functions

In this article the jump problem for monogenic functions (Clifford holomorphicity) on the boundary of a Jordan domain in Euclidean spaces is investigated. We shall establish some criteria that imply the uniqueness of the solution in terms of a natural analogue of removable singularities in the plane to ℝⁿ⁺¹ (n ≥ 2). Sufficient conditions to extend monogenically continuous Clifford algebra valued functions across a hypersurface are proved. Communicated by Jenny Harrison     

Ockham algebras with double pseudocomplementation

The variety dpO consists of those algebras (L; ∧, ∨, f, *, ⁺, 0, 1) with ∧, ∨ binary, f, *, ⁺ unary and 0, 1 nullary, and where (L; ∧, ∨, f, 0, 1) is an Ockham algebra and the unary operations f and * commute, f and⁺ commute. We describe completely the structure of the subdirectly irreducible algebras that belong to the subclass dpK_1,1, characterised by the property f³ = f. This paper is dedicated to Walter Taylor. Received September 29, 2004; accepted in final form September 8, 2005.     

A new class of Zienkiewicz-type non-conforming element in any dimensions

In this paper, a new class of Zienkiewicz-type non-conforming finite element, in n spatial dimensions with n ≥ 2, is proposed. The new finite element is proved to be convergent for the biharmonic equation. The work was supported by the National Natural Science Foundation of China (10571006). This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.     

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).     

Normal families and multiple values

Let be a family of meromorphic functions in a domain D, and let k be a positive integer, and let b be a nonzero complex number. If, for each , f ≠ 0, and the zeros of have multiplicity at least 3 for k = 1 and 2 for k ≥ 2, then is normal in D. Examples show that the multiplicity of the zeros of is best possible. This research is supported by the G.I.F., the German-Israel Foundation for Scientific Research and Development, Grant G-809-234.6/2003, and by the NNSF of China (Grant No. 10471065 and No. 10671073).