Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

SHEAVES OF BANACH SPEACES

Abstract

This paper presents a number of results concerning sheaves on a topological space, with values in the category BAN of Banach spaces, over K = R or Ø, and linear contractions. After showing that these sheaves are reflective in the corresponding category of presheaves (Proposition 1) and that the resulting reflection is stalk preserving (Proposition 2), we concentrate on the approximation sheaves, these being BAN-sheaves satisfying a strong patching condition originally due to Auspitz [1]. The interest in these particular sheaves lies in the fact that they are precisely the BAN-sheaves arising as sheaves of continuous sections of the appropriate kind of Banach fibre spaces [1] and thus central to the representation of Banach spaces by continuous sections. Here, we show that the approximation sheaves on any space are characterized as the BAN-presheaves injective relative to certain maps (Proposition 3) and that, for paracompact spaces X, they are exactly those BAN-sheaves S such that each SU, U open in X, admits a suitable C*U-module structure (Proposition 4). Further, we consider the adjointness between the approximation sheaves on a space X and the Banach modules over C*X (Proposition 5) and investigate its special properties for X being Tychonoff (Proposition 6) and Boolean (Proposition 7). We conclude with some observations regarding the failure of the analogues of Swan's Theorem for vector bundles and the Hahn-Banach Theorem in the present context, and some positive facts concerning injectivity for approximation sheaves on Tychonoff spaces.     

A note on the paper of Kohli and Vashistha

In [1] Kohli and Vashistha gave an analogue of probabilistic version of Pant‘s Theorem ([2], Theorem 1). We note that mappings defined in Examples 3.6 to 3.8 of [1] are not self maps as claimed in the Definitions 3.1 and 3.2. In this context, we provide some relevant examples to complete the interesting results.     

Spatial versus non-spatial dynamics for diffusive Lotka–Volterra competing species models

This paper studies the dynamics of the diffusive Lotka–Volterra competition model under small dispersion rates and Dirichlet boundary conditions. Its main goal is ascertaining the connections between the qualitative behavior of the positive solutions of the parabolic model for small diffusions and the dynamics of its associated non-spatial model given by switching them off. After sharpening very substantially some previous results of Furter and López-Gómez (Proc R Soc Edinb 127A:281–336, 1997), we characterizes the singular limit, as diffusions go to zero, of any sequence of coexistence steady-state solutions. It turns out that they must approximate, point-wise in the inhabiting territory, the global attractor of the non-spatial model, uniformly on compact subsets of the habitat zones where a global hyperbolic attractor exist. As a very special consequence of our general theorem, the Dirichlet counterpart of the singular perturbation theorem of Hutson et al. (World Sci Ser Appl Anal 4:501–533, 1995) holds. Further, a multiplicity result is given when the underlying non-spatial model exhibits a founder control competition somewhere in the territory. This is the first multiplicity theorem available in the literature for small diffusivities.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Thue inequalities with a small number of primitive solutions

Several upper bounds are known for the numbers of primitive solutions (x; y) of the Thue equation (1) j F(x; y) j = m and the more general Thue inequality (3) 0 < j F(x; y) j m. A usual way to derive such an upper bound is to make a distinction between "small" and "large" solutions, according as max( j x j ; j y j ) is smaller or larger than an appropriate explicit constant Y depending on F and m; see e.g. [1], [11], [6] and [2]. As an improvement and generalization of some earlier results we give in Section 1 an upper bound of the form cn for the number of primitive solutions (x; y) of (3) with max( j x j ; j y j )Y0 , wherec 25 is a constant and n denotes the degree of the binary form F involved (cf. Theorem 1). It is important for applications that our lower bound Y0 for the large solutions is much smaller than those in [1], [11], [6] and [4], and is already close to the best possible in terms of m. ByusingTheorem1 we establish in Section 2 similar upper bounds for the total number of primitive solutions of (3), provided that the height or discriminant of F is suficiently large with respect to m (cf. Theorem 2 and its corollaries). These results assert in a quantitative form that, in a certain sense, almost all inequalities of the form (3) have only few primitive solutions. Theorem 2 and its consequences are considerable improvements of the results obtained in this direction in [3], [6], [13] and [4]. The proofs of Theorems 1 and 2 are given in Section 3. In the proofs we use among other things appropriate modifications and refenements of some arguments of [1] and [6].     

混合序列加权和的强收敛性

本文给出混合序列加权和的强收敛性的一些充分条件，这些结论推广和改进了文［１］定理３，文［２］定理３；文［３］定理４．１５以及文［４］定理４．     

Corrigendum to “Toric codes and finite geometries” [Finite Fields Appl. 45 (2017) 203–216]

This correction adds a necessary hypothesis in Proposition 2.4 of [1]. A counterexample to the previous version was pointed out to the author by Melda Görür and the author thanks her for bringing that to his attention. Theorem 3.2 and Corollary 3.3 are also reformulated so they do not refer to the new, more restricted Proposition 2.4.     

ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

A Class of Statistical and σ-Conservative Matrices

In [5] and [10], statistical-conservative and σ-conservative matrices were characterized. In this note we have determined a class of statistical and σ-conservative matrices studying some inequalities which are analogous to Knopp's Core Theorem.     

PROPOSITIONAL LOGIC,FRAMES, AND FUZZY ALGEBRA

Abstract

This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).     

A remark on entire solutions of quasilinear elliptic equations

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below.Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].     

Convergence Theory for AOR Method

In this paper we give some sufficient conditions for the convergence of the AOR method, introduced by Hadjidimos [5], which include the ones from [1], [2], [5], [6], [7], [9], [10], [11], [12] and which show that the necessary condition given in [8] for the convergence of the AOR method is not valid. We give general conditions for the class of H-matrices, but they are not always easy to check in practice. Consequently, we give some more practical conditions concerning some subclasses of H-matrices.     

A note on jets of entire curves in semi-abelian varieties

We prove a product decomposition of the Zariski closure of every jet lift of an entire curve f:CA into a semi-abelian variety A, provided that f is of finite order. On the other hand, by giving an example of f into a three dimensional abelian variety we show that this product decomposition does not hold in general; there was a gap in the proofs of [2], Proposition 1.8 (ii) and of [6], Theorem 2.2. Mathematics Subject Classifications (2000):32H30, 14K20.     

ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

Analyses of subensembles ofM/G/1 queueing sequences

Although many queueing processes of various principles have extensively been investigated, little attention has been paid to the sampling aspect of the theory, by which the nature of sample sequences of finite or infinite length can be examined with respect to some given ensemble of queueing sequences. In this paper we wish to identify classes of sample sequences of an M/G/1 model and investigate several hitherto unknown properties of queueing phenomenon of a given particular service system over a finite or infinite length of time. The method to be used is an extension of both the method of imbedded Markow chains, cf. D. G. Kendall [4], and semi-Markovian processes, Smith [9], Lévy [5], Pyke[7,8], Fabens [2], Neuts [6], etc.     

A generalization of parallel addition

Summary Parallel addition of positive operators, a concept introduced by W. N. Anderson and R. J. Duffin [1] in connection with network theory, has already been studied by several authors. We specifically mention W. N. Anderson and G. E. Trapp [2] and [3], T. Ando [4], K. Nishio [12], E. L. Pekarev and J. L. Smul'jan [13], as well as the article [10] by the present authors.The purpose of this note is to study a generalization of parallel addition. In particular, it will be shown (Theorem 3.2) that the corresponding quasi-units, a concept introduced in [10], are again the extreme points of the convex sets, formed by the positive operators less than or equal to some fixed operator.     

Some common fixed point theorems for selfmappings satisfying two contractive conditions of integral type in a uniform space

In this paper, we establish some common fixed point theorems for selfmappings of a uniform space by employing both the concepts of an A—distance and an E—distance introduced by Aamri and El Moutawakil [1] and two contractive conditions of integral type. Our results are generalizations and extensions of the classical Banach’s fixed point theorem of [2, 3, 19], some results of Aamri and El Moutawakil [1], Theorem 2.1 of Branciari [5] as well as a result of Jungck [7].      

Corrigendum to: When is a sum of annihilator ideals an annihilator ideal?

In this corrigendum, we correct the statement and proof of Lemma 3.2, revise the proofs of Lemma 3.3 and Theorem 3.5, retract Proposition 3.10 and correct several typographical errors.     

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

An integral test (Theorem 5) is established for the dichotomy concerning local extinction and survival (even persistence) at late times for critical multitype spatially homogeneous branching particle systems in continuous time. Our conditions on the branching mechanism are close to the ones known from “classical” processes without motion component. This generalizes and complements results of López-Mimbela and Wakolbinger [LMW96] and others. Our approach is based on some genealogical tree analysis combined with the study of the long-term behavior of L ¹-norms of solutions of related systems of reaction-“diffusion” equations, which is perhaps also of some independent interest. Received: 13 August 1997 / Revised version: 12 May 1998 / Published online: 14 February 2000