Showing papers on "Section (fiber bundle) published in 1987"

New results in binary multiple descriptions

Abstract: An encoder whose input is a binary equiprobable memoryless source produces one output of rate R_{1} and another of rate R_{2} . Let D_{1}, D_{2}, and D_{0} , respectively, denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate R_{l} only, the encoder output of rate R_{2} only, and both encoder outputs. The two-descriptions problem is to determine the region R of all quintuples (R_{1}, R_{2}, D_{1}, D_{2}, D_{0}) that are achievable in thc usual Shannon sense. Let R(D)=1+D \log_{2} D+(1-D) \log_{2}(1-D) denote the error frequency rate-distortion function of the source. The "no excess rate case" prevails when R_{1} + R_{2} = R(D_{0}) , and the "excess rate case" when R_{1} + R_{2} > R(D_{0}) . Denote the section of R at (R_{1}, R_{2}, D_{0}) by D(R_{1} R_{2}, D_{0}) =\{(D_{1},D_{2}): (R_{1}, R_{2}, D_{1},D_{2},D_{0}) \in R} . In the no excess rate case we show that a portion of the boundary of D(R_{1}, R_{2}, D_{0}) coincides with the curve (\frac{1}{2} + D_{1}-2D_{0})(\frac_{1}_{2} + D_{2}-2D_{0})= \frac{1}{2}(1-2D_{0})^{2} . This curve is an extension of Witsenhausen's hyperbola bound to the case D_{0} > 0 . It follows that the projection of R onto the (D_{1}, D_{2}) -plane at fixed D_{0} consists of all D_{1} \geq D_{0} and D_{2} \geq D_{0} that lie on or above this hyperbola. In the excess rate case we show by counterexample that the achievable region of El Gamal and Cover is not tight.

Kac-moody Groups, Topology of the Dirac Determinant Bundle and Fermionization

Abstract: The relation between Kac-Moody groups and algebras and the determinant line bundle of the massless Dirac operator in two dimensions is clarified. Analogous objects are studied in four space-time dimensions and a generalization of Witten's fermionization mechanism is presented in terms of the topology of the Dirac determinant bundle.

Natural Convection From Isothermal Cubical Cavities With a Variety of Side-Facing Apertures

Abstract: An experimental investigation of heat transfer by natural convection from a smooth, isothermal cubic cavity with a variety of side-facing aperatures is described in this paper. The study was motivated by the desire to predict the convective loss from large solar thermal-electric receivers and to understand the mechanisms which control this loss. Hence, emphasis is placed on the large Rayleigh number, Ra, regime with large ratios of the cavity wall temperature T{sub w} to the ambient temperature T{sub {infinity}}. A cryogenic wind tunnel with test section temperatures which are varied between 80 K and 310 K is used to facilitate deduction of the influences of the relevant parameters and to obtain large temperature ratios without masking the results by radiative heat transfer. A 0.4-m cubic cavity, which is mounted in the side wall of this tunnel, is used. The area of the aperture A{sub a} and its location are key variables in this study. The data which are presented cover the ranges: 1 < T{sub w}/T{sub {infinity}} < 3, L{sup 2}/18 {le} A{sub a} {le} L{sup 2}, and 3 {times} 10{sup 10}.

Varieties in positive characteristic with trivial tangent bundle

Abstract: © Foundation Compositio Mathematica, 1987, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces

Abstract: Let H be a Hilbert space and let S be a closed linear relation in H, i.e., S is a * subspace of H 2 such that ScS*cH 2 (for the definition of S and other definitions see Section I). Furthermore, let K be a Krein space such that HcK and the Krein space inner product onKcoincides on Hwith the Hilbert space inner product on H; we denote this situation byH~K. In this case H is anorthocomplemented subspace of K (see [4]). The corresponding orthogonal projection from K onto H is denoted by PH" We consider a selfadjoint relation A in K with a nonempty resolvent set p(A), such that ScA, that a selfadjoint extension of S in K. We define P~2)A-~ ={{PHf,PHg}l{f,g} CA}; it is is, A is clear that S cP(2)A. With A we associate the socalled ~traus extension T of S in H, which is by definition T= (T(£))£E~ U{~}' where T(£) is given by

Metrics of negative curvature on vector bundles

Abstract: It is shown that any vector bundle E over a compact base manifold M admits a complete metric of negative (respectively nonpositive) curvature provided M admits a metric of negative (nonpositive) curvature.

Global moduli for elliptic surfaces with a section

Abstract: © Foundation Compositio Mathematica, 1987, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On moduli of vector bundles on rational surfaces

Abstract: Let S be a smooth, connected, projective surface (over ~) and H an ample divisor on S. Fix ca ~ Pic (S), c 2 ~ H4(S, ~E) ~ ~E, r 6 Z, r > 1. Let MH(r, c a, c2) be the moduli scheme of H-stable vector bundles of rank r with first Chern class c a and second Chern class c 2 (see [16]). If S = ~2, G. Ellingsrud in [8] proved that Mn (r, ca, c2) is irreducible (if not empty). His construction proved also that Mn (r, Cl, c2) is unirational and for some choice of r, c 1, c 2 even stably rational ([1]). The proofs in [8] are based on [6]. The same methods apply for many other surfaces, in particular for the minimal rational surfaces (see Proposition 2.1). The aim of this paper is to show that this happens (at least for suitable H) for every rational surface. It is natural to ask if this is true for all ample line bundles. For the existence of simple vector bundles on certain rational surfaces, see [2], [19]. In the first version of this paper the main result was stated only for r = 2. The referee remarked that the same proof works in general.

Volume estimation of biological objects by systematic sections.

Abstract: The absolute volume of biological objects is often estimated stereologically from an exhaustive set of systematic sections. The usual volume estimator \(\hat V\) is the sum of the section contents times the distance between sections. For systematic sectioning with a random start, it has been recently shown that \(\hat V\) is unbiased when m, the ratio between projected object length and section distance, is an integer number (Cruz-Orive 1985). As this quantity is no integer in the real world, we have explored the properties of \(\hat V\) in the general and realistic situation m e ℝ. The unbiasedness of \(\hat V\) under appropriate sampling conditions is demonstrated for the arbitrary compact set in 3 dimensions by a rigorous proof. Exploration of further properties of \(\hat V\) for the general triaxial ellipsoid leads to a new class of non-elementary real functions with common formal structure which we denote as np-functions. The relative mean square error (CE2) of \(\hat V\) in ellipsoids is an oscillating differentiable np-function, which reduces to the known result CE2= 1/(5m4) for integer m. As a biological example the absolute volumes of 10 left cardiac ventricles and their internal cavities were estimated from systematic sections. Monte Carlo simulation of replicated systematic sectioning is shown to be improved by using m e ℝ instead of m e ℕ. In agreement with the geometric model of ellipsoids with some added shape irregularities, mean empirical CE was proportional to m−1.36 and m−1.73 in the cardiac ventricle and its cavity. The considerable variance reduction by systematic sectioning is shown to be a geometric realization of the principle of antithetic variates.

Vector rank of commuting matrix differential operators. proof of s. p. novikov's criterion

Abstract: The problem of describing a commuting pair of differential operators in terms of its Burchnall–Chaundy curve and the holomorphic bundle over it is considered. A characteristic of the matrix case is the occurrence of vector rank, a bundle having various dimensions over various components of the Burchnall–Chaundy curve. A complete, independent system which determines the pair of operators uniquely is chosen. In the last section, a proof is given of S. P. Novikov's criterion for an operator with periodic coefficients to be an operator of a nontrivial commuting pair. Bibliography: 25 titles.

Integrable almost cotangent structures and Legendrian bundles

Abstract: Recently, the present author together with M. Crampin proved a structure theorem for a certain subclass of geometric objects known as almost tangent structures (Crampin and Thompson [8]). As the name suggests, an almost tangent structure is obtained by abstracting one of the tangent bundle's most important geometrical ingredients, namely its canonical 1–1 tensor, and using it to define a certain class of G-structures. Roughly speaking, the structure theorem referred to above may be paraphrased by saying that, if an almost tangent structure is integrable as a G-structure and satisfies some natural global hypotheses, then it is essentially the tangent bundle of some differentiable manifold. (I shall have a further remark to make about the conclusion of that theorem at the end of Section 2.)

Characteristic classes, singular embeddings, and intersection homology

Abstract: This note announces some results on the relationship between global invariants and local topological structure. The first section gives a local-global formula for Pontrjagin classes or L-classes. The second section describes a corresponding decomposition theorem on the level of complexes of sheaves. A final section mentions some related aspects of “singular knot theory” and the study of nonisolated singularities. Analogous equivariant analogues, with local-global formulas for Atiyah-Singer classes and their relations to G-signatures, will be presented in a future paper.

Velocity-height relation for antimatter meteors

Abstract: A general velocity-height relation for both antimatter and ordinary matter meteor is derived. This relation can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq% aHfpqDdaWgaaWcbaGaamOEaaqabaaakeaacqaHfpqDdaWgaaWcbaGa% eyOhIukabeaaaaGccqGH9aqpcaqGLbGaaeiEaiaabchacaqGGaWaam% WaaeaacqGHsisldaWcaaqaaiaadkeaaeaacaWGHbaaaiaabwgacaqG% 4bGaaeiCaiaabIcacaqGTaGaamyyaiaadQhacaGGPaaacaGLBbGaay% zxaaGaeyOeI0YaaSaaaeaacaWGdbaabaGaamOqaiabew8a1naaBaaa% leaacqGHEisPaeqaaaaakmaacmaabaGaaGymaiabgkHiTiaabwgaca% qG4bGaaeiCamaadmaabaGaeyOeI0YaaSaaaeaacaWGcbaabaGaamyy% aaaacaqGLbGaaeiEaiaabchacaqGOaGaaeylaiaadggacaWG6bGaai% ykaaGaay5waiaaw2faaaGaay5Eaiaaw2haaiaacYcaaaa!64FD!\[\frac{{\upsilon _z }}{{\upsilon _\infty }} = {\text{exp }}\left[ { - \frac{B}{a}{\text{exp( - }}az)} \right] - \frac{C}{{B\upsilon _\infty }}\left\{ {1 - {\text{exp}}\left[ { - \frac{B}{a}{\text{exp( - }}az)} \right]} \right\},\]where υ z is the velocity of the meteoroid at height z, υ∞ its velocity before entrance into the Earth's atmosphere, α is the scale-height, and C parameter proportional to the atom-antiatom annihilation cross- section, which is experimentally unknown. The parameter B (B = DAϱ0/m) is the well known parameter for koinomatter (ordinary matter) meteors, D is the drag factor, ϱ0 is the air density at sea level, A is the cross sectional area of the meteoroid and m its mass.

On the existence of type-6 transonic states in linear elastic media of cubic symmetry

Abstract: Crucial to the understanding of surface-wave propagation in an anisotropic elastic solid is the notion of transonic states, which are defined by sets of parallel tangents to a centred section of the slowness surface. This study points out the previously unrecognized fact that first transonic states of type 6 (tangency at three distinct points on the outer slowness branch S$\_1$) indeed exist and are the rule, rather than the exception, in so-called C$\_3$ cubic media (satisfying the inequalities c$\_{12}$ + c$\_{44}$ > c$\_{11}$ - c$\_{44}$ > 0); simple numerical analysis is used to predict orientations of slowness sections in which type-6 states occur for 21 of the 25 C$_3$ cubic media studied previously by Chadwick & Smith (In Mechanics of solids, pp. 47-100 (1982)). Limiting waves and the composite exceptional limiting wave associated with such type-6 states are discussed.

Activities in the system LiF−NaF−AlF3

Abstract: Measurements have been made of the contents of Na and Li in Al in equilibrium with the molten fluorides at 1020 °C. The theory to calculate the activities of the three constituents is derived. Across the Li3AlF6-Na3AlF6 section the activity coefficients {ie409-1} are given in terms of mol fractionsNi by $$ \begin{gathered} \log \gamma _{{\text{AIF}}_{\text{3}} } = - 3.034 + 3.342N_{LiF} - 0.848(N_{LiF} )^2 \hfill \\ \log \gamma _{{\text{NaF}}} = - 0.246 - 1.114N_{LiF} - 0.283(N_{LiF} )^2 \hfill \\ \log \gamma _{LiF} = 0.158 - 0.266N_{LiF} - 0.283(N_{LiF} )^2 \hfill \\ \end{gathered} $$ Across the LiF−Na2.5AlF5.5 section the activity coefficients for 0≤NLiF≤0.45 are nearly constant at log \GgLiF \t~ 0.1, log \GgNaF \t~ 0.4, and log \GgAIF3 \t~ -2.6.The vapor over these melts is a mixture of LiAlF4 and NaAlF4, the pressures being given bypLiAlF4/bar=0.78aLiF·aAlF3 andpNaAlF4/bar=56.2aNaF·aAlF3. Combination of these equations with those for the activity coefficients reproduces the maximum observed in the total pressure in the Li3AlF6−Na3AlF6 section. The increase in pressure observed when Li3AlF6 is added to Na3AlF6 is due, not to the appearance of LiAlF4 in the gas, but to the increased pressure of NaAlF4 following the rise in AlF3 activity.

Structural body type catalyst

Abstract: PURPOSE:To decrease pressure drop and the closure of through-holes by forming a structural body to have a rectangular section of >=1.5 ratio between the length on the long side and the length on the short side and maintaining the rate of perforation of the structural body at >=80% and the deviation in the crushing strength directions on the side faces of the structural body within 30%. CONSTITUTION:The ratio of the length between the long side 15 and short side 16 of the rectangular section is required to be made >=1.5 and said ratio is more preferably determined at 1.7-3.0. Through-holes 11 of the rectangular section are satisfactory if the holes occupy >=85% of the entire part. The thickness of each block wall 13 is usually about 0.5-2.0mm, more preferably in a 0.7-1.3mm range. The thickness of each side wall 12 is usually about 0.7-2.2mm. The disposition of the holes 11 having the rectangular section in parallel is not enough in order for the structural body type catalyst to have the specified strength. The respective crushing strengths in the two directions perpendicular to the side faces of the structural body type catalyst are measured and are required to maintain the deviation in the crushing strengths in the two directions within 30%.

Paper sheet loosening device

Abstract: PURPOSE:To automate the loosening operation of a paper sheet bundle in order to easily separate paper sheets from each other, by leaping up the one side edge of the paper sheet bundle by means of a leap-up device, then by lifting up both ends of the bundle by a lift-up device to shift the above-mentioned one side end from each other, and by lowering both ends in this condition. CONSTITUTION:A plurality of paper sheet bundles 6 are stacked up in a multiple layers, and are elevated one stage by one stage. The topmost layer bundle 6 is pushed rightward at a predetermined level by a predetermined amount by means of a push-out device. Then an elevating frame 11 is raised, and after a packing paper sheet 91b underneath the paper sheet bundle 6 is sucked up by a sucker 42 which is raised by a cylinder 43, and is then lowered to pull down one edge section of the sheet 91. Then, a cylinder 41 rotates a leap-up plate 40 upward to leap up the center section of the one edge of the paper sheet bundle 6, and simultaneously, a pair of clamp hands 45 is advanced by a cylinder to grip both ends of one side of the paper sheet bundle 6 by means of fingers 49a, and then the hand 49 is lowered in this condition to loosen the papers sheet bundle.

On a Generalization of Bochner's Tube Theorem for Generic CR.Submanifolds

Abstract: The classical Bochner’s tube theorem states that every holomorphic function on a connected tube domain Rn+igcC can be extended holomorphically to the convex hull of the tubeR+ich(2). H. Komatsu [4] has obtained a simple proofo the local versiono this theorem by using Cauchy’s integral formula. By making use of the theory o Fourier-Bros-Iagolnitzer transform, M. S. Baouendi and F. Treves [1] have generalized the result above. In particular they have obtained the microlocal version o Bochner’s tube theorem or generic CR-manifolds. In this paper we shall give a simple proo of this result. In the section 1, we formulate Bochner’s tube theorem for generic CR-submanffolds by employing the notion of specialization o shea o holomorphic unctions (cf. [5]). In the section 2 we give the new proof of the theorem by reducing the problem to the totally real case. 1. Statement of the result. Let N be a real analytic submanifold of a complex mani2old X. For p e N, we denote by H,(N) the complex tangent space to N at p. The submaniold N is said to be generic, i for all p e N, dimeHp(N)=dimcX-codim N. Let us assume hereafter thatN is generic. LetSX be the spherical normal bundle TX-{O}/R with the projection r’SX-N. The disjoint union X=(X--N)]JSX has the structure o real analytic manifold with boundary S.,X. It is cmlled the real monoidal transform o X with center N. Let i (resp. ) be the embedding i" N--+X (resp. (resp. ) be the natural inclusion map ]" X-N-+X (resp. ]" X--NoX).

Recirculation type automatic document feeder

Abstract: PURPOSE:To feed respective documents sequentially and return to the bottom of a bundle of documents in the original order by arranging a curved tray for mounting a bundle of documents which will apply an upward force along a curved edge CONSTITUTION:Taking-out of the uppermost document is executed by means of plural vacuum type belts 16 movable around a pair of rollers 18,20 and a vacuum source 22 The uppermost document 24 in a bundle of documents is taken out in such a manner and fed through paper feed rollers 26, 28 to a copy machine 10 The copied document is allowed to pass through a path 14, then through a portion of the copy machine or an outlet roller and carrying rollers 32, 34 and placed on the bottom of the bundle of documents Upon lifting of the bundle of documents by means of a lifting board 42, the opposite ends of the curved bundle of documents are mounted on the curved shoulder section 52 Through such operation, the return path of document is opened The returning document is collided against a board 120 and finally arranged below the bundle of documents

Mass Transfer in Step Growth Polymerization

Abstract: The progress of reversible step growth polymerization reactions of ARB monomers (in a closed reactor), represented schematically by $$ {P_{n}} + {P_{m}}\underset{{{{k'}_{p}} = {k_{p}}/K}}{\overset{{{k_{p}}}}{\rightleftharpoons}} {P_{{n + m}}} + W $$ (5.1.1) has been discussed in Section 2.9. The variation of the conversion, p A, of functional group A with time was derived and given by Eq. (2.9.8).

Construction of Finite Difference Approximations

Abstract: The FDM considered in the following is to be understood as an “exterior approximation” of the BVP Au = F in the space Rn, \(n: = \left| {\bar \omega } \right|]\), where n tends to infinity as h → 0. The FDSs Ahy = Fh, with Ah ∈ L(Rn, Rn)(family of spaces of linear operators from Rn into Rn) and Fh, y ∈ Rn, are derived by approximating the BVP Au = F under the assumptions \(\text{V(}\bar \Omega \text{)}\), V(A, F) and V(u) from Section 2.1. Then, the properties of the difference operator Ah, a priori estimates and the convergence y → u as h → 0 are studied in the Rn, \(\,\text{n = n(h): = }\left| {\bar \omega } \right|\). This discussion is partially based on some papers of the author, cf. HEINRICH [5,6,7,8,9,10].

Kinematic Dynamo Theory

Abstract: Kinematic dynamo theory is concerned with solutions of the induction equation (7.8), which we rewrite here in the form $$ \frac{{\partial B}} {{\partial t}} - \eta abla ^2 B = B \cdot abla u - u \cdot abla B, $$ (8.1a) for a given velocity field u(r,t). In its broadest sense, the theory seeks to discover, among some classes of admissible u, those functions which lead to “dynamo action”, as discussed in Section 7.3. The precise meaning of dynamo action will generally depend upon the nature of the admissible functions u.

Nonlinear Viscous Materials and the Use of C

Abstract: Creep deformation of metals and alloys at elevated temperatures can be described, to a first approximation, as nonlinear viscous. Viscous behavior is defined by the existence of a unique relationship between the strain rate tensor and the current stress tensor, \( \dot \varepsilon _{ij} = f(\sigma _{ij} ) \). This corresponds to steady-state, or secondary, creep while primary-creep effects and the elastic response of materials are neglected. Although real materials generally do not exhibit exactly nonlinear viscous behavior, the results of this section, in particular the use of C* are often justified as a reasonable approximation within limits to be specified in the following sections.

Interval estimation of the critical value in a general linear model

Abstract: This paper concerns interval estimation of the critical value θ which satisfies\(\mu (\theta ) = \mathop {\sup }\limits_{x \in \mathfrak{X}} \mu (x)\) under the general linear model,Yi=μ(xi)+ei(i=1,2,···), where\(\mu (x) = \sum\limits_{j = 1}^p {\beta _j f_j (x)} \) for\(x \in \mathfrak{X}\) and the functional forms offj′s are known. From an asymptotic expansion it is shown that, under reasonable conditions, the limiting distribution of\(\sqrt n (\hat \theta _n - \theta )\) is normal. Thus in the large-sample case a confidence interval for θ can be obtained. Such a result is useful when one is interested in carrying out a retrospective analysis rather than designing the experiment (as in the Kiefer-Wolfowitz procedure). In Section 3 a sequential procedure is considered for confidence intervals with fixed width 2d. It is shown that, for a given stopping variableN,\(\sqrt n (\hat \theta _n - \theta )\) is also asymptotically normal asd→0. Thus the coverage probability converges to 1−α (preassigned) asd→0. An example of application in estimating the phase parameter in circadian rhythms is given for the purpose of illustration.

The Spectrum of Backscatter for Pulse Reflection from a Randomly Stratified Medium

Abstract: We consider the problem of a plane wave pulse normally incident on a plane stratified acoustic or elastic half space with material parameters which are stationary random functions of position. For the elastic case we study SH waves, so that the problem is completely one dimensional. It is assumed that the wave length is large compared to the typical size of an inhomogeneity, and that sufficient time has elapsed for the pulse to have travelled through many inhomogeneities. More specifically, let ∈ be the ratio of the size of a typical inhomogeneity to a wavelength characteristic of the incident pulse. It is assumed that ∈< <1. We consider a subsection, or “time window”, of the backscattered signal of width 0(l/∈), the order of magnitude of the pulse duration, but centered at a large time τ/σ2 , where τ is 0(1). Then this section of the backscattered process is approximately stationary and Gaussian, with power spectral density Sτ(ω) given by \[{s_\tau }(\omega ) - |\hat f(\omega ){|^2}\frac{1}{\tau }\mu (\sqrt {\alpha \tau } \omega )\] where f is the Fourier transform of the pulse shape, α is a constant which may be computed from the statistics of the random medium, and μ is a universal function, which we characterize.

On the Global Structure of Solutions to Some Semilinear Elliptic Problems

Abstract: In this paper we discuss two problems involving positive solutions of semi linear elliptic equations which are global in the sense that the problem requires the knowledge of all possible solutions. In the first problem, we will study the semi linear elliptic equation $$ \Delta {\text{u + }}\lambda \,{ \sinh }\,{\text{u = 0}} $$ in bounded domains D ⊂ R2 . This equation has sometimes been called the elliptic Sinn-Gordon equation. Of particular interest is the study of the following boundary value problem of “nonlinear eigenvalue” type: $$\begin{array}{*{20}{c}} {\Delta u + \lambda {\text{ sinh u }} = 0{\text{ in }}R} \\ {u = 0{\text{ on }}\partial R} \\ {u \geqslant 0{\text{ in }}R} \\ \end{array}$$ (1) where R is a rectangle in R2 . This problem arises in plasma physics and also statistical mechanics as a way of modeling point vortices. However, it arises in a surprising and central way in the construction of compact surfaces of constant man curvature. This will be explained in the following section. The basic question that we discuss is the behavior of solutions as λ tends to zero.