Topological methods

We show the existence of families of hip-hop solutions in the equal-mass 2N -body problem which are close to highly eccentric planar elliptic homographic motions of 2N bodies plus small perpendicular non-harmonic oscillations. By introducing a parameter , the homographic motion and the small amplitude oscillations can be uncoupled into a purely Keplerian homographic motion of fixed period and a vertical oscillation described by a Hill type equation. Small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise, via a Bolzano argument, to resonant periodic solutions of the uncoupled system in a rotating frame. For small = 0, the topological transversality persists and Brouwer's fixed point theorem shows the existence of this kind of solutions in the full system.

In this paper we prescribe a fourth order curvature -the Q-curvature on the standard n-sphere, n 5. Under flatness condition of order β, n -4 β < n near each critical point of the prescribed Q-curvature function, we characterize the critical points at infinity of the associated variational problem and we prove new existence results through Euler-Hopf formulae type. Our argument gives an upper bound on the Morse index of the obtained solutions. We also give a lower bound on the number of conformal metrics having the same Q-curvature.

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a 3-dimensional CR compact manifold locally conformally CR equivalent to the unit sphere S 3 of C 2 . Due to Kazdan-Warner type obstructions, conditions on the function H to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the same Webster scalar curvature.

Topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps. A topological method is introduced that simplifies the solution of boundary value problems: The activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path. The method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps.

It is known that the Mycielski graph can be generalized to obtain an infinite family of 4-chromatic graphs with no short odd cycles. The first proof of this result, due to Stiebitz, applied the topological method of Lov~sz. The proof presented here is elementary combinatorial.

O objetivo principal deste trabalho consiste em mostrar a existˆencia de m etricas com curva- tura de Ricci positiva na classe conforme de uma m etrica Riemanniana com curvatura escalar positiva em variedades compactas de dimens ao 3 e 4. Catino-Djadli [3] e Gursky-Viaclovsky [13] mostraram que se as curvaturas escalar e de Ricci de uma metrica g satisfazem a uma desigualdade integral em uma variedade compacta tridimensional, entao g e conforme a al- guma metrica de curvatura de Ricci positiva. No primeiro artigo os autores trabalham em variedades tridimensionais e no segundo em variedades de dimensao 4.

We segment the stabilization region in a simulation of a lifted jet flame based on its topology induced by the Y OH field. Our segmentation method yields regions that correspond to the flame base and to potential auto-ignition kernels. We apply a region overlap based tracking method to follow the flame-base and the kernels over time, to study the evolution of kernels, and to detect when the kernels merge with the flame. The combination of our segmentation and tracking methods allow us observe flame stabilization via merging between the flame base and kernels; we also obtain Y CH 2 O histories inside the kernels and detect a distinct decrease in radical concentration during transition to a developed flame.

A series of hydrazide-hydrazones, based on a series of 4-substituted benzoic acid, were synthesized, and their structures were elucidated and screened for the antituberculosis activity against Mycobacterium tuberculosis H37Rv with the help of the BACTEC 460 radiometric system. Compound 3, 4-fluorobenzoic acid [((5-nitro)thiophen-2yl) methylene]hydrazide showed the highest inhibitory activity in this series. The search of pharmacophores was done by means of the Electronic-Topological Method (ETM). The model developed in this study is supposed to be applied to the design, preparation and screening of new compounds of similar structure in order to further test and optimize the model with the eventual goal of preparing new anti-tubercular agents.

These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory, held in Villa de Leyva (Colombia), July 2-20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the Lagrangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar φ 3 theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green's functions given as perturbative series. The last lecture presents the Connes-Kreimer Hopf algebra of renormalization for the scalar theory and its associated proalgebraic group of formal series.

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R m+1 for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations-equivariant Morse and Lusternik-Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere S m , i.e., the space of n-tuples of points (x 1 ,. .. , x n), where x i ∈ S m and x i = x i+1 for i = 1,. .. , n.

These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory, held in Villa de Leyva (Colombia), July 2-20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the Lagrangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar φ 3 theory. The fourth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green functions given as perturbative series. The last lecture presents the Connes-Kreimer Hopf algebra of renormalization for the scalar theory and its associated proalgebraic group of formal series.

These are the notes of five lectures given at the Summer School Geometric and Topological Methods for Quantum Field Theory, held in Villa de Leyva (Colombia), July 2-20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very condensed review of classical and quantum field theory, from the Lagrangian formalism to the Euler-Lagrange equation and the Dyson-Schwinger equation for Green's functions. It poses the main problem of solving some non-linear differential equations for interacting fields. In the third lecture we explain the perturbative solution of the previous equations, expanded on Feynman graphs, in the simplest case of the scalar φ 3 theory. The forth lecture introduces the problem of divergent integrals appearing in quantum field theory, the renormalization procedure for the graphs, and how the renormalization affects the Lagrangian and the Green's functions given as perturbative series. The last lecture presents the Connes-Kreimer Hopf algebra of renormalization for the scalar theory and its associated proalgebraic group of formal series.

Naturally derived antimicrobial peptides have been an area of great interest because of high selectivity against bacterial targets over host cells and the limited development of bacterial resistance to these molecules throughout evolution. There are also a significant number of venom-derived peptides that exhibit antimicrobial activity in addition to activity against mammals or other organisms. Many venom peptides share the same net cationic, amphiphilic nature as host-defense peptides, making them an attractive target for development as potential antibacterial agents. The peptide ponericin L1 derived from Neoponera goeldii was used as a model to investigate the role of cationic residues and net charge on peptide activity. Using a combination of spectroscopic and microbiological approaches, the role of cationic residues and net charge on antibacterial activity, lipid bilayer interactions, and bilayer and membrane permeabilization were investigated. The L1 peptide and derivatives all showed enhanced binding to lipid vesicles containing anionic lipids, but still bound to zwitterionic vesicles. None of the derivatives were especially effective at permeabilizing lipid bilayers in model vesicles, intact Escherichia coli, or human red blood cells. Taken together the results indicate that the lack of facial amphiphilicity regarding charge segregation may impact the ability of the L1 peptides to effectively permeabilize bilayers despite effective binding. Additionally, increasing the net charge of the peptide by replacing the lone anionic residue with either Gln or Lys dramatically improved efficacy against several bacterial strains without increasing hemolytic activity.

A series of hydrazide-hydrazones, based on a series of 4-substituted benzoic acid, were synthesized, and their structures were elucidated and screened for the antituberculosis activity against Mycobacterium tuberculosis H37Rv with the help of the BACTEC 460 radiometric system. Compound 3, 4-fluorobenzoic acid [((5-nitro)thiophen-2yl) methylene]hydrazide showed the highest inhibitory activity in this series. The search of pharmacophores was done by means of the Electronic-Topological Method (ETM). The model developed in this study is supposed to be applied to the design, preparation and screening of new compounds of similar structure in order to further test and optimize the model with the eventual goal of preparing new anti-tubercular agents.

We show that the S 1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S 1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S 1-action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold.

A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

In previous works we proposed a method for the study of systems with one renewable resource. The separation of the individual and the group parameters and the discretization of time led us to scalar linear functional equations with shift. Cyclic models, in which the initial state of the system coincides with the final state, were considered. In this work, we present cyclic models for systems with two renewable resources. In modelling, the interactions and the reciprocal influences between these two resources are taken into account. Analysis of the models is carried out in weighted Holder spaces. For the solution of the balance integral equation with degenerate kernels and inverse operators, a modified Fredholm method is proposed. The modified Fredholm method is applied to the analysis of balance equation. The equilibrium state of the system with renewable resources is found.

In this paper, we analyse the polyharmonic equation (−) m u = a(x)u + f (x) |u| 2 * −2 u in a bounded smooth domain with homogeneous Dirichlet conditions on the boundary and critical exponents. We will obtain a relationship between the topological domain and the multiplicity of solutions. We also give symmetry properties of the solutions and a result about breaking symmetry.

This paper aims at establishing the existence and uniqueness of solutions for a nonstandard variationalhemivariational inequality. The solutions of this inequality are discussed in a subset K of a reflexive Banach space X. Firstly, we prove the existence of solutions in the case of bounded closed and convex subsets. Secondly, we also prove the case when K is compact convex subsets. Finally, we enhance the main results by the application of some differential inclusions.

Algebraic topology brings a powerful framework for studying 

the figure and structure of topological spaces using algebraic methods.

Nowadays, the area of data analysis has seen an emerging interest in applying

Algebraic topology techniques, precisely persistent homology, to prepare

Complex data sets. This thesis examines the mathematical foundations of

Algebraic topology and its application in data analysis with the help of

Persistent homology.

The chemical bond evolution in the isomerization of XNO X s H, Cl has been investigated by a topological method based on the Catastrophe Theory. The different domains of structural stability occurring along the reaction path have been identified as well as the bifurcation catastrophes responsible for the changes in the topology of the systems. The covalent isomerization of HNO requires five steps to break the OH bond and form the NH one. The NO subunit is mostly anionic in the transition state. For ClNO, the process involves the chlorine lone pairs. In the transition state NO has a cationic character.

A series of hydrazide-hydrazones, based on a series of 4-substituted benzoic acid, were synthesized, and their structures were elucidated and screened for the antituberculosis activity against Mycobacterium tuberculosis H37Rv with the help of the BACTEC 460 radiometric system. Compound 3, 4-fluorobenzoic acid [((5-nitro)thiophen-2yl) methylene]hydrazide showed the highest inhibitory activity in this series. The search of pharmacophores was done by means of the Electronic-Topological Method (ETM). The model developed in this study is supposed to be applied to the design, preparation and screening of new compounds of similar structure in order to further test and optimize the model with the eventual goal of preparing new anti-tubercular agents.

This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody.

We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a k-dimensional differentiable manifold M ⊆ R m. We assume that M is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on M to the Brouwer degree of a suitable map in R m. As further applications, we study the set of harmonic solutions to periodic semi-explicit differential-algebraic equations.

In this paper an algebraic method, which shares all the advantages of the topological methods and allows us to obtain the same results as the standard algebraic method with a substantial reduction in memory and cpu requirements, is presented. The main idea consists of writing the link, OD and scanned flows in terms of route instead of OD flows. This alternative permits starting the algebraic and topological processes with identical matrices of zeros and ones. In addition, in most iterations the pivots can be selected in such a way that the resulting matrices after each iteration contain only zeroes, ones and minus ones. This allows us to design a ternary arithmetic which reproduces the algebraic results exactly, requires only two bits to store each matrix entry and have no precision or non-zero pivot identification problems. Only when this process cannot be continued, the pure algebraic method is used, but only with a very reduced size matrix when compared with the size of the initial matrix. The method is illustrated by its application to a very simple network and to a real network example (the city of Cuenca, Spain).

We continue our analysis of geodesics in quenched, random Riemannian environments. In this article, we prove that a geodesic with randomly chosen initial conditions is almost surely not minimizing. To do this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.

For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C * r (T − X) of C *-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.

For a smooth manifold X with boundary we construct a semigroupoid T − X and a continuous field C * r (T − X) of C *-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.

Starting from the canonical symmetroid S(G) associated with a groupoid G, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid S(G), the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid G and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.

Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in this setting, as does Cartesian wedge product. Subspaces of finitely supported Dirac chains and polyhedral chains are both dense, leading to a unification of the discrete with the smooth continuum. We conclude with an application generalizing a simple version of the Reynolds' Transport Theorem to rough domains.

In this paper, we show the existence of a sequences of eigenvalues for an operator homogenous at the infinity, we give his variational formulation and we establish the simplicity of all eigenvalues in the case N = 1. Finally we study the solvability of the problem A(u) := −div(A(x, ∇u)) = f (x, u) + h in Ω, u = 0 on ∂Ω, as well as the spectrum of G ′ 0 (u) = λm|u| p−2 u in Ω, u = 0 on ∂Ω,

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points.

Basic tools of Riemannian geometry In this chapter, we will briefly describe the foundations of Riemannian geometry for readers who are unfamiliar with the subject. Our choice of material is made in view of applications to Carleman estimates and inverse problems. Readers who are interested in a more systematic exposition of Riemannian geometry, can consult, for example, [17], [33]. We provide proofs only if they cannot be found in the standards textbooks. Our interest is focused on Riemannian manifolds which are manifolds equipped with metric structure. 1.1 Manifolds The concept of a manifold is used throughout this lecture note. We start with defining the notion of a coordinate chart. Definition 1.1.1 Let M be a topological space. Then a pair (U, φ) is called a chart (a coordinate system), if φ : U −→ φ(U) ⊂ R n and φ(U) is an open set in R n. The coordinate functions on U are defined as x j : U → R, and φ(a) = (x 1 (a), • • • , x n (a)). Here n is called the dimension of the coordinate system. Definition 1.1.2 A topological space M is called a Hausdorff space if for each two distinct points a 1 , a 2 ∈ M, there are two open sets U 1 , U 2 ⊂ M such that a 1 ∈ U 1 , a 2 ∈ U 2 , U 1 ∩ U 2 = ∅. We now want to consider the case where M is covered by such charts and satisfies some consistency conditions. We have Definition 1.1.3 An n-dimensional atlas on a topological space M is a collection of charts {(U α , φ α)} α∈I with some index set I such that: • M is covered by {U α } α∈I • φ α (U α ∩ U β) is open in R n for each α, β ∈ I. 4 M.Bellassoued and M.Yamamoto, Chapter 1-Riemannian geometry • the map φ β • φ −1 α : φ α (U α ∩ U β) −→ φ β (U α ∩ U β) is smooth. Definition 1.1.4 Two atlases {(U α , φ α)} α∈I and {(V β , ψ β)} β∈J are compatible if their union is an atlas. The set of compatible atlases with a given atlases can be organized by inclusion. The maximal element is called the complete atlas. Finally we come to the definition of a manifold: Definition 1.1.5 A smooth manifold M is called a Hausdorff space with a complete atlas. Definition 1.1.6 An n-dimensional manifold with boundary is defined as a Hausdorff space together with an open cover {U α } and homeomorphism φ α : U α →Ũ α such that eachŨ α is an open set in R n + := {x ∈ R n ; x n ≥ 0} and φ β • φ −1 α : φ α (U α ∩ U β) −→ φ β (U α ∩ U β) is a smooth map whether U α ∩ U β is nonempty. In this lecture note, we deal with smooth manifolds, i.e., C ∞-manifolds, which means that φ β • φ −1 α , α ∈ I, β ∈ J, are C ∞ mappings. We understand that a compact manifold consists of a finite number of pieces of Euclidean spaces which are glued together. The functions φ α (x) = (x 1 (x), • • • , x n (x)) ∈ R n are called local coordinates on U α. Sometimes, when there is no danger of misunderstanding, we also write x = (x 1 , • • • , x n) identifying a point x ∈ M with its representation in some local coordinates. All manifolds in this book are assumed to be compact and connected.

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\int_M f d\mu_{{\it g}_0} 0$ is a suitably small constant. A solution to the equation above can be obtained from a minimizer $u_\lambda$ of certain energy functional associated to the equation. Firstly, we prove that the minimizer $u_\lambda$ exhibits bubbling phenomenon in a certain limit regime as $\lambda \searrow 0$. Then, we show that the analogous phenomenon occurs in the context of $Q$-curvature flow.

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity ∆ m u = ±u α in R n with n 1, m 1, and α ∈ R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and α. Remarkably, we are able to provide necessary and sufficient conditions on the exponent α to guarantee the existence of such solutions in R n. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.

Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, τ = 0. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies a(τ) = τ + o(τ 1+ε) as τ → 0 for some ε > 0. This asymptotic assumption implies a(τ) = τ + o(τ). In this paper, we show that a(τ) = τ + o(τ) is not sufficient to achieve an extension through τ = 0, but it is necessary provided its derivative converges as τ → 0. We also show that the ε in a(τ) = τ + o(τ 1+ε) is not necessary to achieve an extension through τ = 0.

In this paper, by using the topological degree theory for multivalued maps and the bounding function method, we give sufficient conditions for the existence of solutions to a nonlocal problem of differential complementarity systems. An example is given.

Let Gr be a component of the Grassmann manifold of a C *-algebra, presented as the unitary orbit of a given orthogonal projection Gr = Gr(P). There are several natural connections in this manifold, and we first show that they all agree (in the presence of a finite trace in A, when we give Gr the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of P ∈ Gr for the spectral rectifiable distance, and also the conjugate tangent locus of P ∈ Gr along a geodesic. Furthermore, for each tangent vector V at P , we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all. Contents 1. Introduction 2.2. Linear connections 2.3. Levi-Civita connection of the trace 2.4. Complex structure and symplectic form 2.5. Symmetric space structure 2.6. Jacobi fields 3. Cut locus and conjugate locus 3.1. Cut locus 3.2. Conjugate points 3.3. The kernel of D Exp P 3.4. Projective spaces 3.5. Beyond first conjugate point References

We study the optimal control of systems for a class of nonlinear hemivariational inequalities which are in the form of evolutionary inclusions involving Clarke's generalized gradient. The control variables are introduced both in the generalized gradient and in the source terms. We first establish the existence of weak solutions to nonlinear inclusions and prove the upper semicontinuity property of their solution sets. Then, we present the minimization problem and show the existence of optimal admissible state-control pairs. Finally, some examples of our abstract results which appear in applications are discussed.

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

The goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the Dirichlet boundary condition and a convection multivalued term. An existence theorem for positive solutions of the problem is established by applying the method of subsolutionsupersolution, together with truncation and comparison techniques.

In this paper, combining Kirillov's method of orbits with Connes' method in Differential Geometry, we study the so-called MD(5,3C)-foliations, i.e. the orbit foliations of the co-adjoint action of MD(5,3C)-groups. First, we classify topologically MD(5,3C)-foliations based on the classification of all MD(5,3C)-algebras in [22] and the picture of co-adjoint orbits (K-orbits) of all MD(5,3C)-groups in [23]. Finally, we study K-theory for leaf space of MD(5,3C)-foliations and describe analytically or characterize Connes' C *algebras of the considered foliations by KK-functors.