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This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder R n × R + {{\mathbf {R}}^n} \times {{\mathbf {R}}_ + } with initial data in L r {L^r} , n ≥ 3 n \geq 3 , 1 > r > ∞ 1 > r > \infty . An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

We derive a Ray–Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray–Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.

The classical Hartman-Grobman Theorem states that a smooth diffeomorphism F ( x ) F(x) near its hyperbolic fixed point x ¯ \bar x is topological conjugate to its linear part D F ( x ¯ ) DF(\bar x) by a local homeomorphism Φ ( x ) \Phi (x) . In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth F ( x ) F(x) is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a C ∞ C^\infty diffeomorphism F ( x ) F(x) , the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a C 1 C^1 diffeomorphism F ( x ) F(x) with D F ( x ) DF(x) being α \alpha -Hölder continuous at the fixed point that the local homeomorphism Φ ( x ) \Phi (x) is differentiable at the fixed point. Here, α > 0 \alpha >0 depends on the bands of the spectrum of F ′ ( x ¯ ) F’(\bar x) for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on F ( x ) F(x) cannot be lowered to C 1 C^1 .

The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile of the unique generalized traveling wave is of the same recurrence as the media. In particular, if the media is time almost periodic, then so is the wave profile of the unique generalized traveling wave.

In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.

We consider in this paper a general reaction-diffusion equation of the KPP (Kolmogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model will have a family of pulsating waves of constant speed, larger than a critical speed c∗. The family of all supercritical waves attract a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs. In several nontrivial particular cases, we are able to describe the dynamics by an effective equation.

This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 > p > ∞ ) (1>p>\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \widetilde {M}=M_1\times \cdots \times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

In the late 1970s Mumford established Chow stability of smooth unpointed genus g curves embedded by complete linear systems of degree d ≥ 2g + 1, and at about the same time Gieseker established asymptotic Hilbert stability (that is, stability of m th Hilbert points for some large values of m) under the same hypotheses. Both of them then use an indirect argument to show that nodal Deligne-Mumford stable curves are GIT stable. The case of marked points lay untouched until 2006, when Elizabeth Baldwin proved that pointed Deligne-Mumford stable curves are asymptotically Hilbert stable. (Actually, she proved this for stable maps, which includes stable curves as a special case.) Her argument is a delicate induction on g and the number of marked points n; elliptic tails are glued to the marked points one by one, ultimately relating stability of an n-pointed genus g curve to Gieseker’s result for genus g + n unpointed curves. There are three ways one might wish to improve upon Baldwin’s results. First, one might wish to construct moduli spaces of weighted pointed curves or maps; it appears that Baldwin’s proof can accommodate some, but not all, sets of weights. Second, one might wish to study Hilbert stability for small values of m; since Baldwin’s proof uses Gieseker’s proof as the base case, it is not easy to see how it could be modified to yield an approach for small m. Finally, the Minimal Model Program for moduli spaces of curves has generated interest in GIT for 2, 3, or 4-canonical linear systems; due to its use of elliptic tails, Baldwin’s proof cannot be used to study these, as elliptic tails are known to be GIT unstable in these cases. In this paper I give a direct proof that smooth curves with distinct weighted marked points are asymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations. Some of these yield the (coarse) moduli space of Deligne-Mumford stable pointed curves M g,n and Hassett’s moduli spaces of weighted pointed curves M g,A, while other linearizations may give other quotients which are birational to these and which may admit interpretations as moduli spaces. The full construction of the moduli spaces is not contained in this paper, only the proof that smooth curves with distinct weighted marked points are stable, which is the key new result needed for the construction. For this I follow Gieseker’s approach to reduce to the GIT problem to a combinatorial problem, though the solution is very different.

In this paper the "backward-uniqueness property" is ascertained for a two- or three-dimensional, fluid-structure interactive partial differential equation (PDE) system, for which an explicit Co-semigroup formulation was recently given by Avalos and Triggiani (2007) on the natural finite energy space H. (See also their Contemporary Mathematics article of 2007 for a preliminary, simplified, canonical model.) This system of coupled PDEs comprises the parabolic Stokes equations and the hyperbolic Lame system of dynamic elasticity. Each dynamic evolves within its respective domain, while being coupled on the boundary interface between fluid and structure. In terms of said fluid-structure semigroup {e At }, posed on the associated finite energy space H, the backward-uniqueness property can be stated in this way: If for given initial data y 0 ∈ H, e AT y 0 = 0 for some T > 0, then necessarily y 0 = 0. The proof of this property hinges on establishing necessary PDE estimates for a certain static fluid-structure equation in order to invoke the abstract backward-uniqueness resolvent-based criterion by Lasiecka, Renardy, and Triggiani (2001). The backward-uniqueness property for the coupled Stokes-Lame PDE is motivated by, and has positive implications to, the problem of exact controllability (in the hyperbolic state variables {w, w t }) and, simultaneously, approximate controllability (in the parabolic state variable u) of the present coupled PDE model, under boundary control. A similar situation occurred for thermoelastic models as shown in papers by M. Eller, V. Isakov, I. Lasiecka, M. Renardy, and R. Triggiani.

Let Ω C ℝ N be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + 1)-dimensional Riemannian manifold M := {(x, r(x)w): x ∈ ℝ, ω ∈ Ω} with r > 0 and smooth, and the natural metric ds 2 = (1 + r'(x) 2 )dx 2 + r 2 (x)ds 2 Ω . We require that M has conical ends: r(x) = |x| + O(x -1 ) as x → ±∞. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrodinger evolution e itΔ M and the wave evolution e it √ -Δ M are obtained for data of the form f (x, ω) = y n (ω)u(x), where Y n are eigenfunctions of -Δ Ω with eigenvalues u 2 n . In this paper we discuss all cases d + n > 1. If n ≠ 0 there is the following accelerated local decay estimate: with 0 1, ∥ω σ e itΔM Y n f ∥ L ∞ (M) ≤ C (n, M, σ) t - d+1 / 2 -σ ∥w -1 σ f ∥ L 1 (M) , where w σ (x) = 〈x〉 -σ , and similarly for the wave evolution. Our method combines two main ingredients: (A) A detailed scattering analysis of Schrodinger operators of the form -∂ 2 ξ + (v 2 - 1 / 4 )〈ξ〉 -2 + U(ξ) on the line where U is real-valued and smooth with U (l) (ξ) = 0( ξ-3-l ) for all l ≥ 0 as ξ → ±∞ and v > 0. In particular, we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the Wronskian between the outgoing Jost solutions as the energy tends to zero. In particular, the division into Part I and Part II can be explained by the former being resonant at zero energy, where the present paper deals with the nonresonant case. (B) Estimation of oscillatory integrals by (non)stationary phase.

In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map f only needs to be C 1 , without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.

In previous joint work, a theory introduced earlier by Tagare was developed for establishing certain kinds of correspondences, termed bimorphisms, between simple closed regular plane curves of differentiability class at least C 2 C^2 . A class of objective functionals was introduced on the space of bimorphisms between two fixed curves C 1 C_1 and C 2 C_2 , and it was proposed that one define a “best non-rigid match” between C 1 C_1 and C 2 C_2 by minimizing such a functional. In this paper we prove several theorems concerning the nature of the shape-space of plane curves and of spaces of bimorphisms as infinite-dimensional manifolds. In particular, for 2 ≤ j > ∞ 2\leq j>\infty , the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for C ∞ C^\infty curves is the space of bimorphisms an infinite-dimensional manifold, and then only a Fréchet manifold, not a Banach manifold. This paper lays the groundwork for a companion paper in which we use the Nash Inverse Function Theorem and our results on C ∞ C^\infty curves and bimorphisms to show that if Γ \Gamma is strongly convex, if C 1 C_1 and C 2 C_2 are C ∞ C^\infty curves whose shapes are not too dissimilar ( C j C^j -close for a certain finite j j ) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R d . In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in R d can be approximated by a pair of homogeneous polynomials. This conjecture is not resolved yet but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) d = 2, 2) convex surfaces in R d with C 1+ǫ boundary.

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.

Let ( R , m ) (R,\mathfrak {m}) be an n n -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an m \mathfrak {m} -primary ideal a \mathfrak {a} of R R , the relationship between the singularities of the scheme defined by a \mathfrak {a} and those defined by the multiplier ideals J ( a c ) \mathcal {J}(\mathfrak {a}^c) , with c c varying in Q + \mathbb {Q}_+ , are quantified in this paper by showing that the Samuel multiplicity of a \mathfrak {a} satisfies e ( a ) ≥ ( n + k ) n / c n e(\mathfrak {a}) \ge (n+k)^n/c^n whenever J ( a c ) ⊆ m k + 1 \mathcal {J}(\mathfrak {a}^c) \subseteq \mathfrak {m}^{k+1} . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustaţǎ and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

hhIn an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region. This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the Harmonic operator. The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions. In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty. This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty. Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent. In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

Let ( M , ω ) (M, \omega ) be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian S 1 S^1 action such that the fixed point set consists of isolated points or surfaces. Assume dim H 2 ( M ) > 3 H^2(M)>3 . In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six “types”. In this paper, we construct such manifolds with these “types”. As a consequence, we have a precise list of the values of these invariants.

A classical theorem of Boas, Kac, and Krein states that a characteristic function φ with φ(x) = 0 for |x| > T admits a representation of the form φ(x) = ∫u(y)u(y + x) dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complex-valued with u(x) = 0 for |x| ≥ τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If φ is real-valued and even, can the convolution root u be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of φ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on R d is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turan's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if f is a probability density on R d whose characteristic function φ vanishes outside the unit ball, then ∫|x| 2 f(x) dx = -Δφ(0) ≥ 4j 2 (d-2)/2 where j v denotes the first positive zero of the Bessel function J v , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in R 2 does not exist.

We investigate the Dirichlet problem for the parablic equation u t = Δu m ,m > 0, in a non-smooth domain Ω ⊂ R N+1 , N > 2. In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove L 1 -contraction estimation in general non-smooth domains.

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green’s functions for the equations are then random variables. Regularity properties for expectation values of Green’s functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green’s function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial f is smaller than expected. We carry out this study in the case that f is sufficiently non-degenerate with respect to its Newton polyhedron Γ(f), and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial f to the same question about polynomials f μ , where μ are faces of Γ(f) depending on the examined pole and f μ is obtained from f by throwing away all monomials of f whose exponents do not belong to μ. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial f μ , with μ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than expected.

We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any ∈ > 0 and any piece-wise affine map φ ∈ W 1,n (Ω; R n ) with |Dφ(x)| n ≤ Ldet Dφ(x) for almost every x ∈ Ω there exists a map u ∈ W 1,n (Ω; R n ) such that |Du(x)| n = L det Du(x) a.e. x ∈ Ω, u|∂Ω = φ, ∥u-φ∥L n (Ω) < ∈. The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

Let K \mathbf {K} be a field of characteristic 0 0 and let ( G n ( x ) ) n = 0 ∞ (G_{n}(x))_{n=0}^{\infty } be a linear recurring sequence of degree d d in K [ x ] \mathbf {K}[x] defined by the initial terms G 0 , … , G d − 1 ∈ K [ x ] G_0,\ldots ,G_{d-1}\in \mathbf {K}[x] and by the difference equation \[ G n + d ( x ) = A d − 1 ( x ) G n + d − 1 ( x ) + ⋯ + A 0 ( x ) G n ( x ) , for n ≥ 0 , G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots +A_0(x)G_{n}(x), \quad \mbox {for} \,\, n\geq 0, \] with A 0 , … , A d − 1 ∈ K [ x ] A_0,\ldots ,A_{d-1}\in \mathbf {K}[x] . Finally, let P ( x ) P(x) be an element of K [ x ] \mathbf {K}[x] . In this paper we are giving fairly general conditions depending only on G 0 , … , G d − 1 , G_0,\ldots ,G_{d-1}, on P P , and on A 0 , … , A d − 1 A_0,\ldots ,A_{d-1} under which the Diophantine equation \[ G n ( x ) = G m ( P ( x ) ) G_{n}(x)=G_{m}(P(x)) \] has only finitely many solutions ( n , m ) ∈ Z 2 , n , m ≥ 0 (n,m)\in \mathbb {Z}^{2},n,m\geq 0 . Moreover, we are giving an upper bound for the number of solutions, which depends only on d d . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F ( x ) = f ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) F(x)=f(\langle \alpha _{1} , x\rangle , \dots , \langle \alpha _{n} , x\rangle ) where ⟨ α , x ⟩ \langle {\alpha ,x}\rangle denotes the Paley-Wiener-Zygmund stochastic integral ∫ 0 T α ( t ) d x ( t ) \int _{0}^{T} \alpha (t) d x(t) .

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.

Finsler metrics on an open subset in R n with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as Part I. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the initial time is -∞, We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.

In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e. The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in e. 1. THE MODEL AND MAIN RESULT In this paper we present a general technique for establishing analyticity of solutions of systems of partial differential equations which depend analytically on a parameter e. The method works not only for boundary value problems but also for free boundary problems. In this latter context it can be used to establish long time existence of transient solutions, and also to study the existence of spatially asymmetric steady solutions. Since free boundary problems are typically more challenging than their boundary-value counterparts, we shall concentrate here on a free boundary problem from developmental biology, namely, a model of tumor growth. To further exemplify the generality of our approach an instance of a boundary value problem (in a fixed domain) is presented in the last section of the paper. A variety of other problems are amenable to the same analysis, including, in particular, the Hele-Shaw model of fluid flow [11]. Within the last several decades a number of mathematical models have been developed that aimed at describing the evolution of carcinomas (see. e.g., [1, 5, 6, 8, 12, 13] and the references cited there). The main objective of these models has been to qualitatively describe, under various simplifying assumptions, the growth and stability of tumor tissue. Analysis and simulations of such models are helping to assess the relative importance of various mechanisms affecting tumor growth as well as the efficacy of certain cancer treatments. On the other hand, the description of the stationary (dormant) configurations that arise from the models has only been addressed in the case of spherical tumors, but otherwise it remains largely unexplored. In this paper we develop a method for establishing analyticity of Received by the editors August 17, 1999. 1991 Mathematics Subject Classification. Primary 35B32, 35R35; Secondary 35B30, 35B60, 35C10, 35J85, 35Q80, 92C15, 95C15.

We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform

We establish an LP a priori estimate for the Tricomi equation. Our main tool is Mihlin's multiplier theorem combined with well-known estimates of the Newtonian potential. 0. INTRODUCTION The purpose of this paper is to establish an LP a priori estimate for the Tricomi equation in the upper half-space. The Tricomi equation arises in transonic gas dynamics, and is a typical model equation of changing type. It has been extensively investigated from the various viewpoints. The Tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary, which is our viewpoint in this paper. We can formulate the Dirichlet boundary value problem in the upper half-space as follows. 02 + Yacuz=f, in R+ (0.2) u(0,x) = (x), on MT, where R+ {(y, x) : y > 0, x E Rn-} and /x is the Laplacian in the x variable. An L2 a priori estimate can be obtained very easily. Following the presentation in [8], we suppose that u E Co (Rn), and multiply the equation (0.1) by &92U Then, we integrate over R,r to obtain (0.3) IR ( dy2x)2Jj+ V dydx Rn-1 o 9Y2 Rn-1 o a'y 2IRn1l |Vq$ dx + j-1 f ,3 dydx, where Vx stands for the gradient in x E R n-. This yields (0.4) 2U + \F 1 +YX 4Y21 L2(Rn) Dy L2(Rn) L2(Rn) < M(||f 1|L2(Rn) + 11VX01iL2(Rn-1))7 Received by the editors December 30, 1996 and, in revised form, February 10, 1998. 1991 Mathematics Subject Classification. Primary 35J70, 35B45.

The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space. 1. PRELIMINARIES AND STATEMENT OF RESULTS An isometric immersion f: S' -* SV, m > 2, of the Euclidean mr-sphere Sk of constant curvature k into the unit sphere Sv of a Euclidean vector space V is said to be a spherical minimal immersion if f is minimal. By a result of Takahashi [11], f exists if k = m/Ap for some p > 1, where AP = p(p+m1) is the p-th eigenvalue of the spherical Laplacian As7m on Sm := S', and, in this case, each component O f q e V*, of f is a spherical harmonic of order p on Sm; an eigenfunction of Asm with eigenvalue AP. We scale the metric on the domain to curvature one and call f Sm S Sv a spherical minimal immersion of (algebraic) degree p. Because of the scaling, f is a homothetic (minimal) immersion with homothety Ap/m: (1) (f* (X) X f* (Y)) = Ap/m(XI Y), where X, Y are vector fields on Sm. Let HP denote the Euclidean vector space of spherical harmonics of order p on Sm endowed with the L2-scalar product (suitably scaled). The universal example of a spherical minimal immersion is the standard minimal immersion fm,p: Sm SHP whose components (relative to an orthonormal basis in HP ) are orthonormal. For p = 2, fm,2: Sm S-2 is the classical Veronese map. Unless relevant, we suppress m from the notation and write the standard minimal immersion as fp: Sm SEP. A fundamental problem raised by M.DoCarmo and N.Wallach [3] is to find lower bounds for the range dimension of spherical minimal immersions. The main result of this paper, Theorem 4, solves this problem by giving lower bounds in terms of the dimension of the domain, the degree, and differential geometric properties of the immersions such as symmetries and higher order isotropy. The only previously known general lower bound was given by J.D.Moore [10] who proved that, for a spherical minimal immersion f: Sm + Sv of degree > 2, we have dim V > 2m. DoCarmo and Wallach conjectured [3] that the lower bound Received by the editors April 20, 1997. 1991 Mathematics Subject Classification. Primary 53C42. ?)1999 American Mathematical Society

In this paper, we study the topology of spaces of n n -tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the E 2 E^2 term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163–221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.

Motivated by the Blaschke-Santalo inequality, we define for a convex body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52

We study the elliptic equation Δ u + k u − K u p = 0 \Delta u + ku - Ku^{p} = 0 on complete noncompact Riemannian manifolds with K K nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space R n {\mathbf {R}}^{n} and the hyperbolic space H n {\mathbf {H}}^{n} are carried out when k k is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.

An index formula is proved for elliptic systems of P.D.E.’s with boundary values in a simply connected region Ω \Omega in the plane. Let A \mathcal {A} denote the elliptic operator and B \mathcal {B} the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function Δ B + \Delta ^{+}_{{\mathcal {B}}} defined on the unit cotangent bundle of ∂ Ω \partial \Omega was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function Δ B + \Delta ^{+}_{{\mathcal {B}}} have invertible values. In the present paper, the index of ( A , B ) ({\mathcal {A}},{\mathcal {B}}) is expressed in terms of the winding number of the determinant of Δ B + \Delta ^{+}_{{\mathcal {B}}} .

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra. We use jet sheaves of finite order (or equivalently infinitesimal neighbourhood of finite order) and related exterior differential forms to define cohomologies for singularities of mappings and varieties. (See [2], [3],[4], [5],[6],[7] and this paper.) They are new invariances. They were first defined and studied in [2] for smooth functions. In this paper we study jet cohomology HP(QVk. 0) defined by partial differential forms with respect to y for isolated hypersurface singularities.They can be computed by linear algebra (See Theorem 1 and Theorem 2.) and can distinguish singularities whose classical cohomological invariances and Milnor numbers coincide. For example, if X is a quasi-homogeneous hypersurface with isolated singularity 0. QX , is the complex defined by ordinary exterior differential forms. HP(Q> o) = 0, p > 1, and H0(Q0o) = C. (See [1].) But the cohomology groups defined by jets can tell. For example: (1) f = x 5 + x4. Milnor number ,u = 12. If the order of jets k = 3, dimcH0(Q;3-.0) = 6. (2) f = x3 + x7. ,a = 12. If k = 3, dimcHo (Q'V3 .,0) = 7. It is interesting that the cohomological groups defined by jets are determined by higher derivatives of f. (See Theorem 1 and Theorem 2.) All results and proofs in this paper are true not only for complex holomorphic cases but also for real analytic cases. The author would like to express his gratitude to Phillip A. Griffiths for enlightening discussions with him and various supports he and the Institute for Advanced Study offered during the time the author visited IAS. The main results of this paper are as follows. Let F= _~ 1 ak)If | ka! i9xa (

In this paper we consider a family of convex sets in R n \mathbf {R}^{n} , F = { S ( x , t ) } \mathcal {F}= \{S(x,t)\} , x ∈ R n x\in \mathbf {R}^{n} , t > 0 t>0 , satisfying certain axioms of affine invariance, and a Borel measure μ \mu satisfying a doubling condition with respect to the family F . \mathcal {F}. The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of F . \mathcal {F}. This is achieved by showing first a Besicovitch-type covering lemma for the family F \mathcal {F} and then using the doubling property of the measure μ . \mu . The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to F . \mathcal {F}.

This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current I I , of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on I I . These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to ± ∞ \pm \infty , contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165–174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) ∂ t u + u ∂ x u + ∂ x 3 u = f \begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*} on the interval 0 ≤ x ≤ 2 π , t ≥ 0 0\leq x\leq 2\pi , \, t\geq 0 , with periodic boundary conditions (ii) ∂ x k u ( 2 π , t ) = ∂ x k u ( 0 , t ) , k = 0 , 1 , 2 , \begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*} where the distributed control f ≡ f ( x , t ) f\equiv f(x,t) is restricted so that the “volume” ∫ 0 2 π u ( x , t ) d x \int ^{2\pi }_0 u(x,t) dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f f is allowed to act on the whole spatial domain ( 0 , 2 π ) (0,2\pi ) , it is shown that the system is globally exactly controllable, i.e., for given T > 0 T> 0 and functions ϕ ( x ) \phi (x) , ψ ( x ) \psi (x) with the same “volume”, one can alway find a control f f so that the system (i)–(ii) has a solution u ( x , t ) u(x,t) satisfying \[ u ( x , 0 ) = ϕ ( x ) , u ( x , T ) = ψ ( x ) . u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) . \] If the control f f is allowed to act on only a small subset of the domain ( 0 , 2 π ) (0,2\pi ) , then the same result still holds if the initial and terminal states, ψ \psi and ϕ \phi , have small “amplitude” in a certain sense. In the case of closed loop control, the distributed control f f is assumed to be generated by a linear feedback law conserving the “volume” while monotonically reducing ∫ 0 2 π u ( x , t ) 2 d x \int ^{2\pi }_0 u(x,t)^2 dx . The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

Let Ω \Omega be a non-compact complex manifold of dimension n n , ω = ∂ ∂ ¯ Ψ \omega =\partial \overline \partial \Psi a Kähler form on Ω \Omega , and K α ( x , y ¯ ) K_\alpha ( x,\overline y) the reproducing kernel for the Bergman space A α 2 A^2_\alpha of all analytic functions on Ω \Omega square-integrable against the measure e − α Ψ | ω n | e^{-\alpha \Psi } |\omega ^n| . Under the condition \[ K α ( x , x ¯ ) = λ α e α Ψ ( x ) K_\alpha ( x,\overline x)= \lambda _\alpha e^{\alpha \Psi (x)} \] F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on ( Ω , ω ) (\Omega ,\omega ) which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just Ω = C n \Omega = \mathbf {C} ^n and Ω \Omega a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as α → + ∞ \alpha \to +\infty . This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in C n \mathbf {C}^n . Along the way, we also fix two gaps in Berezin’s original paper, and discuss, for Ω \Omega a domain in C n \mathbf {C}^n , a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure | ω n | |\omega ^n| .

This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form \[ ( A ) { m i q ¨ i + ∂ V ∂ q i ( t , q ) = 0 q ( t + T ) = q ( t ) , ∀ t ∈ ℜ . ({\text {A}})\quad {\text { }}\left \{ {\begin {array}{*{20}{c}} {{m_i}{{\ddot q}_i} + \frac {{\partial V}} {{\partial {q_i}}}(t,q) = 0} \\ {q(t + T) = q(t),\quad \forall t \in \Re .} \\ \end {array} } \right . \] where q i ∈ ℜ ℓ , ℓ ⩾ 3 , 1 ⩽ i ⩽ n , q = ( q 1 , … , q n ) {q_i} \in {\Re ^\ell },\ell \geqslant 3,1 \leqslant i \leqslant n,q = ({q_1}, \ldots ,{q_n}) and with V i j ( t , ξ ) {V_{ij}}(t,\xi ) T T -periodic in t t and singular in ξ \xi at ξ = 0 \xi = 0 Under additional hypotheses on V V , when (A) is posed as a variational problem, the corresponding functional, I I , is shown to have an unbounded sequence of critical values if the singularity of V V at 0 0 is strong enough. The critical points of I I are classical T T -periodic solutions of (A). Then, assuming that I I has only non-degenerate critical points, up to translations, Morse type inequalities are proved and used to show that the number of critical points with a fixed Morse index k k grows exponentially with k k , at least when k ≡ 0 , 1 ( mod ℓ − 2 ) k \equiv 0,1( \mod \ell - 2) . The proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by A. Bahri and P. Rabinowitz in their study of the 3 3 -body problem. It uses a recent result of E. Fadell and S. Husseini on the homology of free loop spaces on configuration spaces. The detailed proof is given for the 4 4 -body problem then generalized to the n n -body problem.

Here we study meromorphic foliations with singularities on a smooth compact complex surface. Aim: study the locations of the singularities, using vector bundle techniques and techniques introduced in the cohomological study of projective geometry. Roughly speaking, a meromorphic foliation with singularities on a complex connected manifold S is described by a subsheaf F of the tangent bundle TS with F closed under Lie brackets (see for instance [Gl, G2, G3, G4, G5, GK, GM] or, for the case dim(5') = 2, here in 1.8.1). The singularities of the foliations are the points of S at which F is not a subbundle of TS ; with this definition, the singularities may occur only in codimension at least 2 (but a meromorphic foliation without singularities is not a holomorphic foliation in the classical sense). Note that any rank one subsheaf of TS is involutive and it is contained in a line bundle contained in TS (in general not as subbundle). Thus a rank one meromorphic foliation G with singularities on S is given by a line bundle M on S and a nonzero map M -+ TS (which induces an inclusion as sheaves of M in TS) (up to a scalar factor, of course); one imposes that (s)o has no hypersurface as component (i.e. that M is a saturated subsheaf of TS) ; equivalent^, G is given by M and s e H°(S, TS®M*) with s ? 0 (up to a scalar factor); the singularities of G are the zero locus (s)o of s. Such a foliation is called also a foliation by curves. Thus, for fixed M e Pic(S), we are looking at a linear problem. Many papers are concerned with this case (see e.g. [G2, G3, G4, GK]) from different points of view and with several different kinds of problems in mind. When dim(5) = 2 (as always in this paper) every foliation is a foliation by curves. For the case dim(5) = 2, relevant references are (among others) [G2 and G3]. From now on we assume dim(5) = 2. It happens very seldom that S has a meromorphic foliation without singularities; by definition this happens if and only if TS is an extension of two line bundles. Here we are interested essentially on the possible location (for various M e Pic(iS)) of the singularities associated to some s e H°(S, TS ® M*); very often we will be interested in a general such s. The main problems considered here are: (a) given a zero dimensional scheme Z c S, find if there is seH°{S, TS®M*), 5^0