The manifold structure of maps between open manifolds

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.     

On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ℝn with conical points

In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, D _k ^v u = gk on G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ⁿ (n4) that the Agmon-Miranda maximum principle fails in this cone.     

Smooth approximations without critical points

In any separable Banach space containing c ₀ which admits a C ^k-smooth bump, every continuous function can be approximated by a C ^k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Hermitian and Positive Integrated C-cosine Functions on Banach Spaces

Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on L ^p () (1L ¹(), C₀ , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.     

Lp-Lq DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.     

On the inverse of the generator of a bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let A be the generator of a uniformly bounded C ₀-semigroup in a Banach space B, and let A have a densely defined inverse A ^-1 . We present sufficient conditions on the resolvent (A-I ^-1, Re > 0, under which A ^-1 is also the generator of a uniformly bounded C ₀-semigroup.Dedicated to V. B. Lidskii on the occasion of his eightieth birthdayTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 6–12, 2004Original Russian Text Copyright © by A. M. Gomilko     

The group reduction for bounded cosine functions on UMD spaces

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(−A)^1/2 generates a bounded C ₀-group. The proof uses a transference principle for cosine functions.      

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Four types of bounded wave solutions of CH-γ equation

Recently, many authors have studied the following CH-γ equationut c0ux 3uux - α2(uxxt uuxxx 2uxuxx) γuxxx =0,where α2, c0 and γ are paramters. Its bounded wave solutions have been investigated mainly for the case α2 ＞ 0. For the case α2 ＜ 0, the existence of three bounded waves (regular solitary waves,compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given.In this paper, not only the existence of four types of bounded waves periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case α2 ＜ 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

A. S. Convergence of two-parameter banach space valued martingales and the radon-nikodym property of banach spaces

In this paper, we prove that under theF ₄ conditions, anyL log⁺ L bounded two-parameter Banach spece valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if theF ₄ condition is replaced by the weaker localF ₄ condition. Project supported by the National Natural Science Foundation of China and the State Education Commission Ph. D. Station Foundation     

On the first homology of automorphism groups of manifolds with geometric structures

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category. This research was partially supported by Grant-in-Aid for Scientific Research (No. 16540058), Japan Society for the Promotion of Science. This research was partially supported by Grant-in-Aid for Scientific Research (No. 14540093), Japan Society for the Promotion of Science.     

On the geometry of the space of smooth fibrations

We study geometrical aspects of the space of smooth fibrations between two given manifolds M and B, from the point of view of Fréchet geometry. As a first result, we show that any connected component of this space is the base space of a Fréchet-smooth principal bundle with the identity component of the group of diffeomorphisms of M as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Fréchet principal bundle with structure group the group of diffeomorphisms of the base B.     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

Diffusion times and stability exponents for nearly integrable analytic systems

For a positive integer n and R>0, we set . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H _j ) of the completely integrable Hamiltonian on , with unstable orbits for which we can estimate the time of drift in the action space. These functions H _j are analytic on a fixed complex neighborhood V of , and setting the time of drift of these orbits is smaller than (C(1/ɛ _j )^1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inL ^q () if and only if the set of analytic bounded point evaluations forR ^q (K, ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.     

Residues of higher order and holomorphic vector fields

LetX ₁ andX ₂ be two holomorphic vector fields on a manifoldV with complex dimensionp. Assume that they have the same singular set . For all , it is known (after Chern-Bott) that each of the vector fields defines a residual characteristic classC ₁(V,X ₁)(resp.C ₁(V,X ₂)) inH ^2p (V, V-), which is a lift of the usual characteristic classC ₁ (V) of the tangent bundle. The differenceC ₁ (V,X ₂)-C ₁ (V,X ₁) belongs then to the image of in the exact sequence. In fact, there exists a canonical liftC ₁ (V,X ₁,X ₂) of this difference inH ^2p–1(V-): we will call itthe residual class of order 2 (associated toI, X ₁ andX ₂). This class is localized near the points whereX ₁ andX ₂ are colinear: we will explain this precisely in terms of Grothendieck residues. The formula that we obtain can be interpreted as a generalization of the purely algebraic identity, obtained from the general one as a byproduct: where ( ₁, , _p) and ( ₁,, _p ) denote two families of non-zero complex numbers, such that all denominators in this formula do not vanish. (This identity corresponds in fact to the case whereX ₁ andX ₂ are non-degenerate at the same isolated singular point.)If the _i 's (1ip) depend now differentiably (resp. holomorphically) on a real (resp. complex) parametert then, denoting by the derivative with respect tot, and assuming all numbers lying in a denominator not to be 0, we can deduce from the above identity the following derivation formula:     

R 0-algebras and weak dually residuated lattice ordered semigroups

We introduce the notion of weak dually residuated lattice ordered semi-groups (WDRL-semigroups) and investigate the relation between R ₀-algebras and WDRL-semigroups. We prove that the category of R ₀-algebras is equivalent to the category of some bounded WDRL-semigroups. Moreover, the connection between WDRL-semigroups and DRL-semigroups is studied.     

Feedback operator for a pseudoparabolic optimal control problem

The feedback operator of a linear pseudoparabolic problem with quadratic criterion is obtained by decoupling of the optimality condition. The feedback operator is shown to be related to the solution of a Riccati equation formulated in theB*-algebra of bounded linear operators onL ²(). This approach shows that the linear feedback operator may be considered as a bounded operator fromL ²() intoH ₀ ¹ (). Finally, we give a theorem establishing the convergence behavior for the feedback operators for these problems as they formally approach an analogous problem of parabolic type.This work was supported in part by the National Science Foundation, Grant No. MCS-7902037.