the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

optimal l p l q estimates for parabolic boundary value problems with inhomogeneous data in this paper we investigate vector valued parabolic initial boundary value problems mathcal a t x d mathcal b j t x d subject to general boundary conditions in domains g in mathbb r n with compact c 2m boundary the top order coefficients of mathcal a are assumed to be continuous we characterize optimal l p l q regularity for the solution of such problems in terms of the data we also prove that the normal ellipticity condition on mathcal a and the lopatinskii shapiro condition on mathcal a mathcal b 1 dots mathcal b m are necessary for these l p l q estimates as a byproduct of the techniques being introduced we obtain new trace and extension results for sobolev spaces of mixed order and a characterization of triebel lizorkin spaces by boundary data

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

pseudodifferential operators on bochner spaces and an application we consider pseudodifferential operators with operator valued symbols a x acting on a umd banach space x assuming some regularity of holder type in x and mihlin type in we prove l p n x boundedness of such operators this result is then applied to the study of l p maximal regularity for nonautonomous parabolic evolution equations

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

a family of exact travelling wave solutions to nonlinear evolution and wave equations a family q of the multicomponent special functions is defined for obtaining the exact travelling wave solutions to nonlinear evolution and wave equations it is shown that the functions qn from q for some n 2 4 are closely related to the special unitary groups su n the necessary and sufficient conditions for existence of a family of the exact multicomponent travelling wave solutions to a quasilinear evolution equation are given an efficiency of the method based on q functions is demonstrated on several classes of the nonlinear partial differential equations

sums of regular self adjoint operators in hilbert c modules abstract we introduce a notion of weak anticommutativity for a pair s t of self adjoint regular operators in a hilbert c module e we prove that the sum s t of such pairs is self adjoint and regular on the intersection of their domains a similar result then holds for the sum s 2 t 2 of the squares we show that our definition is closely related to the connes skandalis positivity criterion in kk theory as such we weaken a sufficient condition of kucerovsky for representing the kasparov product our proofs indicate that our conditions are close to optimal

functional calculus for semigroup generators via transference abstract in this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators in particular it is proved that if a generates a c 0 semigroup on a hilbert space then for each 0 the operator a has a bounded calculus for the closed ideal of bounded holomorphic functions on a sufficiently large right half plane that satisfy f z o e re z as z the bound of this calculus grows at most logarithmically as 0 as a consequence f a is a bounded operator for each holomorphic function f on a right half plane with polynomial decay at then we show that each semigroup generator has a so called strong m bounded calculus for all m n and that this property characterizes semigroup generators similar results are obtained if the underlying banach space is a umd space upon restriction to so called bounded semigroups the hilbert space results actually hold in general banach spaces

sharp interface limit for a stokes allen cahn system we consider the sharp interface limit of a coupled stokes allen cahn system when a parameter epsilon 0 that is proportional to the thickness of the diffuse interface tends to zero in a two dimensional bounded domain for sufficiently small times we prove convergence of the solutions of the stokes allen cahn system to solutions of a sharp interface model where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two phase stokes system with an additional contribution to the stress tensor which describes the capillary stress to this end we construct a suitable approximation of the solution of the stokes allen cahn system using three levels of the terms in the formally matched asymptotic calculations and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized allen cahn operator moreover a careful treatment of the coupling terms is needed

on measuring unboundedness of the h calculus for generators of analytic semigroups we investigate the boundedness of the h infty calculus by estimating the bound b e of the mapping h infty b x f f a t e for e near zero here a generates the analytic semigroup t and h infty is the space of bounded analytic functions on a domain strictly containing the spectrum of a we show that b e o loge in general whereas b e o 1 for bounded calculi this generalizes a result by vitse and complements work by haase and rozendaal for non analytic semigroups we discuss the sharpness of our bounds and show that single square function estimates yield b e o loge

a solution to the problem of l p maximal regularity we give a negative solution to the problem of the l p maximal regularity on various classes of banach spaces including l q spaces with 1 q neq 2 infty

finding all maximally matchable edges in a bipartite graph we consider the problem of finding all maximally matchable edges in a bipartite graph g v e i e all edges that are included in some maximum matching we show that given any maximum matching in the graph it is possible to perform this computation in linear time o n m where n v and m e hence the time complexity of finding all maximally matchable edges reduces to that of finding a single maximum matching which is o n 1 2m hopcroft and karp 12 or o n logn 1 2m for dense graphs with m q n 2 alt et al 2 this time complexity improves upon that of the best known algorithms for the problem which is o nm costa 5 for bipartite graphs and carvalho and cheriyan 6 for general graphs other algorithms for solving that problem are randomized algorithms due to rabin and vazirani 15 and cheriyan 3 the runtime of which is o n 2 3 7 6 our algorithm apart from being deterministic improves upon that time complexity for bipartite graphs when m o n r and r 1 876 in addition our algorithm is elementary conceptually simple and easy to implement

matrix solutions of wave equations and clifford algebras we are extending the formation of matrix solutions un for linear and nonlinear wave equations by construction of unitary anti hermitian anti commuting matrices up to the eighth order we use clifford algebras c 0 n with periodicity in modulo 8 to construct coupled matrix solutions we also propose to use the matrix solutions for describing the intrinsic rotations of particles

the flux theory of gravitation v the mathematics of the new physics the mathematics of the new physics developed in the series 11 14 that we summarize here is quite extensive however the reconstruction of the reals without the axiom of choice is the principle focus of this paper combined with the enrichment of the reals by completion in the standard metric algebraic extension and admission of a topological axiom the new reals is established and well defined it is further extended to the new nonstandard analysis which together has the following achievements developed in the series 1 countably infinite counterexamples to fermat s last theorem disproving the conjecture 2 solution of the problem of natural ordering of the new reals proof of goldbach s conjecture 13 resolution of the banach tarski and brouwers paradoxes 13 dynamic modeling of dark matter and the superstring and 3 characterization of undecidable propositions 14 which paved the way for the dynamic methodology the paper also introduces the calculus of set valued functions including the generalized integral and its application to quantum gravity with the dynamic methodology and its main component qualitative mathematics as alternative to the descriptive pragmatic methodology of physics major discoveries are achieved 1 the solution of the turbulence modeling and gravitational n body problems 11 20 23 2 discovery of ambiguous set e g chaos and proof that the real line is chaos 20 23 3 development of generalized fractal 15 23 24 algorithm for macro and quantum interactions 15 18 22 and development of the flux theory of gravitation the solution of modeling problem when the cosmos is the given physical system its adaptation to earth is the theory of turbulence a major achievement of qualitative mathematics the flux theory of gravitation unites all natural forces and interactions solves all the problems of physics and resolves its paradoxes and unanswered questions

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

the flux theory of gravitation xvii the new mathematics and physics the main contribution of the paper is the critique of j m henle s non nonstandard analysis the mathematical intelligencer 21 1 1999 67 the principal criticism of henle s work is the failure to do away with the axiom of choice as he claims he would therefore it suffers from the same flaw that robinson s nonstandard analysis has nonstandard analysis north holland amsterdam 1966 however the paper also summarizes the new mathematics and physics generated by the resolution of fermat s last theorem nonlinear studies 5 2 1998 227 and the solution of the gravitational n body problem nonlinear analysis 30 8 1997 5021 respectively including the resolution of the problems paradoxes contradictions and unanswered questions of mathematics and physics except the bieberbach s conjecture and the riemann hypothesis highlighted in the paper are astonishing results coming from the new real line i e the reconstructed real line as well as the proof of goldbach s conjecture and the natural ordering of the new real line which does not exist in the real line

multilinear operator valued calderon zygmund theory abstract we develop a general theory of multilinear singular integrals with operator valued kernels acting on tuples of umd banach spaces this in particular involves investigating multilinear variants of the r boundedness condition naturally arising in operator valued theory we proceed by establishing a suitable representation of multilinear operator valued singular integrals in terms of operator valued dyadic shifts and paraproducts and studying the boundedness of these model operators via dyadic probabilistic banach space valued analysis in the bilinear case we obtain a t 1 type theorem without any additional assumptions on the banach spaces other than the necessary umd higher degrees of multilinearity are tackled via a new formulation of the rademacher maximal function rmf condition in addition to the natural umd lattice cases our rmf condition covers suitable tuples of non commutative l p spaces we employ our operator valued theory to obtain new multilinear multi parameter operator valued theorems in the natural setting of umd spaces with property

geometrical properties of matrix solutions of the nonlinear klein gordon equation we have constructed some matrix solutions of a nonlinear klein gordon equation and proposed a relation between these solutions and su n matrix groups we have also established the correspondence between the solutions and the rotations around fixed vectors whose endpoints form an octahedron

bounded h infty calculus for a class of nonlocal operators the bidomain operator in the l_q setting the bidomain operator mathbb a a nonlocal operator is studied in a bounded domain omega subset mathbb r d with boundary partial omega of class c 2 within the l q setting for 1 q infty assuming a fairly general framework it is shown that this operator is sectorial invertible on functions with mean zero and admits a bounded h infty calculus with h infty angle 0 in particular it has the property of maximal l p l q regularity and the fractional power domains mathcal d mathbb a alpha are identified furthermore it is shown that mathbb a generates a strongly continuous compact analytic and exponentially stable semigroup on l q 0 omega with q independent purely countable point spectrum sigma mathbb a