presentations of computably enumerable reals we study the relationship between a computably enumerable real and its presentations ways of approximating the real by enumerating a prefix free set of binary strings

schwarz and gruss type inequalities for c seminorms and positive linear functionals on banach modules abstract let a be a unital banach algebra a c seminorm or a positive linear functional on a and x be a semi inner product a module we define a real function on x by x x x 1 2 and show that the schwarz inequality holds therefore x is a semi hilbert a module we also obtain some gruss type inequalities for c seminorms and positive linear functionals on a

a one dimensional bingham flow a bingham fluid is a non newtonian fluid which behaves like a rigid body when the shear stress t is less than a threshold value to while it behaves like a viscous fluid when the stress exceeds ro and for which the relationship between stress and strain rate rr is linear where q is the viscosity a behaviour like that of bingham fluids can be observed in the materials used in the industrial processes for fabrication of ceramics and paper or in many types of slurries a one dimensional evolution model describing the flow of a bingham fluid was formulated in l where the equations governing the flow are the navier stokes equation in the viscous region the field equation and the equation of motion for the rigid part playing the role of a free boundary condition the boundary of the rigid core is the free boundary where we have t ro i e zero strain rate the problem was formulated in a weak form by duvaut and lions 2 as a variational inequality they studied an initial value problem and obtained weak solutions whose regularity was proved in the two dimen sional case in that case they also proved the convergence of the solution of the problem for bingham fluids to the one for newtonian fluids when the shear stress tends to the threshold value the existence and uniqueness in a certain function class of a strong solu tion to the variational inequality associated to the flow in the plane and in a bounded three dimensional domain have been established in 3 4 the steady state motion of a bingham fluid in cylindrical pipes subject to a given constant pressure gradient has been investigated in is 6 7 i27

flow of a bingham fluid in contact with a newtonian fluid abstract we study the problem modeling the flow of a bingham fluid in contact with a newtonian fluid playing the role of lubricant this is a free boundary problem coupled by means of diffraction conditions with a boundary value problem of parabolic type we examine the steady state solutions the evolutive case with particular regard for the asymptotic behaviour of the solution and a regularized model related to the appearance of a new rigid zone

on some refinement of the cauchy schwarz inequality abstract if a and b are positive semidefinite operators on a hilbert space and if is an operator mean in the sense of kubo and ando then the operator inequality a b a b 1 2 a b a b a b a b 1 2 a b b a holds this inequality is a generalization of some refinement of the cauchy schwarz inequality

on the cauchy problem associated with the motion of a bingham fluid in the plane on considere un probleme a valeur initiale pour l inegalite variationnelle qui decrit le mouvement d un fluide de bingham dans le plan on etablit l existence d une solution forte

chaitin ω numbers solovay machines and godel incompleteness computably enumerable c e reals can be coded by chaitin machines through their halting probabilities tuning solovay s construction of a chaitin universal machine for which zfc if arithmetically sound cannot determine any single bit of the binary expansion of its halting probability we show that every c e random real is the halting probability of a universal chaitin machine for which zfc cannot determine more than its initial block of 1 bits as soon as you get a 0 it is all over finally a constructive version of chaitin information theoretic incompleteness theorem is proven

reverse schwarz inequalities and some consequences in inner product spaces abstract necessary and sufficient conditions for a general reverse of schwarz s inequality in inner product spaces are given results are applied to the triangle inequality and gruss type inequality

cauchy schwarz inequality in semi inner product c modules via polar decomposition abstract by virtue of the operator geometric mean and the polar decomposition we present a new cauchy schwarz inequality in the framework of semi inner product c modules over unital c algebras and discuss the equality case we also give several additive and multiplicative type reverses of it as an application we present a kantorovich type inequality on a hilbert c module

on a parabolic free boundary problem arising from a bingham like flow model with a visco elastic core abstract in this paper we study a two phase one dimensional free boundary problem for parabolic equation arising from a mathematical model for bingham like fluids with visco elastic core presented in l fusi a farina a mathematical model for bingham like fluids with visco elastic core z angew math phys 55 2004 826 847 the main feature of this problem consists in the very peculiar structure of the free boundary condition not allowing to use classical tools to prove well posedness local existence is proved using a fixed point argument based on schauder s theorem uniqueness is proved using a non standard technique based on a weak formulation of the problem

a generalization of gruss s inequality in inner product spaces and applications a generalization of the classical gruss s integral inequality in inner product spaces is given some applications for positive linear functionals and integrals are also pointed out

relatively recursive reals and real functions abstract intuitively a real number is recursive if we can get as accurate an approximation as we like using a mechanical procedure such as a turing machine a real function is recursive if its value at a point x in its domain can be approximated effectively given an approximation to x however since there are only countably many turing machines there must be uncountably many non recursive reals and functions in this paper we study some of these non recursive reals and functions using a more recursion theoretic approach via the degree of unsolvability in particular we are interested in reals and real functions that are relatively recursive in where is the jump of the recursive degree inspired by the shoenfield limit lemma we show that a real is recursive if and only if it is the limit of a recursive sequence of rationals we then give three characterizations of a recursive function which are stated in terms of turing machine uniform convergence and sequential computability with uniform continuity a proof of their equivalence using a finite injury priority argument is given with the new definitions we can now give an upper bound to the difficulty of some uncomputable analysis operator such as differentiations and root findings

a treatment of the cauchy schwarz inequality in c modules abstract we study the cauchy schwarz and some related inequalities in a semi inner product module over a c algebra a the key idea is to consider a semi inner product a module as a semi inner product a module with respect to another semi inner product in this way we improve some inequalities such as the ostrowski inequality and an inequality related to the gram matrix the induced semi inner products are also related to the notion of covariance and variance furthermore we obtain a sequence of nested inequalities that emerges from the cauchy schwarz inequality as a consequence we derive some interesting operator theoretical corollaries in particular we show that the sequence arising from our construction when applied to a positive invertible element of a c algebra converges to its inverse

modelling of bingham like fluids with deformable core we present here two extensions of the usual bingham model in which we account for deformations in the region where the shear is below the threshold t o in particular we present two models that consider the material in the above region as a neo hookean solid or as a visco elastic upper convected fluid both models are developed in the general 3 d framework of the natural configurations theory after formulating the basic constitutive equations we focus on two 1 d problems shear flow driven by a constant stress and flow in a channel driven by a constant pressure gradient for both the models the resulting mathematical problems are free boundary problems involving parabolic and hyperbolic equations

from axiomatics to intrinsic characterization some open problems in computable analysis abstract five open problems are presented the framework for these problems is the notion of a computability structure on a banach space which is discussed below the study of banach spaces in particular the study of hilbert space is an important topic in mathematics and its applications problems a and b are concerned with transfer principles for obtaining effective versions of classical theorems of mathematics problems c and d deal with separability and effective separability problem e is concerned with computability for classes of mathematical structures other than banach spaces

on the initial boundary value problem for a bingham fluid in a three dimensional domain the initial boundary value problem associated with the motion of a bingham fluid is considered the existence and uniqueness of strong solution is proved under a certain assumption on the data it is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time periodic o introduction the purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a bingham fluid in a bounded three dimensional domain a bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached since the motion is governed by two entirely different stress strain relations depending on the state of stresses the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the 1 rlglcl zone the initial boundary value problem we shall study is formulated as 0 l it uj u a u w u b u u w j w j u f w a in 0 t for each test function w such that v w 0 in q and w 0 on aq 0 2 v u 0 in q x 0 t 0 3 u 0 on aq x 0 t 0 4 8 x 0 vo x in q here q is a bounded domain in r3 with smooth boundary aq a x t denotes the velocity of the fluid and f x t stands for external force we assume that the density the yield limit and the viscosity are positive constants in particular the received by the editors august 15 1986 1980 mathematics subject classijication 1985 revision primary 35q20 35b10 35b40 35b65 secondary 76a05 76d99 this research was supported by the air force office of scientific research under grant afosr86 0085 and by nsf grant dms 8521848 r 1987 american mathematical society 0002 9947 87 1 00 25 per page

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

an extension of kantorovich inequality to n operators via the geometric mean by ando li mathias abstract in this paper we shall extend kantorovich inequality this is an estimate by using the geometric mean of n operators which have been defined by ando li mathias in t ando c k li r mathias geometric means linear algebra appl 385 2004 305 334 as a related result we obtain a converse of arithmetic geometric means inequality of n operators via kantorovich constant

on the continued fraction representation of computable real numbers abstract the continued fraction representation of real numbers is compared with other types of representations of real numbers in the context of recursive analysis the main result states that a modification of the natural continued fraction representation based on the concept of principal convergents of real numbers is polynomially equivalent to the left cut representation in the sense that for any given real number x the two representations of x are computable from each other in polynomial time following from earlier studies on the left cut representation of real numbers this result verifies the intuition that there is no efficient algorithm for implementing addition of real numbers in the continued fraction form on the other hand when considering computable real functions the continued fraction representation behaves differently from the left cut representation a computable real function must be continuous if it is defined as a mapping on cauchy sequences of real numbers it must be left continuous but is not necessarily continuous if defined as a mapping on left cuts and it ist necessarily left or right continuous if defined as a mapping on continued fractions

an algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees on donne une decomposition definissable du semi treillis superieur des degres recursivement enumerables r comme l union disjointe d un ideal m et un filtre fort nc

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

uniqueness of the bounded solution to a strongly degenerate parabolic problem abstract this paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem the equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies by using holmgren s approach we prove that the bounded solution of the associated initial boundary value problem is unique under some essentially necessary condition on the growth of the convection

dense imbedding of test functions in certain function spaces in a recent paper 1 j u kim studies the cauchy problem for the motion of a bingham fluid in r2 he points out that the extension of his results to three dimensions depends on proving the denseness of c functions with compact support in certain spaces in this note such a result is proved following kim s notation 1 we define the following spaces p rn u e w 22 rn ivu e lp rn f rn u e w 2 r nivu e lp rn nxn divu g r e w 2 r nic u vu vu e lp r x divu 0 s rn u e co rn nidivu 0 according to kim s lemma 1 7 1 fp gp for 1 n9 where on denotes the volume of the unit ball in rn let 1 r oo and v e lr rn if r 1 assume in addition that jrn v 0 then on v o 0 in lr rn as n x o received by the editors november 22 1985 1980 mathematics subject classification primary 46e35