a topos foundation for theories of physics iv categories of systems this paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics the motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos the previous papers in this series are concerned with implementing this program for a single system in the present paper we turn to considering a collection of systems in particular we are interested in the relation between the topos representation for a composite system and the representations for its constituents we also study this problem for the disjoint sum of two systems our approach to these matters is to construct a category of systems and to find a topos representation of the entire category

further inequalities for the gamma function for x 0 and k 0 we present a method which permits us to obtain inequalities of the type k a x 1 r k x r k 1 k x 1 with the usual notation for the gamma function where a and are independent of k some examples are also given which improve well known inequalities finally we are also able to show in some cases that the values a and in the inequalities that we obtain cannot be improved

multiple correction and continued fraction approximation abstract the main aim of this paper is to further develop a multiple correction method formulated in a previous work 6 as applications we find a kind of hybrid type finite continued fraction approximations in two cases of landau constants and lebesgue constants in addition we refine the previous results of lu 29 and xu and you 48 for the euler mascheroni constant

on gamma function inequalities on montre que certaines fonctions comportant les quotients de fonctions gamma sont completement monotones on en deduit des inegalites on etablit aussi la divisibilite infinie de plusieurs distributions de probabilite dont les transformees de laplace comportent des quotients de fonctions gamma

unified treatment of several asymptotic expansions concerning some mathematical constants recently various approximation formulas for some mathematical constants have been investigated and presented by many authors in this paper we first find that the relationship between the coefficients pj and qj is such that xj 0qjxj ln xj 0pjxj x where is the logarithmic derivative of the gamma function often referred to as psi function and p0 q0 1 next by using this result we give a unified treatment of several asymptotic expansions concerning the eulermascheroni constant landau and lebesgue constants glaisherkinkelin constant and choisrivastava constants

kolmogorov landau inequalities for monotone functions where llfll ess su if t kolmogorov 2 suceeded in finding the best possible constants in 1 where the norm is as in 2 except for isolated results no new progress was made until schoenberg and cavaretta 3 found the best possible constants for llfll su if f we propose to prove inequalities about various clases of monotone functions which yield as special cases pointwise inequalities of the form 1 for example

some new asymptotic approximations of the gamma function based on nemes formula ramanujan s formula and burnside s formula in this paper we construct some new approximations of the gamma function based on nemes formula ramanujan s formula and burnside s formula using these approximations some inequalities are established finally for demonstrating the superiority of our new approximation over mortici s formula and other classical ones some numerical computations are also given

non semi regular quantum groups coming from number theory in this paper we study c algebraic quantum groups obtained through the bicrossed product construction examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi regular the crossed product of the quantum group acting on itself by translations does not contain any compact operator we describe all corepresentations of these quantum groups and the associated universal c algebras on the way we provide several remarks on c algebraic properties of quantum groups and their actions

on inequalities and asymptotic expansions for the landau constants we demonstrate how arbitrarily sharp asymptotic expansions for the landau constants can be derived and we prove some related inequalities the main tool used is brouncker s continued fraction formula we also show that the implied series expansion of the landau constants is in fact divergent

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

completely monotonic degrees for a difference between the logarithmic and psi functions abstract in the paper the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions secondly compute completely monotonic degree of the above mentioned function and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical euler gamma function

on stirling s formula remainder let the sequence r n be defined by n 2 n n e n e r n we establish new estimates for stirling s formula remainder r n this improves some known results let x be defined by the relation x 1 2 x e x e x 12 x we prove that x is completely monotonic on 0 this implies the result given by mortici who proved that x 1 x is completely monotonic on 0

sharp bounds for gamma function in terms of x x 1 the aim of this paper is to present new approximations for the gamma function in terms of x x 1 the monotonicity and convexity of some involved functions are studied as applications sharp bounds are presented it is also solved an open problem posed by mortici 2013

inequalities for the harmonic numbers we present various inequalities for the harmonic numbers defined by h n 1 1 2 ldots 1 n n in bf n one of our results states that we have for all integers n 2 alpha frac log log n gamma n 2 leq h n 1 n h n 1 1 n 1 beta frac log log n gamma n 2 with the best possible constant factors alpha frac 6 sqrt 6 2 sqrt 3 396 3 log log 2 gamma 0 0140 ldots quad mbox and quad beta 1 here denotes euler s constant

supplements to a class of logarithmically completely monotonic functions associated with the gamma function abstract in this article the authors present a necessary condition a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic as a consequence of these results supplements to the recent investigation by chen and qi c p chen f qi logarithmically completely monotonic functions relating to the gamma function j math anal appl 321 2006 405 411 are provided and a new keckic vasic type inequality is concluded

multiple correction and continued fraction approximation ii the main aim of this paper is to further develop the multiple correction method formulated in our previous works cao et al 2015 cao 2015 as applications we establish a kind of hybrid type finite continued fraction approximations related to bbp type series of the constant and other classical constants such as catalan constant or 2

a new characterisation of principal actions abstract the notion of a principal group action is generalized to the framework of non commutative geometry through consideration of a certain bimodule morphism this definition is both completely general and immediately applicable indeed we prove that the dual coaction associated to a kac system is principal in our generalized sense and consider some simple non commutative examples

self adjoint operators as functions i lattices galois connections and the spectral order observables of a quantum system described by self adjoint operators in a von neumann algebra or affiliated with it in the unbounded case form a conditionally complete lattice when equipped with the spectral order using this order theoretic structure we develop a new perspective on quantum observables

a topos foundation for theories of physics i formal languages for physics this paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we discuss two different types of language that can be attached to a system s the first is a propositional language pl s the second is a higher order typed language l s both languages provide deductive systems with an intuitionistic logic the reason for introducing pl s is that as shown in paper ii of the series it is the easiest way of understanding and expanding on the earlier work on topos theory and quantum physics however the main thrust o

some properties of the divided difference of psi and polygamma functions abstract let n 1 n 1 n for n 0 where n stands for the psi and polygamma functions for p q r and min p q let d x p x q n 1 n x be the divided difference of the functions n 1 for x in this paper we establish the necessary and sufficient conditions for the function n x n 1 x 2 n x n 2 x to be completely monotonic on in particular we find that the function n 1 2 n n 2 is strictly decreasing from 0 onto n n 1 n 1 n 2 these not only generalize and strengthen some known results but also yield many new and interesting ones

from multiplicative unitaries to quantum groups ii abstract it is shown that all important features of a c algebraic quantum group a defined by a modular multiplicative w depend only on the pair a rather than the multiplicative unitary operator w the proof is based on thorough study of representations of quantum groups as an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary without assuming existence of haar weights

a comparison of two topos theoretic approaches to quantum theory the aim of this paper is to compare the two topos theoretic approaches to quantum mechanics that may be found in the literature to date the first approach which we will call the contravariant approach was originally proposed by isham and butterfield and was later extended by doring and isham the second approach which we will call the covariant approach was developed by heunen landsman and spitters

spectral order for unbounded operators abstract the spectral order a notion originated by olson for bounded operators is investigated here in the context of unbounded operators dissimilarities between bounded and unbounded cases are pointed out new criteria for two operators to be comparable are supplied a way of reducing the study of the spectral order to the case of bounded operators is proposed connections with essential selfadjointness are established integral inequalities for monotonically increasing functions are characterized in terms of distribution functions some illustrative examples are furnished

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

sharp bounds of the landau constants the aim of this paper is to establish new bounds of the landau constants 1 introduction and motivation the landau constants de ned for all positive integers n by gn 1 1 2 2 1 3 2 4 2 1 3 5 2 4 6 2 1 3 2n 1 2 4 2n 2 play an important role in some extremal problems in complex analysis and in the theory of fourier series more precisely in 1913 landau 6 proved that gn is the maximum of the expression j pn k 1 akj with respect to all functions of the form f z p1 k 0 akz k which is analytic in the unit disk and satis es jf z j 1 for every jzj 1 in consequence the problem of approximation of the landau constants have attracted the attention of many authors in particular landau himself studied the asymptotic behaviour of gn and showed that gn 1 lnn then watson 7 established the following asymptotic formula gn c0 1 ln n 1 1 4 n 1 o 1 n2 n 1 here and in what follows c0 1 4 ln 2 1 06627 and 0 577215 is the euler mascheroni constant this problem of approximation of the landau constants was continued in the works of brutman 3 and falaleev 5 who proved that for every non negative integer n 1 1 ln n 1 gn 1 0663 1 ln n 1 respective 1 0662 1 ln n 3 4 gn 1 0916 1 ln n 3 4 we improve the upper bound in the following way that also shows that the constant 3 4 is the best possible date november 18 2009 1991 mathematics subject classi cation primary 41a60 secondary 26d15

some gamma function inequalities a class of completely monotonic functions are presented involving the gamma function as well as the derivative of the psi function as a consequence new upper and lower bounds for the ratio f x 1 f x s are obtained and compared with related bounds given in part by j d keckic and p m vasic our results are further applied to obtain functions which are laplace transforms of infinitely divisible probability measures

exponential gamma and polygamma functions simple proofs of classical and new inequalities abstract some inequalities for the exponential gamma and polygamma functions are established by means of the mean value theorem of differential calculus in particular the cases of the ratio of gamma and polygamma functions are considered in some cases the results are new in some others classical results are recovered or extended

operator equalities related to the quantum e 2 group the paper deals with normal operatorsr ands satisfying simple commutation relations that we encounter investigating the quantum deformation of thee 2 group we show thatr s admits a normal extension if and only ifr 1s satisfies a certain spectral condition a number of related formulae are derived in particular all the functionsf satisfying the character equationf r s f r f s are found

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

a topos foundation for theories of physics ii daseinisation and the liberation of quantum theory this paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we study in depth the topos representation of the propositional language pl s for the case of quantum theory in doing so we make a direct link with and clarify the earlier work on applying topos theory to quantum physics the key step is a process we term daseinisation by which a projection operator is mapped to a subobject of the spectral presheaf the topos quantum analog of a classical state space in the second part of the paper we change gear

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

a new approach to induction and imprimitivity results given a closed quantum subgroup of a locally compact quantum group we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group more generally we study induction given a coaction of the quantum subgroup on a c algebra we prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups this means that we deflne quantum homogeneous spaces as c algebras and that we prove morita equivalence of crossed products and homogeneous spaces we essentially use von neumann algebraic techniques to prove these morita equivalences between c algebras the theory of induced representations of locally compact l c groups was introduced by mackey 21 who discovered the imprimitivity theorem characterizing induced representations through the presence of a covariant representation of a homogeneous space riefiel 26 provided a modern approach using the language of hilbert c modules after the work of riefiel several induction procedures and imprimitivity results have been obtained both for actions and for coactions of groups we shall brie y review them below the purpose of this paper is to develop such an induction and imprimitivity machinery in the setting of locally compact quantum groups 16 17 in this way we provide a unifled approach to several results on actions and coactions of groups on c algebras at the same time our proofs in the setting of l c quantum groups are simpler than the classical proofs dealing with coactions of groups another motivation comes from quantum group theory meyer and nest have undertaken a reformulation of the baum connes conjecture 24 in which induction and restriction play a crucial role the development of induction and imprimitivity in the current paper should play an equally important role in the formulation of a baum connes conjecture for quantum groups

inequalities for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 by gn k 0n1 16k 2 and ln 12 sin n t sin t dt respectively we establish sharp inequalities for gn and ln 2 in terms of the logarithmic derivative of the gamma function further we prove that the sequence gn is completely monotonic we provide best possible upper and lower bounds for the ratios gn 1 gn 1 gn and l n 1 2 l n 1 2 ln 2 and we present sharp bounds for ln 2 gn and ln 2 gn

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

multidimensional spectral order for selfadjoint operators abstract the notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in hilbert space it is shown that the multidimensional spectral order is preserved by transformations represented by spectral integrals of separately increasing borel functions on r in particular the dimensional spectral order is the restriction of product of spectral orders for selfadjoint operators in the context of positive tuples of pairwise commuting selfadjoint operator the relation a b holds if and only if a b for every tuples of nonnegative integer numbers

on a certain approach to quantum homogeneous spaces we propose a definition of a quantum homogeneous space of a locally compact quantum group we show that classically it reduces to the notion of homogeneous spaces giving rise to an operator algebraic characterization of the transitive group actions on the quantum level our definition goes beyond the quotient case providing a framework which besides the vaes quotient of a locally compact quantum group by its closed quantum subgroup our main motivation is also compatible with generically non quotient quantum homogeneous spaces of a compact quantum group studied by p podle as well as the rieffel deformation of g homogeneous spaces finally our definition rules out the paradoxical examples of the non compact quantum homogeneous spaces of a compact quantum group

vigier s theorem for the spectral order and its applications abstract the paper mainly deals with suprema and infima of self adjoint operators in a von neumann algebra m with respect to the spectral order let m s a be the self adjoint part of m and let be the spectral order on m s a we show that a decreasing net in m s a with a lower bound has the infimum equal to the strong operator limit the similar statement is proved for an increasing net bounded above in m s a this version of vigier s theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self adjoint operators in terms of the strong operator limit and operator means as an application of our results on suprema and infima we study the order topology on m s a with respect to the spectral order we show that it is finer than the restriction of the mackey topology

asymptotic expansions for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 in order by gn k 0n116k 2kk 2andln 12 sin n 12 t sin 12t dt which play important roles in the theories of complex analysis and fourier series respectively certain inequalities and asymptotic expansions for the constants gn and ln have been investigated by many authors here we aim at establishing new asymptotic expansions for the constants gn and ln of landau and lebesgue respectively by mainly using bell polynomials and the partition function

limits and inequalities associated with the euler mascheroni constant i we present several limits associated with the euler mascheroni constant ii let c 0 577215 be the euler mascheroni constant and let t n k 1 n1k lnn 12 124n and p n k 1 n22k 1 ln 4n we determine the best possible constants a b a and b such that the inequalities148 n a 3 c t n 148 n b 3and124 n a 2 1

the asymptotic series of the generalized stirling formula the aim of this paper is to construct some asymptotic expansions which produce increasingly accurate approximations of the gamma function

full asymptotic expansions of the landau constants via a difference equation approach abstract we derive full asymptotic expansions for the landau constants g n as n some of the expansions are not new yet all the coefficients of the expansions are given iteratively in an explicit manner and are more efficiently evaluated as compared with the known results we obtain the asymptotic formulas old and new by applying the theory of wong and li for second order linear difference equations in deriving the expansions we have also confirmed a conjecture made by nemes and nemes

continued fraction approximation for the gamma function based on the tri gamma function abstract in this paper we present a continued fraction product approximation for the gamma function via the tri gamma function this approximation is fast in comparison with the recently discovered asymptotic series we also establish the inequalities related to this approximation finally some numerical computations are provided for demonstrating the superiority of our approximation

optimal rate of convergence for sequences of a prescribed form abstract we provide a highly convergent version of the generalized euler sequence and we determine the element with the optimal rate of convergence in several classes of sequences of a prescribed form the new approach extends and unifies many previous efforts in this direction in addition we answer an open problem on the asymptotic expansion of the harmonic number sequence

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

a topos foundation for theories of physics iii the representation of physical quantities with arrows this paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system in paper ii we studied the topos representations of the propositional language pl s for the case of quantum theory and in the present paper we do the same thing for the more extensive local language l s one of the main achievements is to find a topos representation for self adjoint operators this involves showing that for any physical quantity a there is an arrow o a r where r is the quantity value object for this theory the construction of o a is an extension of the daseinisation of projection operators that was discussed in paper ii the object r is a monoid object only in the topos setsv h op of the theory and to enhance the applicability of the formalism we apply to r a topos a

inequalities and monotonicity properties for the gamma function we present a survey of some recent results on the ratio of two gamma functions and prove the following inequalities for the product 1 1 x 1 x 2 1 x 1 1 x 2 0 and x 1 x 2 1 finally we study the possible extension to k variables of the lower bound above

podleś spheres for the braided quantum su 2 abstract starting with the braided quantum group su q 2 for a complex deformation parameter q we perform the construction of the quotient su q 2 t which serves as a model of a quantum sphere then we follow the reasoning of podle who for real q classified quantum spaces with the action of su q 2 with appropriate spectral properties these properties can also be expressed in the context of the braided quantum su q 2 with complex q and we find that they lead to precisely the same family of quantum spaces as found by podle for the real parameter q

completely monotonic functions of positive order and asymptotic expansions of the logarithm of barnes double gamma function and euler s gamma function abstract we introduce completely monotonic functions of order r 0 and show that the remainders in asymptotic expansions of the logarithm of barnes double gamma function and euler s gamma function give rise to completely monotonic functions of any positive integer order

inequalities and asymptotic expansions for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 in order by g n k 0 n 1 16 k 2 k k 2 and l n 1 2 sin n 1 2 t sin 1 2 t d t which play important roles in the theories of complex analysis and fourier series respectively diverse inequalities and approximations for these constants have been investigated and developed by many authors here in this paper we establish new asymptotic expansions for the constants g n and l n 2 of landau and lebesgue respectively in terms of the digamma and polygamma functions based on our expansion for the landau constants g n we present new bounds for the landau constants g n in terms of the digamma and polygamma functions we also establish inequalities for the lebesgue constants l n 2 which are applied to derive an asymptotic expansion for l n 2 in terms of 1 n 1 furthermore by giving numerical calculations to be compared among several developed asymptotic expansions for the constants g n and l n 2 it is shown that our expansions presented here would be best ones

inequalities for the volume of the unit ball in rn abstract let n n 2 1 n 2 be the volume of the unit ball in rn we determine the best possible constants a b a b and such that the inequalities a n n 1 n 1 n b n n 1 n 1 n a 2 n 1 n n b 2 and 1 1 n 2 n n 1 n 1 1 1 n are valid for all integers n 1 our results refine and complement inequalities proved by g d anderson et al k h borgwardt and d a klain and g c rota

new bounds and asymptotic expansions for the constants of landau and lebesgue abstract the constants of landau and lebesgue are defined for all integers n 0 in order by g n k 0 n 1 16 k 2 k k 2 and l n 1 2 sin n 1 2 t sin 1 2 t d t which play important roles in the theories of complex analysis and fourier series respectively in this paper we establish new bounds and asymptotic expansions for the constants of landau and lebesgue

a turan type inequality for the gamma function abstract we prove that the following turan type inequality holds for euler s gamma function for all odd integers n 1 and real numbers x 0 we have n 1 x n 1 x n x 2 with the best possible constant min 1 5 x 2 x 2 x 0 6359

new integral representations for the fox wright functions and its applications abstract our aim in this paper is to derive several new integral representations for the fox wright functions in particular we give new laplace and stieltjes transforms for this special function under some restrictions on parameters from the positivity conditions on the weight in these representations we found sufficient conditions to be imposed on the parameters of the fox wright functions which allow us to conclude that it is completely monotonic as applications we derive a class of functions that are related to the fox h functions and are positive definite moreover we extended the luke s inequalities and we establish new turan type inequalities for the fox wright function finally by appealing to each of the luke s inequalities two sets of two sided bounding inequalities for the generalized mathieu s type series are proved

on ruijsenaars asymptotic expansion of the logarithm of the double gamma function abstract we study the remainder r n x in an asymptotic expansion due to s n m ruijsenaars for the logarithm of the double gamma function we show that for any even number n the function 1 n 2 1 r n x is completely monotonic on 0 this proves a recent conjecture of h l pedersen in addition we give an estimate for the remainder r n x for n even

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

some extensions of w gautschi s inequalities for the gamma function it has been shown by w gautschi that if 0 i xi s f x exp i s x 1 the following closer bounds are proved exp i s 4 x 12 f exp i s x s i f x s 2 and x 2 t x 2 s 4 these are compared with each other and with inequalities given by t erber and j d keckic and p m vasic

completely monotonic functions associated with the gamma function and its q analogues several functions involving the gamma function x and the q gamma function q x are proved to be completely monotonic some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions

some sharp estimates of the constants of landau and lebesgue we establish several sharp two sided inequalities involving the constants of landau and lebesgue which occur and play important roles in two related extremal problems in complex analysis and in fourier analysis respectively the main results improves essentially the watson asymptotic expansion formulas for these constants

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

logarithmically completely monotonic functions and applications in this article we give a sufficient condition and a necessary and sufficient condition for a function class associated with the gamma function to be logarithmically completely monotonic some applications are also presented

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

examples of non compact quantum group actions abstract we present two examples of actions of non regular locally compact quantum groups on their homogeneous spaces the homogeneous spaces are defined in a way specific to these examples but the definitions we use have the advantage of being expressed in purely c algebraic language we also discuss continuity of the obtained actions finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups