solving some q trigonometric conjectures of gosper abstract gosper in 2001 introduced the functions sin q z and cos q z as q analogues for the trigonometric functions sin z and cos z respectively he stated but did not prove many identities involving the functions sin q z and cos q z along with some other related functions and constants in this note we shall provide proofs for some of these identities by means of the classical theory of elliptic functions

proofs for certain πq identities of w gosper abstract in 2001 w gosper introduced constant q and conjectured many intriguing identities on this constant in this paper we employ some knowledge of modular equations with degree 5 to confirm several of gosper s q identities two strange q identities involving q and a lambert series conjectured by w gosper are proved as a consequence a q identity involving q and two lambert series which was conjectured by gosper is proved as an application we confirm an interesting q trigonometric identity of gosper which can be viewed as a theta function analogue for the related well known trigonometric identity our proof relies on an interesting general theta function identity of zhi guo liu

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

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

t norm based addition of fuzzy intervals abstract the aim of this paper is to provide new results regarding the effective practical computation of t norm based additions of lr fuzzy intervals

triangular norm based addition of fuzzy intervals triangular norm based addition of fuzzy intervals on the extended real line is reduced to the addition of shapes on 0 the addition of fuzzy intervals including lr fuzzy intervals lr fuzzy numbers lt and tr fuzzy numbers based on limit t norms is recalled for continuous archimedean t norms special types of incoming summands allowing the exact output computation are introduced several examples are given

real vector space of lr fuzzy intervals with respect to the shape preserving t norm based addition the construction of the membership function of fuzzy intervals is an important problem in vagueness modeling the preservation of the shape of fuzzy sets during the addition is a natural requirement in practical computation the lr fuzzy intervals introduced by dubois and prade satisfy this requirement if the addition is based on the nilpotent t norm generated by l or r shape functions the shortcoming that not any lr fuzzy interval has an opposite inverse related to shape preserving t norm based addition can be solved if the set of lr fuzzy intervals is isomorphically included in an extended set and this extended set forms a group with respect to shape preserving t norm based addition in this paper we construct the extended set of these lr fuzzy intervals we also show that the extended set is a real vector space with scalar product and the modal intervals can be considered as the elements of this extended set finally we present the algebraic form of lr fuzzy intervals and the associated application

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

bounded imaginary powers of cone differential operators on higher order mellin sobolev spaces and applications to the cahn hilliard equation abstract extending earlier results on the existence of bounded imaginary powers for cone differential operators on weighted l p spaces h p 0 b over a manifold with conical singularities we show how the same assumptions also yield the existence of bounded imaginary powers on higher order mellin sobolev spaces h p s b s 0 as an application we consider the cahn hilliard equation on a manifold with possibly warped conical singularities relying on our work for the case of straight cones we first establish r sectoriality and thus maximal regularity for the linearized equation and then deduce the existence of a short time solution with the help of a theorem by clement and li we also obtain the short time asymptotics of the solution near the conical point

shape preserving additions of fuzzy intervals abstract triangular norm based additions preserving the lr shape of lr fuzzy intervals are studied the connection between t norm based additions preserving the linearity of fuzzy intervals and additions preserving lr shape for invertible shapes l and r is given a possibility of controlling the growth of the spread during the calculations with lr fuzzy intervals is shown

on the compositional rule of inference under triangular norms abstract the aim of this paper is to provide a close upper bound of the membership function for the compositional rule of inference under archimedean t norm where both the observation and the relation parts are given by hellendoorn s function 1980 in particular if the left and right spreads of the observation part is the same as those of the relation part then this upper bound is the exact membership function which generalizes the earlier result by fuller and zimmermann 1982 in that assumption of twice differentiability is deleted

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

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

r bounded approximating sequences and applications to semigroups abstract it is shown that on certain banach spaces including c 0 1 and l1 0 1 there is no strongly continuous semigroup tt 0

t sum of sigmoid shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with particular types of fuzzy intervals recently dombi and gyorbiro j dombi n gyorbiro additions of sigmoid shaped fuzzy intervals using the dombi operator and infinite sum theorems fuzzy sets and systems 157 2006 952 963 proved that addition is closed if the dombi t norm is used with sigmoid shaped fuzzy intervals in this paper we define a broader class of sigmoid shaped fuzzy intervals then we study t norms that are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

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

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

maximal regularity of evolution equations on discrete time scales we consider the question of lp maximal regularity for inhomogeneous cauchy problems in banach spaces using operator valued fourier multipliers this follows results by l weis in the continuous time setting and by s blunck for discrete time evolution equations we generalize the later result to the case of some discrete time scales discrete problems with nonconstant step size first we introduce an adequate evolution family of operators to consider the general problem then we consider the case where the step size is a periodic sequence by rewriting the problem on a product space and using operator matrix valued fourier multipliers finally we give a perturbation result allowing to consider a wider class of step sizes

shape preserving multiplications of fuzzy numbers abstract in this note we show that t w the weakest t norm is the only t norm t which induces a shape preserving multiplications of lr fuzzy numbers

γ bounded representations of amenable groups let g be an amenable group let x be a banach space and let g b x be a bounded representation we show that if the set t t g is bounded then extends to a bounded homomorphism w c g b x on the group c algebra of g moreover w is necessarily bounded this extends to the banach space setting a theorem of day and dixmier saying that any bounded representation of an amenable group on hilbert space is unitarizable we obtain additional results and complements when g z r or t and or when x has property

t sum of l r fuzzy numbers abstract the sum of l r fuzzy numbers based on a given archimedian continuous t norm t is studied a series of improving sufficient conditions of several authors ensuring an analytical form of the output sum is completed by a sufficient and necessary condition i e the best sufficient condition some applications are indicated

discrete fourier transform and jacobi θ function identities classical theta function identities are derived using properties of eigenvectors corresponding to the discrete fourier transform 2 in particular an extended watson addition formula is obtained corresponding to 2 and the corresponding result for the discrete fourier transform 3 is obtained

stochastic evolution equations in umd banach spaces we discuss existence uniqueness and space time holder regular ity for solutions of the parabolic stochastic evolution equation du t au t f t u t dt b t u t dwh t t 2 0 t0 u 0 u0 where a generates an analytic c0 semigroup on a umd banach space e and wh is a cylindrical brownian motion with values in a hilbert space h we prove that if the mappings f 0 t e e and b 0 t e l h e satisfy suitable lipschitz conditions and u0 is f0 measurable and bounded then this problem has a unique mild solution which has trajectories in c 0 t d a provided 0 and 0 satisfy 1 2 vari ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations

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

well posedness for semidiscrete fractional cauchy problems with finite delay abstract we address the study of well posedness on lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay 0 1 u n t u n u n f n n n 0 1 r n 0 where t is a bounded linear operator defined on a banach space x typically a space of functions like l p 1 p and corresponds to the time discretization of the continuous riemann liouville fractional derivative by means of the poisson distribution we characterize the existence and uniqueness of solutions in vector valued lebesgue spaces of sequences of the model 0 1 in terms of boundedness of the operator valued symbol z 1 z 1 i z t 1 z 1 z 1 whenever 0 1 and x satisfies a geometrical condition for this purpose we use methods from operator valued fourier multipliers and resolvent operator families associated to the homogeneous problem we apply this result to show a practical and computational criterion in the context of hilbert spaces

on product sum of triangular fuzzy numbers abstract we study the problem if a i i n are fuzzy numbers of triangular form then what is the membership function of the infinite or finite sum a 1 a 2 defined via the sup product norm convolution

necessary and sufficient conditions for the equality of interactive and non interactive extensions of continuous functions abstract in this contribution we find the class of n dimensional joint possibility distributions with the property that the interactive extension principle coincides with the non interactive extension principle as long as the interactive operations are determined by continuous functions strictly increasing in each argument this result completes recent studies by the authors where the particular case of interactive additions and multiplications versus non interactive additions and multiplications were investigated in addition this time we propose results that also cover the cases when we know the fuzzy numbers only from their membership functions it means that we eliminated the limitations that appear when we cannot pass from membership function representation to parametric representation of fuzzy numbers as important new applications we mention the study on the completely correlated fuzzy numbers also of note is that we propose two simple methods to extend bidimensional joint possibility distributions to n dimensional joint possibility distributions one method is based on an inductive construction while the other one is based on a pairwise construction

a parametrized sum of fuzzy numbers with applications to fuzzy initial value problems abstract this paper presents a new parametrized family of joint possibility distributions that can be used to control the pompeiu hausdorff norm of the sum resulting from the extension principle we are able to generate sums with norms covering the entire interval ranging from the smallest possible value to the largest one finally we apply our approach to numerical methods for solving first order fuzzy initial value problems

on the convergence of t sum of l r fuzzy numbers abstract this paper presents the membership function of infinite or finite sum defined by the sup t norm convolution of l r fuzzy numbers under the conditions of the convexity of additive generators and the concavity of l r as an application we shall calculate the membership function of the limit distribution of the hamacher sum hr sum for 0 r 2 which generalizes fuller s results fuzzy sets and systems 41 1991 83 87 42 1991 205 212 in the case r 0 1 2

discrete maximal parabolic regularity for galerkin finite element methods the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the time discontinuous galerkin solution of linear parabolic equations such estimates have many applications they are essential for example in establishing optimal a priori error estimates in non hilbertian norms without unnatural coupling of spatial mesh sizes with time steps

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

extended triangular norms the paper is devoted to classical t norms extended to operations on fuzzy quantities in accordance with the generalized zadeh extension principle such extended t norms are used for calculating intersection of type 2 fuzzy sets analytical expressions for membership functions of some extended t norms are derived assuming special classes of fuzzy quantities i e fuzzy truth intervals or fuzzy truth numbers the possibility of applying these results in the construction of type 2 adaptive network fuzzy inference systems is illustrated on several examples

a t sum bound of lr fuzzy numbers abstract in this paper we provide an upper bound and a lower bound of t sum of lr fuzzy numbers with different spreads where t is archimedean t norm and also show in three examples how close they are to actual membership functions furthermore we study when the membership function of t sum achieves the upper bound

on computation of the compositional rule of inference under triangular norms abstract this paper is devoted to the derivation of exact calculation formulas for the compositional rule of inference under archimedean t norms when both the observation and the relation parts are given by hellendoorn s function 6

analytical expressions for the addition of fuzzy intervals abstract the addition of fuzzy quantities is undeniably the most important operation in fuzzy arithmetic in this paper we consider the general case of the addition of fuzzy intervals based on a continuous triangular norm we start our discussion with an extensive literature review in which we recall explicit formulae for the addition based on the strongest and the weakest triangular norm for the addition based on the algebraic product and several recent results for the addition based on a continuous archimedean triangular norm including specific theorems for strict and nilpotent triangular norms subsequently the addition based on an ordinal sum is studied and it is shown how this addition can be transformed into a series of additions based on the summands of this ordinal sum this important observation implies that the addition based on an arbitrary continuous triangular norm can be practically performed provided the summands of the corresponding ordinal sum representation and the fuzzy intervals involved fulfil the appropriate conditions mentioned in the overview this is illustrated by means of several examples

a note on t norm based addition of fuzzy intervals we generalize a result of fuller and keresztfalvi 1992 regarding the computation of t norm based additions of lr fuzzy intervals the proof of which is new and simple

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

t sum of bell shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with a particular type of fuzzy intervals recently dombi and gyorbiro proved that addition is closed if the dombi t norm is used with two bell shaped fuzzy intervals a result proved by mesiar on a strict t norm based shape preserving additions of lr fuzzy intervals with unbounded support is recalled as applications we define a broader class of bell shaped fuzzy intervals then we study t norms which are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

maximal regularity for perturbed integral equations on periodic lebesgue spaces we characterize the maximal regularity of periodic solutions for an additive perturbed integral equation with infinite delay in the vector valued lebesgue spaces our method is based on operator valued fourier multipliers we also study resonances characterizing the existence of solutions in terms of a compatibility condition on the forcing term

the kalton lancien theorem revisited maximal regularity does not extrapolate abstract we give a new more explicit proof of a result by kalton and lancien stating that on each banach space with an unconditional basis not isomorphic to a hilbert space there exists a generator a of a holomorphic semigroup which does not have maximal regularity in particular we show that there always exists a schauder basis f m such that a can be chosen of the form a m 1 a m f m m 1 2 m a m f m moreover we show that maximal regularity does not extrapolate we construct consistent holomorphic semigroups t p t t 0 on l p r for p 1 which have maximal regularity if and only if p 2 these assertions were both open problems our approach is completely different than the one of kalton and lancien we use the characterization of maximal regularity by r sectoriality for our construction

some algebraic properties of weakly non interactive fuzzy numbers abstract this study focuses on the concept of sup t norm convulution a generalized version of zadeh s extension principle the authors treat the algebraic structure of fuzzy arithmetic based on the sup t norm convolution and especially show some new properties comparing them with those of the conventional fuzzy arithmetic based on sup min convolution

on hamacher sum of triangular fuzzy numbers abstract this paper presents new results concerning the effective practical computation of the membership function of the infinite sum defined via the sup hamacher norm convolution of triangular fuzzy numbers namely we shall calculate the limit distribution of the h sum a 1 a 2 of triangular fuzzy numbers a i i n for 0 1 2

semilinear evolution equations on discrete time and maximal regularity abstract this paper deals with the existence and stability of solutions for semilinear equations on banach spaces by using recent characterizations of discrete maximal regularity as application we examine the asymptotic behavior of discrete control systems

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

a note to the t sum of l r fuzzy numbers abstract we generalize the results of hong and hwang 1994 for the membership function of finite infinite sum of l r fuzzy numbers where the sum is based on some continuous archimedean t norm t

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

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

necessary and sufficient conditions for the equality of the interactive and non interactive sums of two fuzzy numbers this study aims to answer the open question proposed by carlsson et al 2004 4 which asks whether it is possible to find the properties needed by a joint possibility distribution function such that the interactive sum with respect to this function and the standard sum of two fuzzy numbers will coincide we provide the necessary and sufficient conditions such that this equality would hold in addition we find the necessary and sufficient conditions such that the interactive and standard sums will coincide at some point in the domain in order to obtain an interactive addition that does not coincide with the standard addition we show that we need to search for a joint possibility distribution that coincides as little as possible with the independent joint possibility distribution throughout the so called diagonal of the two fuzzy numbers as simple consequences we deduce similar results for triangular norm based additions of fuzzy numbers

weighted norm inequalities off diagonal estimates and elliptic operators part iii harmonic analysis of elliptic operators this is the third part of a series of four articles on weighted norm in equalities o diagonal estimates and elliptic operators for l in some class of elliptic operators we study weighted norm l p inequalities for singular non integral opera tors arising from l those are the operators l for bounded holomorphic functions the riesz transforms rl 1 2 or 1 2 l 1 2 and its inverse l 1 2 1 2 some quadratic functionals gl and gl of littlewood paley stein type and also some vector valued inequalities such as the ones involved for maximal l p regularity for each we obtain sharp or nearly sharp ranges of p using the general theory for bound edness of part i and the o diagonal estimates of part ii we also obtain commutator results with bmo functions

an addition formula for the jacobian theta function and its applications in this paper we prove an addition formula for the jacobian theta function using the theory of elliptic functions it turns out to be a fundamental identity in the theory of theta functions and elliptic function and unifies many important results about theta functions and elliptic functions from this identity we can derive the ramanujan cubic theta function identity winquist s identity a theta function identities with five parameters and many other interesting theta function identities and all of which are as striking as winquist s identity this identity allows us to give a new proof of the addition formula for the weierstrass sigma function a new identity about the ramanujan cubic elliptic function is given the proofs are self contained and elementary

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

on shape preserving additions of fuzzy intervals abstract we prove an open problem suggested by mesiar 1997 fuzzy sets and systems86 73 78 concerning continuous t norm based additions preserving the lr fuzzy intervals

a convexity problem and a new proof for linearity preserving additions of fuzzy intervals in this paper we propose a new sufficient condition for convexity of function this result provides a simple new proof for linearity preserving additions of fuzzy intervals obtained by the author in 2002 which was a conjecture suggested by mesiar in 1997

analogues of jacobi s inversion formula for the incomplete elliptic integral of the first kind abstract in this article we revisit ramanujan s cubic analogue of jacobi s inversion formula for the classical elliptic integral of the first kind our work is motivated by the recent work of milne ramanujan j 6 1 2002 7 149 chan and chua ramanujan j to appear on the representations of integers as sums of even squares

addition of sigmoid shaped fuzzy intervals using the dombi operator and infinite sum theorems the extension principle defines the arithmetic operations on fuzzy intervals in the extension principle one can use any t norm for modeling the conjunction operator it is therefore important to know which t norms are consistent with a particular type of fuzzy intervals we call a t norm consistent if the arithmetic operation is closed in this paper we investigate the addition of sigmoid and two bell shaped membership functions which appear in many natural processes and are used in machine learning applications we prove that the addition is closed if the dombi operator is used the calculation of sum is quite simple and can be used in various applications such as fuzzy neural networks

generalized extended fuzzy implications this paper is devoted to investigate generalized extended fuzzy implications in accordance with the generalized extension principle the algebraic properties of generalized extended fuzzy implications are discussed on different algebras of fuzzy truth values on this basis fuzzy valued fuzzy implications are constructed from arbitrary fuzzy implications and t norms moreover a fuzzy valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t norm

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

on a theta product of jacobi and its applications to q gamma products abstract we give a new proof for a product formula of jacobi which turns out to be equivalent to a q trigonometric product which was stated without proof by gosper we apply this formula to derive a q analogue for the gauss multiplication formula for the gamma function furthermore we give explicit formulas for short products of q gamma functions

some results on the addition of fuzzy intervals abstract a simple new method of computing t sum of fuzzy intervals having the same results as the sum of fuzzy intervals based on the weakest t norm t w is introduced this work extends that of mesiar fuzzy sets and systems 91 1997 231 237 and markova fuzzy sets and systems 91 1997 253 258

a three term theta function identity and its applications in this paper we establish a three term theta function identity using the complex variable theory of elliptic functions this simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities from which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived we also give a new proof of the addition theorem for the weierstrass elliptic function an identity involving the products of four theta functions is given and from which one theta function identity by mccullough and shen is derived the quintuple product identity is used to derive some eisenstein series identities found in ramanujan s lost notebook and our approach is different from that of berndt and yee the proofs are self contained and elementary

degenerate operators of tricomi type in l p spaces and in spaces of continuous functions we study elliptic operators l with dirichlet boundary conditions on a bounded domain whose diffusion coefficients degenerate linearly at in tangential directions we compute the domain of l and establish existence uniqueness and maximal regularity of the elliptic and parabolic problems for l in lp spaces and in spaces of continuous functions moreover the analytic semigroups generated by l are consistent positive compact and exponentially stable

on the theory of elliptic functions based on _ 2 f_ 1 frac 1 3 frac 2 3 frac 1 2 z based on properties of the hypergeometric series 2 f 1 1 3 2 3 1 2 z we develop a theory of elliptic functions that shares many striking similarities with the classical jacobian elliptic functions

a theta function identity and its implications in this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions this identity plays a central role for the cubic theta function identities we use this identity to re derive some important identities of hirschhorn garvan and borwein about cubic theta functions we also prove some other cubic theta function identities a new representation for n 1 1 q n 10 is given the proofs are self contained and elementary

proofs for two q trigonometric identities of gosper abstract in 2001 gosper introduced q analogues for the functions sin z and cos z and stated without proofs many identities involving these q analogues gosper asked whether his formulas are true in this paper we shall use the theory of elliptic functions to confirm two of gosper s identities moreover we shall give two consequences of these identities

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

nilpotent t norm based sum of fuzzy intervals in this paper the sum of l r fuzzy intervals based on a given nilpotent triangular norm t norm t is studied by modifying markova s results which gives the necessary and sufficient condition ensuring an analytical form of the output sum to the form with arbitrary spreads we generalize the results of hong and hwang for the membership function of the sum of l r fuzzy intervals where the sum is based on some archimedean t norm t

the navier stokes equations in exterior lipschitz domains lp theory abstract we show that the stokes operator defined on l p for an exterior lipschitz domain r n n 3 admits maximal regularity provided that p satisfies 1 p 1 2 1 2 n e for some e 0 in particular we prove that the negative of the stokes operator generates a bounded analytic semigroup on l p for such p in addition l p l q mapping properties of the stokes semigroup and its gradient with optimal decay estimates are obtained this enables us to prove the existence of mild solutions to the navier stokes equations in the critical space l 0 t l 3 locally in time and globally in time for small initial data

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

the convergence of t product of fuzzy numbers abstract this paper is to provide new results regarding the effective practical computation of t norm based products of r type fuzzy numbers and their limits