a small world of weak ties provides optimal global integration of self similar modules in functional brain networks the human brain is organized in functional modules such an organization presents a basic conundrum modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple processors for efficient information transfer it is commonly accepted that small world architecture of short paths and large local clustering may solve this problem however there is intrinsic tension between shortcuts generating small worlds and the persistence of modularity a global property unrelated to local clustering here we present a possible solution to this puzzle we first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks these modules are large world self similar structures and therefore are far from being small world however incorporating weaker ties to the network converts it into a small world preserving an underlying backbone of well defined modules remarkably weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost this trade off architecture is reminiscent of the strength of weak ties crucial concept of social networks such a design suggests a natural solution to the paradox of efficient information flow in the highly modular structure of the brain

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

tsallis information dimension of complex networks abstract the fractal and self similarity properties are revealed in many complex networks the information dimension is a useful method to describe the fractal and self similarity properties of the complex networks in order to show the influence of different parts in the complex networks to the information dimension we have proposed a new information dimension based on the tsallis entropy namely the tsallis information dimension the proposed information dimension is changed according to the scale which is described by the nonextensivity parameter q and it is inverse with the nonextensivity parameter q the existing information dimension is a special case of the tsallis information dimension when q 1 the tsallis information dimension is a generalized information dimension of the complex networks

on the construction of some complete metrics with exceptional holonomy on construit explicitement trois metriques completes distinctes d holonomie egale a g 2 une metrique complete d holonomie egale a spin 7 et diverses metriques incompletes d holonomie exceptionnelle

quaternionic reduction and quaternionic orbifolds on considere un processus de reduction pour des varietes de kahler quaternioniques on montre qu il existe dans ce cas une certaine application impulsion qui se reduit a celle classique quand la courbure scalaire k de l espace est zero et les metriques deviennent hyperkahler quand k 0 cette application impulsion est definie de facon unique pour un groupe compact d automorphismes h

church rosser theorem for a rewriting system on categorical combinators abstract this paper develops the church rosser theorem for the rewriting system ccl on type free categorial combinators introduced by curien the system ccl is not confluent however we show that there are various sets d of categorical combinator terms such that each d satisfies the following two conditions 1 d is closed under reduction by ccl 2 ccl is confluent on d moreover we examine the relation among these sets

homogeneous einstein metrics and simplicial complexes on compact homogeneous spaces we investigate the hilbert action restricted to the space of homogeneous metrics of volume one based on a detailed understanding of the high energy levels of this action we assign to a compact homogeneous space a simplicial complex whose non contractibility is a sufficient condition for the existence of homogeneous einstein metrics

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

equational rules for rewriting logic in addition to equations and rules we introduce equational rules that are oriented while having an equational interpretation correspondence between operational behavior and intended semantics is guaranteed by a property of coherence which can be checked by examination of critical pairs and linearity conditions we present applications of this theory to three examples where the rewrite relation is interpreted respectively as equality transition and deduction

empirical regularities of opening call auction in chinese stock market abstract we study the statistical regularities of an opening call auction using the ultra high frequency data of 22 liquid stocks traded on the shenzhen stock exchange in 2003 the distribution of the relative price defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price the detrended fluctuation analysis dfa method is adopted to investigate the long term memory of relative order prices we further study the statistical regularities of order sizes in the opening call auction and observe a phenomenon of number preference known as order size clustering the probability density function pdf of order sizes could be well fitted by a q gamma function and the long term memory also exists in order sizes in addition both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit order book lob established immediately after the opening call auction and a price clustering phenomenon is observed

confluence results for the pure strong categorical logic ccl l calculi as subsystems of ccl abstract the strong categorical combinatory logic ccl ccl sp developed by curien 1986 is when typed and augmented with a rule defining a terminal object a presentation of cartesian closed categories furthermore it is equationally equivalent to the lambda calculus with explicit couples and surjective pairing here we study the confluence propertied of ccl ccl sp and of several of its subsystems and the relationship between untyped lambda calculi and ccl ccl sp as rewriting systems we prove that there exists a subset d of ccl and a subsystem sl of ccl sp confluent on d a very simple isomorphism between the classical lambda calculus and a subset sd of d which is extended between derivations of and a class of derivations of sl substitution which is a one step operation belonging to the metalanguage of is now described by rewritings with sl and calculations between several substitutions launched at the same time may be performed by sl this point is a real increase in the calculation capacities of lambda calculus same results for d the same result holds for the lambda calculus with couples and projection rules without surjective pairing the locally confluent subsystem ccl sp that is sl sp is not confluent this result is obtained by firstly designing a new counter example different from j w klop s one for confluence of the lambda calculus with couples and surjective pairing and then translating it into ccl however ccl sp is shown to be confluent on the set derived from sd these results cannot be obtained with classical methods of confluence and we designed a new method called interpretation method based on this trick a given relation r is confluent on a set x if and only if a relation e r induced by r on a set of regularized terms e x is confluent

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

confluence properties of weak and strong calculi of explicit substitutions categorical combinators curien 1986 1993 hardin 1989 yokouchi 1989 and more recently s calculus abadi 1991 hardin and le vy 1989 have been introduced to provide an explicit treatment of substitutions in the calculus we reintroduce here the ingredients of these calculi in a self contained and stepwise way with a special emphasis on confluence properties the main new results of the paper with respect to curien 1986 1993 hardin 1989 abadi 1991 and hardin and le vy 1989 are the following 1 we present a confluent weak calculus of substitutions where no variable clashes can be feared 2 we solve a conjecture raised in abadi 1991 s calculus is not confluent it is confluent on ground terms only this unfortunate result is repaired by presenting a confluent version of s calculus named the env caldulus in hardin and le vy 1989 called here the confluent s calculus

stability of tsallis entropy and instabilities of renyi and normalized tsallis entropies a basis for q exponential distributions the q exponential distributions which are generalizations of the zipf mandelbrot power law distribution are frequently encountered in complex systems at their stationary states from the viewpoint of the principle of maximum entropy they can apparently be derived from three different generalized entropies the renyi entropy the tsallis entropy and the normalized tsallis entropy accordingly mere fittings of observed data by the q exponential distributions do not lead to identification of the correct physical entropy here stabilities of these entropies i e their behaviors under arbitrary small deformation of a distribution are examined it is shown that among the three the tsallis entropy is stable and can provide an entropic basis for the q exponential distributions whereas the others are unstable and cannot represent any experimentally observable quantities

confluence of extensional and non extensional l calculi with explicit substitutions abstract this paper studies confluence of extensional and non extensional calculi with explicit substitutions where extensionality is interpreted by expansion for that we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence our method makes it possible to treat at the same time many well known calculi such as s v f d and d n

fractal surfaces from simple arithmetic operations fractal surfaces patchwork quilts are shown to arise under most general circumstances involving simple bitwise operations between real numbers a theory is presented for all deterministic bitwise operations on a finite alphabet it is shown that these models give rise to a roughness exponent h that shapes the resulting spatial patterns larger values of the exponent leading to coarser surfaces

a new information dimension of complex network based on renyi entropy abstract with the development of high technology and artificial intelligence it evolves into an open issue to calculate the dimension of the complex network in this paper a new dimension renyi dimension combined with renyi entropy and information dimension is proposed a modified box covering algorithm is introduced to calculate the minimum number and the length of the boxes needed to cover the whole network additionally the self weight factor swf and the positive weight factor pwf are defined to illustrate the change of the dimension value based on the perspective of both topology structure and dynamic property the concept of attractors is proposed to illuminate the physical meaning of the weighted parameter in the formula of renyi entropy pwf and swf finally to demonstrate the efficiency of our method it is applied to calculate the dimension of sierpinski weighted fractal network ba networks and many real world networks the results show that attractors exist in the network researched and can access the attractiveness of attractors as a criterion the swf quantifies the total attractiveness of attractors the comparison results with tsallis dimension indicate the stability of the renyi dimension

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

generalized kolmogorov entropy in the dynamics of multifractal generation abstract we point out that applying a maximization principle on a tsallis like generalized form of the kolmogorov entropy for iterated function systems naturally provides a canonical statistical frame for the description of the multifractal measures generated by such dynamical processes multifractal spectra can then be characterized by usual statistical parameters in particular the temperature we show that in the limit of zero temperature the multifractal measure collapses to a homogeneous distribution over a fractal support for finite temperatures multifractal spectra are studied numerically in an illustrative example

disequilibrium complexity the schottky effect and q entropies in paramagnetism abstract we investigate connections between statistical quantifiers and paramagnetism more concretely we apply the notions of i disequilibrium and ii statistical complexity to a paramagnetic system of non coupled dipoles interesting insights are thereby obtained in particular we encounter a kind of criticality not associated to the temperature but to the disequilibrium

indices for testing neural codes one of the most critical challenges in systems neuroscience is determining the neural code a principled framework for addressing this can be found in information theory with this approach one can determine whether a proposed code can account for the stimulus response relationship specifically one can compare the transmitted information between the stimulus and the hypothesized neural code with the transmitted information between the stimulus and the behavioral response if the former is smaller than the latter i e if the code cannot account for the behavior the code can be ruled out the information theoretic index most widely used in this context is shannon s mutual information the shannon test however is not ideal for this purpose while the codes it will rule out are truly nonviable there will be some nonviable codes that it will fail to rule out here we describe a wide range of alternative indices that can be used for ruling codes out the range includes a continuum from shannon information to measures of the performance of a bayesian decoder we analyze the relationship of these indices to each other and their complementary strengths and weaknesses for addressing this problem

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

convergence in reaction diffusion systems an information theory approach abstract we have applied an information theory approach in order to study the problem of convergence to point like or extended attractors in reaction diffusion systems a distance between two states based on the kullback leibler relative information was defined different forms of the probability distribution some of them based on the knowledge of the nonequilibrium potential when accesible give the possibility to look for a faster and or more accurate convergence this approach offers the chance to estimate the attraction basins of the different attractors as well as detecting limit circles together with an easy evaluation of their periods

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

student s t and r distributions unified derivation from an entropic variational principle abstract the optimization of the recently generalized entropy of the form s q 1 dx p x q q 1 with the constraints dx p x 1 and x 2 q dx x 2 p x q 1 yields the student s t distribution for q 1 and the r distribution for q

proof of termination of the rewriting system subst on ccl abstract curien 1985 defines a translation of the c calculus in the pure combinatory categorical logic and establishes an equivalence theorem between these two theories the rewriting system subst simulates in particular the substitution of the c calculus this system is locally confluent we shall show here that it is also noetherian

lambda calculi for strict parallel functions we introduce two calculi and show that they are expressive for two canonical domains of parallel functions the first calculus is an enrichment of the lazy call by name calculus with call by value abstractions and parallel composition while in the second the usual call by name abstractions are disallowed the corresponding domains are respectively abramsky s domain d d d a lifted function space and d d d a lifted domain of strict functions these domains are lattices and we show that the parallelism is adequately represented by the join operator while call by value abstractions correspond to strict functions the proofs of the results rely on a completeness theorem for the logical presentation of the semantics

zeroth law compatibility of nonadditive thermodynamics nonextensive thermodynamics is criticized by the statement that the zeroth law cannot be satisfied with nonadditive composition rules corrected in this paper we determine the general functional form of those nonadditive composition rules that are compatible with the zeroth law of thermodynamics we find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these our result offers a possible solution to the long standing questions about equilibrium between extensive and nonextensive systems or systems with different nonextensivity parameters

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

higher order unification via explicit substitutions higher order unification is equational unification for conversion but it is not first order equational unification as substitution has to avoid capture thus the methods for equational unification such as narrowing built upon grafting i e substitution without renaming cannot be used for higher order unification which needs specific algorithms our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory this is achieved by replacing substitution by grafting but this replacement is not straightforward as it raises two major problems first some unification problems have solutions with grafting but no solution with substitution then equational unification algorithms rest upon the fact that grafting and reduction commute but grafting and reduction do not commute in calculus and reducing an equation may change the set of its solutions this difficulty comes from the interaction between the substitutions initiated by reduction and the ones initiated by the unification process two kinds of variables are involved those of conversion and those of unification so we need to set up a calculus which distinguishes between these two kinds of variables and such that reduction and grafting commute for this purpose the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated such a separation and delay are provided by a calculus of explicit substitutions unification in such a calculus can be performed by well known algorithms such as narrowing but we present a specialised algorithm for greater efficiency at last we show how to relate unification in calculus and in a calculus with explicit substitutions thus we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms in particular the functional handling of scopes huet s algorithm can be seen as a specific strategy for our algorithm since each of its steps can be decomposed into elementary ones leading to a more atomic description of the unification process also solved forms in calculus can easily be computed from solved forms in calculus

einstein metrics on s 3 r 3 and r 4 bundles starting from a 4n dimensional quaternionic kahler base space we construct metrics of cohomogeneity one in 4n 3 dimensions whose level surfaces are thes 2 bundle space of almost complex structures on the base manifold we derive the conditions on the metric functions that follow from imposing the einstein equation and obtain solutions both for compact and non compact 4n 3 dimensional spaces included in the non compact solutions are two ricci flat 7 dimensional metrics withg 2 holonomy we also discuss two other ricci flat solutions one on ther 4 bundle overs 3 and the other on anr 4 bundle overs 4 these haveg 2 and spin 7 holonomy respectively

generalization of shannon s theorem for tsallis entropy by using the assumptions that the entropy must i be a continuous function of the probabilities pi pi 0 1 i only ii be a monotonic increasing function of the number of states w in the case of equiprobability iii satisfy the pseudoadditivity relation sq a b k sq a k sq b k 1 q sq a sq b k2 a and b being two independent systems q r and k a positive constant and iv satisfy the relation sq pi sq pl pm pl qsq pi pl pm qsq pi pm where pl pm 1 pl i 1wlpi and pm i wl 1wpi we prove along shannon s lines that the unique function that satisfies all these properties is the generalized tsallis entropy sq k 1 i 1wpiq q 1

small world to fractal transition in complex networks a renormalization group approach we show that renormalization group rg theory applied to complex networks is useful to classify network topologies into universality classes in the space of configurations the rg flow readily identifies a small world fractal transition by finding i a trivial stable fixed point of a complete graph ii a nontrivial point of a pure fractal topology that is stable or unstable according to the amount of long range links in the network and iii another stable point of a fractal with shortcuts that exist exactly at the small world fractal transition as a collateral the rg technique explains the coexistence of the seemingly contradicting fractal and small world phases and allows us to extract information on the distribution of shortcuts in real world networks a problem of importance for information flow in the system

on the derivation of power law distributions within classical statistical mechanics far from the thermodynamic limit we show that within classical statistical mechanics without taking the thermodynamic limit the most general boltzmann factor for the canonical ensemble is a q exponential function the only assumption here is that microcanonical distributions have to be separated from the total system energy which is the prerequisite for any sensible measurement we derive that all separable distributions are parametrized by a mathematical separation constant q which can be related to the non extensivity q parameter in tsallis distributions we further demonstrate that nature fixes the separation constant q to 1 for large dimensionality of gibbs phase space our results will be relevant for systems with a low dimensional space for example nanosystems comprised of a small number of particles or for systems with a dimensionally collapsed phase space which might be the case for a large class of complex systems

extracting features of gaussian self similar stochastic processes via the bandt pompe approach by recourse to appropriate information theory quantifiers normalized shannon entropy and martin plastino rosso intensive statistical complexity measure we revisit the characterization of gaussian self similar stochastic processes from a bandt pompe viewpoint we show that the ensuing approach exhibits considerable advantages with respect to other treatments in particular clear quantifiers gaps are found in the transition between the continuous processes and their associated noises

measure the structure similarity of nodes in complex networks based on relative entropy abstract similarity of nodes is a basic structure quantification in complex networks lots of methods in research on complex networks are based on nodes similarity such as node s classification network s community structure detection network s link prediction and so on therefore how to measure nodes similarity is an important problem in complex networks in this paper a new method is proposed to measure nodes structure similarity based on relative entropy and each node s local structure in the new method each node s structure feature can be quantified as a special kind of information the quantification of similarity between different pair of nodes can be replaced as the quantification of similarity in structural information then relative entropy is used to measure the difference between each pair of nodes structural information at last the value of relative entropy between each pair of nodes is used to measure nodes structure similarity in complex networks comparing with existing methods the new method is more accuracy to measure nodes structure similarity

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

a rewriting system for categorical combinators with multiple arguments categorical combinators have been derived from the study of categorical semantics of lambda calculus and it has been found that they may be used in implementation of functional languages in this paper categorical combinators are extended so that functions with multiple arguments can be directly handled thus making them more suitable for practical computation a rewriting system named operatorname cclm beta is formulated for these combinators this system naturally corresponds to the type free lambda beta calculus the relationship between these two systems is established and as a result of this the church rosser property of operatorname cclm beta is proved a similar relationship is also established between the original operatorname ccl beta by curien and the type free lambda beta calculus with product finally the embedding theorem of operatorname cclm beta into operatorname ccl beta is shown