constructing general orthogonal fractional factorial split plot designs while the orthogonal design of split plot fractional factorial experiments has received much attention already there are still major voids in the literature first designs with one or more factors acting at more than two levels have not yet been considered second published work on nonregular fractional factorial split plot designs was either based only on plackett burman designs or on small nonregular designs with limited numbers of factors in this article we present a novel approach to designing general orthogonal fractional factorial split plot designs one key feature of our approach is that it can be used to construct two level designs as well as designs involving one or more factors with more than two levels moreover the approach can be used to create two level designs that match or outperform alternative designs in the literature and to create two level designs that cannot be constructed using existing methodology our new approach involves the use of integer linear programming and mixed integer linear programming and for large design problems it combines integer linear programming with variable neighborhood search we demonstrate the usefulness of our approach by constructing two level split plot designs of 16 96 runs an 81 run three level split plot design and a 48 run mixed level split plot design supplementary materials for this article are available online 2015 american statistical association and the american society for quality

rotation and gray scale transform invariant texture identification using wavelet decomposition and hidden markov model in this correspondence we have presented a rotation and gray scale transform invariant texture recognition scheme using the combination of quadrature mirror filter qmf bank and hidden markov model hmm in the first stage the qmf bank is used as the wavelet transform to decompose the texture image into subbands the gray scale transform invariant features derived from the statistics based on first order distribution of gray levels are then extracted from each subband image in the second stage the sequence of subbands is modeled as a hidden markov model hmm and one hmm is designed for each class of textures the hmm is used to exploit the dependence among these subbands and is able to capture the trend of changes caused by rotation during recognition the unknown texture is matched against all the models the best matched model identifies the texture class up to 93 33 classification accuracy is reported

multi class boosting with asymmetric binary weak learners we introduce a multi class generalization of adaboost with binary weak learners we use a vectorial codification to represent class labels and a multi class exponential loss function to evaluate classifier responses this representation produces a set of margin values that provide a range of punishments for failures and rewards for successes moreover the stage wise optimization of this model introduces an asymmetric boosting procedure whose costs depend on the number of classes separated by each weak learner in this way the boosting algorithm takes into account class imbalances when building the ensemble the experiments performed compare this new approach favorably to adaboost mh gentleboost and the samme algorithms

extensions and generalizations of a theorem of widder and of the theory of symmetric local semigroups the theory of symmetric local semigroups due to a klein and l landau j funct anal 44 1981 121 136 is generalized to semigroups indexed by subsets of rn for n 1 the result implies a similar result of a e nussbaum j funct anal 48 1982 213 223 it is further generalized to semigroups that are symmetric local in some directions and unitary in others the results are used to give a simple proof of a devinatz s duke math j 22 1955 185 192 and n i akhiezer s the classical moment problem and some related questions hafner new york 1965 generalization of a theorem of widder concerning the representation of functions as laplace integrals this result is extended to the representation as a laplace integral of a function taking values in b r the set of bounded linear operators on a hilbert space r also a theorem is proved encompassing both the result of devinatz and akhiezer and bochner s theorem on the representation of positive definite functions as fourier integrals

rotation invariant texture classification using lbp variance lbpv with global matching local or global rotation invariant feature extraction has been widely used in texture classification local invariant features e g local binary pattern lbp have the drawback of losing global spatial information while global features preserve little local texture information this paper proposes an alternative hybrid scheme globally rotation invariant matching with locally variant lbp texture features using lbp distribution we first estimate the principal orientations of the texture image and then use them to align lbp histograms the aligned histograms are then in turn used to measure the dissimilarity between images a new texture descriptor lbp variance lbpv is proposed to characterize the local contrast information into the one dimensional lbp histogram lbpv does not need any quantization and it is totally training free to further speed up the proposed matching scheme we propose a method to reduce feature dimensions using distance measurement the experimental results on representative databases show that the proposed lbpv operator and global matching scheme can achieve significant improvement sometimes more than 10 in terms of classification accuracy over traditional locally rotation invariant lbp method

rotation invariant texture classification using a complete space frequency model a method of rotation invariant texture classification based on a complete space frequency model is introduced a polar analytic form of a two dimensional 2 d gabor wavelet is developed and a multiresolution family of these wavelets is used to compute information conserving microfeatures from these microfeatures a micromodel which characterizes spatially localized amplitude frequency and directional behavior of the texture is formed the essential characteristics of a texture sample its macrofeatures are derived from the estimated selected parameters of the micromodel classification of texture samples is based on the macromodel derived from a rotation invariant subset of macrofeatures in experiments comparatively high correct classification rates were obtained using large sample sets

a double mapping framework for extraction of shape invariant features based on multi scale partitions with adaboost for video smoke detection traditional methods for video smoke detection can easily achieve very low training errors but their generalization performances are not good due to arbitrary shapes of smoke intra class variations occlusions and clutters to overcome these problems a double mapping framework is proposed to extract partition based features with adaboost the first mapping is from an original image to block features a feature vector is presented by concatenating histograms of edge orientation edge magnitude and local binary pattern lbp bit and densities of edge magnitude lbp bit color intensity and saturation each component of the feature vector produces a feature image to obtain shape invariant features a detection window is partitioned into a set of small blocks called a partition and many multi scale partitions are generated by changing block sizes and partition schemes the sum of each feature image within each block of each partition is computed to generate block features the second mapping is from the block features to statistical features the statistical features of the block features such as mean variance skewness kurtosis and hu moments are computed on all partitions to form a feature pool adaboost is used to select discriminative shape invariant features from the feature pool experiments show that the proposed method has better generalization performance and less insensitivity to geometry transform than traditional methods

a statistical approach to texture classification from single images we investigate texture classification from single images obtained under unknown viewpoint and illumination a statistical approach is developed where textures are modelled by the joint probability distribution of filter responses this distribution is represented by the frequency histogram of filter response cluster centres textons recognition proceeds from single uncalibrated images and the novelty here is that rotationally invariant filters are used and the filter response space is low dimensional classification performance is compared with the filter banks and methods of leung and malik ijcv 2001 schmid cvpr 2001 and cula and dana ijcv 2004 and it is demonstrated that superior performance is achieved here classification results are presented for all 61 materials in the columbia utrecht texture database we also discuss the effects of various parameters on our classification algorithm such as the choice of filter bank and rotational invariance the size of the texton dictionary as well as the number of training images used finally we present a method of reliably measuring relative orientation co occurrence statistics in a rotationally invariant manner and discuss whether incorporating such information can enhance the classifier s performance

reflection positivity and monotonicity we prove general reflection positivity results on riemannian manifolds for both scalar fields and dirac fields and comment on applications to quantum field theory as another application we prove the inequality cd cn between dirichlet and neumann covariance operators on a manifold with a reflection

multi parameter local semi groups of hermetian operators abstract it is proven that if a weakly continuous multi parameter family of hermetian operators in a hilbert space whose domains are not necessarily dense but whose union is dense and which has the semi group property for the parameters belonging to an open convex set in euclidean n space with 0 a limit point then the operators have self adjoint extensions which have a joint spectral representation

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

reconstruction of kubo martin schwinger structure from euclidean green functions the explicit form of the poisson kernel for the strip t z c 0 im z is used to derive some conditions for a state on a given c algebra dynamical system a t to be a kubo martin schwinger kms state reflection positivity and generalized periodicity of the euclidean green functions corresponding to are established and used for an analytic continuation to real time reflection positivity is also used for the construction of a cyclic and faithful representation of a t on some hilbert space a sufficient condition for the identification of the constructed dynamics with the modular group of a von neumann algebra generated by the representation is formulated in terms of the euclidean green functions

gaussian mrf rotation invariant features for image classification features based on markov random field mrf models are sensitive to texture rotation this paper develops an anisotropic circular gaussian mrf acgmrf model for retrieving rotation invariant texture features to overcome the singularity problem of the least squares estimate method an approximate least squares estimate method is designed and implemented rotation invariant features are obtained from the acgmrf model parameters using the discrete fourier transform the acgmrf model is demonstrated to be a statistical improvement over three published methods the three methods include a laplacian pyramid an isotropic circular gmrf icgmrf and gray level cooccurrence probability features

continuous integral kernels for unbounded schrodinger semigroups and their spectral projections abstract by suitably extending a feynman kac formula of simon canad math soc conf proc 28 2000 317 we study one parameter semigroups generated by the negative of rather general schrodinger operators which may be unbounded from below and include a magnetic vector potential in particular a common domain of essential self adjointness for such a semigroup is specified moreover each member of the semigroup is proven to be a maximal carleman operator with a continuous integral kernel given by a brownian bridge expectation the results are used to show that the spectral projections of the generating schrodinger operator also act as carleman operators with continuous integral kernels applications to schrodinger operators with rather general random scalar potentials include a rigorous justification of an integral kernel representation of their integrated density of states a relation frequently used in the physics literature on disordered solids

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

w kms structure from multi time euclidean green functions an axiomatic approach to the problem of reconstruction of a dynamics from the euclidean multi time green functions is proposed the existence of a w algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown a condition under which the reconstruction from the one time and multi time green functions are equivalent is formulated

spectral representations for unbounded operators with real spectrum soit t un operateur agissant dans un espace de banach reflexif x avec domaine d t et spectre reel alors il existe une mesure spectrale e sur z la variete de semisimplicite pour t telle que 1 pour chaque b r e commute avec chaque u b x qui commute avec t 2 d t z x z r se ds x existe et appartient a z et 3 tx r se ds x pour tout x d t z

a compact local binary pattern using maximization of mutual information for face analysis although many variants of local binary patterns lbp are widely used for face analysis due to their satisfactory classification performance they have not yet been proven compact we propose an effective code selection method that obtain a compact lbp clbp using the maximization of mutual information mmi between features and class labels the derived clbp is effective because it provides better classification performance with smaller number of codes we demonstrate the effectiveness of the proposed clbp by several experiments of face recognition and facial expression recognition our experimental results show that the clbp outperforms other lbp variants such as lbp ulbp and mct in terms of smaller number of codes and better recognition performance

the generation of weakly integrable semigroups abstract a characterization of the generators of a class of weakly integrable semigroups on a locally convex space is proved it bears a resemblance to the hille yosida phillips theorem on the generators of strongly continuous semigroups on a banach space

from the euclidean group to the poincare group via osterwalder schrader positivity given a continuous representation of the euclidean group inn 1 dimensions together with a covariant system of subspaces which satisfies osterwalder schrader positivity we construct a continuous unitary representation of the orthochronous poincare group inn 1 dimensions satisfying the spectral condition a similar result holds for the covering groups of the euclidean and poincare group

locally rotation contrast and scale invariant descriptors for texture analysis textures within real images vary in brightness contrast scale and skew as imaging conditions change to enable recognition of textures in real images it is necessary to employ a similarity measure that is invariant to these properties furthermore since textures often appear on undulating surfaces such invariances must necessarily be local rather than global despite these requirements it is only relatively recently that texture recognition algorithms with local scale and affine invariance properties have begun to be reported typically they comprise detecting feature points followed by geometric normalization prior to description we describe a method based on invariant combinations of linear filters unlike previous methods we introduce a novel family of filters which provides scale invariance resulting in a texture description invariant to local changes in orientation contrast and scale and robust to local skew significantly the family of filters enables local scale invariants to be defined without using a scale selection principle or a large number of filters a texture discrimination method based on the chi2 similarity measure applied to histograms derived from our filter responses outperforms existing methods for retrieval and classification results for both the brodatz textures and the university of illinois urbana champaign uiuc database which has been designed to require local invariance

texture classification by wavelet packet signatures this correspondence introduces a new approach to characterize textures at multiple scales the performance of wavelet packet spaces are measured in terms of sensitivity and selectivity for the classification of twenty five natural textures both energy and entropy metrics were computed for each wavelet packet and incorporated into distinct scale space representations where each wavelet packet channel reflected a specific scale and orientation sensitivity wavelet packet representations for twenty five natural textures were classified without error by a simple two layer network classifier an analyzing function of large regularity d sub 20 was shown to be slightly more efficient in representation and discrimination than a similar function with fewer vanishing moments d sub 6 in addition energy representations computed from the standard wavelet decomposition alone 17 features provided classification without error for the twenty five textures included in our study the reliability exhibited by texture signatures based on wavelet packets analysis suggest that the multiresolution properties of such transforms are beneficial for accomplishing segmentation classification and subtle discrimination of texture

on virtual representations of symmetric spaces and their analytic continuation in this paper we study a problem in group representation theory motivated by relativistic quantum field theory one of the most powerful approaches to constructing models of relativistic quantum fields relies on euclidean field theory a description of quantum field theory in which time is purely imaginary the basic objects of euclidean field theory the euclidean green s or schwinger functions can often be expressed in terms of functional integrals the construction of quantum field models is thereby reduced to quadratures a well known simple example of this method is the reformulation of quantum mechanisms by means of wiener integrals the fundamental solution of the schrodinger equation analytically continued to imaginary time is given by the well known feynman kac formula see e g 15 19 in this approach one faces the problem of analytic continuation back to real time while this problem is elementary in non relativistic quantum mechanics it is rather intricate in relativistic quantum field theory for some class of relativistic quantum field theories a fairly general solution to this problem was given in papers by osterwalder and

dominant local binary patterns for texture classification this paper proposes a novel approach to extract image features for texture classification the proposed features are robust to image rotation less sensitive to histogram equalization and noise it comprises of two sets of features dominant local binary patterns dlbp in a texture image and the supplementary features extracted by using the circularly symmetric gabor filter responses the dominant local binary pattern method makes use of the most frequently occurred patterns to capture descriptive textural information while the gabor based features aim at supplying additional global textural information to the dlbp features through experiments the proposed approach has been intensively evaluated by applying a large number of classification tests to histogram equalized randomly rotated and noise corrupted images in outex brodatz meastex and curet texture image databases our method has also been compared with six published texture features in the experiments it is experimentally demonstrated that the proposed method achieves the highest classification accuracy in various texture databases and image conditions

design selection and classification for hadamard matrices using generalized minimum aberration criteria deng and tang 1999 and tang and deng 1999 proposed and justified two criteria of generalized minimum aberration for general two level fractional factorial designs the criteria are defined using a set of values called j characteristics in this article we examine the practical use of the criteria in design selection specifically we consider the problem of classifying and ranking designs that are based on hadamard matrices a theoretical result on j characteristics is developed to facilitate the computation the issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity our studies reveal that generalized aberration performs quite well under these familiar criteria

stochastically positive structures on weyl algebras the case of quasi free states we consider quasi free stochastically positive ground and thermal states on weyl algebras in the imaginary time formulation in particular we obtain a new derivation of a general form of thermal quasi free state and give conditions when such a state is stochastically positive i e when it defines a periodic stochastic process with respect to imaginary time a so called thermal process then we show that the thermal process completely determines modular structure canonically associated with the quasi free thermal state on weyl algebra we discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space times and models of quantum statistical mechanics

multiresolution gray scale and rotation invariant texture classification with local binary patterns presents a theoretically very simple yet efficient multiresolution approach to gray scale and rotation invariant texture classification based on local binary patterns and nonparametric discrimination of sample and prototype distributions the method is based on recognizing that certain local binary patterns termed uniform are fundamental properties of local image texture and their occurrence histogram is proven to be a very powerful texture feature we derive a generalized gray scale and rotation invariant operator presentation that allows for detecting the uniform patterns for any quantization of the angular space and for any spatial resolution and presents a method for combining multiple operators for multiresolution analysis the proposed approach is very robust in terms of gray scale variations since the operator is by definition invariant against any monotonic transformation of the gray scale another advantage is computational simplicity as the operator can be realized with a few operations in a small neighborhood and a lookup table experimental results demonstrate that good discrimination can be achieved with the occurrence statistics of simple rotation invariant local binary patterns

high order local ternary patterns with locality preserving projection for smoke detection and image classification it is a challenging task to recognize smoke from visual scenes due to large variations in the color texture shapes of smoke to improve detection accuracy we propose a novel feature extraction method by encoding high order directional derivatives at each pixel we first quantize the directional derivatives into ternary values to generate local ternary patterns ltp for the sake of simplification each ltp code is usually decomposed into an upper lbp code and a lower lbp code but this leads to loss of information hence we use joint histograms to preserve the co occurrence of upper and lower lbp codes for each order ltp then we concatenate all joint histograms from different orders to propose high order local ternary patterns hltp to improve computational efficiency we apply locality preserving projection lpp to reduce the dimension of hltp to further improve performance we present a noise resistant mechanism to remove noisy derivatives and then propose hltp based on magnitudes of noise removed derivatives and values of center pixels hltpmc finally the support vector machine svm is used for training and classification experiments on large scale smoke data sets show that our method can achieve detection rates above 94 with false alarm rates below 1 33 experiments on a multi class brodatz texture data set also achieved good performance with low dimensional features so our method has powerful discriminative capabilities and compact feature representation for multi class image classification

thermal quantum fields with spatially cutoff interactions in 1 1 space time dimensions we construct interacting quantum fields in 1 1 space time dimensions representing charged or neutral scalar bosons at positive temperature and zero chemical potential our work is based on prior work by klein and landau and hoegh krohn generalized path space methods are used to add a spatially cut off interaction to the free system which is described in the araki woods representation it is shown that the interacting kms state is normal w r t the araki woods representation the observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting liouvillean is described in terms of the free liouvillean and the interaction

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

reflection positivity and conformal symmetry abstract a reflection positive hilbert space is a triple e e where e is a hilbert space a unitary involution and e a closed subspace on which the hermitian form v w v w is positive semidefinite for a triple g s where g is a lie group an involutive automorphism of g and s a subsemigroup invariant under the involution s s s 1 a unitary representation of g on e e is called reflection positive if g g and s e e this is the first in a series of papers in which we develop a new and systematic approach to reflection positive representations based on reflection positive distributions and reflection positive distribution vectors this approach is most natural to obtain classification results in particular in the abelian case among the tools we develop is a generalization of the bochner schwartz theorem to positive definite distributions on open convex cones we further illustrate our techniques with a non abelian example by constructing reflection positive distribution vectors for complementary series representations of the conformal group o 1 n 1 r of the sphere s n

filtering for texture classification a comparative study in this paper we review most major filtering approaches to texture feature extraction and perform a comparative study filtering approaches included are laws masks 1980 ring wedge filters dyadic gabor filter banks wavelet transforms wavelet packets and wavelet frames quadrature mirror filters discrete cosine transform eigenfilters optimized gabor filters linear predictors and optimized finite impulse response filters the features are computed as the local energy of the filter responses the effect of the filtering is highlighted keeping the local energy function and the classification algorithm identical for most approaches for reference comparisons with two classical nonfiltering approaches co occurrence statistical and autoregressive model based features are given we present a ranking of the tested approaches based on extensive experiments

diffusion semigroups in spaces of continuous functions with mixed topology abstract we study transition semigroups and kolmogorov equations corresponding to stochastic semilinear equations on a hilbert space h it is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on h when endowed with the so called mixed topology as a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups

weakly integrable semigroups on locally convex spaces abstract a class of weakly integrable semigroups on locally convex spaces is introduced and studied the results are illustrated by examples of semigroups of unbounded operators on a banach space which include fractional powers of a closed operator and spectral local semigroups

blocking orthogonal designs with mixed integer linear programming we present a mixed integer linear programming approach to orthogonally block two level multilevel and mixed level orthogonal designs the approach involves an exact optimization technique which guarantees an optimal solution it can be applied to many problems where combinatorial methods for blocking orthogonal designs cannot be used by means of 54 run and 64 run examples we demonstrate that our approach outperforms two benchmark techniques in terms of the number of estimable two factor interaction contrasts and in terms of the d efficiency for models with main effects and some two factor interaction contrasts we demonstrate the generic nature of our approach by applying it to the most challenging instances in a catalog of all orthogonal designs of strength 3 with up to 81 runs as well as a small catalog of strength 4 designs the approach can also be applied to strength 2 designs but for these cases alternative methods described in the literature may perform equally well supplementary materials for this article are available online 2015 american statistical association and the american society for quality

correction to rotation and gray scale transform invariant texture classification using spiral resampling subband decomposition and hidden markov model this paper proposes a new texture classification algorithm that is invariant to rotation and gray scale transformation first we convert two dimensional 2 d texture images to one dimensional 1 d signals by spiral resampling then we use a quadrature mirror filter qmf bank to decompose sampled signals into subbands in each band we take high order autocorrelation functions as features features in different bands which form a vector sequence are then modeled as a hidden markov model bmm during classification the unknown texture is matched against all the models and the best match is taken as the classification result simulations showed that the highest correct classification rate for 16 kinds of texture was 95 14

computer aided blocking of factorial and response surface designs we consider the problem of blocking response surface and factorial designs when block sizes are prespecified often rendering standard approaches inapplicable a generalization of harville s 1974

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

ergodic properties of operator semigroups in general weak topologies abstract the aim of this paper is to study ergodic properties i e properties about the limit of cesaro averages of a semigroup of bounded linear operators on a banach space x which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of x then the results are applied to particular examples such as locally strongly integrable semigroups their dual semigroups and the tensor product semigroup of two c 0 semigroups