super lie groups global topology and local structure a general mathematical framework for the super lie groups of supersymmetric theories is presented the definition of super lie group is given in terms of supermanifolds and two theorems analogous to theorems in classical lie group theory are proved the relationship of the super lie groups defined here to the formal groups of berezin and kac and the graded lie groups of kostant is analyzed

the hyperbolic class of quadratic time frequency representations i constant q warping the hyperbolic paradigm properties and members the time frequency tf version of the wavelet transform and the affine quadratic bilinear tf representations can be used for a tf analysis with constant q characteristic the paper considers a new approach to constant q tf analysis a specific tf warping transform is applied to cohen s class of quadratic tf representations which results in a new class of quadratic tf representations with constant q characteristic the new class is related to a hyperbolic tf geometry and is thus called the hyperbolic class hc two prominent tf representations previously considered in the literature the bertrand p sub 0 distribution and the altes marinovic q distribution are members of the new hc the authors show that any hyperbolic tf representation is related to both the wideband ambiguity function and a hyperbolic ambiguity function it is also shown that the hc is the class of all quadratic tf representations which are invariant to hyperbolic time shifts and tf scalings operations which are important in the analysis of doppler invariant signals and self similar random processes the paper discusses the definition of the hc via constant q warping some signal theoretic fundamentals of the hyperbolic tf geometry and the description of the hc by 2d kernel functions several members of the hc are considered and a list of desirable properties of hyperbolic tf representations is given together with the associated kernel constraints

instantaneous frequency estimation using the wigner distribution with varying and data driven window length the estimation of the instantaneous frequency if of a harmonic complex valued signal with an additive noise using the wigner distribution is considered if the if is a nonlinear function of time the bias of the estimate depends on the window length the optimal choice of the window length based on the asymptotic formulae for the variance and bias can be used in order to resolve the bias variance tradeoff however the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the if the goal of this paper is to develop an adaptive if estimator with a time varying and data driven window length which is able to provide quality close to what could be achieved if the smoothness of the if were known in advance the algorithm uses the asymptotic formula for the variance of the estimator only its value may be easily obtained in the case of white noise and relatively high sampling rate simulation shows good accuracy for the proposed adaptive algorithm

even sectors of lie superalgebras as locally convex lie algebras let g be a finite dimensional lie superalgebra over a graded commutative local augmented arens michael ground algebra a lie group is associated with a number of pleasant properties to the even sector g0 viewed as an ordinary complete locally convex lie algebra which is in general infinite dimensional

invariant solutions of the supersymmetric sine gordon equation a comprehensive symmetry analysis of the supersymmetric sine gordon equation is performed two different forms of the supersymmetric system are considered we begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables next we consider the super sine gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables in each case a lie super algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed the method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model several types of algebraic hyperbolic and doubly periodic solutions are obtained in explicit form

baker campbell hausdorff relations for the orthosymplectic group osp 1 2 baker campbell hausdorff relations are presented for the connected supergroup associated with the orthosymplectic algebra osp 1 2

superspaces and supersymmetries a theory of graded banach modules over a banach grassmann algebra is developed and applied to differential geometry of super manifolds the explicit structure of superspaces carrying poincare supersymmetry and extended supersymmetry including central charges is described

a representation formula for maps on supermanifolds we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces i e following berezin leites manin etc we establish a representation formula for all pull back morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of a superspace we then derive two consequences of this result the first one is that we can integrate the data associated with a morphism in order to get a nonunique map defined on an ordinary space and uniqueness can be achieved by restriction to a scheme the second one is a simple and intuitive recipe to compute pull back images of a function on a manifold m by a map from a superspace to m

invariance of the magnitude squared coherence estimate with respect to second channel statistics properties of the magnitude squared coherence msc estimate in the presence of spherically symmetric noise are examined from a geometric point of view invariance to second channel statistics is shown and the distribution function of the estimate is derived a relationship between spherically symmetric and gaussian sequences is also given

supermanifolds and automorphisms of super lie groups the geometrical theory of supermanifolds is developed and applied to the super lie groups the resulting structural equations and the adjoint representation are studied and used to find the automorphisms of super lie groups the n 1 supergravity symmetry group the so called graded poincare group is explicitly studied and possible applications are outlined

baker campbell hausdorff relations for supergroups one parameter subgroups the exponential mapping and canonical coordinates have previously been defined for connected supergroups in this paper a technique for obtaining analytical expressions linking different coordinate schemes for supergroups is presented these are the baker campbell hausdorff relations the method is illustrated in detail with the examples of supergroups based on the supersymmetric quantum mechanical superalgebra sqm 2 and on the inonu wigner contraction isop 1 2 of the simple superalgebra osp 1 2

global properties of supermanifolds we construct new examples of supermanifolds and determine the vector bundle structure of the supermanifolds commonly used in physics we show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts and which is one of a nested sequence of foliations we point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds for example a compact supermanifold with a single even dimension must have vanishing euler characteristic we also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations by this mechanism the global topology of a supermanifold can be used to impose physically interesting constraints on superfields as an example we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral

nonparametric estimation of instantaneous frequency the local polynomial approximation of time varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex valued harmonic signal given by discrete time observations with a noise the considered estimators are high order nonparametric generalizations of the short time fourier transform and the wigner ville distribution the asymptotic variance and bias of the estimates are obtained

orthogonality relations and supercharacter formulas of u m n representations in this paper we obtain the orthogonality relations for the supergroup u m n which are remarkably different from the ones for the u n case we extend our results for ordinary representations obtained some time ago to the case of complex conjugated and mixed representations our results are expressed in terms of the young tableaux notation for irreducible representations we use the supersymmetric harish chandra itzykson zuber integral and the character expansion technique as mathematical tools for deriving these relations as a byproduct we also obtain closed expressions for the supercharacters and dimensions of some particular irreducible u m n representations a new way of labeling the u m n irreducible representations in terms of m n numbers is proposed finally as a corollary of our results new identities among the dimensions of the irreducible representations of the unitary group u n are presented

instantaneous frequency estimation of polynomial fm signals using the peak of the pwvd statistical performance in the presence of additive gaussian noise the peak of the polynomial wigner ville distribution pwvd has been previously proposed as an estimator of the instantaneous frequency if for a monocomponent polynomial frequency modulated fm signal in this paper we evaluate the statistical performance of this estimator in the case of additive white gaussian noise and provide an analytical expression for the variance we show that for a given pwvd order the estimator performance can be improved by a proper choice of the kernel coefficients in the pwvd a performance comparison between the pwvd based if estimator and another previously proposed one based on the high order ambiguity function haf is also provided simulation results show that for a signal to noise ratio larger than 3 db the proposed sixth order pwvd outperforms the haf in estimating the if of a third or fourth order polynomial phase signal evaluated at the central point of the observation interval

graded gauge theory the mathematical background for a graded extension of gauge theories is investigated after discussing the general properties of graded lie algebras and what may serve as a model for a graded lie group the graded fiber bundle is constructed its basis manifold is supposed to be the so called superspace i e the product of the minkowskian space time with the grassmann algebra spanned by the anticommuting lorentz spinors the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the su n lie algebra the connection and curvature are defined then on this bundle the two different gradings are either independent of each other or may be unified in one common grading which is equivalent to the choice of the spin statistics dependence the yang mills lagrangian is investigated in the simplified case the conformal symmetry breaking is discussed as well as some other physical consequences of the model

on the theory of one parameter subgroups of supergroups the one parameter subgroups of supergroups are defined and their primary properties are delineated parametrization schemes analogous to those used in lie group theory are investigated in particular exponentiation is introduced and related to the one parameter subgroups canonical coordinates of three kinds are investigated the third of which has no analog in conventional lie theory we illustrate our results with the examples of the supersymmetric quantum mechanical superalgebra sqm 2 and the simple superalgebra osp 1 2

variational calculus on supermanifolds and invariance properties of superspace field theories the foundations of a variational calculus on fibered supermanifolds are outlined giving applications to the formulation of superspace field theories utiyama theorems and conservation laws related to local gauge and general invariance of the theory are proved

chern simons forms on principal superfiber bundles a graded weil homomorphism is defined for principal superfiber bundles and the related transgression or chern simons forms are introduced as an example of the application of these concepts a superextension of the dirac monopole is discussed

asymptotic analysis of the generalized coherence estimate the generalized coherence gc estimate has shown promise as a multiple channel detection statistic but analysis of its performance in this role has been hampered because its probability density function is difficult to evaluate under signal absent hypotheses and is unknown under signal present hypotheses this paper presents an as asymptotic analysis of the gc estimate that provides tractable closed form expressions for the density of the cc estimate under useful signal absent and signal present hypotheses these expressions are valid as the number of data samples employed in the estimate approaches infinity theoretical predictions of the detection performance based on these expressions are shown to match closely results obtained by monte carlo simulation even when relatively small numbers of samples are used in the gc estimate these results are used to further examine the merits of the cc estimate as a multiple channel detection statistic

the haar integral for lie supergroups each supermanifold can be considered as equivalent to a certain family of real manifolds the haar integral of a lie supergroup is defined using this equivalence a useful general formula is derived together with explicit construction methods of the simple lie supergroups of types spl n m el osp n 2r el and b n el and the extended poincare group

wide band proportional bandwidth wigner ville analysis the wigner ville w v distribution is a time frequency representation that yields a highly accurate estimate of instantaneous frequency it is related to the narrow band ambiguity function by an internal transform and it can be used in a variety of detection and estimation problems a spectrogram constructed with constant bandwidth filters can be obtained by convolving two w v distributions or by forming a magnitude squared narrow band cross ambiguity function the wide band ambiguity function represents the doppler effect with dilation or compression rather than with frequency shift as in the narrow band approximation the q distributions a modified w v representation that is related to the wide band ambiguity function by an integral transform is defined a spectrogram constructed with proportional bandwidth or constant q filters can be obtained by a convolution like operation involving two q distributions or by forming a magnitude squared wide band cross ambiguity function the q distribution is thus a wide band version of the w v distribution properties of the q distribution indicate that it may prove useful for detection and parameter estimation as well as for tomographic measurement of wideband scattering functions with relatively few transmitted waveforms

geometry and representations of the quantum supergroup ospq 1 2n the quantum supergroup ospq 1 2n is studied systematically a haar functional is constructed and an algebraic version of the peter weyl theory is extended to this quantum supergroup quantum homogeneous superspaces and quantum homogeneous supervector bundles are defined following the strategy of connes theory parabolic induction is developed by employing the quantum homogeneous supervector bundles quantum frobenius reciprocity and a generalized borel weil theorem are established for the induced representations

audio visual group recognition using diffusion maps data fusion is a natural and common approach to recovering the state of physical systems but the dissimilar appearance of different sensors remains a fundamental obstacle we propose a unified embedding scheme for multisensory data based on the spectral diffusion framework which addresses this issue our scheme is purely data driven and assumes no a priori statistical or deterministic models of the data sources to extract the underlying structure we first embed separately each input channel the resultant structures are then combined in diffusion coordinates in particular as different sensors sample similar phenomena with different sampling densities we apply the density invariant laplace beltrami embedding this is a fundamental issue in multisensor acquisition and processing overlooked in prior approaches we extend previous work on group recognition and suggest a novel approach to the selection of diffusion coordinates to verify our approach we demonstrate performance improvements in audio visual speech recognition

a four parameter atomic decomposition of chirplets a new four parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components the four parameter chirplet atom is obtained from the unit gaussian function by successive applications of scaling fractional fourier transform frft and time shift and frequency shift operators the application of the frft operator results in a rotation of the wigner distribution of the gaussian in the time frequency plane by a specified angle the decomposition is realized by using the matching pursuit algorithm for this purpose the four parameter space is discretized to obtain a small but complete subset in the hilbert space a time frequency distribution tfd is developed for clear and readable visualization of the signal components it is observed that the chirplet decomposition and the related tfd provide more compact and precise representation of signal inner structures compared with the commonly used time frequency representations

analysis on the p adic superspace i generalized functions gaussian distribution the theory of generalized functions on the p adic superspace qn mp over the super commutative banach superalgebra 0 1 with the trivial 1 annihilator is proposed the p adic gaussian supermeasure is defined as a generalized function on the p adic superspace the connection between the p adic gaussian integration and the p adic gamma function of morita is considered

differential equations frobenius theorem and local flows on supermanifolds the classical frobenius theorem both in its local and global formulations is generalised to superanalytic supermanifolds as an application it is proved that a coset space g h where both g and h are super lie groups is a supermanifold existence and uniqueness of local flows of tangent vector fields is proved

spin network states in gauge theory abstract given a real analytic manifoldm a compact connected lie groupgand a principalg bundlep m there is a canonical generalized measure on the space a g of smooth connections onpmodulo gauge transformations this allows one to define a hilbert spacel2 a g here we construct a set of vectors spanningl2 a g these vectors are described in terms of spin networks graphs embedded inm with oriented edges labelled by irreducible unitary representations ofgand with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges we also describe an orthonormal basis of spin network states associated to any fixed graph we conclude with a discussion of spin networks in the loop representation of quantum gravity and give a category theoretic interpretation of the spin network states

super twistor realization of unitary super representations of u p q n this paper is concerned with demonstrating that the twistor realization of the ladder representation of u p q can be generalized to super representations of u p q n this is achieved by considering the generalization of twistor elementary states to a super algebraic category unitarity of these super representations follows from the positive definiteness of a super twistor scalar product constructed in this paper although generalizations of the ladder representations have been well studied by other means it will be shown that the super twistor generalization is especially natural and merits special investigation

p adic interpolation of square roots of central l values of modular forms we construct a meromorphic function on part of the eigencurve that interpolates a square root of a ratio of quadratic twists of the central modular l value

introduction to supersymmetric spin networks in this article we study the construction of supersymmetric spin networks which has a direct interpretation in context of the representation theory of the superalgebra in particular we analyze a special kind of spin network associated with superalgebra osp 1 2n it turns out that the set of corresponding spin network states forms an orthogonal basis of the hilbert space l2 a g and this argument holds even in the q deformed case the osp n 2 spin networks are also discussed briefly we expect they could provide useful techniques to quantum supergravity and gauge field theories from the point of nonperturbative view

supersymmetric distributions hilbert spaces of supersymmetric functions and quantum fields the recently investigated hilbert krein and other positivity structures of the superspace are considered in the framework of superdistributions these tools are applied to problems raised by rigorous supersymmetric quantum field theory

dimension and character formulas for lie supergroups a character formula is derived for lie supergroups the basic technique is that of symmetrization and antisymmetrization associated with young tableaux generalized to supergroups we rewrite the characters of the ordinary lie groups u n o n and sp 2n in terms of traces in the fundamental representation it is then shown that by simply replacing traces with supertraces the characters of certain representations for u n m and osp n 2m are obtained dimension formulas are derived by calculating the characters of a special diagonal supergroup element with 1 and 1 eigenvalues formulas for the eigenvalues of the quadratic casimir operators are given as applications the decomposition of a representation into representations of subgroups is discussed examples are given for the lie supergroup su 6 4 which has physical applications as a dynamical supersymmetry in nuclei

a formulation of supergravity based on cartan s connection the supergravity is formulated as a gauge theory with the yang mills lagrangian quadratic in curvature to arrive at such a description the super fiber bundle based on a superspace with the osp n 4 symmetry group is introduced the connection in this bundle becomes the cartan connection when osp n 4 is broken to sl 2 c so n in the reduced bundle this leads automatically to curvature and torsion explicit calculations of the lagrangian are carried out in terms of the cartan killing metrics and the curvature form for n 1 and n 8 the final theory describes the coupling of geometry to fermion fields these fields emerge as solder forms of the cartan connection

harmonic decomposition of audio signals with matching pursuit we introduce a dictionary of elementary waveforms called harmonic atoms that extends the gabor dictionary and fits well the natural harmonic structures of audio signals by modifying the standard matching pursuit we define a new pursuit along with a fast algorithm namely the fast harmonic matching pursuit to approximate n dimensional audio signals with a linear combination of m harmonic atoms our algorithm has a computational complexity of o mkn where k is the number of partials in a given harmonic atom the decomposition method is demonstrated on musical recordings and we describe a simple note detection algorithm that shows how one could use a harmonic matching pursuit to detect notes even in difficult situations e g very different note durations lots of reverberation and overlapping notes

characteristic classes of super vector bundles a theory of characteristic classes for complex super vector bundles over supermanifolds is developed such characteristic classes are proved to fulfill the usual properties and it is shown that under suitable conditions they can be represented in terms of supersmooth curvature forms

a geometric approach to multiple channel signal detection the paper introduces the generalized coherence gc estimate and examines its application as a statistic for detecting the presence of a common but unknown signal on several noisy channels the gc estimate is developed as a natural generalization of the magnitude squared coherence msc estimate a widely used statistic for nonparametric detection of a common signal on two noisy channels the geometrical nature of the gc estimate is exploited to derive its distribution under the h sub 0 hypothesis that the data channels contain independent white gaussian noise sequences detection thresholds corresponding to a range of false alarm probabilities are calculated from this distribution the relationship of the h sub 0 distribution of the gc estimate to that of the determinant of a complex wishart distributed matrix is noted the detection performance of the three channel gc estimate is evaluated by simulation using a white gaussian signal sequence in white gaussian noise its performance is compared with that of the multiple coherence mc estimate another nonparametric multiple channel detection statistic the gc approach is found to provide better detection performance than the mc approach in terms of the minimum signal to noise ratio on all data channels necessary to achieve desired combinations of detection and false alarm probabilities

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

spectrographic measurement of instantaneous frequency and the time dependent weighted average instantaneous frequency instantaneous frequency if is an important signal characteristic arising in many fields it is a concept intimately linked to time frequency analysis where it can be obtained from a time frequency distribution tfd as the first conditional moment in frequency suggesting that the if is the average frequency at each time however this interpretation is questionable since it is well known that the if often ranges beyond the spectral support of the signal in addition to obtain the if from a spectrogram which is one possible tfd a very wideband and thus severely spectrally distorted spectrogram must be used more reasonable bandwidths are investigated and in particular give the conditions by which the first conditional moment of a spectrogram can indeed be interpreted as the average frequency at each time under these conditions namely when the spectrogram bandwidth is less than the frequency separation of the signal components in the time frequency plane the spectrogram yields not the usual if but a time dependent weighted average instantaneous frequency waif of the signal also while the if and waif are generally different sometimes they are the same in particular when there is symmetry in the time frequency spectrum of the signal in that case the first conditional spectral moments of both wideband and narrow band spectrograms are the same and interpretable as the average frequency at each time

time varying techniques for multisensor signal detection in source detection and localization the presence of a common but unknown signal can be detected from several noisy sensor measurements using the generalized coherence gc estimate we propose to improve the performance of the gc estimate for multisensor detection using noise suppressed signal estimates obtained from time varying techniques if one of the sensors has a significantly higher signal to noise ratio snr than the other sensors then it could be preprocessed prior to the gc estimate to improve detection performance for the remaining lower snr sensors we perform this processing by estimating time varying signals of interest with nonlinear phase functions using two methods a a modified matching pursuit decomposition mmpd algorithm whose dictionary is similar in time frequency structure to the signal and b an instantaneous frequency if estimation method using highly localized time frequency representations the mmpd can yield signal estimates with lower mean square errors than the if estimation technique but at the expense of higher computational cost and memory requirements using simulations we compare the performance of the gc estimate with the significantly improved performance of the gc estimate that employs the signal estimate from the high snr sensor for the two sensor detection the estimated signal is also used with a generalized likelihood ratio test statistic to further improve performance

superspace first order partial differential equations through the cartan kahler integration theorem in the paper first order partial differential equations are studied on superspace the key point is the use of a suitable generalization of the cartan kahler integration theorem on superspace by means of this result it is possible to investigate the characteristic fields and the structure of the solutions

invariant integration on classical and quantum lie supergroups invariant integrals on hopf superalgebras in particular the classical and quantum lie supergroups are studied the uniqueness up to scalar multiples of a left integral is proved and a z2 graded version of maschke s theorem is discussed a construction of left integrals is developed for classical and quantum lie supergroups applied to several classes of examples the construction yields the left integrals in explicit form

supermanifolds and brs transformations we use a supermanifold formalism to support a geometrical interpretation of a recently proposed superfield formulation of brs becchi ronet stora and anti brs transformations

supergroups have no extra topology the global topology of supergroups is studied and it is found that they are equivalent as far as homotopy is concerned to the product of the underlying ordinary lie groups

lie groups and lie algebras with generalized supersymmetric parameters matrices with symmetric parameters the most general extension of supersymmetric parameters are investigated the superdeterminants of such matrices are defined lie groups consisting of these matrices and their lie algebras are studied

lattices supersymmetry and kahler fermions it is shown that a graded extension of the space group of a generalised simple cubic lattice exists in any space dimension d the fermionic variables which arise admit a kahlerian interpretation each graded space group is a subgroup of a graded extension of the appropriate euclidean group e d the relevance of this to the construction of lattice theories is discussed

baker campbell hausdorff relations for the super poincare group baker campbell hausdorff relations are obtained for the connected supergroup associated with the super poincare algebra iosp 1 4

variational calculus and poincare cartan formalism on supermanifolds in this work we show how the poincare cartan form can be used to describe the symmetries of lagrangian supersymmetric possibly p adic field theories we show that the existence of the poincare cartan form in supermanifold theory is ensured only in a relevant class of lagrangian densities moreover we give an abstract characterization of supersymmetric invariance based on the poincare cartan form

enlargeable graded lie algebras of supersymmetry a criterion to enlarge infinite dimensional lie algebras to analytic lie groups is used here for the extension of graded lie algebras of physical interest to super lie groups