maximum likelihood estimation of a structured covariance matrix with a condition number constraint in this paper we deal with the problem of estimating the disturbance covariance matrix for radar signal processing applications when a limited number of training data is present we determine the maximum likelihood ml estimator of the covariance matrix starting from a set of secondary data assuming a special covariance structure i e the sum of a positive semi definite matrix plus a term proportional to the identity and a condition number upper bound constraint we show that the formulated constrained optimization problem falls within the class of maxdet problems and develop an efficient procedure for its solution in closed form remarkably the computational complexity of the algorithm is of the same order as the eigenvalue decomposition of the sample covariance matrix at the analysis stage we assess the performance of the proposed algorithm in terms of achievable signal to interference plus noise ratio sinr both for a spatial and a doppler processing the results show that interesting sinr improvements with respect to some existing covariance matrix estimation techniques can be achieved

geodesic convexity and covariance estimation geodesic convexity is a generalization of classical convexity which guarantees that all local minima of g convex functions are globally optimal we consider g convex functions with positive definite matrix variables and prove that kronecker products and logarithms of determinants are g convex we apply these results to two modern covariance estimation problems robust estimation in scaled gaussian distributions and kronecker structured models maximum likelihood estimation in these settings involves non convex minimizations we show that these problems are in fact g convex this leads to straight forward analysis allows the use of standard optimization methods and paves the road to various extensions via additional g convex regularization

a note on set theoretic solutions of the yang baxter equation abstract this paper shows that every finite non degenerate involutive set theoretic solution x r of the yang baxter equation whose permutation group g x r has cardinality which is a cube free number is a multipermutation solution some properties of finite braces are also investigated it is also shown that if a is a left brace whose cardinality is an odd number and a b a b for all a b a then a is a two sided brace and hence a jacobson radical ring it is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level

optimization of quiescent response in partially adaptive beamformers the adaptive weights utilize only a fraction of the available degrees of freedom in partially adaptive lcmv linearly constrained minimum invariance beamformers two procedures are presented for incorporating unused degrees of freedom into the beamformer s quiescent response in one case the quiescent response is chosen to approximate a desired quiescent response and in the other case it is chosen to maximize the average interference cancellation over a set of likely interference scenarios a sidelobe canceling interpretation indicates that maximization of average interference cancellation results in quiescent weights that optimize the sum of the white noise gain and the error between the quiescent response and the class of responses which the adaptive branch can synthesize simulations illustrate the utility of the methods

matched pairs approach to set theoretic solutions of the yang baxter equation we study set theoretic solutions x r of the yang baxter equations on a set x in terms of the induced left and right actions of x on itself we give a characterisation of involutive square free solutions in terms of cyclicity conditions we characterise general solutions in terms of abstract matched pair properties of the associated monoid s x r and we show that r extends as a solution rs on s x r as a set finally we study extensions of solutions both directly and in terms of matched pairs of their associated monoids we also prove several general results about matched pairs of monoids s of the required type including iterated products s s s equivalent to rs a solution and extensions s t rs t examples include a general double construction s s rs s and some concrete extensions their actions and graphs based on small sets

a cfar adaptive subspace detector based on a single observation in system dependent clutter background in this paper the problem of detecting target in system dependent clutter sdc background with a single observation from the test cell is researched classical detectors such as the generalized likelihood ratio detectors glrds and the adaptive matched filters amfs etc usually deal with the clutter and the noise as a whole the low rank detectors lrds make use of the low rank property of the clutter to improve the detection performance however the performance of lrds degrades when the signal is not orthogonal with respect to w r t the clutter in this paper an adaptive subspace detector for sdc sdc asd background which deals with the clutter and the noise separately is proposed the sdc asd designs the test statistic by replacing the signal and the clutter covariance matrix with their maximum likelihood estimations mles its theoretical false alarm probability and detection probability are analytically deduced analytical results show that the test statistic has the form of non central distribution besides it is shown that the sdc asd has constant false alarm rate cfar performance w r t the clutter and the noise numerical experiments are provided to validate the detection performance of the sdc asd in dealing with the target detection in sdc background less

fast converging adaptive detection of doppler shifted range distributed targets a generalized likelihood ratio test glrt for the adaptive detection of a target or targets that are doppler shifted and distributed in range is derived the unknown parameters associated with the hypothesis test are the complex amplitudes in range of the desired target and the unknown covariance matrix of the additive interference which is assumed to be characterized as complex zero mean correlated gaussian random variables the target s or targets complex amplitudes are assumed to be distributed across the entire input data block sensor spl times range the unknown covariance matrix is constrained to have the reasonable form of the identity matrix the internal noise contribution plus an unknown positive semidefinite psdh matrix the external interference contribution it is shown via simulation for a variety of interference scenarios that the new detector has the characteristic of having a bounded constant false alarm rate cfar i e for our problem the probability of false alarm p sub f for a given detection threshold is bounded by the p sub f that results when no external interference is present it is also shown via simulation that the new detector converges relatively fast with respect to the number of sample vectors k necessary in order to achieve a given probability of detection p sub d

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

a new parametric glrt for multichannel adaptive signal detection a parametric generalized likelihood ratio test glrt for multichannel signal detection in spatially and temporally colored disturbance was recently introduced by modeling the disturbance as a multichannel autoregressive ar process the detector however involves a highly nonlinear maximum likelihood estimation procedure which was solved via a two dimensional iterative search method initialized by a suboptimal estimator in this paper we present a simplified glrt along with a new estimator for the problem both the estimator and the glrt are derived in closed form at considerably lower complexity with adequate training data the new glrt achieves a similar detection performance as the original one however for the more interesting case of limited training the original glrt may become inferior due to poor initialization because of its simpler form the new glrt also offers additional insight into the parametric multichannel signal detection problem the performance of the proposed detector is assessed using both a simulated dataset which was generated using multichannel ar models and the kassper dataset a widely used dataset with challenging heterogeneous effects found in real world environments

on using a priori knowledge in space time adaptive processing in space time adaptive processing stap the clutter covariance matrix is routinely estimated from secondary ldquotarget freerdquo data because this type of data is more often than not rather scarce the so obtained estimates of the clutter covariance matrix are typically rather poor in knowledge aided ka stap an a priori guess of the clutter covariance matrix e g derived from knowledge of the terrain probed by the radar is available in this note we describe a computationally simple and fully automatic method for combining this prior guess with secondary data to obtain a theoretically optimal in the mean squared error sense estimate of the clutter covariance matrix the authors apply the proposed method to the kassper data set to illustrate the type of achievable performance

a cfar adaptive subspace detector for second order gaussian signals we study the problem of detecting subspace signals described by the second order gaussian sog model in the presence of noise whose covariance structure and level are both unknown such a detection problem is often called gauss gauss problem in that both the signal and the noise are assumed to have gaussian distributions we propose adaptive detectors for the sog model signals based on a single observation and multiple observations with a single observation the detector can be derived in a manner similar to that of the generalized likelihood ratio test glrt but the unknown covariance structure is replaced by sample covariance matrix based on training data the proposed detectors are constant false alarm rate cfar detectors as a comparison we also derive adaptive detectors for the first order gaussian fog model based on multiple observations under the same noise condition as for the sog model with a single observation the seemingly ad hoc cfar detector for the sog model is a true glrt in that it has the same form as the glrt cfar detector for the fog model we give an approximate closed form of the probability of detection and false alarm in this case furthermore we study the proposed cfar detectors and compute the performance curves

computationally efficient maximum likelihood estimation of structured covariance matrices by invoking the extended invariance principle exip we present herein a computationally efficient method that provides asymptotic for large samples maximum likelihood aml estimation for structured covariance matrices and is referred to as the aml algorithm a closed form formula for estimating the hermitian toeplitz covariance matrices that makes aml computationally simpler than most existing hermitian toeplitz matrix estimation algorithms is derived although the aml covariance matrix estimator can be used in a variety of applications we focus on array processing our simulation study shows that aml enhances the performance of angle estimation algorithms such as music by making them very close to the corresponding cramer rao bound crb for uncorrelated signals numerical comparisons with several structured and unstructured covariance matrix estimators are also presented

glrt based direction detectors in homogeneous noise and subspace interference in this paper we derive and assess decision schemes to discriminate resorting to an array of sensors between the h0 hypothesis that data under test contain disturbance only i e noise plus interference and the h1 hypothesis that they also contain signal components along a direction which is a priori unknown but constrained to belong to a given subspace of the observables the disturbance is modeled in terms of complex normal random vectors plus deterministic interference assumed to belong to a known subspace we assume that a set of noise only secondary data is available which possess the same statistical characterization of noise in the cells under test at the design stage we resort to either the plain generalized likelihood ratio test glrt or the two step glrt based design procedure the performance analysis conducted resorting to simulated data shows that the one step glrt performs better than the detector relying on the two step design procedure when the number of secondary data is comparable to the number of sensors moreover it outperforms a one step glrt based subspace detector when the dimension of the signal subspace is sufficiently high

adaptive subspace detectors we use the theory of generalized likelihood ratio tests glrts to adapt the matched subspace detectors msds of scharf 1991 and of scharf and frielander 1994 to unknown noise covariance matrices in so doing we produce adaptive msds that may be applied to signal detection for radar sonar and data communication we call the resulting detectors adaptive subspace detectors asds these include kelly s 1987 glrt and the adaptive cosine estimator ace of kaurt and scharh see ibid vol 47 p 2538 41 1999 and of scharf and mcwhorter see proc 30th asilomar conf signals syst comput pacific grove ca 1996 for scenarios in which the scaling of the test data may deviate from that of the training data we then present a unified analysis of the statistical behavior of the entire class of asds obtaining statistically identical decompositions in which each asd is simply decomposed into the nonadaptive matched filter the nonadaptive cosine or t statistic and three other statistically independent random variables that account for the performance degrading effects of limited training data

affine structures of decomposable solvable groups abstract bi crossed products of groups are closely related to braces a ring like structure connected with solutions to the yang baxter equation the asymmetric product of braces which has become useful for the construction of regular affine groups and simple braces as well as several other product constructions are unified extended and simplified all bi crossed products of bicyclic braces are classified

radar detection of distributed targets in homogeneous interference whose inverse covariance structure is defined via unitary invariant functions in this paper we deal with the problem of detecting an extended target embedded in homogeneous gaussian interference with unknown but structured covariance matrix we model the possible target echo from each range bin under test as a deterministic signal with an unknown scaling factor accounting for the target response at the design stage we exploit some a priori knowledge about the operating environment enforcing the inverse interference plus noise covariance matrix to belong to a set described via unitary invariant continuous functions hence we derive the constrained maximum likelihood ml estimates of the unknown parameters under both the h0 and h1 hypotheses and design the generalized likelihood ratio test glrt for the considered decision problem at the analysis stage we assess the performance of the devised glrt for some covariance matrix uncertainty sets of practical relevance both for spatial and doppler processing the results highlight that correct use of the a priori knowledge can lead to a detection performance quite close to the optimum receiver which supposes the perfect knowledge of the interference plus noise covariance matrix

a unifying framework for adaptive radar detection in homogeneous plus structured interference part ii detectors design this paper deals with the problem of adaptive multidimensional multichannel signal detection in homogeneous gaussian disturbance with unknown covariance matrix and structured unknown deterministic interference the aforementioned problem extends the well known generalized multivariate analysis of variance gmanova tackled in the open literature in part i of this paper we have obtained the maximal invariant statistic mis for the problem under consideration as an enabling tool for the design of suitable detectors which possess the constant false alarm rate cfar property herein we focus on the development of several theoretically founded detectors for the problem under consideration first all the considered detectors are shown to be function of the mis thus proving their cfarness property second coincidence or statistical equivalence among some of them in such a general signal model is proved third strong connections to well known simpler scenarios analyzed in adaptive detection literature are established finally simulation results are provided for a comparison of the proposed receivers

multipermutation solutions of the yang baxter equation set theoretic solutions of the yang baxter equation form a meeting ground of mathematical physics algebra and combinatorics such a solution consists of a set x and a function r x x x x which satisfies the braid relation

on finite complete rewriting systems finite derivation type and automaticity for homogeneous monoids this paper investigates the class of finitely presented monoids defined by homogeneous length preserving relations from a computational perspective the properties of admitting a finite complete rewriting system having finite derivation type being automatic and being biautomatic are investigated for this class of monoids the first main result shows that for any consistent combination of these properties and their negations there is a homogeneous monoid with exactly this combination of properties we then introduce the new concept of abstract rees commensurability an analogue of the notion of abstract commensurability for groups in order to extend this result to show that the same statement holds even if one restricts attention to the class of n ary homogeneous monoids where every side of every relation has fixed length n we then introduce a new encoding technique that allows us to extend the result partially to the class of n ary multihomogenous monoids

joint estimation of inverse covariance matrices lying in an unknown subspace we consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices we perform the estimation using groups of measurements with different covariances assuming the inverse covariances span a low dimensional subspace our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation we develop a novel optimization algorithm discovering and exploiting the underlying low dimensional subspace we provide a computationally efficient algorithm and derive a tight upper performance bound numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm

subspace based adaptive generalized likelihood ratio detection subspace based adaptive detection performance is examined for the generalized likelihood ratio detector based on wilks spl lambda statistic the problem considered here is detecting the presence of one or more signals of known shape embedded in gaussian distributed noise with unknown covariance structure the data is mapped into a subspace prior to detection the probability of false alarm is independent of the subspace transformation and depends only on subspace dimension the probability of detection depends on the subspace transformation through a nonadaptive signal to noise ratio snr parameter subspace processing results in an snr loss that tends to decrease performance and a gain in statistical stability that tends to increase performance it is shown that the statistical stability effect dominates the snr loss for short data records and subspace detectors can require substantially less snr than full space detectors for equivalent performance a method for designing the subspace transformation to minimize the snr loss is proposed and illustrated through simulations

on the spectrum of six vertex corner transfer matrices we consider the eigenstates of the six vertex corner transfer matrix using its generator which is an operator of heisenberg xxz type in the ordered regime we construct a complete set of fundamental eigenstates and some bethe ansatz eigenstates as well in the disordered critical regime the spectrum of the finite size system collapses in the thermodynamic limit and we show how this depends on in m the size of the system and

adaptive array detection of uncertain rank one waveforms adaptive array detection of known within a complex scaling rank one space time waveforms in unknown spatial noise has received considerable attention the two published solutions are the adaptive matched filter and the glrt we expand on this work to consider the case of rank one waveforms that are uncertain i e only partially known more precisely we model the space time steering vector as the kronecker product of two vectors each of which is unknown but is known to lie in a known subspace applications for such a model include detection in the presence of multipath and spectral or polarization diversity in both radar processing and wireless communication using the principle of invariance we construct detectors based on the maximal invariant we show that the snr required to achieve a given detection probability for a given false alarm rate is only weakly impacted by waveform uncertainty thus our detector approaches the performance of earlier detectors which entail known waveforms

knowledge aided parametric adaptive matched filter with automatic combining for covariance estimation in this paper a knowledge aided parametric adap tive matched filter ka pamf is proposed that utilizing both observations including the test and training signals and ap riori knowledge of the spatial covariance matrix unlike existing ka pamf methods the proposed ka pamf is able to automat ically adjust the combining weight of ap rioricovariance matrix thus gaining enhanced robustness against uncertainty in the prior knowledge meanwhile the proposed ka pamf is significantly more efficient than its ka nonparametric counterparts when the amount of training signals is limited one distinct feature of the proposed ka pamf is the inclusion of both the test and training signals for automatic determination of the combining weights for the prior spatial covariance matrix and observations numerical results are presented to demonstrate the effectiveness of the proposed ka pamf especially in the limited training scenarios

robust design of radar doppler filters this paper considers the design of robust filters for radar pulse doppler processing when the interference is a wide sense stationary random process the figure of merit which is optimized is the signal to interference plus noise ratio sinr at the filter output under a multitude of constraints accounting for doppler filter sidelobes as well as uncertainties both in the received useful signal component and interference covariance matrix the design is analytically formulated as a constrained optimization problem whose solvability is thoroughly studied precisely a polynomial time solution technique to get the optimal filter is proposed exploiting the representation of non negative trigonometric polynomials via linear matrix inequalities the spectral factorization theorem and the duality theory last but not least a detailed analysis of the optimum filter performance is provided showing the tradeoffs involved in the design and the gain achievable over some already known counterparts

on groups of i type and involutive yang baxter groups abstract we suggest a cohomological framework to describe groups of i type and involutive yang baxter groups these groups are key in the study of involutive non degenerate set theoretic solutions of the quantum yang baxter equation our main tool is a lifting criterion for 1 cocycles established here in a general non abelian setting

shrinkage algorithms for mmse covariance estimation we address covariance estimation in the sense of minimum mean squared error mmse when the samples are gaussian distributed specifically we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples large p small n first we improve on the ledoit wolf lw method by conditioning on a sufficient statistic by the rao blackwell theorem this yields a new estimator called rblw whose mean squared error dominates that of lw for gaussian variables second to further reduce the estimation error we propose an iterative approach which approximates the clairvoyant shrinkage estimator convergence of this iterative method is established and a closed form expression for the limit is determined which is referred to as the oracle approximating shrinkage oas estimator both rblw and oas estimators have simple expressions and are easily implemented although the two methods are developed from different perspectives their structure is identical up to specified constants the rblw estimator provably dominates the lw method for gaussian samples numerical simulations demonstrate that the oas approach can perform even better than rblw especially when n is much less than p we also demonstrate the performance of these techniques in the context of adaptive beamforming

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

factorizations of skew braces we introduce strong left ideals of skew braces and prove that they produce non trivial decomposition of set theoretic solutions of the yang baxter equation we study factorization of skew left braces through strong left ideals and we prove analogs of ito s theorem in the context of skew left braces as a corollary we obtain applications to the retractability problem of involutive non degenerate solutions of the yang baxter equation finally we classify skew braces that contain no non trivial proper characteristic ideals

robust gaussian and non gaussian matched subspace detection we address the problem of matched filter and subspace detection in the presence of arbitrary noise and interference or interfering signals that may lie in an arbitrary unknown subspace of the measurement space a minmax methodology developed to deal with this uncertainty can also be adapted to situations where partial information on the interference or other uncertainties is available this methodology leads to a hypothesis test with adequate levels of false alarm robustness and signal detection sensitivity the robust test is applicable to a large class of noise density functions in addition generalized likelihood ratio glr detectors are derived for the class of generalized gaussian noise the detectors are generalizations of the spl chi sup 2 t and f statistics used with gaussian noise which are themselves motivated in a new way by the robust test for matched filter detection these expressions are simpler and computationally efficient the robust test reduces to the conventional test when unlearned subspace interference is known to be absent the results demonstrate that when compared with the conventional detector the robust one trades off some detection performance in the absence of interference for the sake of robustness in its presence

complex amplitude estimation in the known steering matrix and generalized waveform case we consider a special growth curve sgc model with a known steering matrix and generalized waveform in the presence of unknown interference and noise several estimators of the complex amplitude based on this model are derived including the methods of approximate maximum likelihood aml minimum variance distortionless response mvdr and amplitude and phase estimation apes we analyze the statistical properties of these estimators and show that in the presence of temporally white but spatially correlated noise and interference aml is asymptotically statistically efficient for a large snapshot number while mvdr and apes are asymptotically equivalent but not statistically efficient via several numerical examples we also show that when the noise and interference are both spatially and temporally correlated the apes estimator can achieve better estimation accuracy and exhibit greater robustness than the other methods

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

corner transfer matrix of generalized free fermion vertex systems the hamiltonian limit of the corner transfer matrix ctm of a generalized free fermion vertex system of finite size leads to a quantum spin hamiltonian of the particular form hn sigma n 1n 1 n sigma nx sigma n 1x lambda sigma ny sigma n 1y h sigma nz sigma sub n 1z diagonalization may be achieved for all pairs of parameters lambda h with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of ctm

fast non searching method for maneuvering target detection and motion parameters estimation this paper considers the coherent integration problem for maneuvering target detection and motion parameters estimation involving range migration rm and doppler frequency migration dfm within the coherent pulse interval a fast non searching method based on adjacent cross correlation function accf and lv s distribution lvd is proposed where the adjacent correlation operation is first employed to remove the rm and reduce the order of dfm after that lvd is applied to realize the coherent integration target detection and parameters estimation in addition at the cost of some performance loss another fast method via accf iteratively is also introduced to further reduce the computational complexity and obtain the motion parameters estimation the proposed two methods are fast in that they can be easily implemented by using complex multiplications the fast fourier transform fft and inverse fft ifft compared with the existing methods the presented algorithms can obtain the motion parameters estimation without any searching procedure and can achieve a good balance between the computational cost and the detection ability as well as parameters estimation performance finally several simulation experiments are provided to demonstrate the effectiveness

positive definite toeplitz completion in doa estimation for nonuniform linear antenna arrays ii partially augmentable arrays this paper considers the problem of direction of arrival doa estimation for multiple uncorrelated plane waves incident on so called fully augmentable sparse linear arrays in situations where a decision is made on the number of existing signal sources m prior to the estimation stage we investigate the conditions under which doa estimation accuracy is effective in the maximum likelihood sense in the case where m is less than the number of antenna sensors m a new approach called music maximum entropy equalization is proposed to improve doa estimation performance in the preasymptotic region of finite sample size n and signal to noise ratio a full sized positive definite p d toeplitz matrix is constructed from the m spl times m direct data covariance matrix and then alternating projections are applied to find a p d toeplitz matrix with m variate signal eigensubspace signal subspace truncations when m spl ges m cramer rao bound analysis suggests that the minimal useful sample size n is rather large even for arbitrarily strong signals it is demonstrated that the well known direct augmentation approach daa cannot approach the accuracy of the corresponding cramer rao bound even asymptotically as n spl rarr spl infin and therefore needs to be improved we present a new estimation method whereby signal subspace truncation of the daa augmented matrix is used for initialization and is followed by a local maximum likelihood optimization routine the accuracy of this method is demonstrated to be asymptotically optimal for the various superior scenarios m spl ges m presented

diagonalisation of corner transfer matrix by orthogonal polynomials the authors discuss the generator of baxter s corner transfer matrix 1982 for a critical ising model of finite size they perform its diagonalisation in terms of special orthogonal polynomials and give an asymptotic expression of the eigenvalues for large size which agrees with the conformal predictions

recursive estimation of the covariance matrix of a compound gaussian process and its application to adaptive cfar detection adaptive detection of signals embedded in gaussian or non gaussian noise is a problem of primary concern among radar engineers we propose a recursive algorithm to estimate the structure of the covariance matrix of either a set of gaussian vectors that share the spectral properties up to a multiplicative factor or a set of spherically invariant random vectors sirvs with the same covariance matrix and possibly correlated texture components we also assess the performance of an adaptive implementation of the normalized matched filter nmf relying on the newly introduced estimate in the presence of compound gaussian clutter dominated disturbance in particular it is shown that a proper initialization of the recursive procedure leads to an adaptive nmf with the constant false alarm rate cfar property and that it is very effective to operate in heterogeneous environments of relevant practical interest

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

fractional qcqp with applications in ml steering direction estimation for radar detection this paper deals with the problem of estimating the steering direction of a signal embedded in gaussian disturbance under a general quadratic inequality constraint representing the uncertainty region of the steering we resort to the maximum likelihood ml criterion and focus on two scenarios the former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot the latter supposes that the useful signal keeps a constant amplitude within all the snapshots we prove that the ml criterion leads in both cases to a fractional quadratically constrained quadratic problem qcqp in order to solve it we first relax the problem into a constrained fractional semidefinite programming sdp problem which is shown equivalent via the charnes cooper transformation to an sdp problem then exploiting a suitable rank one decomposition we show that the sdp relaxation is tight and give a procedure to construct in polynomial time an optimal solution of the original problem from an optimal solution of the fractional sdp we also assess the quality of the derived estimator through a comparison between its performance and the constrained cramer rao lower bound crb finally we give two applications of the proposed theoretical framework in the context of radar detection

adaptive double subspace signal detection in gaussian background part i homogeneous environments in this two part paper we consider the problem of adaptive multidimensional multichannel signal detection in gaussian noise with unknown covariance matrix the test data primary data is assumed as a collection of sample vectors arranged as the columns of a rectangular data array the rows and columns of the signal matrix are both assumed to lie in known subspaces but with unknown coordinates due to this feature of the signal structure we name this kind of signal as the double subspace signal part i of this paper focuses on the adaptive detection in homogeneous environments while part ii deals with the adaptive detection in partially homogeneous environments precisely in this part we derive the generalized likelihood ratio test glrt rao test wald test as well as their two step variations in homogeneous environments three types of spectral norm tests snts are also introduced all these detectors are shown to possess the constant false alarm rate cfar property moreover we discuss the differences between them and show how they work another contribution is that we investigate various special cases of these detectors remarkably some of them are well known existing detectors while some others are still new at the stage of performance evaluation conducted by monte carlo simulations both matched and mismatched signals are dealt with for each case more than one scenario is considered

solving the yang baxter like matrix equation for a class of elementary matrices let u and v be two n dimensional vectors in the complex field such that v t u 0 and let a i u v t we find all the solutions of the quadratic matrix equation a x a x a x in various situations with the help of a spectral perturbation result for rank one updated matrices two numerical examples are given to demonstrate the effectiveness of the method including an application of the result to a 4 4 equation related to the classic yang baxter equation

finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids abstract we consider algebras over a field k with generators x 1 x 2 x n subject to n 2 quadratic relations of the form x i x j x k x l with i j k l and moreover every monomial x i x j appears at most once in one of the defining relations if these relations are non degenerate then it is shown that the algebra is left and right noetherian satisfies a polynomial identity and has gelfand kirillov dimension at most n in case the defining relations are square free this was already established by gateva ivanova jespers and okninski to prove these results we investigate the structure of the underlying monoid s defined by the same presentation it is called a quadratic monoid we show that there is a strong link with the divisibility monoids and monoids of i type also referred to as yb monoids monoids of i type are examples of non degenerate quadratic monoids they are a monoid interpretation of non degenerate involutive set theoretic solutions of the yang baxter equation and have been studied quite intensively in recent years by gateva ivanova and van den bergh etingof schedler and soloviev jespers and okninski divisibility monoids have been introduced by kuske these are cancellative monoids that include the class of monoids of i type we show that they have a presentation with at most n 2 relations and if they have precisely n 2 defining relations then they are monoids of i type

a diagonal growth curve model and some signal processing applications we consider a variation of the growth curve gc model referred to as the diagonal growth curve dgc model where the steering vectors and waveforms are both known and the complex amplitude matrix is constrained to be diagonal a closed form approximate maximum likelihood aml estimator for this model is derived based on the maximum likelihood principle we analyze the statistical properties of this method theoretically and show that the aml estimate is unbiased and asymptotically statistically efficient for a large snapshot number via several numerical examples in array signal processing and spectral analysis we also show that the proposed aml estimator can achieve better estimation accuracy and exhibit greater robustness than the best existing methods

complete commuting solutions of the yang baxter like matrix equation for diagonalizable matrices let a be a square matrix that is diagonalizable we find all the commuting solutions of the quadratic matrix equation a x a x a x by taking advantage of the jordan form structure of a together with the help of a well known theorem on the uniqueness of a solution to sylvester s equation two special classes of the given matrix a are further investigated including circular matrices and those that are equal to some of their powers moreover all the non commuting solutions are constructed when a is a householder matrix based on a spectral perturbation result

adaptive radar detectors based on the observed fim modified versions of rao wald and durbin tests are considered exploiting an estimator of the fisher information matrix fim in place of the exact one they are asymptotically equivalent under some technical conditions to the standard counterparts and rely on the use of the observed fim ofim which is proportional to the negative hessian of the log likelihood the developed framework is applied to the problem of adaptive radar detection of a point like target in homogeneous or partially homogeneous interference remarkably for both the scenarios it is shown that rao wald and durbin tests with ofim are statistically equivalent to the generalized likelihood ratio test glrt for the specific detection problem namely kelly s detector for the homogeneous environment and the adaptive coherence estimator ace s kraut and l l scharf the cfar adaptive subspace detector is a scale invariant glrt ieee trans signal process vol 47 no 9 pp 2538 2541 sep 1999 also known as adaptive normalized matched filter anmf e conte m lops and g ricci asymptotically optimum radar detection in compound gaussian clutter ieee trans aerosp electron syst vol 31 no 2 pp 617 625 apr 1995 for the partially homogeneous scenario this provides a new interpretation of the mentioned glrts laying the foundations for a better understanding of their theoretical validity

a physically constrained maximum likelihood method for snapshot deficient adaptive array processing this paper presents a physically constrained maximum likelihood pcml method for spatial covariance matrix and power spectral density estimation as a reduced rank adaptive array processing algorithm the physical constraints of propagating energy imposed by the wave equation and the statistical nature of the snapshots are exploited to estimate the ldquotruerdquo maximum likelihood covariance matrix that is full rank and physically realizable the resultant matrix may then be used in adaptive processing for interference cancellation and improved power estimation in nonstationary environments where the amount of available data is limited minimum variance distortionless response mvdr power estimates are computed for a given environment at different levels of snapshot support using the pcml method and several other reduced rank techniques the mvdr power estimates from the pcml method are shown to have less bias and lower standard deviation at a given level of snapshot support than any of the other reduced rank methods used furthermore the estimated power spectral density from the pcml method is shown to offer better low level source detection than the mvdr power estimates

on commuting solutions of the yang baxter like matrix equation abstract let a be an arbitrary square matrix and its jordan canonical form is p 1 a p j diag j 1 1 j q q with p an invertible matrix 1 q are different eigenvalues of matrix a the commuting solution problem of the matrix equation a x a x a x is equivalent to the problem j i i y i y i j i i j i i 2 y i j i i y i 2 with y p 1 x p diag y 1 y q we give the structures of the commuting solutions y i in special toeplitz forms based on them we construct new matrices h 1 2 related to the commuting solutions then we propose a method of solving all the commuting yang baxter like solutions by which all solutions can be obtained step by step by recursively solving matrix equations in two cases i 0 or i 0 with respect to the i th jordan block j i i

retractability of set theoretic solutions of the yang baxter equation abstract it is shown that square free set theoretic involutive non degenerate solutions of the yang baxter equation whose associated permutation group referred to as an involutive yang baxter group is abelian are retractable in the sense of etingof schedler and soloviev this solves a problem of gateva ivanova in the case of abelian iyb groups it also implies that the corresponding finitely presented abelian by finite groups called the structure groups are poly z groups secondly an example of a solution with an abelian involutive yang baxter group which is not a generalized twisted union is constructed this answers in the negative another problem of gateva ivanova the constructed solution is of multipermutation level 3 retractability of solutions is also proved in the case where the natural generators of the iyb group are cyclic permutations moreover it is shown that such solutions are generalized twisted unions

detecting influential measurements in a growth curves model this paper comprises a study of the influence of deleting measurements on growth curve estimates the problem of influential measurements is considered both at the inference level and at the design level a measure of detecting influential vs uninfluential measurements is proposed the influence analysis envisaged also serves as a means for comparing the robustness of various models to missing measurements and to different study designs the paper concludes with an illustrative application

efficient algorithms for solving condition number constrained matrix minimization problems abstract well conditioned matrices are often required in science and engineering such as signal processing and finance problem to find the nearest positive definite matrix by explicitly imposing a constraint on the condition number are considered in this paper a new algorithm based on geometric perspective is proposed for getting the required well conditioned matrix based on these a condition number constrained matrix minimization problem is further considered where the constraints are imposed for avoiding degenerate solutions in which parameter matrices become rank deficient an inexact version of alternating direction method with truly implementable inexactness criteria is proposed for solving this problem numerical experiments including comparison with some existing methods are performed to illustrate the efficiency of the proposed algorithms

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

diffuse multipath exploitation for adaptive radar detection we deal with the problem of detecting point like targets in diffuse multipath environments modeling the target echo as the superposition of a deterministic signal with an unknown scaling factor due to the direct path plus a zero mean complex circular symmetric gaussian random vector with an unknown covariance matrix accounting for the echoes from the glistening surface we devise a constrained generalized likelihood ratio test glrt for the resulting hypothesis testing problem enforcing the primary data covariance matrix due to both interference and multipath echoes to belong to a neighborhood of the secondary data sample covariance matrix remarkably the proposed decision scheme ensures the desirable constant false alarm rate cfar property with respect to the unknown parameters of the interference the performance assessment conducted on simulated data in terms of detection probability also in comparison with existing solutions highlights the effectiveness of the new approach to cope with diffuse multipath phenomena

solutions of the yang baxter equation associated with a left brace abstract given a left brace b a method is given to construct explicitly all the non degenerate involutive set theoretic solutions x r of the yang baxter equation such that the associated permutation group g x r is isomorphic as a left brace to b this method depends entirely on the brace structure of b

selection of covariables in the growth curve model summary the growth curve model introduced by potthoff roy 1964 is a generalization of the multivariate analysis of variance model and is often employed to analyze longitudinal data in which a single characteristic has been measured at p different occasions on each individual inferential problems of this model are studied by using analysis of covariance i e partitioning the p measurements into the measurements of q response variables and p q covariables rao 1965 1966 and grizzle allen 1969 discuss the possibility of using fewer than p q covariables in this paper we propose two types of formulation for the hypotheses of redundancy of a given set of covariables the likelihood ratio criteria are obtained for testing the hypotheses these results permit the use of information criteria such as aic for selection of the best subset of covariables

matched subspace detectors we formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise we derive the generalized likelihood ratio glr for each problem in the class we then establish the invariances for the glr and argue that these are the natural invariances for the problem in each case the glr is a maximal invariant statistic and the distribution of the maximal invariant statistic is monotone this means that the glr test glrt is the uniformly most powerful invariant detector we illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise in each case we give the distribution for the detector and compute performance curves

adaptive detection of range distributed targets a modified generalized likelihood ratio test mglrt for the adaptive detection of a target or targets that are distributed in range is derived the unknown parameters associated with the hypothesis test are the complex amplitudes in range of the desired target and the unknown covariance matrix of the additive interference which is assumed to be characterized as complex zero mean correlated gaussian random variables the target s or targets complex amplitudes are assumed to be distributed across the entire input data block sensor spl times range results on probabilities of false alarm and detection are derived and a bounded constant false alarm rate cfar detector is developed simulation results are presented it is shown that the derived mglrt of range distributed targets is much more effective in detecting targets distributed in range than an m out of k detector which is cascaded with a single point target kelly 1986 detector

braces and the yang baxter equation several aspects of relations between braces and non degenerate involutive set theoretic solutions of the yang baxter equation are discussed and many consequences are derived in particular for each positive integer n a finite square free multipermutation solution of the yang baxter equation with multipermutation level n and an abelian involutive yang baxter group is constructed this answers a problem of gateva ivanova and cameron it is proved that finite non degenerate involutive set theoretic solutions of the yang baxter equation whose associated involutive yang baxter group is abelian are multipermutation solutions earlier the authors proved this with the additional square free hypothesis on the solutions it is also proved that finite square free non degenerate involutive set theoretic solutions associated to a left brace are multipermutation solutions

the contribution of individual variables to hotelling s t2 wilks lambda and r2 we examine the effect of each variable on the following statistics the one sample and two sample hotelling s t2 wilks lambda for multivariate analysis of variance and r2 in multiple regression for t2 the net effect of each variable is an increase in the multivariate statistic and the particular factors determining the amount of increase are i the multiple correlation of the variable with all other variables and ii how well the variable s contribution to falsifying the hypothesis can be linearly predicted from the other variables the effect of each predictor variable on r2 is similar to the effect of each variable on t2 for wilks lambda each variable induces a decrease due to i the f for that variable alone and ii the change in multiple correlation from within sample to total sample

prediction and estimation of growth curves with special covariance structures abstract the model considered here is a generalized multivariate analysis of variance model useful especially for many types of growth curve problems including biological growth and technology substitutions it is defined as yp n xp m m rar n p n where is unknown and x and a are known design matrices of ranks m p and r n respectively further the columns of are independent p variate normal with mean vector 0 and common covariance matrix in general p is the number of time or spatial points observed on each of the n cases m 1 is the degree of polynomial and r is the number of groups this article focuses mainly on predicting future observations including partially observed vectors and future values on a given number of cases using their past observations these prediction problems as well as the estimation of parameters are considered for two parsimonious covariance structures that are very important in practice the results are illustrated with three real data sets som

performance of a class of adaptive detection algorithms in nonhomogeneous environments a two dimensional 2 d adaptive sidelobe blanker asb detection algorithm was developed through experimentation as an extenuate for false alarms caused by undernulled interference encountered when applying the adaptive matched filter amf in nonhomogeneous environments the algorithm s utility has been demonstrated empirically considering theoretic performance analyses of the asb detection algorithm as well as the amf generalized likelihood ratio test glrt and the adaptive cosine estimator ace under nonideal conditions can become fairly intractable rather quickly especially in an adaptive processing context involving covariance estimation in this paper however we have developed and exploited a theoretic framework through which the performance of these algorithms under nonhomogeneous conditions can be examined theoretically it is demonstrated through theoretic analysis that in the presence of undernulled interference the asb is a pliable false alarm regulatory far detector that maintains good target sensitivity a viable method of asb threshold selection is also presented and demonstrated

finite size scaling of corner transfer matrices for the two dimensional ising model the eigenvalue spectra of generalised corner transfer matrices are calculated for an ising model on a square lattice the predictions of conformal invariance at the critical point are thereby verified

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

unified framework to regularized covariance estimation in scaled gaussian models we consider regularized covariance estimation in scaled gaussian settings e g elliptical distributions compound gaussian processes and spherically invariant random vectors asymptotically in the number of samples the classical maximum likelihood ml estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex we propose a unified framework for regularizing this estimate in order to improve its finite sample performance our approach is based on the discovery of hidden convexity within the ml objective we begin by restricting the attention to diagonal covariance matrices using a simple change of variables we transform the problem into a convex optimization that can be efficiently solved we then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices we regularize the problem using appropriately convex penalties these allow for shrinkage towards the identity matrix shrinkage towards a diagonal matrix shrinkage towards a given positive definite matrix and regularization of the condition number we demonstrate the advantages of these estimators using numerical simulations

involutive yang baxter groups in 1992 drinfeld posed the question of finding the set theoretic solutions of the yang baxter equation recently gateva ivanova and van den bergh and etingof schedler and soloviev have shown a group theoretical interpretation of involutive non degenerate solutions namely there is a one to one correspondence between involutive non degenerate solutions on finite sets and groups of i type a group g of i type is a group isomorphic to a subgroup of fa n sym n so that the projection onto the first component is a bijective map where fa n is the free abelian group of rank n and sym n is the symmetric group of degree n the projection of g onto the second component sym n we call an involutive yang baxter group iyb group this suggests the following strategy to attack drinfeld s problem for involutive non degenerate set theoretic solutions first classify the iyb groups and second for a given iyb group g classify the groups of i type with g as associated iyb group it is known that every iyb group is solvable in this paper some results supporting the converse of this property are obtained more precisely we show that some classes of groups are iyb groups we also give a non obvious method to construct infinitely many groups of i type and hence infinitely many involutive non degenerate set theoretic solutions of the yang baxter equation with a prescribed associated iyb group

conformal invariance and critical spectrum of corner transfer matrices the finite size scaling spectra of ising model corner transfer matrices and their generators are studied at criticality the generators are diagonalised using fermion algebra the low lying eigenvalues given by the zeros of meixner polynomials are equally spaced and collapse like 1 log n for large n as predicted by conformal invariance the asymptotics are obtained using a generalised euler maclaurin summation formula the shift in the largest eigenvalue is given analytically as pi c 6 log n with central charge c 1 2 the spectrum generating functions for both fixed and free boundary conditions are expressed simply in terms of the c 1 2 virasoro characters chi delta q with modular parameter q exp pi log n and conformal dimensions delta 0 1 2 1 16

cramer rao lower bounds on covariance matrix estimation for complex elliptically symmetric distributions this paper introduces the cramer rao lower bounds crlbs for the scatter matrix of complex elliptically symmetric distributions and compares them to the performance of the constrained ml estimators in the particular cases of complex gaussian generalized gaussian gg and t distributed observation vectors numerical results confirm the goodness of the ml estimators and the advantage of taking into proper account a constraint on the matrix trace for small data size the work is completed with the comparison with the performance of tyler s matrix estimator that shows a very robust behavior in almost all the analyzed cases and with the crlbs for the complex angular elliptical distributions whose tyler s estimator is the ml one

rao test for adaptive detection in gaussian interference with unknown covariance matrix this paper deals with the problem of detecting a signal known up to a scaling factor in the presence of gaussian disturbance with unknown covariance matrix a new detector based on the rao test criterion is introduced and its invariance properties and constant false alarm rate cfar behavior are studied at the analysis stage the performance of the new receiver is assessed also in comparison with some classical adaptive radar detectors both in the matched as well as in the mismatched signal case remarkably the rao test may achieve a matched detection performance which is commensurate with that of the generalized likelihood ratio test glrt based detectors if a sufficient number of training data is available moreover it also exhibits better rejection capabilities of mismatched signals than the counterparts in the last part of the work a two stage detector whose second stage coincides with the rao test is devised it represents a suitable means to restore the detection performance of the plain rao test in the presence of a small number of training data finally the performance of the aforementioned two stage processor is analyzed in closed form and the cfar behavior is proved

set theoretic solutions of the yang baxter equation braces and symmetric groups abstract we involve simultaneously the theory of braided groups and the theory of braces to study set theoretic solutions of the yang baxter equation ybe we show the intimate relation between the notions of a symmetric group in the sense of takeuchi i e a braided involutive group and a left brace we find new results on symmetric groups of finite multipermutation level and the corresponding braces we introduce a new invariant of a symmetric group g r the derived chain of ideals of g which gives a precise information about the recursive process of retraction of g we prove that every symmetric group g r of finite multipermutation level m is a solvable group of solvable length m to each set theoretic solution x r of ybe we associate two invariant sequences of involutive braided groups i the sequence of its derived symmetric groups ii the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction of x r we find new criteria necessary and sufficient to claim that x r is a multipermutation solution

glrt based adaptive detection algorithms for range spread targets we address adaptive detection of a range spread target or targets embedded in gaussian noise with unknown covariance matrix to this end we assume that cells referred to in the following as secondary data that are free of signal components are available those secondary data are supposed to possess either the same covariance matrix or the same structure of the covariance matrix of the cells under test in this context we design detectors relying on the generalized likelihood ratio test glrt and on a two step glrt based design procedure remarkably both criteria lead to receivers ensuring the constant false alarm rate cfar property with respect to the unknown quantities a thorough performance assessment of the proposed detection strategies together with the evaluation of their processing cost highlights that the two step design procedure is to be preferred with respect to the plain glrt in fact the former leads to detectors that achieve satisfactory performance under several situations of practical interest and are simpler to implement than those designed resorting to the latter

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

commuting solutions of the yang baxter like matrix equation for a class of rank two updated matrices abstract let a i p q t where p and q are two n 2 complex matrices of full column rank and det q t p 0 all the commuting solutions with a of the quadratic matrix equation a x a x a x have been founded

corner transfer matrix of a critical free fermion system in order to diagonalize the generator of the corner transfer matrix of a critical free fermion 8 vertex system with magnetic field the authors derive an integral representation for the eigenvector components of the generator at large n they numerically verify the predictions of conformal invariance for critical systems