nondegenerate 2d complex euclidean superintegrable systems and algebraic varieties a classical or quantum superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion polynomial in the momenta the maximum possible if the constants are all quadratic the system is second order superintegrable the kepler coulomb system is the best known example such systems have remarkable properties multi integrability and multi separability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with qes systems for complex riemannian spaces with n 2 the structure and classification of second order superintegrable systems is complete here however we present a new and conceptually simpler approach to the classification for complex euclidean 2 space in which the possible superintegrable systems with nondegenerate potentials correspond to points on an algebraic variety specifically we determine a variety in six variables subject to two cubic and one quartic polynomial constraints each point on the variety corresponds to a superintegrable system the euclidean group e 2 acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation

separable coordinates integrability and the niven equations finite polynomial solutions of laplace s equation are investigated the unifying features of this study are the so called niven equations which yield the dimension of the space of such solutions in carrying out this study complete sets of solutions are obtained on the n dimensional sphere in terms of ellipsoidal coordinates this corresponds to an integrable system having all the integrals of motion given by quadratic orbits of the universal enveloping algebra of o n 1 they call this system the n dimensional euler top the spectrum of the integrals of motion has been recently computed for n 3 by komarov and kuznetsov 1991 using results originally due to niven 1891 these calculations are extended to arbitrary dimension

the smallest eigenvalue of the signless laplacian recently the signless laplacian matrix of graphs has been intensively investigated while there are many results about the largest eigenvalue of the signless laplacian the properties of its smallest eigenvalue are less well studied the present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless laplacian

minimax mse ratio estimation with signal covariance uncertainties in continuation to an earlier work we further consider the problem of robust estimation of a random vector or signal with an uncertain covariance matrix that is observed through a known linear transformation and corrupted by additive noise with a known covariance matrix while in the earlier work we developed and proposed a competitive minimax approach of minimizing the worst case mean squared error mse difference regret criterion here we study in the same spirit the minimum worst case mse ratio regret criterion namely the worst case ratio rather than difference between the mse attainable using a linear estimator ignorant of the exact signal covariance and the minimum mse mmse attainable by optimum linear estimation with a known signal covariance we present the optimal linear estimator under this criterion in two ways the first is as a solution to a certain semidefinite programming sdp problem and the second is as an expression that is of closed form up to a single parameter whose value can be found by a simple line search procedure we then show that the linear minimax ratio regret estimator can also be interpreted as the mmse estimator that minimizes the mse for a certain choice of signal covariance that depends on the uncertainty region we demonstrate that in applications the proposed minimax mse ratio regret approach may outperform the well known minimax mse approach the minimax mse difference regret approach and the plug in approach where in the latter one uses the mmse estimator with an estimated covariance matrix replacing the true unknown covariance

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

completeness of multiseparable superintegrability on the complex 2 sphere the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

a maximally superintegrable system on an n dimensional space of nonconstant curvature abstract we present a novel hamiltonian system in n dimensions which admits the maximal number 2 n 1 of functionally independent quadratic first integrals this system turns out to be the first example of a maximally superintegrable hamiltonian on an n dimensional riemannian space of nonconstant curvature and it can be interpreted as the intrinsic smorodinsky winternitz system on such a space moreover we provide three different complete sets of integrals in involution and solve the equations of motion in closed form

image sequence filtering in quantum limited noise with applications to low dose fluoroscopy clinical angiography requires hundreds of x ray images putting the patients and particularly the medical staff at risk dosage reduction involves an inevitable sacrifice in image quality in this work the latter problem is addressed by first modeling the signal dependent poisson distributed noise that arises as a result of this dosage reduction the commonly utilized noise model for single images is shown to be obtainable from the new model stochastic temporal filtering techniques are proposed to enhance clinical fluoroscopy sequences corrupted by quantum mottle the temporal versions of these filters as developed here are more suitable for filtering image sequences as correlations along the time axis can be utilized for these dynamic sequences the problem of displacement field estimation is treated in conjunction with the filtering stage to ensure that the temporal correlations are taken along the direction of motion to prevent object blur

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

separation of variables and subgroup bases on n dimensional hyperboloids a graphical formalism is introduced for describing subgroup type coordinates on n dimensional lorentzian hyperboloids imbedded into n 1 dimensional minkowski spaces the o n 1 group element is parametrized according to different subgroup chains involving lorentz rotation and euclidean subgroups the coordinates are then induced by the corresponding group action eigenfunctions of the laplace beltrami operator are obtained as products of jacobi functions associated legendre functions and modified bessel functions

the least eigenvalue of signless laplacian of graphs under perturbation abstract we investigate how the least eigenvalue of the signless laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex and minimize the least eigenvalue of the signless laplacian among the class of connected graphs with fixed order which contains a given non bipartite graph as an induced subgraph

n dimensional integrability from two photon coalgebra symmetry a wide class of hamiltonian systems with n degrees of freedom and endowed with at least n 2 functionally independent integrals of motion in involution is constructed by making use of the two photon lie poisson coalgebra h6 the set of n 2 constants of the motion is shown to be a universal one for all these hamiltonians irrespective of the dependence of the latter on several arbitrary functions and n free parameters within this large class of quasi integrable n dimensional hamiltonians new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub coalgebra structure of h6 in particular new completely integrable n dimensional hamiltonians describing natural systems geodesic flows and static electromagnetic hamiltonians are presented

quasiseparation of variables in the schrodinger equation with a magnetic field we consider a two dimensional integrable hamiltonian system with a vector and scalar potential in quantum mechanics contrary to the case of a pure scalar potential the existence of a second order integral of motion does not guarantee the separation of variables in the schrodinger equation we introduce the concept of quasiseparation of variables and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra

bipartiteness and the least eigenvalue of signless laplacian of graphs let g be a simple graph and let b g the least eigenvalue of the signless laplacian of the graph g in this paper we focus on the relations between the least eigenvalue and some parameters reflecting the graph bipartiteness we introduce two parameters the vertex bipartiteness b g and the edge bipartiteness b g and show that b g b g b g we also define another parameter g involved with a cut set and prove that b g g g 2 g 2 where g is the maximum degree of the graph g the above two inequalities are very similar in form to those given by fiedler and mohar respectively with respect to the algebraic connectivity of laplacian of graphs which is used to characterize the connectedness of graphs

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

completeness of multiseparable superintegrability in e2 c the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability examples of this type of system are well known in this paper we demonstrate how to establish a complete list of such potentials using essentially algebraic means our approach is to classify all nondegenerate potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters this is carried out for two dimensional complex euclidean space though the method generalizes to other spaces and dimensions we show that all these superintegrable systems correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

the domination number and the least q eigenvalue abstract in this paper for a nonbipartite graph with both order n and domination number we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number we also present some results about the changing of the domination number when the structure of a graph is changed by investigating the relation between the domination number and the least q eigenvalue of a graph we minimize the least q eigenvalue among all the nonbipartite graphs with both order n 4 and given domination number n 1 3

on the least eigenvalue of aα matrix of graphs abstract let g be a graph of order n with m edges and chromatic number let a g be the adjacency matrix and d g be the diagonal matrix of vertex degrees of g nikiforov defined the matrix a g as a g d g 1 a g where 0 1 then a 1 2 g 1 2 d g a g 1 2 q g where q g is the signless laplacian matrix of g let q n g and n a be the least eigenvalue of q g and a g respectively lima et al 2011 8 proposed some open problems on q n g two of which are as follows 1 to characterize the graphs for which q n g 2 m n 1 2 to characterize the graphs for which q n g 1 1 1 2 m n in this paper we present an upper bound on n a in terms of n m and 1 2 1 and characterize the extremal graphs as an application we completely solve problem 1 moreover we examine the more generalized result of problem 2 on a g when 1 we obtain some necessary conditions for n a 1 2 m 1 n and as a corollary for the equality in problem 2

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

r separation of variables for the time dependent hamilton jacobi and schrodinger equations the theory of r separation of variables is developed for the time dependent hamilton jacobi and schrodinger equations on a riemannian manifold v n where time dependent vector and scalar potentials are permitted as an application it is shown how to obtain all r separable coordinates for the n sphere and euclidean n space

least squares restoration of multichannel images multichannel restoration using both within and between channel deterministic information is considered a multichannel image is a set of image planes that exhibit cross plane similarity existing optimal restoration filters for single plane images yield suboptimal results when applied to multichannel images since between channel information is not utilized multichannel least squares restoration filters are developed using the set theoretic and the constrained optimization approaches a geometric interpretation of the estimates of both filters is given color images three channel imagery with red green and blue components are considered constraints that capture the within and between channel properties of color images are developed issues associated with the computation of the two estimates are addressed a spatially adaptive multichannel least squares filter that utilizes local within and between channel image properties is proposed experiments using color images are described

multichannel restoration of single channel images using a wavelet based subband decomposition we present a new matrix vector formulation of a wavelet based subband decomposition this formulation allows for the decomposition of both the convolution operator and the signal in the subband domain with this approach any single channel linear space invariant filtering problem can be cast into a multichannel framework we apply this decomposition to the linear space invariant image restoration problem and propose a family of multichannel linear minimum mean square error lmmse restoration filters these filters explicitly incorporate both within and between subband channel relations of the decomposed image since only within channel stationarity is assumed in the image model this approach presents a new method for modeling the nonstationarity of images experimental results are presented which test the proposed multichannel lmmse filters these experiments show that if accurate estimates of the subband statistics are available the proposed multichannel filters provide major improvements over the traditional single channel filters

simultaneous recursive displacement estimation and restoration of noisy blurred image sequences we develop a recursive model based maximum a posteriori map estimator that simultaneously estimates the displacement vector field dvf and the intensity field from a noisy blurred image sequence current motion compensated spatio temporal noise filters treat the estimation of the dvf as a preprocessing step generally no attempt is made to verify the accuracy of these estimates prior to their use in the filter by simultaneously estimating these two fields we establish a link between the two estimators it is through this link that the dvf estimate and its corresponding accuracy information are shared with the other intensity estimator and vice versa to model the dvf and the intensity field we use coupled gauss markov cgm models a cgm model consists of two levels an upper level which is made up of several submodels with various characteristics and a lower level or line field which governs the transitions between the submodels the cgm models are well suited for estimating the displacement and intensity fields since the resulting estimates preserve the boundaries between the stationary areas present in both fields detailed line fields are proposed for the modeling of these boundaries which also take into account the correlations that exist between these two fields a kalman type estimator results followed by a decision criterion for choosing the appropriate set of line fields several experiments using noisy and noisy blurred image sequences demonstrate the superior performance of the proposed algorithm with respect to prediction error and mean square error

a sharp lower bound for the least eigenvalue of the signless laplacian of a non bipartite graph we prove that the minimum value of the least eigenvalue of the signless laplacian of a connected non bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices

a general framework for frequency domain multi channel signal processing the authors provide a general framework for performing processing of stationary multichannel mc signals that is linear shift invariant within channel and shift varying across channels emphasis is given to the restoration of degraded signals it is shown that by utilizing the special structure of semiblock circulant and block diagonal matrices mc signal processing can be easily carried out in the frequency domain the generalization of many frequency domain single channel sc signal processing techniques to the mc case is presented it is shown that in mc signal processing each frequency component of a signal and system is presented respectively by a small vector and a matrix of size equal to the number of channels while in sc signal processing each frequency component in both cases is a scalar

nondegenerate three dimensional complex euclidean superintegrable systems and algebraic varieties a classical or quantum second order superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent second order constants of the motion polynomial in the momenta the maximum possible such systems have remarkable properties multi integrability and multiseparability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with quasiexactly solvable systems here we announce a complete classification of nondegenerate i e four parameter potentials for complex euclidean 3 space we characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints the euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation there are exactly ten nondegenerat

spherical geometry zernike s separability and interbasis expansion coefficients free motion on a 3 sphere properly projected on the 2 dimensional manifold of a disk yields the zernike system which exhibits the fundamental properties of superintegrability these include separability in a variety of coordinate systems polynomial solutions and a particular subset of clebsch gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the askey scheme deriving these results from the initial formulation in spherical geometry provides the zernike system with interest beyond its optical applications

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

multichannel blind deconvolution of spatially misaligned images existing multichannel blind restoration techniques assume perfect spatial alignment of channels correct estimation of blur size and are prone to noise we developed an alternating minimization scheme based on a maximum a posteriori estimation with a priori distribution of blurs derived from the multichannel framework and a priori distribution of original images defined by the variational integral this stochastic approach enables us to recover the blurs and the original image from channels severely corrupted by noise we observe that the exact knowledge of the blur size is not necessary and we prove that translation misregistration up to a certain extent can be automatically removed in the restoration process

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

quantum superintegrable system with a novel chain structure of quadratic algebras we analyse the n dimensional superintegrable kepler coulomb system with non central terms we find a novel underlying chain structure of quadratic algebras formed by the integrals of motion we identify the elements for each sub structure and obtain the algebra relations satisfied by them and the corresponding casimir operators these quadratic sub algebras are realized in terms of a chain of deformed oscillators with factorized structure functions we construct the finite dimensional unitary representations of the deformed oscillators and give an algebraic derivation of the energy spectrum of the superintegrable system

the least q eigenvalue with fixed domination number abstract denote by lg l the lollipop graph obtained by attaching a pendant path p v g v g 1 v g l l 1 to a cycle c v 1 v 2 v g v 1 g 3 a f g l g r a p h of order n g 1 is defined to be the graph obtained by attaching n g l pendent vertices to some of the nonpendant vertices of lg l in which each vertex other than v g l 1 is attached at most one pendant vertex a f g l graph is a f g l g r a p h in which vg is attached with pendant vertex denote by qmin the least q e i g e n v a l u e of a graph in this paper we proceed on considering the domination number the least q eigenvalue of a graph as well as their relation further results obtained are as follows i some results about the changing of the domination number under the structural perturbation of a graph are represented ii among all nonbipartite unicyclic graphs of order n with both domination number and girth g g n 1 the minimum qmin attains at a f g l graph for some l iii among the nonbipartite graphs of order n and with given domination number which contain a f g l graph as a subgraph some lower bounds for qmin are represented iv among the nonbipartite graphs of order n and with given domination number n 2

parameter estimation in bayesian high resolution image reconstruction with multisensors we consider the estimation of the unknown parameters for the problem of reconstructing a high resolution image from multiple undersampled shifted degraded frames with subpixel displacement errors we derive mathematical expressions for the iterative calculation of the maximum likelihood estimate of the unknown parameters given the low resolution observed images these iterative procedures require the manipulation of block semi circulant bsc matrices that is block matrices with circulant blocks we show how these bsc matrices can be easily manipulated in order to calculate the unknown parameters finally the proposed method is tested on real and synthetic images

maximum likelihood estimation methods for multispectral random field image models considers the problem of estimating parameters of multispectral random field rf image models using maximum likelihood ml methods for images with an assumed gaussian distribution analytical results are developed for multispectral simultaneous autoregressive msar and markov random field mmrf models which lead to practical procedures for calculating ml estimates although previous work has provided least squares methods for parameter estimation the superiority of the ml method is evidenced by experimental results provided in this work the effectiveness of multispectral rf models using ml estimates in modeling color texture images is also demonstrated

on superintegrable symmetry breaking potentials in n dimensional euclidean space we give a graphical prescription for obtaining and characterizing all separable coordinates for which the schr dinger equation admits separable solutions for one of the superintegrable potentials here xn 1 is a distinguished cartesian variable the algebra of second order symmetries of the resulting schr dinger equation is given and for the first potential the closure relations of the corresponding quadratic algebra these potentials are particularly interesting because they occur in all dimensions n 1 the separation of variables problem is highly nontrivial for them and many other potentials are limiting cases