differential stackel matrices we show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential stackel matrices generalizations of the classical stackel matrices used for multiplicative separation of second order schrodinger equations and additive separation of hamilton jacobi equations we work out the principal properties of these matrices and demonstrate that even for second order laplace equations additive separation may occur when multiplicative separation does not

homotopy groups of suspended classifying spaces an experimental approach when the results of a computer program are confronted to some theorems proved on a theoretical basis three situations can occur there can be an agreement between both approaches the computer program can obtain calculations not covered by the theorems or a discrepancy can be found between both methods in this note we report on a work where the three above mentioned situations happen we have enhanced the computer algebra called kenzo to deal with the computation of homotopy groups of suspended classifying spaces a problem tackled by mikhailov and wu in a paper published in the journal algebraic and geometric topology our experimental approach based on completely different methods from those by mikhailov and wu has allowed us in particular to detect an error in one of their published theorems 1 motivation and statement of the problem it is well known that the suspension functor applied to a topological space x shifts its homology groups that is to say hn x hn 1 x however in the case of homotopy groups the situation is not so favorable and in general there is not a direct relation between x and x let us consider for instance a circle s whose homotopy type is very simple the homology groups of the suspension s s are obvious but the problem of computing the homotopy groups of s is a difficult problem in algebraic topology given a group g its classifying space that is to say the eilenberg maclane spacek g 1 10 has trivial homotopy groups 1 k g 1 g and n k g 1 0 for each n 6 1 but when applying the suspension functor the new homotopy groups k g 1 are in general unknown in 11 several groups n k g 1 are obtained for some particular cases of g and n making use of different results and techniques from group theory and homotopy theory more concretely the main results of the article by mikhailov and wu are descriptions of the groups 4 k a 1 and 5 k a 1 when a is any finitely generated abelian group as applications they also determine n k g 1 with n 4 and 5 for some nonabelian groups as g 3 the 3 th symmetric group and g sl z the standard linear group and 4 k a4 1 for a4 the 4 th alternating group the goal of this work consists in trying to obtain a computer program calculating k g 1 for a general group g which will allow one to obtain as particular computations some theoretical results of 11 to this aim we have developed 2010 mathematics subject classification 68w05 55 04 55q99 partially supported by ministerio de ciencia e innovacion spain project mtm2009 13842c02 01 c xxxx american mathematical society

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

wythoff polytopes and low dimensional homology of mathieu groups abstract we describe two methods for computing the low dimensional integral homology of the mathieu simple groups and use them to make computations such as h 5 m 23 z z 7 and h 3 m 24 z z 12 one method works via sylow subgroups the other method uses a wythoff polytope and perturbation techniques to produce an explicit free z m n resolution both methods apply in principle to arbitrary finite groups

variational approximations to breather modes in the discrete sine gordon equation abstract long lived breather like modes have been observed in the numerical simulations of discrete sine gordon systems however little analytical work has been carried out on these systems we show how to use the variational form of nonlinear klein gordon equations to find corrections to breather modes caused by discreteness

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

shape reconstruction from po multifrequency scattered fields via the singular value decomposition approach this paper deals with the problem of determining the shape of unknown perfectly conducting infinitely long cylinders starting from the knowledge of the scattered electric far field under the incidence of plane waves with a fixed angle of incidence and varying frequency the problem is formulated as a nonlinear inverse one by searching for a compact support distribution accounting for the objects contour the nonlinear unknown to data mapping is then linearized by means of the kirchhoff approximation which reduces it into a fourier transform relationship then the fourier transform inversion from incomplete data is dealt with by means of the singular value decomposition svd approach and the features of the reconstructable unknowns are investigated finally numerical results confirm the performed analysis

symmetry analysis and exact solutions of the 2 1 dimensional sine gordon system according to the space variable exchange symmetry we change the 2 1 dimensional sine gordon system obtained by konopelchenko and rogers kr to a variant form some types of similarity reductions are obtained by using some lie symmetry analysis from these similarity reductions we find that the soliton structure of the system possesses quite a rich structure the line solitons parallel to the lines x y 0 and x y 0 may have an arbitrary shape in addition to the well known dromion solution which is constructed by two line solitons one may find many other kinds of soliton solutions localized in all directions for instance some kinds of ring type basin like plateau like and bowl like and instanton type soliton solutions can be found directly by selecting the arbitrary functions included in the single soliton solution the dromion solutions may also be constructed by straight line and curved line solitons

from the special 2 1 toda lattice to the kadomtsev petviashvili equation the nonlinearization of the eigenvalue problems associated with the toda hierarchy and the coupled korteweg de vries kdv hierarchy leads to an integrable symplectic map s and an integrable hamiltonian system h0 respectively it is proved that s and h0 have the same integrals hk the quasi periodic solution of the 2 1 dimensional kadomtsev petviashvili equation is split into three hamiltonian systems h0 h1 h2 while that of the special 2 1 dimensional toda equation is separated into h0 h1 plus the discrete flow generated by the symplectic map s a clear evolution picture of various flows is given through the window of abel jacobi coordinates the explicit theta function solutions are obtained by resorting to this separation technique

generalized stackel matrices stackel and differential stackel matrices are generalized so that the matrix elements may be functions of the derivatives of the dependent variable as well as the independent variable the inverses of these matrices are characterized and it is shown that for significant classes of linear and nonlinear partial differential equations variable separation is accomplished via this generalized stackel mechanism

the iterative time reversal process analysis of the convergence the efficiency of a time reversal acoustic mirror to focus on a reflective target through an inhomogeneous media has been demonstrated in a multitarget media the ability of such a mirror to work in an iterative mode in order to focus selectively on the strongest target was shown c prada f wu and m fink j acoust soc am 90 1119 1991 the theory of how the iterative time reversal process is built is based on a matrix formalism and treats the array of l transducers in a given medium as a linear system of l inputs l outputs the system is characterized at each frequency by its transfer matrix k and the time reversal iterative process is then described by a time reversal operator k k because of the reciprocity principle this operator is hermitian the following result is shown if the scattering medium is a set of well resolved targets of different reflectivities then each eigenvector of the operator k k with nonzero eigenvalue corresponds to one of the targets in the set and provides the opt

2 1 dimensional compacton solutions with and without completely elastic interaction properties taking the 2 1 dimensional broer kaup kupershmidt system as a simple example some special types of 2 1 dimensional compacton solutions are constructed it is shown that there is quite rich interaction behaviour between two travelling compactons for some types of compactons the interactions among them may not be completely elastic for some others the interactions are completely elastic there is no phase shift for the interactions of the 2 1 dimensional compactons discussed in this paper

beyond physical optics svd shape reconstruction of metallic cylinders we consider the problem of reconstructing the shape of perfectly conducting cylinders under the physical optics approximation for direct scattering physical optics provides small model errors when the scatterers are convex smooth have large radii of curvature as compared to the impinging wavelength and are noninteracting in this paper we investigate whether these hypotheses can be relaxed for the inverse problem in particular we consider three classes of scatterers circular cylinder with radius smaller than the wavelength multiple cylinders and a concave cylinder and we show that for them the model error is provided by a multiplicative almost frequency independent factor since this factor can be incorporated within the unknown distribution representing the scatterers illuminated boundaries then we are able to show that successful reconstructions also beyond the physical optics limits can be obtained

localizing thin metallic cylinders by a 2 5 d linear distributional approach experimental results the problem of detecting and localizing thin metallic scatterers from electromagnetic scattered field measurements is addressed the scatterers are assumed to be infinitely long and a 2 5 dimensional mathematical model is adopted according to the distributional approach the problem is cast as the inversion of a linear integral operator acting on 6 distributions whose supports are representative of the scatterers locations in this paper we present the experimental validation of such an approach to this end we perform the reconstructions by exploiting experimental scattered field data collected in controlled conditions at the electromagnetic diagnostics laboratory of the second university of naples the experiments are conducted in an anechoic chamber where the scatterers are embedded in a homogenous and lossless background medium the multistatic single view multifrequency and the multistatic multiview multifrequency configurations are both considered

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

localized solutions with chaotic and fractal behaviours in a 2 1 dimensional dispersive long wave system it is known that in high dimensions there exist abundant localized excitations such as dromions lumps ring soliton solutions and so on in this paper the possible chaotic and fractal localized structures are revealed for the 2 1 dimensional dispersive long wave equation the chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower dimensional chaotic and fractal patterns

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

generalized symmetries and w infinity algebras in 3 dimensional toda field theory after establishing a formal theory for getting solutions of one type of high dimensional partial differential equation two sets of generalized symmetries of the 3d toda theory which arises from a particular reduction of the 4d self dual gravity equation are obtained concretely by a simple formula each set of symmetries constitutes a generalized w algebra which contains three types of the usual w algebras as special cases some open questions are discussed

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

new exact jacobi elliptic functions solutions for the generalized coupled hirota satsuma kdv system abstract in this paper an generalized jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact jacobi elliptic functions solutions of the generalized coupled hirota satsuma kdv system as a result eight families of new doubly periodic solutions are obtained by using this method some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the jacobi elliptic functions m 1 or 0 which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics

noniterative analytical formula for inverse scattering of multiply scattering point targets this paper derives in the exact framework of multiple scattering theory for point targets a noniterative analytical formula for the nonlinear inversion of the target scattering strengths from the scattering or response matrix that can be applied after the target positions have been estimated in a previous step via e g time reversal multiple signal classification or another approach the new formula provides a noniterative analytical alternative to the iterative numerical solution approach for the same problem presented in a recent paper a j devaney e a marengo and f k gruber time reversal based imaging and inverse scattering of multiply scattering point targets j acoust soc am 118 3129 3138 2005 the two methods noniterative versus iterative are comparatively investigated with two numerical examples

a robust noniterative method for obtaining scattering strengths of multiply scattering point targets a noniterative method based on the least squares technique is presented in this letter to obtain the scattering strengths of multiply scattering point targets benefiting from the least squares optimization property this new method achieves a good estimation of the scattering strengths compared with the noniterative method proposed in a recent paper e a marengo and f k gruber noniterative analytical formula for inverse scattering of multiply scattering point targets j acoust soc am 120 3782 3788 2006 the new method produces more accurate results in the presence of noise

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

on permutation polytopes abstract in this paper we lay the foundations for the study of permutation polytopes the convex hull of a group of permutation matrices we clarify the relevant notions of equivalence prove a product theorem and discuss centrally symmetric permutation polytopes we provide a number of combinatorial properties of faces of permutation polytopes as an application we classify 4 dimensional permutation polytopes and the corresponding permutation groups classification results and further examples are made available online we conclude with several questions suggested by a general finiteness result

remarks on several 2 1 dimensional lattices in this paper five 2 1 dimensional lattices considered by several authors are revisited again first of all we will show that two lattices proposed by blaszak and szum j math phys 42 225 2001 become the so called differential difference kp equation due to date jimbo and miwa j phys soc jpn 51 4116 1982 51 4125 1982 52 388 1983 52 761 1983 52 766 1983 by simple variable transformations while another lattice found by blaszak and szum can be viewed as a higher dimensional generalization of a lattice given by wu and hu j phys a 32 1515 1999 some integrable properties on these three lattices are derived second it is shown that a 2 1 dimensional toda like lattice studied by cao geng and wu j phys a 32 8059 1999 can be transformed into the bilinear equation given by hu clarkson and bullough j phys a 30 l669 1997 for this bilinear version we also present some rational solutions and lie symmetries finally a lattice due to levi ragnisco and shabat can j phys 72 439 1994 is transformed into coupled bilinear equations it is shown that these coupled bilinear equations do not have two soliton solutions this further confirms that the lattice under consideration is not completely integrable

experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix the d o r t method french acronym for decomposition of the time reversal operator is an active remote sensing technique using arrays of antennas for the detection and localization of scatterers prada et al j acoust soc am 99 2067 2076 1996 the analogy between the time reversal operator and the covariance matrix used for classical sources separation in passive remote sensing bienvenu et al ieee trans assp 31 1235 1247 1983 is established then an experiment of subwavelength detection and localization of point like scatterers with a linear array of transducers is presented using classical estimators in reception like maximum likelihood and multiple signal characterization music two point like scatterers separated by 3 and placed at 100 from the array of transducers are resolved in these experiments the role of multiple scattering and the existence of additional eigenvectors associated with dipolar and monopolar radiation of each scatterer is discussed

shape reconstruction by the spectral data of the far field operator analysis and performances the second version of the linear sampling method is considered with reference to perfectly conducting cylinders and its performances analyzed it is pointed out that for nonconvex objects the shape of the object under test is not retrieved whereas in its place the convex hull is recovered we also point out that in the case of aspect limited data neither the convex hull of the object is reconstructed

a new algebraic procedure to construct exact solutions of nonlinear differential difference equations this paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential difference equations the discrete sine gordon equation and differential difference asymmetric nizhnik novikov veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure where various types of exact travelling wave solutions for these nonlinear differential difference equations have been constructed it is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential difference equations

affine symmetries of orbit polytopes abstract an orbit polytope is the convex hull of an orbit under a finite group g gl d r we develop a general theory of possible affine symmetry groups of orbit polytopes for every group we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups we prove that the symmetry group of a generic orbit polytope is again g if g is itself the affine symmetry group of some orbit polytope or if g is absolutely irreducible on the other hand we describe some general cases where the affine symmetry group grows we apply our theory to representation polytopes the convex hull of a finite matrix group and show that their affine symmetries can be computed effectively from a certain character we use this to construct counterexamples to a conjecture of baumeister et al on permutation polytopes baumeister et al 2009 4 conjecture 5 4

time reversal imaging with multiple signal classification considering multiple scattering between the targets the time reversal imaging with multiple signal classification method for the location of point targets developed within the framework of the born approximation in lehman and devaney transmission mode time reversal super resolution imaging j acoust soc am 113 2742 2753 2003 is generalized to incorporate multiple scattering between the targets it is shown how the same method can be used in the location of point targets even if there is multiple scattering between them on the other hand both the conventional images and the calculated values of the target scattering amplitudes are scattering model dependent

separation of variables in 1 2 dimensional schrodinger equations using our classification of separable schrodinger equations with two space dimensions published in j math phys 36 5506 1995 we give an exhaustive description of the coordinate systems providing their separability furthermore we apply these results to separate variables in the heat hamilton jacobi and fokker planck equations

decomposition of the time reversal operator detection and selective focusing on two scatterers the decomposition of the time reversal operator dort method is a selective detection and focusing technique using an array of transmit receive transducers it relies on the theory of iterative time reversal mirrors which was presented by prada et al c prada j l thomas and m fink j acoust soc am 97 62 71 1995 the time reversal operator was defined as k k where is the frequency means complex conjugate and k is the transfer matrix of the array of l transducers insonifying a time invariant scattering medium it was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers the dort method consists in determining these eigenvectors and using them for the selective focusing this paper presents a complete analysis of this method in the case of two scatterers the

linear distribution imaging of thin metallic cylinders under mutual scattering in this paper we address the problem of imaging thin metallic cylinders from the knowledge of the electric far field scattered under the incidence of plane waves a scattering model based on a multipole expansion accounting for the mutual scattering is introduced and then simplified under the single and second order scattering hypotheses the locations of the unknown cylinders for the inverse problem are described in terms of support of spl delta functions the performances of a truncated singular value decomposition based linear inversion algorithm in the presence of model errors due to mutual scattering is investigated by both analytical and numerical arguments in three different measurement configurations single frequency multiview multifrequency single view and multifrequency multiview the different effects of the angular and frequency diversities are pointed out

morse theory from an algebraic viewpoint forman s discrete morse theory is studied from an algebraic viewpoint and we show how this theory can be extended to chain complexes of modules over arbitrary rings as applications we compute the homologies of a certain family of nilpotent lie algebras and show how the algebraic morse theory can be used to derive the classical anick resolution as well as a new two sided anick resolution