desingularization of quasi excellent schemes in characteristic zero grothendieck proved in ega iv that if any integral scheme of finite type over a locally noetherian scheme x admits a desingularization then x is quasi excellent and conjectured that the converse is probably true we prove this conjecture for noetherian schemes of characteristic zero namely starting with the resolution of singularities for algebraic varieties of characteristic zero we prove the resolution of singularities for noetherian quasi excellent q schemes

the godbillon vey invariant of a transversely homogeneous foliation a real projective foliation is a foliation 3 with a system of local coordinates transverse to 5 modelled on rp1 so that the coordinate changes are real linear fractional transformations given a closed manifold m there is but a finite set of values in f 3 m r which the godbillon vey invariant of such foliations may assume a bound on the possible values in terms of the fundamental group is computed for m an oriented circle bundle over a surface this finite set is explicitly computed in 5 c godbillon and j vey introduced a characteristic class associated to a codimension one foliation which lies in the third real cohomology group of the ambient manifold the question was then raised whether one could assign some ge ometric significance to this invariant the problem of understanding the godbillon vey invariant in such a manner was underscored by a result of thurston 12 who showed that for any closed 3 manifold there are families of codimension one folia tions whose godbillon vey invariant varies continuously in a parameter and which take on arbitrarily large values it is the main goal of this paper to show that this situation changes when one restricts to foliations which are transversely homogeneous see 1 to explain the geometric picture which emerges it is worthwhile reviewing briefly gromov s notion

dynamical systems defining jacobi s ϑ constants we propose a system of equations that defines weierstrass jacobi s eta and theta constant series in a differentially closed way this system is shown to have a direct relationship to a little known dynamical system obtained by jacobi the classically known differential equations by darboux halphen chazy and ramanujan are the differential consequences or reductions of these systems the proposed system is shown to admit the lagrangian hamiltonian and nambu formulations we explicitly construct a pencil of nonlinear poisson brackets and complete set of involutive conserved quantities as byproducts of the theory we exemplify conserved quantities for the ramamani dynamical system and quadratic system of halphen brioschi

bi hamiltonian structure of the kermack mckendrick model for epidemics the dynamical system proposed by kermack and mckendrick 1933 to model the spread of epidemics is shown to admit bi hamiltonian structure without any restrictions on the rate constants these two inequivalent hamiltonian structures are compatible

sur une conjecture de kato et kuzumaki concernant les hypersurfaces de fano resume nous montrons que le corps qp et les corps de nombres totalement imaginaires verifient la propriete c 1 conjecturee par kato et kuzumaki en 1986 autrement dit si k est l un de ces corps et f k x0 xn est un polynome homogene de degre d 6 n tout element de k s ecrit comme produit de normes depuis des extensions finies de k dans lesquelles f admet un zero non trivial nous etablissons aussi la conjecture d ax sur les corps pseudo algebriquement clos parfaits pour les corps dont le groupe de galois absolu est un pro p groupe

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

hamiltonians and variational principles for alfven simple waves the evolution equations for the magnetic field induction b with the wave phase for alfven simple waves are expressed as variational principles and in the hamiltonian form the evolution of b with the phase which is a function of the space and time variables depends on the generalized frenet serret equations in which the wave normal n which is a function of the phase is taken to be tangent to a curve x in a 3d cartesian geometry vector space the physical variables the gas density fluid velocity gas pressure and magnetic field induction in the wave depend only on the phase three approaches are developed one approach exploits the fact that the frenet equations may be written as a 3d hamiltonian system which can be described using the nambu bracket it is shown that b as a function of the phase satisfies a modified version of the frenet equations and hence the magnetic field evolution equations can be expressed in the hamiltonian form a second approach develops an euler poincare variational formulation a third approach uses the frenet frame formulation in which the hodograph of b moves on a sphere of constant radius and uses a stereographic projection transformation due to darboux the equations for the projected field components reduce to a complex riccati equation by using a cole hopf transformation the riccati equation reduces to a linear second order differential equation for the new variable a hamiltonian formulation of the second order differential equation then allows the system to be written in the hamiltonian form alignment dynamics equations for alfven simple waves give rise to a complex riccati equation or equivalently to a quaternionic riccati equation which can be mapped onto the riccati equation obtained by stereographic projection

generalization of solutions of the jacobi pdes associated to time reparametrizations of poisson systems the determination of solutions of the jacobi partial differential equations pdes for finite dimensional poisson systems is considered in particular a novel procedure for the construction of solution families is developed such a procedure is based on the use of time reparametrizations preserving the existence of a poisson structure as a result a method which is valid for arbitrary values of the dimension and the rank of the poisson structure under consideration is obtained in this article two main families of time reparametrizations of this kind are characterized in addition these results lead to a novel application which is also developed namely the global and constructive determination of the darboux canonical form for poisson systems of arbitrary dimension and rank two thus improving the local result provided by darboux theorem for such a case

new solution family of the jacobi equations characterization invariants and global darboux analysis a new family of skew symmetric solutions of the jacobi partial differential equations for finite dimensional poisson systems is characterized and analyzed such family has some remarkable properties firstly it is defined for arbitrary values of the dimension and the rank secondly it is described in terms of arbitrary differentiable functions namely it is not limited to a given degree of nonlinearity additionally it is possible to determine explicitly the fundamental properties of those solutions such as their casimir invariants and the algorithm for the reduction to the darboux canonical form which have been reported only for a very limited sample of finite dimensional poisson structures moreover such analysis is carried out globally in phase space thus improving the usual local scope of the darboux theorem

a new approach to collaborative filtering operator estimation with spectral regularization we present a general approach for collaborative filtering cf using spectral regularization to learn linear operators mapping a set of users to a set of possibly desired objects in particular several recent low rank type matrix completion methods for cf are shown to be special cases of our proposed framework unlike existing regularization based cf our approach can be used to incorporate additional information such as attributes of the users objects a feature currently lacking in existing regularization based cf approaches using popular and well known kernel methods we provide novel representer theorems that we use to develop new estimation methods we then provide learning algorithms based on low rank decompositions and test them on a standard cf data set the experiments indicate the advantages of generalizing the existing regularization based cf methods to incorporate related information about users and objects finally we show that certain multi task learning methods can be also seen as special cases of our proposed approach

nonequilibrium statistical mechanics of a susceptible infected recovered epidemic model the liouville and stochastic liouville equations have been investigated with respect to the susceptible infected recovered sir epidemic model described in terms of extended nambu mechanics it has been shown that along with the equilibrium distribution functions the kubo formula can be derived for the sir model although the theory undergoes a few restrictions imposed by the nonlinear nature of the model

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

consistency of trace norm minimization regularization by the sum of singular values also referred to as the trace norm is a popular technique for estimating low rank rectangular matrices in this paper we extend some of the consistency results of the lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss we also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled

quantization with maximally degenerate poisson brackets the harmonic oscillator nambu s construction of multi linear brackets for super integrable systems can be thought of as degenerate poisson brackets with a maximal set of casimirs in their kernel by introducing privileged coordinates in phase space these degenerate poisson brackets are brought to the form of heisenberg s equations we propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate poisson brackets into operators they pose a set of eigenvalue problems for a new state vector the requirement of the single valuedness of this eigenfunction leads to quantization the example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super integrable systems

unramified class field theory of arithmetical surfaces let k be an algebraic number field 9k its ring of integers and v a non empty open subscheme of spec ck let x be a projective smooth geometrically irreducible scheme over k and x a regular proper flat scheme over v such that x x vk x the purpose of this paper is to give an explicit description of the abelian fundamental groups ab x and 7t b x using some idele class groups attached to x and x by a k theoretical method its outline is as follows in general for a noetherian scheme z and i 0 or 1 we define the group ski z to be the cokernel of

hamiltonian structure of compartmental epidemiological models any epidemiological compartmental model with constant population is shown to be a hamiltonian dynamical system in which the total population plays the role of the hamiltonian function moreover some particular cases within this large class of models are shown to be bi hamiltonian new interacting compartmental models among different populations which are endowed with a hamiltonian structure are introduced the poisson structures underlying the hamiltonian description of all these dynamical systems are explicitly presented and their associated casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological models such as the ones describing the dynamics of the covid 19 pandemic

new global solutions of the jacobi partial differential equations abstract a new family of solutions of the jacobi partial differential equations for finite dimensional poisson systems is investigated this family is mathematically remarkable as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves this kind of poisson structure termed distinguished solutions or d solutions is defined for every nontrivial combination of values of the dimension and the rank and is also determined in terms of functions of arbitrary nonlinearity properties usually not present simultaneously in the already known solution families in addition d solutions display several properties allowing the generation of an infinity of d solutions from a given one which is an uncommon feature in the framework of the jacobi equations furthermore a special family of d solutions complying with the previous requirements is constructively characterized and analyzed examples are discussed focusing on physical implications and including an application for the global construction of the darboux canonical form

hamiltonian and quasi hamiltonian systems nambu poisson structures and symmetries the theory of hamiltonian and quasi hamiltonian systems with respect to nambu poisson structures is studied it is proved that if a dynamical system is endowed with certain properties related to the theory of symmetries then it can be considered as a quasi hamiltonian or hamiltonian system with respect to an appropriate nambu poisson structure several examples of this construction are presented these examples are related to integrability and also to superintegrability

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

perturbed euler top and bifurcation of limit cycles on invariant casimir surfaces abstract analytical perturbations of the euler top are considered the perturbations are based on the poisson structure for such a dynamical system in such a way that the casimir invariants of the system remain invariant for the perturbed flow by means of the poincare pontryagin theory the existence of limit cycles on the invariant casimir surfaces for the perturbed system is investigated up to first order of perturbation providing sharp bounds for their number examples are given

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

composite integrators for bi hamiltonian systems abstract symmetric composition methods are applied to dynamical systems in bi hamiltonian form to lotka volterra equations to the completely integrable lorenz equation and to the periodic toda lattice the numerical results obtained show that the hamiltonians are preserved with high accuracy and the periodicity of the solutions are retained

first integrals of a generalized darboux halphen system a third order system of nonlinear ordinary differential equations depending on three arbitrary parameters is analyzed the system arises in the study of su 2 invariant hypercomplex manifolds and is a dimensional reduction of the self dual yang mills equation the general solution first integrals and the nambu poisson structure of the system are explicitly derived it is shown that the first integrals are multi valued on the phase space even though the general solution of the system is single valued for special choices of parameters

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

line integral methods which preserve all invariants of conservative problems recently the class of hamiltonian boundary value methods hbvms has been introduced with the aim of preserving the energy associated with polynomial hamiltonian systems and more in general with all suitably regular hamiltonian systems however many interesting problems admit other invariants besides the hamiltonian function it would be therefore useful to have methods able to preserve any number of independent invariants this goal is achieved by generalizing the line integral approach which hbvms rely on thus obtaining a number of generalizations which we collectively name line integral methods in fact it turns out that this approach is quite general so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial such as a collocation method as well as to any conservative problem in particular a completely conservative variant of both hbvms and gauss collocation methods is presented numerical experiments confirm the effectiveness of the proposed methods

new four dimensional solutions of the jacobi equations for poisson structures a four dimensional family of skew symmetric solutions of the jacobi equations for poisson structures is characterized as a consequence previously known types of poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of new family of solutions additionally it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions such as their casimir invariants symplectic structure and the algorithm for the reduction to the darboux canonical form which have been reported only for a limited sample of known finite dimensional poisson structures moreover the results developed are valid globally in phase space thus ameliorating the usual scope of darboux theorem which is of local nature

on l _ q optimization and sparse inverse covariance selection graphical models are well established in providing meaningful conditional probability descriptions of complex multivariable interactions in the gaussian case the conditional independencies between different variables correspond to zero entries in the precision inverse covariance matrix hence there has been much recent interest in sparse precision matrix estimation in areas such as statistics machine learning computer vision pattern recognition and signal processing a popular estimation method involves optimizing a penalized log likelihood problem the penalty is responsible for inducing sparsity and a common choice is the convex l1 norm even though the l0 penalty is the natural choice guaranteeing maximum sparsity it has been avoided due to lack of convexity as a result in this paper we bridge the gap between these two penalties and propose the non concave lq penalized log likelihood problem for sparse precision matrix estimation where 0 q 1 a novel algorithm is developed for the optimization and we provide some of its theoretic properties that are useful in sparse linear regression we illustrate on synthetic and real data showing reconstruction quality comparisons of sparsity inducing penalties l0 lq with 0 q 1 l1 and scad

sparsenet coordinate descent with nonconvex penalties we address the problem of sparse selection in linear models a number of nonconvex penalties have been proposed in the literature for this purpose along with a variety of convex relaxation algorithms for finding good solutions in this article we pursue a coordinate descent approach for optimization and study its convergence properties we characterize the properties of penalties suitable for this approach study their corresponding threshold functions and describe a df standardizing reparametrization that assists our pathwise algorithm the mc penalty is ideally suited to this task and we use it to demonstrate the performance of our algorithm certain technical derivations and experiments related to this article are included in the supplementary materials section

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

bi hamiltonian dynamical systems related to low dimensional lie algebras abstract a method of generation of bi hamiltonian systems based on the use of low dimensional lie algebras is presented new nontrivial 3 4 and 5 dimensional bi hamiltonian systems are found

hamiltonian equations in r3 the hamiltonian formulation of n 3 systems is considered in general the most general solution of the jacobi equation in r3 is proposed the form of the solution is shown to be valid also in the neighborhood of some irregular points compatible poisson structures and corresponding bi hamiltonian systems are also discussed hamiltonian structures the classification of irregular points and the corresponding reduced first order differential equations of several examples are given

quantum algebras in classical mechanics it is shown that the equations of motion do not determine their hamiltonian descriptions uniquely in particular different poisson bracket relations are admissible for a given physical problem the free symmetric top and a charged particle moving on a homogeneous and constant magnetic field for instance may be described by a classical hamiltonian theory with an su 2 q poisson bracket structure

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

bi hamiltonian structure in 2 d field theory we exhibit the bi hamiltonian structure of the equations of associativity witten dijkgraaf verlinde verlinde dubrovin equations in 2 d topological field theory which reduce to a single equation of monge ampere type f ttt f xxt 2 f xxx f xtt in the case of three primary fields the first hamiltonian structure of this equation is based on its representation as a 3 component system of hydrodynamic type and the second hamiltonian structure follows from its formulation in terms of a variational principle with a degenerate lagrangian

quadratic poisson algebras and their infinite dimensional extensions a certain generalization of the classical modified yang baxter equation is presented also certain poisson algebras arising in the calculus of variations are investigated

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

bi hamiltonian structure in frenet serret frame abstract we reduce the problem of constructing bi hamiltonian structure in three dimensions to the solutions of a riccati equation in moving coordinates of frenet serret frame all explicitly constructed examples in the literature are exhausted by constant solutions we conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi hamiltonian

jacobi structures in r3 the most general jacobi brackets in r3 are constructed after solving the equations imposed by the jacobi identity two classes of jacobi brackets were identified according to the rank of the jacobi structures the associated hamiltonian vector fields are also constructed

poisson lie groups bi hamiltonian systems and integrable deformations given a lie poisson completely integrable bi hamiltonian system on we present a method which allows us to construct under certain conditions a completely integrable bi hamiltonian deformation of the initial lie poisson system on a non abelian poisson lie group of dimension n where is the deformation parameter moreover we show that from the two multiplicative poisson lie hamiltonian structures on that underly the dynamics of the deformed system and by making use of the group law on one may obtain two completely integrable hamiltonian systems on by construction both systems admit reduction via the multiplication in to the deformed bi hamiltonian system in the previous approach is applied to two relevant lie poisson completely integrable bi hamiltonian systems the lorenz and euler top systems

robust elastic net subspace representation recently finding the low dimensional structure of high dimensional data has gained much attention given a set of data points sampled from a single subspace or a union of subspaces the goal is to learn or capture the underlying subspace structure of the data set in this paper we propose elastic net subspace representation a new subspace representation framework using elastic net regularization of singular values due to the strong convexity enforced by elastic net the proposed method is more stable and robust in the presence of heavy corruptions compared with existing lasso type rank minimization approaches for discovering a single low dimensional subspace we propose a computationally efficient low rank factorization algorithm called facten using a property of the nuclear norm and the augmented lagrangian method then clusten is proposed to handle the general case in which the data samples are drawn from a union of multiple subspaces for joint subspace clustering and estimation the proposed algorithms are applied to a number of subspace representation problems to evaluate the robustness and efficiency under various noisy conditions and experimental results show the benefits of the proposed method compared with existing methods