the discriminance for a special class of circulant matrices and their diagonalization abstract this paper deals with a new type of matrices the structure of this class of matrices is similar to circulant matrix we call it h circulant matrix furthermore the sum the difference the product the inverse and the adjoint matrix of these matrices are also h circulant matrixes we not only give six discriminance for h circulant matrices but also discuss the diagonalization and nonsingularity of these matrices finally author gives a numerical example

complex network dimension and path counts large complex networks occur in many applications of computer science the complex network zeta function and the graph surface function have been used to characterize these networks and to define a dimension for complex networks in this work we present three new results related to the complex network dimension first we show the relationship of the concept to kolmogorov complexity second we show how the definition of complex network dimension can be reformulated by defining the concept for a single node and then defining the complex network dimension as the supremum over all nodes this makes the concept work better for formally infinite graphs third we study interesting parallels to zeta dimension a notion originally from number theory which has found connections to theoretical computer science these parallels lead to a deeper insight into the complex network dimension e g the formulation in terms of the entropy and a theorem relating dimension to connectivity

on the symmetric doubly stochastic inverse eigenvalue problem abstract let 1 2 n be a list of real numbers the symmetric doubly stochastic inverse eigenvalue problem hereafter sdiep is to determine the list which can occur as the spectrum of an n n symmetric doubly stochastic matrix a if the matrix a is positive we can necessarily obtain a subproblem symmetric positive doubly stochastic inverse eigenvalue problem hereafter spdiep of the sdiep in this paper we give some sufficient conditions for the sdiep and spdiep and prove that the set formed by the spectra of all n n symmetric positive doubly stochastic matrices is non convex for n 4

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

generic density and small span theorem we refine the genericity concept of ambos spies by assigning a real number in 0 1 to every generic set called its generic density we construct sets of generic density any e computable real in 0 1 and show a relationship between generic density and lutz resource bounded dimension we also introduce strong generic density and show that it is related to packing dimension we show that all four notions are different we show that whereas dimension notions depend on the underlying probability measure generic density does not which implies that every dimension result proved by generic density arguments simultaneously holds under any biased coin based probability measure we prove such a result we improve the small span theorem of juedes and lutz to the packing dimension setting for k bounded truth table reductions under any biased coin probability measure

hexagonal circle patterns and integrable systems patterns with constant angles hexagonal circle patterns with constant intersection angles are introduced and studied it is shown that they are described by discrete integrable systems of toda type conformally symmetric patterns are classified circle pattern analogs of holomorphic mappings z sp c and log z are constructed as special isomonodromic solutions circle patterns studied in the paper include schramm s circle patterns with the combinatorics of the square grid as a special case

on the canonical forms of the multi dimensional averaged poisson brackets we consider here special poisson brackets given by the averaging of local multi dimensional poisson brackets in the whitham method for the brackets of this kind it is natural to ask about their canonical forms which can be obtained after transformations preserving the physical meaning of the field variables we show here that the averaged bracket can always be written in the canonical form after a transformation of hydrodynamic type in the case of absence of annihilators of initial bracket however in general case the situation is more complicated as we show here in more general case the averaged bracket can be transformed to a pseudo canonical form under some special physical requirements on the initial bracket

bounding the dimension of points on a line abstract we use kolmogorov complexity methods to give a lower bound on the effective hausdorff dimension of the point x a x b given real numbers a b and x we apply our main theorem to a problem in fractal geometry giving an improved lower bound on the classical hausdorff dimension of generalized sets of furstenberg type

tensor robust principal component analysis with a new tensor nuclear norm in this paper we consider the tensor robust principal component analysis trpca problem which aims to exactly recover the low rank and sparse components from their sum our model is based on the recently proposed tensor tensor product or t product 14 induced by the t product we first rigorously deduce the tensor spectral norm tensor nuclear norm and tensor average rank and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm these definitions their relationships and properties are consistent with matrix cases equipped with the new tensor nuclear norm we then solve the trpca problem by solving a convex program and provide the theoretical guarantee for the exact recovery our trpca model and recovery guarantee include matrix rpca as a special case numerical experiments verify our results and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

on lie groups and toda lattices we extend the construction of the relativistic toda chains as integrable systems on the poisson submanifolds in lie groups beyond the case of the a series for the simply laced case this is just a direct generalization of the well known relativistic toda chains procedure and we construct explicitly the set of ad invariant integrals of motion on symplectic leaves which can be described using poisson quivers which are just blown up dynkin diagrams we also demonstrate how to get the set of minimal integrals of motion using the co multiplication rules for the corresponding lie algebras in the non simply laced case the corresponding bogoyavlensky coxeter toda systems are constructed using the fock goncharov folding of the corresponding poisson submanifolds we discuss also how this procedure can be extended for the affine case beyond the a series and consider explicitly an example from the affine d series

nonlinear quantum dynamical system based on the kadomtsev petviashvili ii equation the structure of soliton solutions of classical integrable nonlinear evolution equations which can be solved through the hirota transformation suggests a new way for the construction of nonlinear quantum dynamical systems that are based on the classical equations in the new approach the classical soliton solution is mapped into an operator u which is a nonlinear functional of the particle number operators over a fock space of quantum particles u obeys the evolution equation the classical soliton solutions are the eigenvalues of u in multi particle states in the fock space the construction easily allows for the incorporation of particle interactions which generate soliton effects that do not have a classical analog in this paper this new approach is applied to the case of the kadomtsev petviashvili ii equation the nonlinear quantum dynamical system describes a set of m 2s 1 particles with intrinsic spin s which interact in clusters of 1 n m 1 particles

finite state incompressible infinite sequences in this paper we define and study finite state complexity of finite strings and infinite sequences as well as connections between these complexity notions to randomness and normality we show that the finite state complexity does not only depend on the codes for finite transducers but also on how the codes are mapped to transducers as a consequence we relate the finite state complexity to the plain kolmogorov complexity to the process complexity and to prefix free complexity working with prefix free sets of codes we characterise martin lof random sequences in terms of finite state complexity the weak power of finite transducers is compensated by the high complexity of enumeration of finite transducers we also prove that every finite state incompressible sequence is normal but the converse implication is not true these results also show that our definition of finite state incompressibility is stronger than all other known forms of finite automata based incompressibility in particular the notion related to finite automaton based betting systems introduced by schnorr and stimm the paper concludes with a discussion of open questions

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

effective strong dimension in algorithmic information and computational complexity the two most important notions of fractal dimension are hausdorff dimension developed by hausdorff math ann 79 1919 pp 157 179 and packing dimension developed independently by tricot math proc cambridge philos soc 91 1982 pp 57 74 and sullivan acta math 153 1984 pp 259 277 both dimensions have the mathematical advantage of being defined from measures and both have yielded extensive applications in fractal geometry and dynamical systems lutz proceedings of the 15th ieee conference on computational complexity florence italy 2000 ieee computer society press piscataway nj 2000 pp 158 169 has recently proven a simple characterization of hausdorff dimension in terms of gales which are betting strategies that generalize martingales imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of hausdorff dimension including constructive computable polynomial space polynomial time and finite state dimensions work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science in this paper we show that packing dimension can also be characterized in terms of gales moreover even though the usual definition of packing dimension is considerably more complex than that of hausdorff dimension our gale characterization of packing dimension is an exact dual of and every bit as simple as the gale characterization of hausdorff dimension effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions which are exact duals of the effective dimensions mentioned above in general and in analogy with the classical fractal dimensions the effective strong dimension of a set or sequence is at least as great as its effective dimension with equality for sets or sequences that are sufficiently regular we develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness kolmogorov complexity prediction boolean circuit size complexity polynomial time degrees and data compression aside from the above characterization of packing dimension our two main theorems are the following 1 if vec beta beta 0 beta 1 ldots is a computable sequence of biases that are bounded away from 0 and r is random with respect to vec beta then the dimension and strong dimension of r are the lower and upper average entropies respectively of vec beta 2 for each pair of delta 0 2 computable real numbers 0 alpha le beta le 1 there exists a in rm e such that the polynomial time many one degree of a has dimension alpha in e and strong dimension beta in e our proofs of these theorems use a new large deviation theorem for self information with respect to a bias sequence vec beta that need not be convergent

the kolmogorov complexity of infinite words we present a brief survey of results on relations between the kolmogorov complexity of infinite strings and several measures of information content dimensions known from dimension theory information theory or fractal geometry special emphasis is placed on bounds on the complexity of strings in constructively given subsets of the cantor space finally we compare the kolmogorov complexity to the subword complexity of infinite strings

surfaces and the sklyanin bracket we discuss the lie poisson group structures associated to splittings of the loop group lgl n due to sklyanin concentrating on the finite dimensional leaves of the associated poisson structure we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a invariant poisson structure in particular sklyanin s lie poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve the sklyanin structure is then equivalent to one considered by mukai tyurin and bottacin on a moduli space of sheaves on the poisson surface the abelianization procedure gives rise to natural darboux coordinates for these leaves as well as separation of variables for the integrable hamiltonian systems associated to invariant functions on the group

symplectic structures and volume elements in the function space for the cubic schrodinger equation we consider various trace formulas for the cubic schrodinger equa tion in the space of infinitely smooth functions subject to periodic boundary conditions the formulas relate conventional integrals of motion to the periods of some abelian differentials holomorphic one forms on the spectral curve we show that the periods of abelian differentials are global coordinates on the moduli space of spectral curves the exterior derivatives of the holomorphic one forms are the basic and higher symplectic structures on the phase space we write ex plicitly these symplectic structures in qp coordinates we compute the ratio of two symplectic volume elements in the infinite genus limit

holomorphic bundles and many body systems we show that spin generalization of elliptic calogero moser system elliptic extension of gaudin model and their cousins are the degenerations of hitchin systems applications to the constructions of integrals of motion angle action variables and quantum systems are discussed the constructions of classical systems are motivated by conformal field theory and their quantum counterparts can be thought of as being the degenerations of the critical level knizhnik zamolodchikov bernard equations

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

on the quantum inverse problem for the closed toda chain we reconstruct the canonical operators pi qi of the quantum closed toda chain in terms of sklyanin s separated variables

relaxation of nonlinear oscillations in bcs superconductivity the diagonal case of the sl 2 richardson gaudin quantum pairing model is known to be solvable as an abel jacobi inversion problem the effect of random time dependent perturbations of the single particle spectrum on the exact solution is an open question of considerable physical relevance weak perturbations introduce a new slow time scale while preserving the nonlinear character of the dynamics in this paper such perturbations are considered it is shown that the long time asymptotics can be obtained by a deformation of the original integrable system equivalent to phase averaging over the fast time scale

eulerian entropy and non repetitive subword complexity we consider continuous self maps of compact metric spaces and for each point of the space we define the notion of eulerian entropy by considering the exponential growth rate of complexity in the initial chunks of the orbit of the point we show that eulerian entropy is constant on a residual subset for transitive dynamical systems for elements in the shift dynamical system we define an equivalent notion named non repetitive subword complexity and show that for a large class of mixing subshifts of finite type the set of points for which the non repetitive subword complexity is equal to the topological entropy is residual if f is either a transitive interval map or an infinite transitive subshift of finite type we establish that there is email protected n such that the eulerian entropy of f t is a positive constant that is attained on a residual set of points

a new algebra of toeplitz plus hankel matrices and applications we introduce a new simultaneously diagonalizable real algebra a of symmetrical centrosymmetrical matrices having a toeplitz plus hankel structure we give the corresponding orthonormal basis of eigenvectors which are alternately symmetrical and skewsymmetrical vectors an application is the construction of a symmetrical toeplitz plus centrosymmetrical hankel matrix of equal row sums having a prescribed real spectrum this matrix can be used as the starting matrix for symmetrical centrosymmetrical isospectral flows in particular for the isospectral flow corresponding to the construction of a regular toeplitz matrix having prescribed eigenvalues moreover if a is a noise representation of an unknown matrix in a of rank k then we give a procedure to approximate a by a matrix in a of rank k

lagrangian configurations and symplectic cross ratios we consider moduli spaces of cyclic configurations of n lines in a 2n dimensional symplectic vector space such that every set of n consecutive lines generates a lagrangian subspace we study geometric and combinatorial problems related to these moduli spaces and prove that they are isomorphic to quotients of spaces of symmetric linear difference operators with monodromy 1 the symplectic cross ratio is an invariant of two pairs of 1 dimensional subspaces of a symplectic vector space for n 2n 2 the moduli space of lagrangian configurations is parametrized by n 1 symplectic cross ratios these cross ratios satisfy a single remarkable relation related to tridiagonal determinants and continuants given by the pfaffian of a gram matrix

guo perturbation for symmetric nonnegative circulant matrices abstract guo w guo eigenvalues of nonnegative matrices linear algebra appl 266 1997 261 270 sets the question if the list 1 2 n is symmetrically realizable that is is the spectrum of a symmetric nonnegative matrix and t 0 whether or not the list t 1 t 2 t 3 n is also symmetrically realizable in this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant we also give a necessary and sufficient condition for to be the spectrum of a nonnegative left circulant matrix

the computation of the square roots of circulant matrices abstract in the first part of this paper we investigate the reduced forms of circulant matrices and quasi skew circulant matrices by using their properties we present two efficient algorithms to compute the square roots of circulant matrices and quasi skew circulant matrices respectively those methods are faster than the traditional algorithm which is based on the schur decomposition in the second part we further consider circulant h matrices with positive diagonal entries and develop two algorithms for computing their principal square roots those two algorithms have the common advantage that is they only need matrix matrix multiplications in their iterative sequences an operation which can be done very efficiently on modern high performance computers

classical r matrices and the feigin odesskii algebra via hamiltonian and poisson reductions we present a formula for a classical r matrix of an integrable system obtained by hamiltonian reduction of some free field theories using pure gauge symmetries the framework of the reduction is restricted only by the assumption that the respective gauge transformations are lie group ones our formula is in terms of dirac brackets and some new observations on these brackets are made we apply our method to derive a classical r matrix for the elliptic calogero moser system with spin starting from the higgs bundle over an elliptic curve with marked points in the paper we also derive a classical feigin odesskii algebra by a poisson reduction of some modification of the higgs bundle over an elliptic curve this allows us to include integrable lattice models in a hitchin type construction

existence and construction of nonnegative matrices with complex spectrum abstract the following inverse spectrum problem for nonnegative matrices is considered given a set of complex numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum our work is motivated by a relevant theoretical result of guo wuwen linear algebra appl 266 1997 261 theorem 2 1 there exists a real parameter 0 max 2 j n j such that the problem has a solution if and only if 1 0 in particular we discuss how to compute 0 and the solution matrix a for certain class of matrices a sufficient condition for the problem to have a solution is also derived

connectedness properties of dimension level sets we prove that the set of all points of effective hausdorff dimension 1 in r n n 2 is connected and simultaneously that the complement of this set is not path connected when n 2

the dimensions of individual strings and sequences a constructive version of hausdorff dimension is developed using constructive supergales which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences this constructive dimension is used to assign every individual infinite binary sequence s a dimension which is a real number dim s in the interval 0 1 sequences that are random in the sense of martin lof have dimension 1 while sequences that are decidable 10 or 10 have dimension 0 it is shown that for every 20 computable real number in 0 1 there is a 20 sequence s such that dim s a discrete version of constructive dimension is also developed using termgales which are supergale like functions that bet on the terminations of finite binary strings as well as on their successive bits this discrete dimension is used to assign each individual string w a dimension which is a nonnegative real number dim w the dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes the kolmogorov complexity of a string is proven to be the product of its length and its dimension this gives a new characterization of algorithmic information and a new proof of mayordomo s recent theorem stating that the dimension of a sequence is the limit inferior of the average kolmogorov complexity of its first n bits every sequence that is random relative to any computable sequence of coin toss biases that converge to a real number in 0 1 is shown to have dimension h the binary entropy of

backward errors for eigenproblem of two kinds of structured matrices in this paper we consider backward errors in the eigenproblem of symmetric centrosymmetric and symmetric skew centrosymmetric matrices by making use of the properties of symmetric centrosymmetric and symmetric skew centrosymmetric matrices we derive explicit formulae for the backward errors of approximate eigenpairs

integrable cluster dynamics of directed networks and pentagram maps abstract the pentagram map was introduced by r schwartz more than 20 years ago in 2009 v ovsienko r schwartz and s tabachnikov established liouville complete integrability of this discrete dynamical system in 2011 m glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver using compatibility of poisson and cluster structures and poisson geometry of directed networks on surfaces we generalize glick s construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations the appendix relates the simplest of these discrete maps to the toda lattice and its tri hamiltonian structure

constructing symmetric nonnegative matrices via the fast fourier transform abstract we derive an algorithm based on the fast fourier transform to construct a real symmetric matrix s with eigenvalues 1 2 n with eigenvector e 1 1 1 t belonging to the eigenvalue 1 we find simple conditions on the eigenvalues such that the algorithm constructs an irreducible matrix s 1 e where e is a symmetric doubly stochastic matrix

realizable lists on a class of nonnegative matrices abstract a square matrix of order n with n 2 is called permutative matrix when all its rows are permutations of the first row in this paper recalling spectral results for partitioned into 2 by 2 symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given in particular we study niep and pniep when some complex elements in the lists under consideration have non zero imaginary part realizability regions for nonnegative permutative matrices are obtained a guo s realizability preserving perturbations result is obtained

lagrangian multiform structure for the lattice gel fand dikii hierarchy the lattice gel fand dikii hierarchy was introduced by nijhoff et al 1992 inverse problems 8 597 621 as the family of partial difference equations generalizing to higher rank the lattice korteweg de vries systems and includes in particular the lattice boussinesq system we present a lagrangian for the generic member of the lattice gel fand dikii hierarchy and show that it can be considered as a lagrangian 2 form when embedded in a higher dimensional lattice obeying a closure relation thus the multiform structure proposed by lobb and nijhoff 2009 j phys a math theor 42 454013 is extended to a multi component system

on the integrability of the shift map on twisted pentagram spirals in this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type possesses a non standard lax representation with an associated monodromy whose conjugation class is preserved by the map we prove this by finding a coordinate system in the moduli space of twisted spirals writing the map in terms of the coordinates and associating a natural parameter free non standard lax representation we then show that the map is invariant under the action of a one parameter group on the moduli space of twisted spirals which allows us to construct the lax pair we also show that the monodromy defines an associated riemann surface that is preserved by the map we use this fact to generate invariants of the shift map

on the bisymmetric nonnegative inverse eigenvalue problem abstract the problem of finding conditions for a list of n complex numbers to be the eigenvalues of an n n nonnegative matrix is called the nonnegative inverse eigenvalue problem niep the sniep bniep is the niep when the desired matrices are symmetric bisymmetric recently some sufficient conditions for the bniep are given by julio and soto in 6 in this article we study the bniep for n 4 and prove that the bniep and sniep are different for n 6 then we give some sufficient conditions for the bniep and for the niep for normal centrosymmetric matrices related to the sufficient conditions of julio and soto and we also provide certain sufficient conditions for the bniep analogous to the sufficient conditions for the niep given by borobia in 1 finally we give sufficient conditions for the bniep when the prescribed diagonal entries are required

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

algebraic varieties in the birkhoff strata of the grassmannian gr 2 harrison cohomology and integrable systems the local properties of the families of algebraic subsets wg in the birkhoff strata 2g of gr 2 containing the hyperelliptic curves of genus g are studied it is shown that the tangent spaces tg for wg are isomorphic to the linear spaces of 2 coboundaries particular subsets in wg are described by the integrable dispersionless coupled kdv systems of hydrodynamical type defining a special class of 2 cocycles and 2 coboundaries in tg it is demonstrated that the blows ups of such 2 cocycles and 2 coboundaries and gradient catastrophes for associated integrable systems are interrelated

hitchin systems symplectic hecke correspondence and two dimensional version the aim of this paper is two fold first we define symplectic maps between hitchin systems related to holomorphic bundles of different degrees we call these maps the symplectic hecke correspondence shc of the corresponding higgs bundles they are constructed by means of the modification of the underlying holomorphic bundles shc allows to construct backlund transformations in the hitchin systems defined over riemann curves with marked points we apply the general scheme to the elliptic calogero moser cm system and construct shc to an integrable sl n euler arnold top the elliptic sl n rotator next we propose a generalization of the hitchin approach to 2d integrable theories related to the higgs bundles of infinite rank the main example is an integrable two dimensional version of the two body elliptic cm system the previous construction allows us to define shc between the two dimensional elliptic cm system and the landau lifshitz equation

beyond aztec castles toric cascades in the dp 3 quiver given one of an infinite class of supersymmetric quiver gauge theories string theorists can associate a corresponding toric variety which is a calabi yau 3 fold as well as an associated combinatorial model known as a brane tiling in combinatorial language a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike a cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra known as cluster variables for the quiver associated to the cone over the del pezzo surface d p 3 in particular mutation sequences involving mutations exclusively at vertices with two in coming arrows and two out going arrows are referred to as toric cascades in the string theory literature such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics we provide an explicit algebraic formula for all cluster variables that are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the d p 3 brane tiling for these formulas in most cases

toric networks geometric r matrices and generalized discrete toda lattices we use the combinatorics of toric networks and the double affine geometric r matrix to define a three parameter family of generalizations of the discrete toda lattice we construct the integrals of motion and a spectral map for this system the family of commuting time evolutions arising from the action of the r matrix is explicitly linearized on the jacobian of the spectral curve the solution to the initial value problem is constructed using riemann theta functions

fast construction of a symmetric nonnegative matrix with a prescribed spectrum abstract in this paper for a prescribed real spectrum using properties of the circulant matrices and of the symmetric persymmetric matrices we derive a fast and stable algorithm to construct a symmetric nonnegative matrix which realizes the spectrum the algorithm is based on the fast fourier transform

vector bundles and lax equations on algebraic curves the hamiltonian theory of zero curvature equations with spectral parameter on an arbitrary compact riemann surface is constructed it is shown that the equations can be seen as commuting flows of an infinite dimensional field generalization of the hitchin system the field analog of the elliptic calogero moser system is proposed an explicit parameterization of hitchin system based on the tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained

the pentagram map and y patterns abstract the pentagram map introduced by r schwartz is defined by the following construction given a polygon as input draw all of its shortest diagonals and output the smaller polygon which they cut out we employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map

factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference painleve equations we study factorizations of rational matrix functions with simple poles on the riemann sphere for the quadratic case two poles we show using multiplicative representations of such matrix functions that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference painleve equations in particular in these coordinates basic isomonodromic transformations take the form of the discrete euler lagrange equations secondly we show that dpv equations previously obtained in this context by d arinkin and a borodin can be understood as simple relationships between the residues of such matrices and their inverses

finite state dimension and real arithmetic we use entropy rates and schur concavity to prove that for every integer k 2 every nonzero rational number q and every real number a the base k expansions of a q a and q a all have the same finite state dimension and the same finite state strong dimension this extends and gives a new proof of wall s 1949 theorem stating that the sum or product of a nonzero rational number and a borel normal number is always borel normal

on the integrable geometry of soliton equations and n 2 supersymmetric gauge theories we provide a unified construction of the symplectic forms which arise in the solution of both n 2 supersymmetric yang mills theories and soliton equations their phase spaces are jacobian type bundles over the leaves of a foliation in a universal configuration space on one hand imbedded into finite gap solutions of soliton equations these symplectic forms assume explicit expressions in terms of the auxiliary lax pair expressions which generalize the well known gardner faddeev zakharov bracket for kdv to a vast class of 2d integrable models on the other hand they determine completely the effective lagrangian and bps spectrum when the leaves are identified with the moduli space of vacua of an n 2 supersymmetric gauge theory for su n c with n f leq n c 1 flavors the spectral curves we obtain this way agree with the ones derived by hanany and oz and others from physical considerations

on complexity functions of infinite words associated with generalized dyck languages in this article we construct a family of infinite words generated by countable automata and also generated by substitutions over infinite alphabets closely related to parenthesis languages and we study their complexity functions we obtain a family of binary infinite words m b indexed on the number b 1 of parenthesis types such that the growth order of the complexity function of m b is n logn 2 if b 1 and n 1 l o g 2 b b if b 2

some results on symmetric circulant matrices and on symmetric centrosymmetric matrices we derive a readily computable sufficient condition for the existence of a nonnegative symmetric circulant matrix having a prescribed spectrum moreover we prove that any set 1 2 n is the spectrum of real symmetric centrosymmetric matrices s1 and s2 such that for s1 an eigenvector corresponding to 1 is the all ones vector and for s2 an eigenvector corresponding to 1 is the vector with components 1 and 1 alternately the proof is constructive then we derive an improved condition on k k 1n such that s1 1e where e is a stochastic symmetric centrosymmetric matrix finally we propose an algorithm to compute the eigenvalues of some real symmetric centrosymmetric matrices all the numerical procedures are based on the use of the fast fourier transform

on the minimal set of conservation laws and the hamiltonian structure of the whitham equations we consider the questions connected with the hamiltonian properties of the whitham equations in case of several spatial dimensions an essential point of our approach here is a connection of the hamiltonian structure of the whitham system with the finite dimensional poisson bracket defined on the space of periodic or quasi periodic solutions from our point of view this approach gives a possibility to construct the hamiltonian structure of the whitham equations under minimal requirements on the properties of the initial system the poisson bracket for the whitham system can be considered here as a deformation of the finite dimensional bracket with the aid of the dubrovin novikov procedure of bracket averaging at the end we consider the examples where the constructions of the paper play an essential role for the construction of the poisson bracket for the whitham system

quasi hamiltonian geometry of meromorphic connections for each connected complex reductive group g we find a family of new examples of complex quasi hamiltonian g spaces with g valued moment maps these spaces arise naturally as moduli spaces of suitably framed meromorphic connections on principal g bundles over a disk and they generalise the conjugacy class example of alekseev malkin and meinrenken 3 which appears in the simple pole case using the fusion product in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy stokes data of meromorphic connections over arbitrary genus riemann surfaces together with a new proof of the symplectic nature of isomonodromic deformations of such connections

r matrix structure of hitchin system in tyurin parameterization we present a classical r matrix for the hitchin system without marked points on an arbitrary non degenerate algebraic curve of genus g 2 using tyurin parameterization of holomorphic vector bundles

non commutative integrability of the grassmann pentagram map abstract the pentagram map is a discrete dynamical system first introduced by schwartz in 1992 it was proved to be integrable by schwartz ovsienko and tabachnikov in 2010 gekhtman shapiro and vainshtein studied poisson geometry associated to certain networks embedded in a disc or annulus and its relation to cluster algebras later gekhtman et al and tabachnikov reinterpreted the pentagram map in terms of these networks and used the associated poisson structures to give a new proof of integrability in 2011 mari beffa and felipe introduced a generalization of the pentagram map to certain grassmannians and proved it had a lax representation we reinterpret this grassmann pentagram map in terms of noncommutative algebra in particular the double brackets of van den bergh and generalize the approach of gekhtman et al to establish a noncommutative version of integrability

the method of finite gap integration in classical and semi classical string theory in view of proving the ads cft correspondence one day a deeper understanding of string theory on certain curved backgrounds such as ads5 s5 is required in this review we make a step in this direction by focusing on it was discovered in recent years that string theory on ads5 s5 admits a lax formulation however the complete statement of integrability requires not only the existence of a lax formulation but also that the resulting integrals of motion are in pairwise involution this idea is central to the first part of this review exploiting this integrability we apply algebro geometric methods to string theory on and obtain the general finite gap solution the construction is based on an invariant algebraic curve previously found in the ads5 s5 case however encoding the dynamics of the solution requires specification of additional marked points by restricting the symplectic structure of the string to these algebro geometric data we derive the action angle variables of the system we then perform a first principle semiclassical quantization of string theory on as a toy model for strings on ads5 s5 the result is exactly what one expects from the dual gauge theory perspective namely the underlying algebraic curve discretizes in a natural way we also derive a general formula for the fluctuation energies around the generic finite gap solution the ideas used can be generalized to ads5 s5

topological properties of omega context free languages abstract this paper is a study of topological properties of omega context free languages cfl we first extend some decidability results for the deterministic ones dcfl proving that one can decide whether an dcfl is in a given borel class or in the wadge class of a given regular language we prove that cfl exhaust the hierarchy of borel sets of finite rank and that one cannot decide the borel class of an cfl giving an answer to a question of lescow and thomas a decade of concurrency springer lecture notes in computer science vol 803 springer berlin 1994 pp 583 621 we give also a partial answer to a question of simmonet automates et theorie descriptive ph d thesis universite paris 7 march 1992 about omega powers of finitary languages we show that buchi landweber s theorem cannot be extended to even closed cfl in a gale stewart game with a closed cfl winning set one cannot decide which player has a winning strategy from the proof of topological properties we derive some arithmetical properties of cfl

liouville arnold integrability of the pentagram map on closed polygons the pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane this map has recently attracted a considerable interest mostly because its connection to a number of different domains such as classical projective geometry algebraic combinatorics moduli spaces cluster algebras and integrable systems integrability of the pentagram map was conjectured by schwartz and proved by the present authors for a larger space of twisted polygons in this article we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons in the case of convex polygons in the real projective plane this result implies the existence of a toric foliation on the moduli space the leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic our proof is based on an invariant poisson structure on the space of twisted polygons we prove that the hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants

integrability of higher pentagram maps we define higher pentagram maps on polygons in mathbb p d for any dimension d which extend r schwartz s definition of the 2d pentagram map we prove their integrability by presenting lax representations with a spectral parameter for scale invariant maps the corresponding continuous limit of the pentagram map in dimension d is shown to be the 2 d 1 equation of the kdv hierarchy generalizing the boussinesq equation in 2d we also study in detail the 3d case where we prove integrability for both closed and twisted polygons and describe the spectral curve first integrals the corresponding tori and the motion along them as well as an invariant symplectic structure

kolmogorov complexity and hausdorff dimension abstract in this paper various relationships between the kolmogorov complexity of infinite strings and measures of information content are given the general approach taken here is to bound the complexity of a maximally complex string in a given set of strings by the hausdorff dimension or the entropy of that set it turns out that hausdorff dimension yields lower bounds to the kolmogorov complexity whereas under certain recursiveness constraints on the structure of the respective sets their entropy yields upper bounds more detailed investigations result in a generalization of p martin lof s theorems on the kolmogorov complexity of random strings to maximally complex strings in regularly structured sets of infinite strings

applications of a brauer theorem in the nonnegative inverse eigenvalue problem abstract a result by brauer which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues plays a relevant role in the study of the nonnegative inverse eigenvalue problem niep perfect in a long time ignored paper 1955 presents an extension of this result which shows how to modify r eigenvalues of a matrix of order n r n via a rank r perturbation without changing any of the n r remaining eigenvalues by using this extension perfect gives a realizability criterion for the real niep which is not contained in soto s realizability criterion in this work by extending perfect s result we give a new realizability criterion for the real niep which contains soto s criterion thus this new realizability criterion appears to be the most general sufficient condition for the real niep so far we also contribute to the solution of the symmetric niep for n 5

determinants and alternating sign matrices let m be an n by n matrix by a connected minor of m of size k we mean a minor formed from k consecutive rows and k consecutive columns we give formulas for det m in terms of connected minors one involving minors of two consecutive sizes and one involving minors of three consecutive sizes the formulas express det m as sums indexed by sets of alternating sign matrices these matrices are described here and by w h mills d p robbins and h rumsey jr invent math 66 1982 73 87 j combin theory ser a 34 1983 340 359 the former study has led to the solution of macdonald s conjecture on cyclically symmetric plane partitions g e andrews invent math 53 1979 193 225 i g macdonald symmetric functions and hall polynomials p 53 oxford univ press clarendon oxford 1979

a divergence formula for randomness and dimension if s is an infinite sequence over a finite alphabet s and b is a probability measure on s then the dimension of s with respect to b written dim b s is a constructive version of billingsley dimension that coincides with the constructive hausdorff dimension dim s when b is the uniform probability measure this paper shows that dim b s and its dual dim b s the strong dimension of s with respect to b can be used in conjunction with randomness to measure the similarity of two probability measures a and b on s specifically we prove that the divergence formuladim b r dim b r h a h a d email protected email protected holds whenever a and b are computable positive probability measures on s and email protected email protected is random with respect to a in this formula h a is the shannon entropy of a and d email protected email protected is the kullback leibler divergence between a and b we also show that the above formula holds for all sequences r that are a normal in the sense of borel when dim b r and dim b r are replaced by the more effective finite state dimensions dim f s b r and dim f s b r in the course of proving this we also prove finite state compression characterizations of dim f s b s and dim f s b s

on the complexity of a family of k context free sequences we introduce a large family of k context free sequences for which we give a polynomial upper bound on their complexity functions

kolmogorov structure functions for automatic complexity abstract for a finite word w of length n we define and study the kolmogorov structure function h w for nondeterministic automatic complexity we prove upper bounds on h w that appear to be quite sharp based on numerical evidence

the stackel systems and algebraic curves we show how the abel jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves we prove that derivative of the abel jacobi map is just the stackel matrix which determines n orthogonal curvilinear coordinate systems in a flat space the lax pairs r matrix algebras and explicit form of the flat coordinates are constructed an application of the weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space

dimension in complexity classes a theory of resource bounded dimension is developed using gales which are natural generalizations of martingales when the resource bound delta a parameter of the theory is unrestricted the resulting dimension is precisely the classical hausdorff dimension sometimes called fractal dimension other choices of the parameter delta yield internal dimension theories in e e2 espace and other complexity classes and in the class of all decidable problems in general if mathcal c is such a class then every set x of languages has a dimension in mathcal c which is a real number dim x mid mathcal c in 0 1 along with the elements of this theory two preliminary applications are presented for every real number 0 le alpha le frac 1 2 the set rm freq le alpha consisting of all languages that asymptotically contain at most alpha of all strings has dimension mathcal h alpha the binary entropy of alpha in e and in e2 for every real number 0 le alpha le 1 the set rm size alpha frac 2 n n consisting of all languages decidable by boolean circuits of at most alpha frac 2 n n gates has dimension alpha in espace

on tables of random numbers the set theoretic axioms of the calculus of probability in formulating which i had the opportunity of playing some part kolmogorov 1950 had solved the majority of formal difficulties in the construction of a mathematical apparatus which is useful for a very large number of applications of probabilistic methods so successfully that the problem of finding the basis of real applications of the results of the mathematical theory of probability became rather secondary to many investigators i have already expressed the view see kolmogorov 1950 chapter i that the basis for the applicability of the results of the mathematical theory of probabi lity to real random phenomena must depend on some form of the frequency concept of probability the unavoidable nature of which has been established by von mises in a spirited manner however for a long time i had the following views 1 the frequency concept based on the notion of limiting frequency as the number of trials increases to infinity does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials 2 the frequency concept applied to a large but finite number of trials does not admit a rigorous formal exposition within the framework of pure mathematics accordingly i have sometimes put forward the frequency concept which involves the conscious use of certain not rigorously formal ideas about practical reliability approximate stability of the frequency in a long series of trials without the precise definition of the series which are sufficiently large etc see foundation of the theory of probability chapter i and for more details great soviet encyclo paedia section on probability and mathematika iou metod i znachenye chapter on probability theory i still maintain the first of the two theses mentioned above as regards the second however i have come to realise that the concept of random distribution of a property in a large finite population can have a strict formal mathematical exposition in fact we can show that in sufficiently large populations the distribution of the pro perty may be such that the frequency of its occurrence will be almost the same for all sufficiently large sub populations when the law of choosing these is sufficiently simple such a conception in its full development requires the introduction of a measure of the complexity of the algorithm i propose to discuss this question in another article in the present article however i shall use the fact that there cannot be a very large number of simple algorithms 369 3

integrable deformations of a polygon abstract a new integrable mapping with a simple geometric interpretation is presented this mapping arises from the nonlinear superposition principle for the backlund transformations of some vector evolution equation