covering relations for multidimensional dynamical systems abstract we present a topological technique for analyzing dynamical systems with complex behavior based on the general notion of covering relations our method can be used to study multidimensional dynamical systems with an arbitrary number of topologically expanding directions

existence and instability of steady states for a triangular cross diffusion system a computer assisted proof abstract in this paper we present and apply a computer assisted method to study steady states of a triangular cross diffusion system our approach consist in an a posteriori validation procedure that is based on using a fixed point argument around a numerically computed solution in the spirit of the newton kantorovich theorem it allows to prove the existence of various non homogeneous steady states for different parameter values in some situations we obtain as many as 13 coexisting steady states we also apply the a posteriori validation procedure to study the linear stability of the obtained steady states proving that many of them are in fact unstable

coexistence theorem of steady states for nonlinear self cross diffusion systems with competitive dynamics in this paper we discuss the existence of positive solutions to certain nonlinear elliptic systems representing competitive interaction with self cross diffusions between two species the method employed is the fixed point index theory in a positive cone sufficient conditions for the existence of positive solutions are provided

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

inverses of infinite sign regular matrices let a be an infinite sign regular sr matrix which can be viewed as a bounded linear operator from lx to itself it is proved here that if the range of a contains the sequence 1 1 1 1 then a is onto if a exists then da d is also sr where d is the diagonal matrix with diagonal entries alternately 1 and 1 in case a is totally positive tp then da d is also tp under additional assumptions on a 0 introduction if the problem of spline interpolation is expressed in terms of b splines then the question of existence of a bounded spline interpolant to bounded data is seen to be equivalent to the question of whether a certain bounded band matrix has all bounded sequences in its range in 5 c a micchelli conjectured that there exists a unique bounded spline interpolant of a given order and a given knot sequence to any data sequence t y z in the plane with t strictly increasing and y bounded provided only that it is possible to interpolate the particular data sequence t l ez by sucn a sp ine there is apparently nothing special about the particular spline problem other than that it leads to a banded totally positive matrix therefore one of us quoted this conjecture in 3 p 319 prob lem 4 as a bi infinite banded totally positive matrix a is boundedly invert ible if and only if the linear system ax 1 has a bounded solution 2 micchelli gave a simple argument for the case when a is a toeplitz matrix cavaretta dahmen micchelli and smith 2 recently proved the conjecture in case a is a block toeplitz matrix this is all the more remarkable since it is easy to see in hindsight that the conjecture is faulty even in the original context of spline interpolation for example interpolation by bounded broken lines with breakpoint sequence z at the sequence t z 0 is possible to any bounded ordinate sequence y but not uniquely so since the value of the interpolant at 0 is freely choosable in matrix terms this corresponds to the matrix obtained from the bi infinite identity matrix by dropping one row but with the condition changed to has a unique bounded solution the conjecture was proved in 1

rigorous computation of smooth branches of equilibria for the three dimensional cahn hilliard equation in this paper we propose a new general method to compute rigorously global smooth branches of equilibria of higher dimensional partial differential equations the theoretical framework is based on a combination of the theory introduced in global smooth solution curves using rigorous branch following van den berg et al math comput 79 271 1565 1584 2010 and in analytic estimates and rigorous continuation for equilibria of higher dimensional pdes gameiro and lessard j diff equ 249 9 2237 2268 2010 using this method one can obtain proofs of existence of global smooth solution curves of equilibria for large continuous parameter ranges and about local uniqueness of the solutions on the curve as an application we compute several smooth branches of equilibria for the three dimensional cahn hilliard equation

surjectivity and invertibility properties of totally positive matrices abstract a new finite section type theorem is used to show that the members of an interesting class of bounded totally positive matrices map l onto l if and only if their range contains a vector which alternates in sign and has coordinates bounded away from zero the class of matrices studied contains all banded totally positive matrices and thus all infinite spline collocation matrices connections to related work and extension to matrices which are not sign regular are indicated

covariance matrix estimation using repeated measurements when data are incomplete modeling repeated measurements data has been studied extensively lately in the parametric situation however it is significant to study the effect of various treatments over a period of time where the repeated measurements on the same subject are expected to be correlated the correlation among the repeated measurements for all the subjects will be studied via the covariance matrix the positive definite constraint is one of obstacles that encounters modeling the covariance structure however the cholesky decomposition removes this constraint and allows modeling the components of the covariance matrix m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 in this paper we adopt the estimation procedures introduced by diggle and verbyla nonparametric estimation of covariance structure in longitudinal data biometrics 54 1998 401 415 where the variogram cloud as well as the squared residuals are used to estimate the variogram and the variances via the kernel smoothing selecting the appropriate bandwidth value is one of the important steps in the estimation process thus in our data analysis we choose the bandwidth using one of the most simple straight forward methods which is the cross validation method developed by rice and silverman estimating the mean and covariance structure nonparametrically when the data are curves j roy statist soc b 53 1991 233 243 and adapted by others finally we apply these nonparametric techniques as well as a graphical method m pourahmadi joint mean covariance models with applications to longitudinal data unconstrained parameterization biometrika 86 1999 677 690 to a real life data and use the penalized likelihood criterion like aic and bic to compare models of our interest our interest

structure of invertible bi infinite totally positive matrices an invertible nonfinite totally positive matrix a is shown to have one and only one main diagonal this means that exactly one diagonal of a has the property that all finite sections of a principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to a i this is shown to imply restrictions on the possible shapes of such a matrix in the proof such a matrix is also shown to have an la invertible ldu factorization in addition decay of the entries of such a matrix away from the main diagonal is demonstrated it is also shown that a bounded sign regular matrix carrying some bounded sequence to a uniformly alternating sequence must have all its columns in co

covering relations cone conditions and the stable manifold theorem abstract we show how to effectively link covering relations with cone conditions we give a new geometric proof of the stable manifold theorem for hyperbolic fixed point of a map

analytic estimates and rigorous continuation for equilibria of higher dimensional pdes abstract in this paper we extend the ideas of the so called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher dimensional spatial domains for that effect we present a new set of general analytic estimates these estimates are valid for any dimension and are used together with rigorous computations to construct a finite number of radii polynomials these polynomials provide a computationally efficient method to prove via a contraction argument the existence and local uniqueness of solutions for a rather large class of nonlinear problems we apply this technique to prove existence and local uniqueness of equilibrium solutions for the cahn hilliard and the swift hohenberg equations defined on two and three dimensional spatial domains

multidimensional lines ii proximity and applications in inselberg and dimsdale siam j appl math 54 1994 pp 559 577 it was shown that in coordinates any line in r n can be represented by n 1 planar points indexed by a pair of distinct integers from 0 1 2 cdots n based on this representation proximity properties and construction algorithms are obtained they include i an algorithm constructing the intersection between two lines and displaying the common point when it exists and ii construction and display of the points where the minimum l 1 distance between two lines occurs from this representation tight estimates for the analogous results with the euclidean l 2 are provided to study line proximity line neighborhoods for a topology are proposed results indicate that there are ambiguities in line detection for orthogonal coordinates that are eliminated in parallel coordinates an application to air traffic control for an information display and an algorithm for collision avoidance are illustrated

invariant densities and escape rates rigorous and computable approximations in the l norm abstract in this article we study piecewise linear discretization schemes for transfer operators perron frobenius operators associated with interval maps we show how these can be used to provide rigorous pointwise approximations for invariant densities of markov interval maps we also derive the order of convergence of the approximate invariant density to the real one in the l norm the outcome of this paper complements recent results on the formulae of escape rates of open dynamical systems keller and liverani 2009 7 in particular the novelty of our work over previous results on b v and l approximations is that it provides a method for explicit computation of the approximation error

recent advances about the uniqueness of the slowly oscillating periodic solutions of wright s equation abstract an old conjecture in delay equations states that wright s equation y t y t 1 1 y t r has a unique slowly oscillating periodic solution sops for every parameter value 2 we reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of sops of wright s equation using this method we show that a part of this branch does not have any fold point partially answering the new reformulated conjecture

the inverse of a totally positive biinfinite band matrix it is shown that a bounded bi infinite banded totally positive matrix a is boundedly invertible iff there is one and only one bounded sequence mapped by a to the sequence the argument shows that such a matrix has a main diagonal i e the inverse of a is the bounded pointwise limit of inverses of finite sections of a principal with respect to a particular diagonal hence 4 j 7 or its inverse is again totally positive 0 introduction this paper is a further step in a continuing effort to understand certain linear spline approximation schemes convergence of such processes is intimately tied to their stability i e to their boundedness as maps on c say use of the b spline basis shows this question to be equivalent to bounding the inverse of certain totally positive band matrices the calculation of bounds on the inverse of a given matrix is in general a difficult task it is hoped that the present investigation into the consequences of bandedness and total positivity for the structure of the inverse may ultimately prove helpful in obtaining such bounds the results in this paper were obtained in the study of a conjecture due to c a micchelli 7 in connection with his work on the specific approximation scheme of interpolation at a strictly increasing point sequence t by elements of sm t i e by splines of some order m with some knot sequence t micchelli became convinced that every bounded function has one and only one bounded spline interpolant iff the particular function which satisfies t all has a bounded spline interpolant in sm if ni ni mt denotes the corresponding b spline basis for sm then micchelli s conjecture can be phrased thus the matrix

interpreting plots of a multidimensional dose response surface in a parallel coordinate system the dose response surface for a combination of drugs is a multidimensional figure consequently it is not possible to view such a surface using orthogonal axes when the number of dimensions exceeds 3 parallel axes have been used to represent hyperdimensional figures this paper reports on the use of parallel coordinate axes to plot the dose response surface and its contours of constant response isobols for a combination of drugs it is shown that patterns formed by intersecting line segments in the parallel system can aid in the interpretation of the fitted dose response surface more generally analytic results are developed that permit the ready visualization and characterization of interaction effects of a polynomial model

computer assisted enclosure methods for elliptic differential equations abstract we report on methods for computing enclosures of solutions of second order nonlinear elliptic boundary value problems simultaneously proving the existence of a solution in the enclosing set the old fashioned monotonicity methods are well suited for this task but only for a restricted class of problems therefore we propose a new approach which is based on a suitable fixed point formulation of the problem and uses as an essential ingredient norm bounds for the inverse of the linearization of the given problem at some approximate solution which is computed numerically these norm bounds are obtained via eigenvalue enclosures we also give a brief description of an alternative method proposed by m t nakao

a uniqueness result for a semilinear elliptic problem a computer assisted proof abstract starting with the famous article a gidas w m ni l nirenberg symmetry and related properties via the maximum principle comm math phys 68 1979 209 243 many papers have been devoted to the uniqueness question for positive solutions of u u u p in u 0 on where p 1 and ranges between 0 and the first dirichlet eigenvalue 1 of for the case when is a ball uniqueness could be proved mainly by ode techniques but very little is known when is not a ball and then only for 0 in this article we prove uniqueness for all 0 1 in the case 0 1 2 and p 2 this constitutes the first positive answer to the uniqueness question in a domain different from a ball our proof makes heavy use of computer assistance we compute a branch of approximate solutions and prove existence of a true solution branch close to it using fixed point techniques by eigenvalue enclosure methods and an additional analytical argument for close to 1 we deduce the non degeneracy of all solutions along this branch whence uniqueness follows from the known bifurcation structure of the problem

topological invariants mulitvalued maps and computer assisted proofs in dynamics abstract we introduce the concept of inheritable properties of multivalued maps and show how this concept when combined with interval arithmetic and representable multivalued approximations may be used in adapting topological methods to computer assisted proofs in dynamics as our first example we propose a simple geometric criterion for chaos a more sophisticated example is the conley index theory we show how the computation for the conley index for multivalued representable maps may be reduced to a finite computation this reduction was used in the computer assisted proof of chaos in the lorenz equations represented in 1

a simple method for error bounds of eigenvalues of symmetric matrices abstract we propose a simple method for validated computation of eigenvalues of symmetric matrices the method is based on ldl t decomposition and its error estimation the indices of eigenvalues with respect to magnitude can also be obtained by this method

validated continuation for equilibria of pdes one of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor corrector continuation techniques in the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output we introduce a new technique that combines the information obtained from the predictor corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation using the cahn hilliard and swift hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone

l boundedness of l 2 projections on splines for a multiple geometric mesh this paper concerns the l2 projectors from loo to the normed linear space of polynomial splines it is shown that for the multiple geometric meshes the loo norms of the corresponding l2 projectors are bounded independently of the mesh ratio 0 introduction let us begin with some notation let k be a positive integer and x x i z a real nondecreasing knot sequence with xi xi k all i set x 00 lim xi i x o lim xji i x oo x00 i 0 i o00o fori e z and x e i define mi k x k xi xi k x i ni k x xi k xi mi k x k where po pr f denotes the rth divided difference of the function f at the points po 0 pr then ni k is called the i th b spline of order k on the knot sequence x for a e rz the rule f x a i niak x defines a function on i if we take the sum to be pointwise every such function is called a polynomial spline of order k with the knot sequence x and their collection is denoted by skx further let denote the normed linear space of bounded polynomial splines of order k with the knot sequence x and norm fll ii sup i f x ix xei we shall be concerned with p the orthogonal projector onto with respect to the ordinary inner product f g ff x g x dx but restricted to l i we want to bound its norm 11 p ii sup iipsf ii 1 iii o fel j received march 26 1984 revised december 16 1985 1980 mathematics subject classification primary 41a15 15a60 this is a part of the author s thesis which was submitted to the graduate school university of wisconsin madison in partial fullfillment of the requirement for ph d degree present address department of mathematics zhejiang university hangzhou zhejiang the people s republic of china 1987 american mathematical society 0025 5718 87 1 00 25 per page 675 this content downloaded from 207 46 13 178 on wed 15 jun 2016 07 00 20 utc all use subject to http about jstor org terms