noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales the paper focuses on studying the noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales first the hamilton equations of nonholonomic nonconservative systems on time scales are established which is based on the lagrange equations for nonholonomic systems on time scales then based upon the quasi invariance of hamilton action of systems under the infinitesimal transformations with respect to the time and generalized coordinate on time scale the noether identity and the conserved quantity of nonholonomic nonconservative systems on time scales are obtained finally an example is presented to illustrate the application of the results

breakpoint detection using covariance propagation presents a statistical approach for detecting breakpoints from chain encoded digital arcs an arc point is declared as a breakpoint if the estimated orientations of the two fitted lines of the two arc segments immediately to the right and left of the arc point are significantly statistically different the major contributions of this research include developing a method for analytically estimating the covariance matrix of the fitted line parameters and proposing a perturbation model to characterize the perturbation associated with each arc point

second order periodic boundary value problems on time scales in this paper we study the existence of solutions of periodic boundary value problem for second order nonlinear dynamic equations on time scales for this purpose we use two methods here first sign properties of the green s function are investigated and existence results of bounded solutions of nonlinear boundary value problems are established as a result of schauder fixed point theorem second the monotone method is discussed to assure the existence and uniqueness of solutions of such periodic boundary value problems on time scales

noether s theorem on time scales abstract we show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective euler lagrange extremals

segmentation of two dimensional curve contours abstract segmentation of digitized planar curves is one of the most important elements in early image processing because a segmented image can describe the object profile in a compact form to facilitate higher level vision processing in many applications it is necessary to decompose an object boundary contour into several primitives such as segments and curves in this paper a two stage hybrid technique for the segmentation of two dimensional 2d curves is presented in which the number of segments is assumed to be known first the boundary is iteratively approximated using a split and merge method next an end point adjustment procedure is applied to reach the best fitting polygonal approximation a computational comparison with two existing methods shows that the proposed technique is fast and accurate an application of the new segmentation technique to industrial part inspection is also provided

nonparametric segmentation of curves into various representations this paper describes and demonstrates the operation and performance of an algorithm for segmenting connected points into a combination of representations such as lines circular elliptical and superelliptical arcs and polynomials the algorithm has a number of interesting properties including being scale invariant nonparametric general purpose and efficient

reduced resolution and scale space for dominant feature detection in contours abstract analysis of contours or curves at various scales or levels of smoothing scale space is an important tool in curve segmentation and feature detection in this paper we propose the application of the hierarchical discrete correlation algorithm for efficiently calculating and creating a scale space of curves in conjunction with this procedure we also investigate the use of the reduced resolution or gaussian pyramid representation of the set of smoothed curves as the basis for initially localizing and detecting features we also create an approximation to the above algorithms that is computationally less expensive finally we propose a new inter scale method for curve segmentation and feature detection based on the motion of a curve through scale space

controller design for discrete time stochastic processes with nonquadratic loss the problem of optimal controller synthesis for discrete time stochastic nonlinear processes is examined when the objective for design is specified as the expected value of a nonsymmetric nonquadratic loss functional it is necessary to relate the closed loop process dynamics to the stationary probability density function pdf because explicit relationships are available for only a few special cases a general solution to the problem is not possible in this work a technique for developing approximations of the optimal control law by application of a dynamics pdf approximation is presented numerical simulations illustrate application of the proposed technique

pseudospherical surfaces on time scales a geometric definition and the spectral approach we define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains e g the cantor set in particular we present a new expression for the discrete gaussian curvature which turns out to be valid for asymptotic nets on any time scale we show that asymptotic chebyshev nets on an arbitrary time scale have constant negative gaussian curvature we present also the quaternion valued spectral problem the lax pair and the darboux backlund transformation for pseudospherical surfaces in asymptotic coordinates on arbitrary time scales

hamiltonian systems on time scales linear and nonlinear hamiltonian systems are studied on time scales t we unify symplectic flow properties of discrete and continuous hamiltonian systems a chain rule which unifies discrete and continuous settings is presented for our so called alpha derivatives on generalized time scales this chain rule allows transformation of linear hamiltonian systems on time scales under simultaneous change of independent and dependent variables thus extending the change of dependent variables recently obtained by dosl and hilscher we also give the legendre transformation for nonlinear euler lagrange equations on time scales to hamiltonian systems on time scales

time scale symplectic systems without normality abstract we present a theory of the definiteness nonnegativity and positivity of a quadratic functional f over a bounded time scale the results are given in terms of a time scale symplectic system s which is a time scale linear system that generalizes and unifies the linear hamiltonian differential system and discrete symplectic system the novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of f and in establishing equivalent conditions for the nonnegativity of f without any normality assumption to reach this goal a new notion of generalized focal points for conjoined bases x u of s is introduced results on the piecewise constant kernel of x t are obtained and various picone type identities are derived under the piecewise constant kernel condition the results of this paper generalize and unify recent ones in each of the discrete and continuous time setting and constitute a keystone for further development in this theory

smoothness of invariant manifolds in this paper we present some functional analytic tools that allow us to prove a theorem on ck smoothness of general discrete and continuous invariant manifolds for ck systems a crucial role in this proof will be played by a theorem on ck parameter dependence of a fixed point of a contraction defined on a scale of banach spaces this fixed point principle is stated independently and may be of interest in itself further functional analytic considerations concerning smoothness properties of the nemytski operator are presented in a rather general form throughout the paper we treat the two cases of continuous and discrete dynamical systems in a unified manner thereby we use some concepts and results from the so called calculus on measure chains which was developed by the author in res in math 18 1990 18 56 knowledge of this calculus is not necessary for understanding the paper

comparison theorem for a nonlinear boundary value problem on time scales we prove a comparison theorem for the lower and upper solutions of a nonlinear two point boundary value problem on time scales this theorem plays an important role in the development of the method of generalized quasilinearization on time scales

a new measurement for assessing polygonal approximation of curves this paper presents a novel method for assessing the accuracy of unsupervised polygonal approximation algorithms this measurement relies on a polygonal approximation called the reference approximation the reference approximation is obtained using the method of perez and vidal 11 by an iterative method that optimizes an objective function then the proposed measurement is calculated by comparing the reference approximation with the approximation to be evaluated taking into account the similarity between the polygonal approximation and the original contour and penalizing polygonal approximations with an excessive number of points a comparative experiment by using polygonal approximations obtained with commonly used algorithms showed that the proposed measurement is more efficient than other proposed measurements at comparing polygonal approximations with different number of points

assessing the behaviour of polygonal approximation algorithms abstract in a recent paper we described a method for assessing the accuracy of polygonal approximation algorithms ieee trans pami 19 6 1997 659 here we develop several measures to assess the stability of such approximation algorithms under two classes of variations namely perturbations of the data and changes in the algorithms scale parameters in order to quantify the former a break point stability index is defined and tested for the latter two indices are introduced 1 a monotonicity index is applied to analyse the change in the approximation error or the number of line segments against increasing scale and 2 a consistency index quantifies the variation in results produced at the same scale by an algorithm but with different input parameter values finally the previously developed accuracy figure of merit is calculated and averaged over 21 test curves for different parameter values to obtain more reliable scores

lyapunov inequality for linear hamiltonian systems on time scales the principal aim of this paper is to state and prove some lyapunov inequalities for linear hamiltonian system on an arbitrary time scale t so that the well known case of differential linear hamiltonian systems t r and the recently developed case of discrete hamiltonian systems t z are unified applying these inequalities a disconjugacy criterion for hamiltonian systems on time scales is obtained

on the detection of dominant points on digital curves a parallel algorithm is presented for detecting dominant points on a digital closed curve the procedure requires no input parameter and remains reliable even when features of multiple sizes are present on the digital curve the procedure first determines the region of support for each point based on its local properties then computes measures of relative significance e g curvature of each point and finally detects dominant points by a process of nonmaximum suppression this procedure leads to the observation that the performance of dominant points detection depends not only on the accuracy of the measure of significance but also on the precise determination of the region of support this solves the fundamental problem of scale factor selection encountered in various dominant point detection algorithms the inherent nature of scale space filtering in the procedure is addressed and the performance of the procedure is compared to those of several other dominant point detection algorithms using a number of examples

new definitions of exponential hyperbolic and trigonometric functions on time scales abstract we propose two new definitions of the exponential function on time scales the first definition is based on the cayley transformation while the second one is a natural extension of exact discretizations our exponential functions map the imaginary axis into the unit circle therefore it is possible to define hyperbolic and trigonometric functions on time scales in a standard way the resulting functions preserve most of the qualitative properties of the corresponding continuous functions in particular pythagorean trigonometric identities hold exactly on any time scale dynamic equations satisfied by cayley motivated functions have a natural similarity to the corresponding differential equations the exact discretization is less convenient as far as dynamic equations and differentiation are concerned

optimum uniform piecewise linear approximation of planar curves two dimensional digital curves are often uniformly approximated by polygons or piecewise linear curves several algorithms have been proposed in the literature to find such curves we present an algorithm that finds a piecewise linear curve with the minimal number of segments required to approximate a curve within a uniform error with fixed initial and final points we compare our optimal algorithm to several suboptimal algorithms with respect to the number of linear segments required in the approximation and the execution time of the algorithm

matrix lie theory and measure chains in this paper we investigate the relation between lie matrix groups mg mg n k of n n matrices with k entries and their so called lie h algebras lh mg a lie h algebra for a matrix group mg mg n k is defined as the set a mat n k ah i mg if h 0 and a mat n k exp at mg for all t r if h 0 where mat n k is the set of n n matrices with k entries and i mat n k is the identity matrix we investigate the structural relations between the matrix group mg and its lie h algebras there are results concerning the group and topological isomorphy of the group and its lie h algebra for h 0 the subsequent h 0 limiting behaviour hereby we can reveal the exact relation between the commutator in mg and the lie bracket in l0 mg in a final result we give a condition on the linear operator function in x a t x x i which keeps the solution trajectory in mg

existence of weak solutions of two point boundary value problems for second order dynamic equations on time scales abstract in this paper by using critical point theory we establish the existence of weak solutions of the two point boundary value problem for a second order dynamic equation on an arbitrary time scale t so that the well known case of differential dynamic systems t r and the recently developed case of discrete dynamic systems t z are unified to the best of our knowledge this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

qualitative properties of hamiltonian difference systems abstract we study some qualitative properties of the hamiltonian difference system y t b t y t 1 c t z t z t a t y t 1 b t z t the methods used involve both variational and riccati type arguments

the sine gordon equation on time scales abstract we formulate and discuss integrable analogue of the sine gordon equation on arbitrary time scales this unification contains the sine gordon equation discrete sine gordon equation and the hirota equation doubly discrete sine gordon equation as special cases we present the lax pair check compatibility conditions and construct the darboux backlund transformation finally we obtain a soliton solution on arbitrary time scale the solution is expressed by the so called cayley exponential function

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

on lyapunov type inequalities for nonlinear dynamic systems on time scales in this paper by using elementary analysis we establish several new lyapunov type inequalities for the following nonlinear dynamic system on an arbitrary time scale t x d t email protected t x s t email protected t y t p 2y t y d t email protected t x s t q 2x s t email protected t y t when the end points are not necessarily usual zeros but rather generalized zeros which generalize and improve all related existing ones including the continuous and discrete cases

the quasilinearization method for boundary value problems on time scales abstract in this paper we apply the method of quasilinearization to a family of boundary value problems for second order dynamic equations y q t y h t y on time scales the results include a variety of possible cases when h is either convex or a splitting of convex and concave parts and whether lower and upper solutions are of natural form or of natural coupled form

dynamics symmetries of hamiltonian system on time scales in this paper the dynamics symmetries of hamiltonian system on time scales are studied we study the symmetries and quantities based on the calculation of variation and lie transformation group particular focus lies in the noether symmetry leads to the noether conserved quantity and the lie symmetry leads to the noether conserved quantity if the infinitesimal transformations satisfy the structure equation as the new application of result at end of the article we give a simple example of noether symmetry and lie symmetry on time scales

limit point and limit circle classification for symplectic systems on time scales abstract in this paper we study the limit point and limit circle classification for symplectic systems on time scales which depend linearly on the spectral parameter in a broader context we develop a unified weyl titchmarsh theory for continuous and discrete linear hamiltonian and symplectic systems both separated and coupled boundary conditions are allowed our results include the study of the weyl disks and circles and their limiting behavior as well as a precise analysis of the number of linearly independent square integrable solutions we also prove an analogue of the famous weyl alternative we connect and unify many known results in the weyl titchmarsh theory for continuous discrete and special time scales systems and explain the differences between them some of our statements in particular those connected with coupled endpoints or the weyl alternative are new even in the continuous time setting

optimal matching problem in detection and recognition performance evaluation abstract this paper proposes a principle of one to one correspondence in performance evaluation of a general class of detection and recognition algorithms such a correspondence between ground truth entities and algorithm declared entities is essential in accurately computing objective performance measures such as the detection recognition and false alarm rates we mathematically define the correspondence by formulating a combinatorial optimal matching problem in addition to evaluating detection performance this methodology is also capable of evaluating recognition performance our study shows that the proposed principle for detection performance evaluation is simple general and mathematically sound the derived performance evaluation technique is widely applicable precise consistent and efficient

extraction of dominant points by estimation of the contour fluctuations a new method of dominant points dps extraction is presented for each boundary point the method assigns a support region whose length is individually limited two steps are used to detect dps firstly points having higher curvature value than the estimated fluctuations over the support region are extracted as candidates secondly adjacent candidates are merged and a final elimination step is performed on each region additionally a new error measurement scheme is introduced which takes into account the error due to the polygonalisation and the deviation brought by each dp this error is used to compare our algorithm to other methods