subspace based and direct algorithms for distorted circular contour estimation circular features are commonly sought in digital image processing the subspace based line detection slide method proposed to estimate the center and the radius of a single circle in this paper we introduce a novel method for estimating several radii while extending the circle estimation to retrieve circular like distorted contours particularly we develop and validate a new model for virtual signal generation by simulating a circular antenna the circle center is estimated by the slide method a variable speed propagation scheme toward the circular antenna yields a linear phase signal therefore a high resolution method provides the radius either the gradient method or the more robust combination of dividing rectangles and spline interpolation can extend this method for free form object segmentation the retrieval of multiple non concentric circles and rotated ellipses is also considered to evaluate the performance of the proposed methods we compare them with a least squares method hough transform and gradient vector flow we apply the proposed method to hand made images while considering some real world images

controlling chaotic dynamical systems we review the major ideas involved in the control of chaos we present the ott grebogi yorke ogy method of controlling chaos which is a particular case of the pole placement technique but which is the one leading to the shortest time to achieve the control of chaotic systems implementation using only measured time series in experimental settings is also described

g 1 approximation of conic sections by quartic bezier curves this paper proposes a new approximation method for conic sections by quartic bezier curves with g 1 end point continuity we give an upper bound on the hausdorff distance between the conic section and the quartic bezier approximation curve and also show that the approximation order is eight moreover using the subdivision scheme at the shoulder point of the conic section the resulting approximation curve has g 2 curvature continuity with the conic section at the endpoints finally we prove that our approximation has a smaller error bound than previous quartic bezier approximations and also present some numerical examples to demonstrate the validity and effectiveness of our method

an approximation method to circular arcs abstract modern cad cam systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function parametric rational polynomials are used to approximate the circular arcs a new approximation method by means of quartic bezier curves is presented to effectively approximate the circular arcs it is shown that our method provides the approximation of order eight and yields a smaller error than previous methods by equally spaced subdivision of circular arcs the curvature continuous spline approximation algorithm of the circular arcs is given

a characterization of all elliptic algebro geometric solutions of the akns hierarchy here 9 x p x wl w3 denotes the elliptic weierstrass function with fundamental periods 2wl and 2w3 im w3 wl 0 in the special case where wl is real and w3 is purely imaginary the potential q x in 1 1 is real valued and ince s striking result 51 in modern spectral theoretic terminology yields that the spectrum of the unique self adjoint operator associated with the differential expression l2 d2 dx 2 q x in l2 r exhibits finitely many bands and gaps respectively that is

optimal parameter values for approximating conic sections by the quartic bzier curves the previous approximation curves of conic section by quartic bzier curves interpolate the conic section at the specified parameter values in this paper by solving the minimax problem we present the optimal parameter values for approximating conic sections by the quartic bzier curves the upper bound on the hausdorff distance between the conic section and the approximation curves is minimized the method of solving the minimax problem is changed to solve a quartic algebraic equation by delicate reasoning

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

adaptive tracking control of a class of uncertain chaotic systems in the presence of random perturbations considering the effect of random perturbations on the chaotic system a new adaptive tracking control is presented for a large class of uncertain chaotic systems using the invariance principle of differential equations where the bound of random perturbations is not necessarily known in advance and it is estimated through an adaptive control process it is theoretically proved that this approach can make the perturbed chaotic system track any desired reference signal in addition we can see that this method can apply to almost all uncertain chaotic systems and it is simpler and easier to implement in practical application in the end we take the perturbed lorenz system as an example to illustrate that the proposed scheme is effective

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

constrained polynomial approximation of quadric surfaces quadric surfaces are one of the geometric elements most commonly used for shape expression and mechanical accessory cartography in three dimensions since quadric surfaces cannot be represented exactly using polynomials it is not suitable to deal with most problems encountered in engineering application for this reason we present a new method for polynomial approximation of quadric surfaces the approximation tensor product surface obtained by the proposed method has no limit on its degree i e arbitrary degree it has c k h k h 0 1 2 continuity at four corners with the original quadric surface meanwhile the method has an optimal approximation result in the l2 norm and it can also be applied to piecewise continuous quadric surface patches or a quadric surface patch combined with surface subdivision scheme and the resulting piecewise approximation patches are globally c0 continuous finally error estimation for the approximation method is given and numerical examples are presented to validate the feasibility and effectiveness of our method

completeness and spectral synthesis of nonselfadjoint one dimensional perturbations of selfadjoint operators abstract we study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators the problems under consideration include completeness of eigenvectors relations between completeness of the perturbed operator and its adjoint and the spectral synthesis problem we obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is in general unexpectedly rich and complicated a parallel spectral theory is developed for one dimensional singular perturbations of unbounded selfadjoint operators our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic entire functions

on the shape of spectra for non self adjoint periodic schrodinger operators the spectra of the schrodinger operators with periodic potentials are studied when the potential is real and periodic the spectrum consists of at most countably many line segments energy bands on the real line while when the potential is complex and periodic the spectrum consists of at most countably many analytic arcs in the complex plane in some recent papers such operators with complex symmetric periodic potentials have been studied in particular the authors argued that some energy bands would appear and disappear under perturbations here we show that the appearance and disappearance of such energy bands imply existence of nonreal spectra this is a consequence of a more general result describing the local shape of the spectrum

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

approximation of conic sections by curvature continuous quartic bezier curves in this paper we propose two approximation methods of conic section by quartic bezier curves these are the extensions of the quartic bezier approximations of circular arcs presented in ahn and kim 1997 1 and kim and ahn 2007 10 to conic cases we also give the error bounds of the hausdorff distances between the conic section and the approximation curves and show that the error bounds have the approximation order eight our methods yield quartic g 2 curvature continuous spline approximations of conic sections using the subdivision scheme stated in floater 1995 1997 5 11 we illustrate our results by some numerical examples

trace formulas for non self adjoint periodic schrodinger operators and some applications abstract recently a trace formula for non self adjoint periodic schrodinger operators in l 2 r associated with dirichlet eigenvalues was proved in differential integral equations 14 2001 671 700 here we prove a corresponding trace formula associated with neumann eigenvalues in addition we investigate dirichlet and neumann eigenvalues of such operators in particular using the dirichlet and neumann trace formulas we provide detailed information on location of the dirichlet and neumann eigenvalues for the model operator with the potential k e 2 i x where k c

brief converting chaos into periodic motion by state feedback control a simple control method is presented to convert chaos into periodic motion by using linear state feedback of an available system variable the low periodic orbits can be derived from the chaotic attractor with slight change of the dynamical structure in the original dynamics melnikov s perturbation method in generalized hamiltonian systems is introduced to explain the control mechanism of directing chaotic motion towards low periodic motion in the lorenz equations moreover the existence of periodic orbits is examined in the perturbed lorenz system simulation results verify the effectiveness of melnikov s analysis

approximation of circular arcs and offset curves by bezier curves of high degree in this paper we present an exact error analysis for circle approximation by bezier curve the approximation method is a special case of floater s conic approximation method comput aided geom design 8 1991 135 using this we propose an approximation method of the offset curve of given plane bezier curve by bezier curve of the same degree

circle approximation by g2 bezier curves of degree n with 2n 1 extreme points abstract in this paper we present a new circle approximation method using g 2 bezier curve of degree n the signed error function of which is an alternant or equioscillating with 2 n 1 extreme points for n 2 3 4 the new circle approximation curve is obtained in explicit form as is the hausdorff distance between the curve and a circular arc compared with previous g 2 quartic circle approximation methods the proposed g 2 quartic approximation yields the smallest hausdorff distance

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

a general framework for the optimal approximation of circular arcs by parametric polynomial curves abstract we propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves the approach is based on a constrained uniform approximation of an error function by scalar polynomials the system of nonlinear equations for the unknown control points of the approximating polynomial given in the bezier form is derived and a detailed analysis provided for some low degree cases which were not studied yet at least for these cases the solutions can be in principal written in a closed form and provide the best known approximants according to the simplified radial distance a general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given

curvatures of the quadratic rational bezier curves abstract we find necessary and sufficient conditions for the curvature of a quadratic rational bezier curve to be monotone in 0 1 to have a unique local minimum to have a unique local maximum and to have both extrema in 0 1 and we also visualize them in figures as an application we present a necessary and sufficient condition for the offset curve to be regular and to have the same tangent direction with the given quadratic rational bezier curve and give a simple algorithm to find it

spectral theory of two point differential operators determined by d2 i spectral properties abstract the necessary tools for the complete resolution of the spectral theory for all linear two point differential operators l in l2 0 1 determined by d2 and by boundary values b1 b2 are developed all work is based on six numerical parameters defined in terms of the coefficients of b1 b2 these quantities are utilized in the development of the characteristic determinant of l and the green s function of i l and in the development of the projections of l2 0 1 onto the generalized eigenspaces of l they are used in the study of the decay rates of the resolvent operator r l as and they are used to identify the 13 cases treated in developing the spectral theory for d2 all results pertaining to the spectrum l of l to the algebraic multiplicities of the eigenvalues l to the ascents of the operators i l to the boundedness of the family of all finite sums of the projections associated with l to the denseness of the generalized eigenfunctions and to bases consisting of generalized eigenfunctions are summarized in a table the proofs of these results are the subject of the sequel part ii

on the riesz basis property of root vectors system for 2 2 dirac type operators abstract the paper is concerned with the riesz basis property of a boundary value problem associated in l 2 0 1 c 2 with the following 2 2 dirac type equation 0 1 l y i b 1 y q x y y b b 1 0 0 b 2 y col y 1 y 2 with a summable potential matrix q l 1 0 1 c 2 2 and b 1 0 b 2 if b 2 b 1 1 this equation is equivalent to one dimensional dirac equation it is proved that the system of root functions of a linear boundary value problem constitutes a riesz basis in l 2 0 1 c 2 provided that the boundary conditions are strictly regular by analogy with the case of ordinary differential equations boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed q 0 operator are asymptotically simple and separated in opposite to the dirac case there is no simple algebraic criterion of the strict regularity whenever b 1 b 2 0 however under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case in particular it is shown that regular separated boundary conditions are always strictly regular while antiperiodic boundary conditions are strictly regular if b 1 b 2 are coprime integers of different parity the proof of the main result is based on existence of triangular transformation operators for system 0 1 their existence is also established here in the case of a summable q in the case of regular but not strictly regular boundary conditions we prove the riesz basis property with parentheses the main results are applied to establish the riesz basis property of the dynamic generator of spatially non homogeneous damped timoshenko beam model

equiconvergence of spectral decompositions of hill schrodinger operators abstract we study in various functional spaces the equiconvergence of spectral decompositions of the hill operator l d 2 d x 2 v x x 0 with h per 1 potential and the free operator l 0 d 2 d x 2 subject to periodic antiperiodic or dirichlet boundary conditions in particular we prove that s n s n 0 l a l b 0 if 1 a b 1 a 1 b 1 2 where s n and s n 0 are the n th partial sums of the spectral decompositions of l and l 0 moreover if v h with 1 2 1 and 1 a 3 2 then we obtain uniform equiconvergence s n s n 0 l a l 0 as n

on hill s equation with a singular complex valued potential 1 introduction the differential equation y00 qy ey is called hill s equation if q is a realvalued periodic locally integrable function of a real variable it is very well known that the spectrum of the operator associated with this equation consists of a countably in nite number of compact intervals of the real line these intervals are called spectral bands and are generally separated by open intervals called gaps in some instances however all but a nite number of gaps are empty resulting in a spectrum consisting of a nite number of compact bands and one closed interval extending to negative in nity in this case one calls q a nite band or nite gap potential the relationship of nite band potentials with the hierarchy of korteweg de vries equations has been the reason for a heightened interest in them during the past twenty years for recent accounts see for instance the monographs by dickey 8 and belokolos et al 3 the rst non trivial example of a nite band potential was given by ince in 1940 18 consider the lame potentials q x yg g 1 x c jq q0 where g 2n and jq q0 denotes weierstrass s elliptic function with fundamental half periods q and q0 if q is real q0 is pure imaginary and c q0 then q is realvalued periodic and continuous ince s result when translated to the present language shows that q then is a nite band potential with g non empty gaps between the bands this however raises immediately the question of what happens when c 6 q0 or when the half periods are more general in these cases one encounters potentials which are complex valued but continuous as well as potentials which have singularities of the form yg g 1 x and are realor complex valued before giving an overview of the content of the paper i want to give a brief account of the history of the subject let q be a real locally integrable periodic function with period p l the differential expression d dx q and h the self adjoint operator in l r associated with l then consider the following four statements a there are precisely 2g 1 real numbers e2g e0 such that the spectrum of h is given by

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

brief notch filter feedback control in a class of chaotic systems the paper focuses on the problem of the notch filter feedback control in the perturbed planar hamiltonian systems by melnikov s method a suitable range of parameters in the notch filter controller can be obtained to convert chaotic motions into desired low period motions the averaging method is introduced to analyze the stability of the low period orbits the subharmonic orbits inside the homoclinic loop in the control systems as a typical example the design procedure for controlling duffing s oscillator and its stability analysis are derived in detail and the thorough simulation results are presented to demonstrate the effectiveness of the theoretical analysis finally the further examples show the applicability of the notch filter controller in a wide range of chaotic systems and hyper chaotic systems

asymptotics of instability zones of the hill operator with a two term potential let n denote the length of the nth zone of instability of the hill operator ly y 4t cos2x 2 2cos4x y where 0 and either both t are real or both are pure imaginary numbers for even n we prove if t n are fixed then for 0 n 8 n2n n 1 2 k 1n 2 t2 2k 1 2 1 o and if t are fixed then for n n 8 2 n 2 4 n 2 2 cos 2t 1 o lognn the asymptotics for 0 for n 2m imply the following identities for squares of integers s 1k m2 is2 1 j1 jk m s 1k 2js 1 2 where 1 k m and the left sum is over all indices i1 ik such that m i1 ik m is ir 2if s r similar formulae see theorems 7 9 hold for odd n

on the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations the paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations we introduce and investigate the class of weakly regular boundary conditions we show that this class is much broader than the class of regular boundary conditions introduced by g d birkhoff and r e langer our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular moreover we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness also we investigate the completeness for 2 2 dirac type equations subject to irregular boundary conditions emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions

some new g1 quartic parametric approximants of circular arcs abstract in this paper a method for approximation of circular arc by quartic bezier curve is presented interpolation by geometrically continuous g 1 parametric polynomials is considered and necessary and sufficient conditions for the existence of the solution are established optimal asymptotic approximation order eight is also confirmed for the obtained approximant application of our new solution to approximation of conic sections is presented and some numerical examples provided

refined asymptotics of the spectral gap for the mathieu operator abstract the mathieu operator l y y 2 a cos 2 x y a c a 0 considered with periodic or anti periodic boundary conditions has close to n 2 for large enough n two periodic if n is even or anti periodic if n is odd eigenvalues n n for fixed a we show that n n 8 a 4 n n 1 2 1 a 2 4 n 3 o 1 n 4 n this result extends the asymptotic formula of harrell avron simon by providing more asymptotic terms

new treatment on bifurcations of periodic solutions and homoclinic orbits at high r in the lorenz equations in this paper the existence of periodic solutions and homoclinic orbits in the lorenz equations with high r is rigorously proved the paper deals with the lorenz model as a three dimensional perturbed hamiltonian system generated by the three dimensional lie algebra by using the method of melnikov vector the explicit parametric conditions can be determined

on polynomial approximation of circular arcs and helices we present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two point taylor expansion we obtain the degree 2n 1 polynomial for the approximation problem in an efficient way which is very convenient to increase the degree of polynomial by adding new terms an upper bound on the approximation error is available so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user prescribed error tolerance

the best g 1 cubic and g 2 quartic bezier approximations of circular arcs we obtain cubic and quartic bezier approximations of circular arcs that respectively satisfy g 1 and g 2 end point interpolation conditions we identify the necessary and sufficient conditions for such approximations to be the best in the sense that they have the minimum hausdorff distance to the circular arc we then establish the existence and uniqueness of these best approximations and present practical methods to calculate them which are verified by examples

a note on quintic polynomial approximation of generalized cornu spiral segments in two recent papers cripps et al 2010 3 and cross and cripps 2012 2 quintic polynomial approximations of the generalized cornu spirals have been studied by considering g 2 continuity and g 3 continuity at the end points respectively the quintic curve is constructed so that the maximum curvature error is within the specified tolerance in this paper we provide corrections to the typing errors in cross and cripps 2012 2 and propose a simpler and more efficient method for the g 2 constrained quintic polynomial approximation where the g 2 conditions are always satisfied by four free variables also we introduce a new error measure of the maximum curvature error which can greatly reduce the computational time when looking for the solution numerical experiments demonstrate the effectiveness of the new measure

eigenvalue asymptotics for sturm liouville operators with singular potentials abstract we derive eigenvalue asymptotics for sturm liouville operators with singular complex valued potentials from the space w 2 1 0 1 0 1 and dirichlet or neumann dirichlet boundary conditions we also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra

a schauder and riesz basis criterion for non self adjoint schrodinger operators with periodic and antiperiodic boundary conditions under the assumption that v l2 0 dx we derive necessary and sufficient conditions in terms of spectral data for non self adjoint schrodinger operators d2 dx2 v in l2 0 dx with periodic and antiperiodic boundary conditions to possess a riesz basis of root vectors i e eigenvectors and generalized eigenvectors spanning the range of the riesz projection associated with the corresponding periodic and antiperiodic eigenvalues we also discuss the case of a schauder basis for periodic and antiperiodic schrodinger operators d2 dx2 v in lp 0 dx p 1

criteria for existence of riesz bases consisting of root functions of hill and 1d dirac operators abstract we study the system of root functions srf of hill operator l y y v y with a singular complex valued potential v h per 1 and the srf of 1d dirac operator l y i 1 0 0 1 d y d x v y with matrix l 2 potential v 0 p q 0 subject to periodic or anti periodic boundary conditions series of necessary and sufficient conditions in terms of fourier coefficients of the potentials and related spectral gaps and deviations for srf to contain a riesz basis are proven equiconvergence theorems are used to explain basis property of srf in l p spaces and other rearrangement invariant function spaces

non linear dynamics and chaos control for an electromagnetic system a non linear mathematical model has been obtained by applying a modified conventional identification technique based on the principle of harmonic balance in this study analytical work is carried out on this identified non linear model by applying the first harmonic approximation solution and the floquet theory the resulting criteria for bifurcations can be used to evaluate the operational range of a system employing such a non linear actuator we also employ the method of lyapunov exponents to show the occurrence of chaotic motion and to verify the above analyses finally various methods such as the state feedback control and injection of dither signal control are used to control chaos effectively

approximating conic sections by constrained bezier curves of arbitrary degree in this paper an algorithm for approximating conic sections by constrained bezier curves of arbitrary degree is proposed first using the eigenvalues of recurrence equations and the method of undetermined coefficients some exact integral formulas for the product of two bernstein basis functions and the denominator of rational quadratic form expressing conic section are given then using the least squares method a matrix based representation of the control points of the optimal bezier approximation curve is deduced this algorithm yields an explicit arbitrary degree bezier approximation of conic sections which has function value and derivatives at the endpoints that match the function value and the derivatives of the conic section up to second order and is optimal in the l 2 norm to reduce error the method can be combined with a curve subdivision scheme computational examples are presented to validate the feasibility and effectiveness of the algorithm for a whole curve or its part generated by a subdivision

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

smoothness of schrodinger operator potential in the case of gevrey type asymptotics of the gaps abstract consider the schrodinger equation y v x y y with a periodic real valued l 2 potential v of period 1 v x m v m exp 2 imx let n n be its zones of instability i e n n are pairs of periodic and antiperiodic eigenvalues and b 0 1 determines gevrey classes theorem if n 1 n n 2 1 n 2s e 2an b s 0 a0 then v x is a gevrey function and moreover m v m 2 1 m 2s e 2a m b

on a theorem of hochstadt in fact hochstadt gives an explicit formula for the discriminant which will be reproduced below since hochstadt s work there has been a considerable interest in finite band potentials i e potentials q such that the spectrum of the operator associated with d2 dx 2 q consists of finitely many regular analytic arcs due to their close relationship with the korteweg de vries hierarchy and hence with integrable systems however the interest is not restricted to real valued potentials for instance

circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials abstract the quality of the approximation of circular arcs by parametric polynomials is usually measured by the hausdorff distance it is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property in this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs a general approach to this problem is considered and corresponding nonlinear equations are derived for the approximants possessing the maximal order of geometric smoothness a scalar nonlinear equation is analyzed in detail for the parabolic the cubic and the quartic case the existence of the admissible solution is confirmed moreover the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved theoretical results are confirmed by numerical examples

the spectral expansion for a nonself adjoint hill operator with a locally integrable potential abstract we construct the spectral expansion for the one dimensional schrodinger operator l d 2 dx 2 q x x in l 2 where q x is a 1 periodic lebesgue integrable on 0 1 and complex valued potential we obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator l t for t 0 generated by this operation in l 2 0 1 and the t periodic boundary conditions using it we prove that the eigenfunctions and associated functions of l t form a riesz basis in l 2 0 1 for t 0 then we find the spectral expansion for the operator l

quintic polynomial approximation of log aesthetic curves by curvature deviation log aesthetic curves lacs possessing monotone curvature and including many classical curves have been widely used to describe fair shapes in geometric modeling however they are generally represented in non polynomial form and are thus not compatible with current cad systems in this paper we present quintic polynomial approximation of lac segments for a given lac segment a quintic g 2 interpolating bezier curve is obtained by minimizing a curvature based error metric with the advantage of being more likely to preserve the monotone curvature property numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures

a study of a nonlinear vibration isolator with a quasi zero stiffness characteristic a vibration isolator consisting of a vertical linear spring and two nonlinear pre stressed oblique springs is considered in this paper the system has both geometrical and physical nonlinearity firstly a static analysis is carried out the softening parameter leading to quasi zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry pre stress and the stiffness of the springs the optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone it also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position for these values a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities it includes considering the possibilities of the appearance of period doubling bifurcation and its development into chaotic motion for this purpose approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots poincare maps and lyapunov exponents are used finally the frequency at which the first period doubling bifurcation appears is found and the effect of damping on this frequency determined

circle approximation using ln bezier curves of even degree and its application abstract we present an approximation method of circular arcs using linear normal ln bezier curves of even degree four and higher our method achieves g m continuity for endpoint interpolation of a circular arc by a ln bezier curve of degree 2m for m 2 3 we also present the exact hausdorff distance between the circular arc and the approximating ln bezier curve we show that the ln curve has an approximation order of 2 m 2 for m 2 3 our approximation method can be applied to offset approximation so obtaining a rational bezier curve as an offset approximant we derive an algorithm for offset approximation based on the ln circle approximation and illustrate our method with some numerical examples

steady state self oscillations and chaotic behavior of a controlled electromechanical device by using the first lyapunov value and the melnikov theory abstract in this paper regular and chaotic oscillations in a controlled electromechanical transducer are investigated the nonlinear control laws are defined by an electric tension excitation and an external force applied to the mobile piece of the transducer the paper shows that an andronov poincare hopf bifurcation appears as long as adequate parameters are chosen for the nonlinear control laws the stability of the weak focuses associated to such bifurcation is examined according to the sign of the first lyapunov value showing that chaotic behavior can arise when the first lyapunov value is varied harmonically the appearance of a homoclinic orbit is investigated assuming an approximated model for the device on the basis of the parametric equations of the homoclinic orbit and the presence of harmonic disturbances on the platform it is demonstrated that chaotic oscillations can also appear and they have been examined by means of the melnikov theory chaotic behavior is corroborated by means of the sensitive dependence lyapunov exponents and power spectral density and it is applied to drive the transducer mobile piece to a predefined set point assuming that noise due to the measurement process can appear in the control signals the steady state error associated to such random noise is eliminated by adding a pi linear controller to the control force numerical simulations are used to corroborate the analytical results

multiplicities of the eigenvalues of periodic dirac operators abstract let us consider the dirac operator l ij d dx u j 1 0 0 1 u 0 a cos 2 x a cos 2 x 0 where a 0 is real on i 0 1 with boundary conditions bc per i e f 1 f 0 and bc per i e f 1 f 0 f f 1 f 2 h 1 i then l bc l bc and all per l u are of multiplicity 2 while per l u are simple theorem 15 this is an analogue of ince s statement for mathieu hill operator links between the spectra of dirac and hill operators lead to detailed information about the spectra of hill operators with potentials of the ricatti form v p p 2 section 3 it helps to get analogues of grigis results ann sci ecole norm sup 4 20 1987 641 on the zones of instability of hill operators with polynomial potentials and their asymptotics for the case of dirac operators as well section 4 2

impulsive control of lorenz system abstract in this paper an impulsive control scheme of the lorenz system is presented first we use the theory of impulsive differential equation to find conditions under which impulsively controlled lorenz system is asymptotically stable then we give the estimate of the upper bound of impulse interval for asymptotically stable control numerical experimental results are presented

new infinite families of exact sums of squares formulas jacobi elliptic functions and ramanujan s tau function in this paper we give two infinite families of explicit exact formulas that generalize jacobi s 1829 4 and 8 squares identities to 4n2 or 4n n 1 squares respectively without using cusp forms our 24 squares identity leads to a different formula for ramanujan s tau function n when n is odd these results arise in the setting of jacobi elliptic functions jacobi continued fractions hankel or turanian determinants fourier series lambert series inclusion exclusion laplace expansion formula for determinants and schur functions we have also obtained many additional infinite families of identities in this same setting that are analogous to the function identities in appendix i of macdonald s work macdonald i g 1972 invent math 15 91 143 a special case of our methods yields a proof of the two conjectured kac v g and wakimoto m 1994 in progress in mathematics eds brylinski j l brylinski r guillemin v kac v birkhauser boston boston ma vol 123 pp 415 456 identities involving representing a positive integer by sums of 4n2 or 4n n 1 triangular numbers respectively our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series gustafson s cl nonterminating 6 5 summation theorem and andrews basic hypergeometric series proof of jacobi s 4 and 8 squares identities we have elsewhere applied symmetry and schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n n 1 squares respectively our sums of more than 8 squares identities are not the same as the formulas of mathews 1895 glaisher 1907 ramanujan 1916 mordell 1917 1919 hardy 1918 1920 kac and wakimoto and many others

inverse spectral theory for uniformly open gaps in a weighted sturm liouville problem abstract motivated by a pde existence problem we study the inverse problem for a weighted sturm liouville operator l s associated with the eigenvalue problem y s x y 0 where s is a real valued periodic even function that is bounded from below by a positive constant and belongs to the l 2 based sobolev space h r 0 1 r 1 choosing gap lengths and gap midpoints as coordinates we define a spectral map g that assigns to a coefficient s the structure of the spectrum of l s we find that g is a real analytic isomorphism locally around s 1 which in particular implies the existence of coefficients s h r 0 1 r 3 2 whose spectrum features band structure with all gaps uniformly open around the gap midpoints this result paves the way for the construction of so called breathers in nonlinear wave equations with such coefficients s apart from the novelty of treating the inverse spectral problem for the full banach scale h r 0 1 r 1 the local nature of our result allows more concise and transparent proofs in particular instead of using any preliminary transformations we treat the weighted problem directly by adapting techniques used for schrodinger operators with distribution potentials

sequence of gn ln polynomial curves approximating circular arcs abstract in this paper we derive a sequence of linear normal ln curves b 2 n of degree 2 n which are g n endpoint interpolations of a circular arc and have approximation order 2 n 2 this is an extension of the circle approximation method by ln bezier curves given in ahn and hoffmann 2014 to all even degrees we also extend the circle approximation to an ellipse approximation by g n ln curves of degree 2 n an upper bound of the hausdorff distance between the ellipse and its ln approximation is obtained we illustrate our results through an ln approximation of convolution curves of ellipses and a spline curve

divergence of spectral decompositions of hill operators with two exponential term potentials abstract we consider the hill operator l y y v x y 0 x subject to periodic or antiperiodic boundary conditions bc with potentials of the form v x a e 2 i r x b e 2 i s x a b 0 r s n r s it is shown that the system of root functions does not contain a basis in l 2 0 c if bc are periodic or if bc are antiperiodic and r s are odd or r 1 and s 3

approximation of circular arcs by bezier curves abstract for the circular arc of angle 0 we present the explicit form of the best gc 3 quartic approximation and the best gc 2 quartic approximations of various types and give the explicit form of the hausdorff distance between the circular arc and the approximate bezier curves for each case we also show the existence of the gc 4 quintic approximations to the arc and find the explicit form of the best gc 3 quintic approximation in certain constraints and their distances from the arc all approximations we construct in this paper have the optimal order of approximation twice of the degree of approximate bezier curves

spectra of non selfadjoint hill s operators and a class of riemann surfaces i am grateful to a boutet de monvel for the invitation to laboratory of mathematical physics and geometry university paris 7 where the initial version of this paper was written it gives me a special pleasure to thank h mckean for his permanent interest and support of my study of nonselfadjoint periodic operators i am thankful to the referees of this paper whose criticism helped me to improve the exposition the final version of the paper was written at the ben gurion university of the negev the financial support of the ministry of science of israel is gratefully appreciated

oscillation problems for hill s equation with periodic damping abstract this paper deals with the second order linear differential equation x a t x b t x 0 where a and b are periodic coefficients the main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory our nonoscillation theorem and oscillation theorem are proved by using the riccati technique in our theorem the composite function of an indefinite integral of b and a suitable multiple valued continuously differentiable function are focused and the composite function of them plays an important role the results obtained here include a result by kwong and wong 15 and a result by sugie and matsumura 26 an application to a equation of whittaker hill type is given to show the usefulness of our results finally simulations are also attached to illustrate that our oscillation criterion is sharp

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

resolvent growth and birkhoff regularity abstract we prove a long standing conjecture in the theory of two point boundary value problems that unconditional basisness implies birkhoff regularity it is a corollary of our two main results minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant and sparseness of nth order roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis considerations are based on a new direct method exploiting almost orthogonality of birkhoff s solutions of the equation l y y this property was discovered earlier by the author

convergence of spectral decompositions of hill operators with trigonometric polynomial potentials we consider the hill operator ly y prime prime v x y quad0 leq x leq pi subject to periodic or antiperiodic boundary conditions with potentials v which are trigonometric polynomials with nonzero coefficients of the form then the system of eigenfunctions and at most finitely many associated functions is complete but it is not a basis in l 2 0 pi mathbb c if a b in the case i if a b and neither b 2 4b nor a 2 4a is an integer square in the case iii and it is never a basis in the case ii subject to periodic boundary conditions

controlling hamiltonian chaos the method for stabilizing an unstable periodic orbit in chaotic dynamical systems originally formulated by ott grebogi and yorke ogy is not directly applicable to chaotic hamiltonian systems the reason is that an unstable periodic orbit in such systems often exhibits complex conjugate eigenvalues at one or more of its orbit points in this paper we extend the ogy stabilization method to control hamiltonian chaos by incorporating the notion of stable and unstable directions at each periodic point we also present an algorithm to calculate the stable and unstable directions other issues specific to the control of hamiltonian chaos are also discussed

characterization of the potential smoothness of one dimensional dirac operator subject to general boundary conditions and its riesz basis property abstract the one dimensional dirac operator with periodic potential v 0 p x q x 0 where p q l 2 0 subject to periodic antiperiodic or a general strictly regular boundary condition bc has discrete spectrums it is known that for large enough n in the disk centered at n of radius 1 2 the operator has exactly two periodic if n is even or antiperiodic if n is odd eigenvalues n and n counted according to multiplicity and one eigenvalue n b c corresponding to the boundary condition b c we prove that the smoothness of the potential could be characterized by the decay rate of the sequence n b c n where n b c n b c n and n n n furthermore it is shown that the dirac operator with periodic or antiperiodic boundary condition has the riesz basis property if and only if sup n 0 n b c n is finite

characterization of potential smoothness and the riesz basis property of the hill schrodinger operator in terms of periodic antiperiodic and neumann spectra abstract the hill operators l y y v x y considered with complex valued periodic potentials v and subject to periodic antiperiodic or neumann boundary conditions have discrete spectra for sufficiently large n close to n 2 there are two periodic if n is even or antiperiodic if n is odd eigenvalues n n and one neumann eigenvalue n we study the geometry of the spectral triangle with vertices n n n and show that the rate of decay of triangle size characterizes the potential smoothness moreover it is proved for v l p 0 p 1 that the set of periodic antiperiodic root functions contains a riesz basis if and only if for even respectively odd n sup n n n n n n

spectral parametrization of non selfadjoint hill s operators abstract for the set of hill operators l d 2 dx 2 q x with periodic potentials q x l 2 0 a spectral parametrization is given using a complete system of independent parameters

whittaker hill equation and semifinite gap schrodinger operators a periodic one dimensional schrodinger operator is called semifinite gap if every second gap in its spectrum is eventually closed we construct explicit examples of semifinite gap schrodinger operators in trigonometric functions by applying darboux transformations to the whittaker hill equation we give a criterion for the regularity of the corresponding potentials and investigate the spectral properties of the new operators