symplectic geometry and dissipative differential operators abstract everitt and markus characterized the domains of self adjoint differential operators in terms of lagrangian subspaces of complex symplectic spaces in this paper we define dissipative and strictly dissipative subspaces for complex symplectic spaces and characterize the domains of dissipative and strictly dissipative differential operators in terms of these subspaces

secondary resonances and approximate models of routes to chaotic motion in non linear oscillators in two types of non linear oscillators with single equilibrium positions the question of transition to chaos is investigated by making use of a computer simulation and approximate analytical methods to study periodic solutions and their stability the zone of chaotic motion is found as a transition zone between qt periodic sub or subultraharmonic solution and the t periodic solution thus replacing the classical jump phenomenon two routes to chaos are studied the gentle one via a cascade of period doubling bifurcations which appears to be characteristic in unsymmetrical oscillators and a sharp route which proves to accompany a stability limit of a symmetrical periodic solution an approximate mathematical model of the pre chaotic motion is proposed

realizing rcc8 networks using convex regions rcc8 is a popular fragment of the region connection calculus in which qualitative spatial relations between regions such as adjacency overlap and parthood can be expressed while rcc8 is essentially dimensionless most current applications are confined to reasoning about two dimensional or three dimensional physical space in this paper however we are mainly interested in conceptual spaces which typically are high dimensional euclidean spaces in which the meaning of natural language concepts can be represented using convex regions the aim of this paper is to analyze how the restriction to convex regions constrains the realizability of networks of rcc8 relations first we identify all ways in which the set of rcc8 base relations can be restricted to guarantee that consistent networks can be convexly realized in respectively 1d 2d 3d and 4d most surprisingly we find that if the relation partially overlaps is disallowed all consistent atomic rcc8 networks can be convexly realized in 4d if instead refinements of the relation part of are disallowed all consistent atomic rcc8 relations can be convexly realized in 3d we furthermore show among others that any consistent rcc8 network with 2 n 1 variables can be realized using convex regions in the n dimensional euclidean space

self adjoint domains symplectic geometry and limit circle solutions abstract everitt and markus characterized the domains of self adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of lagrangian subspaces of symplectic spaces recently for the even order case with real coefficients wang sun and zettl constructed limit circle lc solutions and hao wang sun and zettl characterized the self adjoint domains in terms of lc solutions these lc solutions are higher order analogues of the celebrated titchmarsh weyl limit circle solutions in the second order case this lc characterization has been used to obtain information about the discrete continuous and essential spectra of these operators in this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self adjoint operators

characterization of domains of self adjoint ordinary differential operators article history the gkn glazman krein naimark theorem characterizes all self adjoint realizations of linear symmetric formally self adjoint ordinary differential equations in terms of maximal domain functions these functions depend on the coefficients and this dependence is implicit and complicated in the regular case an explicit characterization in terms of two point boundary conditions can be given in the singular case when the deficiency index d is maximal the gkn characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter i n the much more difficult intermediate cases not all solutions contribute to the singular self adjoint conditions in 1986 sun found a representation of the self adjoint singular conditions in terms of certain solutions for nonreal values of in this paper we give a representation in terms of certain solutions for real this leads to a classification of solutions as limit point lp or limit circle lc in analogy with the celebrated weyl classification in the second order case the lc solutions contribute to the singular boundary conditions the lp solutions do not the advantage of using real is not only because it is in general easier to find explicit solutions but more importantly it yields information about the spectrum

bifurcations of approximate harmonic balance solutions and transition to chaos in an oscillator with inertial and elastic symmetric nonlinearities abstract the local stability of approximate periodic solutions and period doubling bifurcations in a harmonically forced non linear oscillator with symmetric elastic and inertia non linearities are studied analytically and numerically approximate principal resonance solutions are first obtained using a two term harmonic balance and then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict zones of symmetry breaking leading to period doubling bifurcation and chaos the results of the present work which follows the analysis approach presented by szemplinska stupnika 1986 international journal of nonlinear mechanics 23 257 277 1987 journal of sound and vibration 113 155 172 are verified for selected system parameters by numerical simulations using methods of qualitative theory and good agreement was obtained between the analytical and numerical results finally a criterion for the period doubling bifurcation is proposed analytically for this type of oscillator and compared with computer simulation results that predict the true period doubling bifurcation and chaos boundaries

continuous spectrum and square integrable solutions of differential operators with intermediate deficiency index we explore the connection between square integrable solutions for real values of the spectral parameter and the continuous spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d we show that if for all in an open interval i there are d of linearly independent square integrable solutions then for every extension of dmin the point spectrum is nowhere dense in i and there is a self adjoint extension of smin which has no continuous spectrum in i this analysis is based on our construction of limit point lp and limit circle lc solutions obtained recently in an earlier paper

the classification of self adjoint boundary conditions separated coupled and mixed there are three basic types of self adjoint regular and singular boundary conditions separated coupled and mixed for even order problems with real coefficients one regular endpoint and arbitrary deficiency index d we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square integrable solutions for some real value of the spectral parameter in the separated case our construction yields non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available

an unusual self adjoint linear partial differential operator hhin an american mathematical society memoir published in 2003 the authors everitt and markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions and the generation of self adjoint differential operators in sobolev hilbert spaces in the case when the differential expression has smooth coefficients on the closure of a bounded open region in euclidean space and when the region has a smooth boundary this theory leads to the construction of certain self adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region this present paper concerns the spectral properties of one of these unusual self adjoint operators sometimes called the harmonic operator the boundary value problems considered in the memoir see above and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region also the region is bounded and has a smooth boundary under these and some additional technical conditions it is shown in the memoir and emphasized in this present paper that all the self adjoint operators considered are explicitly determined on their domains by the partial differential expression this property makes a remarkable comparison with the case of symmetric ordinary differential expressions in the regular ordinary case the spectrum of all the self adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation and each eigenvalue is of finite multiplicity thus the essential spectrum of all these operators is empty this spectral property extends to the present partial differential case for the classical dirichlet and neumann operators but not to the harmonic operator it is shown in this paper that the harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane thus the essential spectrum of this operator is not empty both the weak and strong formulations of the harmonic boundary value problem are considered these two formulations are shown to be equivalent in the final section of the paper examples are considered which show that the harmonic operator defined by the methods of symplectic algebra has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region

a solution procedure based on the ateb function for a two degree of freedom oscillator abstract in this paper vibration of a two mass system with two degrees of freedom is considered two equal harmonic oscillators are coupled with a strong nonlinear viscoelastic connection mathematical model of the system is two coupled second order strong nonlinear differential equations introducing new variables the system transforms into two uncoupled equations one of them is linear and the other with a strong nonlinearity in the paper a method for solving the strong nonlinear equation is developed based on the exact solution of a pure nonlinear differential equation we assumed a perturbed version of the solution with time variable parameters due to the fact that the solution is periodical the averaging procedure is introduced as a special case vibrations of harmonic oscillators with fraction order nonlinear connection are considered depending on the order and coefficient of nonlinearities bounded and unbounded motion of masses is determined besides the conditions for steady state periodical solution are discussed the procedure given in the paper is applied for investigation of the vibration of a vocal cord which is modeled with two harmonic oscillators with strong nonlinear fraction order viscoelastic connection using the experimental data for the vocal cord the parameters for the steady state solution which describes the flexural vibration of the vocal cord is analyzed the influence of the order of nonlinearity on the amplitude and frequency of vibration of the vocal cord is obtained the analytical results are close to those obtained experimentally

a logic framework for reasoning with movement based on fuzzy qualitative representation abstract we present a logic approach to reason with moving objects under fuzzy qualitative representation this way we can deal both with qualitative and quantitative information and consequently to obtain more accurate results the proposed logic system is introduced as an extension of propositional dynamic logic this choice on the one hand simplifies the theoretical study concerning soundness completeness and decidability on the other hand provides the possibility of constructing complex relations from simpler ones and the use of a language very close to programming languages

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

resonant response functions for nonlinear oscillators with polynomial type nonlinearities in this paper we consider the steady state response of forced damped weakly nonlinear oscillators with polynomial type nonlinearities in particular we define general expressions that can be used to compute resonant response functions which define the steady state constant amplitude oscillatory response at the primary resonance and the associated harmonics the resonant response functions are derived using a normal form transformation which is carried out directly on the second order nonlinear oscillator the example of a forced van der pol oscillator with an additional cubic stiffness nonlinearity is used to demonstrate how the general analysis can be applied

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

spatial reasoning in a fuzzy region connection calculus although the region connection calculus rcc offers an appealing framework for modelling topological relations its application in real world scenarios is hampered when spatial phenomena are affected by vagueness to cope with this we present a generalization of the rcc based on fuzzy set theory and discuss how reasoning tasks such as satisfiability and entailment checking can be cast into linear programming problems we furthermore reveal that reasoning in our fuzzy rcc is np complete thus preserving the computational complexity of reasoning in the rcc and we identify an important tractable subfragment moreover we show how reasoning tasks in our fuzzy rcc can also be reduced to reasoning tasks in the original rcc while this link with the rcc could be exploited in practical reasoning algorithms we mainly focus on the theoretical consequences in particular using this link we establish a close relationship with the egg yolk calculus and we demonstrate that satisfiable knowledge bases can be realized by fuzzy regions in any dimension

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

computing improved optimal solutions to max min flexible constraint satisfaction problems abstract the formal framework for decision making in a fuzzy environment is based on a general max min bottleneck like optimization problem proposed by zadeh it is also the basis for extending the constraint satisfaction paradigm of artificial intelligence to accommodating flexible or prioritized constraints this paper surveys refinements of the ordering of solutions supplied by the max min formulation namely the discrimin partial ordering and the leximin complete preordering a general algorithm is given which computes all maximal solutions in the sense of these relations it also sheds light on the structure of the set of best solutions moreover classes of problems for which there is a unique best discrimin and leximin solution are exhibited namely continuous problems with convex domains and so called isotonic problems noticeable examples of such problems are fuzzy linear programming problems and fuzzy pert like scheduling problems

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

some observations and remarks on differential operators generated by first order boundary value problems this paper deals with the study of the set of all self adjoint differential operators which are generated from first order linear ordinary boundary value problems these operators are defined on a weighted hilbert function space and are examined as an application of the result obtained by everitt and markus in their paper in 1997 an investigation is given so that first order self adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self adjoint operators however the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum and consequently to the determination of conditions under which the construction of kramer analytic kernels from the above boundary value problems can be accomplished

on the complexity of qualitative spatial reasoning a maximal tractable fragment of the region connection calculus the computational properties of qualitative spatial reasoning have been investigated to some degree however the question for the boundary between polynomial and np hard reasoning problems has not been addressed yet in this paper we explore this boundary in the region connection calculus rcc 8 we extend bennett s encoding of rcc 8 in modal logic based on this encoding we prove that reasoning is np complete in general and identify a maximal tractable subset of the relations in rcc 8 that contains all base relations further we show that for this subset path consistency is sufficient for deciding consistency

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

a unifying approach to temporal constraint reasoning abstract we present a formalism disjunctive linear relations dlrs for reasoning about temporal constraints dlrs subsume most of the formalisms for temporal constraint reasoning proposed in the literature and is therefore computationally expensive we also present a restricted type of dlrs horn dlrs which have a polynomial time satisfiability problem we prove that most approaches to tractable temporal constraint reasoning can be encoded as horn dlrs including the ord horn algebra by nebel and burckert and the simple temporal constraints by dechter et al thus dlrs is a suitable unifying formalism for reasoning about temporal constraints

dissipative operators symplectic geometry and limit circle solutions for a general symmetric ordinary differential expression m and a positive weight function w the authors characterized the dissipative and strictly dissipative extensions of the minimal operator generated by m in terms of subspaces of a symplectic geometry space here we characterize these subspaces in terms of lc solutions namely we characterize the boundary conditions of the dissipative and strictly dissipative extensions by lc solutions in a hilbert space

on a weyl titchmarsh theory for discrete symplectic systems on a half line recently bohner and sun introduced basic elements of a weyl titchmarsh theory into the study of discrete symplectic systems we extend this development through the introduction of weyl titchmarsh functions together with a preliminary study of their properties a limit point criterion is described and characterized green s function for the half line is introduced as a limit of such functions in the regular case and half line solutions obtained are seen to satisfy lambda dependent boundary conditions at infinity

generalized region connection calculus the region connection calculus rcc is one of the most widely referenced system of high level qualitative spatial reasoning rcc assumes a continuous representation of space this contrasts sharply with the fact that spatial information obtained from physical recording devices is nowadays invariably digital in form and therefore implicitly uses a discrete representation of space recently galton developed a theory of discrete space that parallels rcc but question still lies in that can we have a theory of qualitative spatial reasoning admitting models of discrete spaces as well as continuous spaces in this paper we aim at establishing a formal theory which accommodates both discrete and continuous spatial information and a generalization of region connection calculus is introduced grcc the new theory takes two primitives the mereological notion of part and the topological notion of connection rcc and galton s theory for discrete space are both extensions of grcc the relation between continuous models and discrete ones is also clarified by introducing some operations on models of grcc in particular we propose a general approach for constructing countable rcc models as direct limits of collections of finite models compared with standard rcc models given rise from regular connected spaces these countable models have the nice property that each region can be constructed in finite steps from basic regions two interesting countable rcc models are also given one is a minimal rcc model the other is a countable sub model of the continuous space r 2

on the third superharmonic resonance in the duffing oscillator abstract the third order superharmonic resonance in two versions of a harmonically excited duffing oscillator is investigated by analytical and numerical methods it is shown that even in situations for which a non linear oscillator may be thought a priori to satisfy the small perturbation requirements the analytical results obtained by the amplitude expansion method of perturbation analysis may lead to incomplete and or erroneous results for harmonically excited non linear oscillators equivalence of the approximate solutions obtained by two fundamentally different perturbation schemes is questioned and for the duffing ueda oscillator it is shown that a sudden transition into chaos is the result of a saddle node bifurcation of the third superharmonic response and a sudden transition out of chaos is the result of a boundary crisis

a fuzzy sets theoretic approach to approximate spatial reasoning relational composition based reasoning has become the most prevalent method for qualitative reasoning since allen s 1983 work on temporal intervals underlying this reasoning technique is the concept of a jointly exhaustive and pairwise disjoint set of relations systems of relations such as rcc5 and rcc8 were originally developed for ideal regions not subject to imperfections such as vagueness or fuzziness which are found in many applications in geographic analysis and image understanding this paper however presents a general method for classifying binary topological relations involving fuzzy regions using the rcc5 or the rcc8 theory our approach is based on fuzzy set theory and the theory of consonant random set some complete classifications of topological relations between fuzzy regions are also given furthermore two composition operators on spatial relations between fuzzy regions are introduced in this paper these composition operators provide reasonable relational composition based reasoning engine for spatial reasoning involving fuzzy regions

solving conflicts in information merging by a flexible interpretation of atomic propositions although many techniques for merging conflicting propositional knowledge bases have already been proposed most existing work is based on the idea that inconsistency results from the presence of incorrect pieces of information which should be identified and removed in contrast we take the view in this paper that conflicts are often caused by statements that are inaccurate rather than completely false suggesting to restore consistency by interpreting certain statements in a flexible way rather than ignoring them completely in accordance with this view we propose a novel approach to merging which exploits extra logical background information about the semantic relatedness of atomic propositions several merging operators are presented which are based on different formalizations of this background knowledge ranging from purely qualitative approaches related to possibilistic logic to quantitative approaches with a probabilistic flavor both syntactic and semantic characterizations are provided for each merging operator and the computational complexity is analyzed