eigenvalue distribution of second order dynamic equations on time scales considered as fractals abstract let t a b be a time scale with a b t in this paper we study the asymptotic distribution of eigenvalues of the following linear problem u u with mixed boundary conditions u a u a 0 u b u b it is known that there exists a sequence of simple eigenvalues k k we consider the spectral counting function n t k k and we seek for its asymptotic expansion as a power of let d be the minkowski or box dimension of t which gives the order of growth of the number k t e of intervals of length e needed to cover t namely k t e e d we prove an upper bound of n which involves the minkowski dimension n t c d 2 where c is a positive constant depending only on the minkowski content of t roughly speaking its d volume although the minkowski content is not a measure we also consider certain limiting cases d 0 infinite minkowski content and we show a family of self similar fractal sets where n t admits two side estimates

positive solutions for a class of boundary value problems on time scales by obtaining intervals of the parameter l this paper is concerned with the existence and nonexistence of positive solution of the second order nonlinear dynamic equation on time scales p t x d t q t x t lw t f t x ax r a bx d r a 0 cx b dx d b 0 for t a b t where t is a time scale a 0 c 0 b 0 d 0 with a c 0 the arguments are based upon fixed point theorems in a cone

lyapunov inequalities for discrete linear hamiltonian systems abstract in this paper we present some lyapunov type inequalities for discrete linear scalar hamiltonian systems when the coefficient c t is not necessarily nonnegative valued and when the end points are not necessarily usual zeros but rather generalized zeros applying these inequalities we obtain some disconjugacy and stability criteria for discrete hamiltonian systems

methods one switch conditions for multiattribute utility functions we introduce a variety of new independence conditions for multiattribute utility functions that permit preference dependencies among the attributes of a decision problem the hierarchy of new conditions varies in the degree to which it specifies the functional form ranging from more general solutions with weaker constraints to more specific solutions with stronger constraints this formulation provides a wealth of new functional forms that a decision maker may use in a multiattribute decision problem in addition it may be used to tailor the utility elicitation process to the comfort level of the decision maker the new conditions and the corresponding functional forms are based on the idea of limiting the number of switches that a decision maker may make between two decision alternatives as a parameter of the problem varies we show how this formulation also relates many widely used concepts in single and multiattribute utility theory

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

oscillation of second order delay differential equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for a second order delay differential equation on time scales in terms of the coefficients

asymptotic behavior of solutions of a third order nonlinear dynamic equation on time scales in this paper we will establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation c t a t x t q t f x t 0 t t0 oscillates or converges to zero

correlations and copulas for decision and risk analysis the construction of a probabilistic model is a key step in most decision and risk analyses typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model s random variables we describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables the approach is designed specifically to permit the use of an expert s subjective judgments of marginal distributions and correlations the copula that underlies the multivariate normal distribution provides the basis for modeling dependence but arbitrary marginals are allowed we discuss how correlations can be assessed using techniques that are familiar to decision analysts and we report the results of an empirical study of the accuracy of the assessment methods the approach is demonstrated in the context of a simple example including a study of the sensitivity of the results to the assessed correlations

vibration analysis of timoshenko beams with non homogeneity and varying cross section abstract in this paper an analytic solution for free and forced vibrations of stepped timoshenko beams is presented and used for the approximate analysis of generally non uniform timoshenko beams in the case of free vibrations the frequency equation is expressed in terms of some initial parameters at one end of the beam while in the case of forced vibrations the solution may be obtained by solving a set of algebraic equations with only two unknowns several examples are presented to illustrate the validity and accuracy of the analysis

multiattribute utility copulas we introduce the notion of a multiattribute utility copula that expresses any i continuous ii bounded multiattribute utility function that is iii nondecreasing with each of its arguments and iv strictly increasing with each argument for at least one reference value of the complement attributes in terms of single attribute utility assessments this formulation provides a wealth of new functional forms that can be used to model preferences over utility dependent attributes and enables sensitivity analyses to some of the widely used functional forms of utility independence we introduce a class of utility copulas called archimedean utility copulas and discuss the conditions under which it yields the additive and multiplicative forms we also discuss linear and composite transformations of utility copulas that construct utility functions with partial utility independence we conclude with the risk aversion functions that are induced by utility copula formulations and work through several examples to illustrate the approach

calculus of variations on time scales weak local piecewise crd1 solutions with variable endpoints abstract a nonlinear calculus of variations problem on time scales with variable endpoints is considered the space of functions employed is that of piecewise rd continuously differentiable functions c1prd for this problem the euler lagrange equation the transversality condition and the accessory problem are derived as necessary conditions for weak local optimality assuming the coercivity of the second variation a corresponding second order sufficiency criterion is established

application of chebyshev series to solution of non prismatic beam vibration problems the problem of the vibration of a non prismatic beam resting on a two parameter elastic foundation has been solved by applying the approximation by chebyshev series as a result closed analytical formulas defining the coefficients of the sought solutions were obtained the method was used to solve the eigenproblem for a simply supported beam and a cantilever beam the obtained results were compared with the results reported by other authors

dynamic stability problem of a non prismatic rod a dynamic stiffness matrix for a non prismatic rod finite element resting on a two parameter non homogenous elastic foundation has been determined to obtain the solution the shape function was approximated by chebyshev series this yielded closed analytical formulae for the coefficients of the matrices sought the finite element obtained was used to solve the dynamic stability problem for a non prismatic cantilever column the results were compared with those reported by other authors

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

free vibrations of uniform timoshenko beams with attachments the problem of free transverse vibrations of timoshenko beams with attachments like translational and rotational springs concentrated mass including the moment of inertia linear undamped oscillators and additional supports is considered the frequency equation for the combined system is derived by means of the lagrange multiplier formalism the exact solution of the free vibration problem of the beam without attachments is taken into account for the formulation of the free vibration problem of the combined system numerical examples show the separate or coupling influences of the additional elements on the combined system s frequencies the comparison of results obtained by using the present approach with results of the exact solution indicates a good agreement

oscillation of nonlinear dynamic equations on time scales by means of riccati transformation techniques we establish some new oscillation criteria for the second order nonlinear dynamic equation p t x d t d q t f x s t 0 email protected a b on time scales in terms of the coefficients and the graininess function we illustrate our results with a nontrivial example

the application of chebyshev polynomials to the solution of the nonprismatic timoshenko beam vibration problem abstract chebyshev series approximation is applied to solve the problem of vibration of the nonprismatic timoshenko beam resting on a two parameter elastic foundation as a result a system of equations whose coefficients have a closed form for calculating the coefficients of the sought solution is obtained the method is used to solve the free vibration problem for simple supported and clamped free nonprismatic beams the results are compared with the results obtained by other authors also the nonprismatic beam stability problem is solved and the results are compared with those obtained for euler beams to demonstrate the method s applicability to more complex systems the problem of stability of a frame system is solved

a spectral tchebychev technique for solving linear and nonlinear beam equations abstract this paper presents a spectral tchebychev technique for solving linear and nonlinear beam problems the technique uses tchebychev polynomials as spatial basis functions and applies galerkin s method to obtain the spatially discretized equations of motion unlike alternative techniques that require different admissible functions for each different set of boundary conditions the spectral tchebychev technique incorporates the boundary conditions into the derivation and thereby enables the utilization of the solution for any linear boundary conditions without re derivation furthermore the proposed technique produces symmetric system matrices for self adjoint problems in this work the spectral tchebychev solutions for euler bernoulli and timoshenko beams are derived the convergence and accuracy characteristics of the spectral tchebychev technique is studied by solving eigenvalue problems with different boundary conditions it is found that the convergence is exponential and a small number of polynomials is sufficient to obtain machine precision accuracy the application of the technique is demonstrated by solving 1 eigenvalue problems for tapered timoshenko beams with different boundary conditions taper ratios and beam lengths 2 an euler bernoulli beam problem with spatially and temporally varying forcing elastic boundary and damping 3 large deflection nonlinear euler bernoulli beam problems with different boundary conditions and 4 a micro beam problem with nonlinear electrostatic excitation the results obtained from the spectral tchebychev solutions are seen to be in excellent agreement with those presented in the literature

monotone comparative statics the authors derive a necessary and sufficient condition for the solution set of an optimization problem to be monotonic in the parameters of the problem in addition they develop practical methods for checking the condition and demonstrate its applications to the classical theories of the competitive firm the monopolist the bertrand oligopolist consumer and growth theory game theory and general equilibrium analysis copyright 1994 by the econometric society

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

integration on time scales in this paper we study the process of riemann and lebesgue integration on time scales the relationship of the riemann and lebesgue integrals is considered and a criterion for riemann integrability is established

non conservative instability of a timoshenko beam resting on winkler elastic foundation abstract the transfer matrix method is used to investigate the influence of a winkler elastic foundation on the non conservative instability of uniform timoshenko beams it is found that the critical flutter load for a cantilever timoshenko beam subjected to an end concentrated or linearly distributed tangential follower force will first decrease as the elastic foundation modulus is increased and as the elastic foundation modulus increases beyond the corresponding critical value which corresponds to the lowest critical flutter load it will increase instead it is also observed that when the elastic foundation modulus is large enough the critical flutter load for a cantilever timoshenko beam can be greater than that of a bernoulli euler beam and there exists a corresponding critical turning point for the critical flutter load at this critical turning point the instability mechanism changes the instability mechanisms for a beam subjected to end concentrated and linearly distributed tangential follower force are different

vibration and stability analysis of non uniform timoshenko beams under axial and distributed tangential loads two sets of governing equations for transverse vibration of non uniform timoshenko beam subjected to both axial and tangential loads have been presented in the first set the axial and tangential loads were taken perpendicular to the shearing force i e normal to the cross section inclined at an angle while in the second set the axial force is assumed to be tangential to the axis of the beam column for each case there exist a pair of differential equations coupled in terms of the flexural displacement and the angle of rotation due to bending the two coupled second order governing differential equations were combined into one fourth order ordinary differential equation with variable coefficients the parameters of the frequency equation were determined for different boundary conditions the exact fundamental solutions could be found by expressing the coefficients of the reduced differential equation in a polynomial form before applying the frobenius method several illustrative examples of uniform and non uniform beams with various boundary conditions such as clamped supported elastically supported and free end mass and pinned end mass have been presented the stability analysis for the variation of the natural frequencies of the uniform and non uniform beams with the axial force has also been investigated moreover the present work illustrates the frequency behavior of the beam under a tangential load

ordinal one switch utility functions we study the problem of finding ordinal utility functions to rank the outcomes of a decision when preference independence between the attributes is not present we propose that the next level of complexity is to assume that preferences over one attribute can switch at most once as another attribute varies from low to high we refer to this property as ordinal one switch independence we present both necessary and sufficient conditions for this ordinal property to hold and provide families of functions that satisfy this new property

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

utility copula functions matching all boundary assessments the construction of a multiattribute utility function is an important step in decision analysis and can be a challenging task unless some decomposition of the utility function is performed when every attribute is utility independent of its complement the utility elicitation task is significantly simplified because the functional form of the utility function requires only one conditional utility function for each attribute and some normalizing constants when utility independence conditions do not hold the conditional utility function of an attribute may vary across the domain of the complement attributes and therefore a single conditional utility assessment for each attribute may not be sufficient to capture the decision maker s preferences this paper proposes a method to construct utility functions that have the flexibility to match the variations in the conditional utility function across the domain of the attributes using univariate utility assessments at the boundary values the approach incor

higher order dynamic equations on measure chains wronskians disconjugacy and interpolating families of functions this paper introduces generalized zeros and hence disconjugacy of nth order linear dynamic equations which cover simultaneously as special cases among others both differential equations and difference equations we also define markov fekete and descartes interpolating systems of functions the main result of this paper states that disconjugacy is equivalent to the existence of any of the above interpolating systems of solutions and that it is also equivalent to a certain factorization representation of the operator the results in this paper unify the corresponding theories of disconjugacy for nth order linear ordinary differential equations and for nth order linear difference equations

on nonuniform euler bernoulli and timoshenko beams with jump discontinuities application of distribution theory in this article bending of nonuniform euler bernoulli and timoshenko beams with jump discontinuities in the slope deflection and mechanical properties are studied the governing equations are obtained in the space of generalized functions and the expression of its governing differential equations in terms of a single displacement function and a single rotation function is shown always to be possible in contrast for a nonuniform euler bernoulli beam with jump discontinuities in slope and deflection and abrupt changes in flexural stiffness the governing equation can be written in terms of a single displacement function only under certain conditions it is observed that for most discontinuous nonuniform euler bernoulli beams we cannot write the governing differential equation in terms of a single displacement function usually if there are n discontinuity points on a nonuniform euler bernoulli beam n 1 displacement functions appear in the governing equilibrium equation

green s functions for uniform timoshenko beams abstract the vibration and control of distributed parameter systems is a topic of ongoing interest the availability of closed form expressions for the green s functions or equivalently transfer functions of individual elements facilitates the analysis of these systems in this paper a concise formulation is given for the green s functions of uniform timoshenko beams it is shown that green s functions for uniform euler bernoulli beams both with and without constant axial loads can be expressed in the same form