explicit solutions to the singular discrete finite time linear estimation problem the problem of optimal filtering of linear discrete time processes is considered it is assumed that the driving and measurement noise signals are stationary white and that the number of measurements is finite explicit expressions for the filtering error covariance and the corresponding kalman gain matrices are obtained for the limiting case where the measurement noise intensity tends uniformly to zero similar expressions are derived for the limiting singular smoothing problem with finite number of measurements

on the solution of strong nonlinear oscillators by applying a rational elliptic balance method a rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using jacobi elliptic functions to illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators we first investigate the exact solution of the non homogenous undamped duffing equation then we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators at the end of the paper we compare the numerical integration values of the angular frequencies with approximate solution results based on the proposed rational elliptic balance method

solution of a mixed parity nonlinear oscillator harmonic balance abstract the method of harmonic balance is used to calculate first order approximations to the periodic solutions of a mixed parity nonlinear oscillator first the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction then the two auxiliary equations where the restoring force functions are odd are solved by using a first harmonic term without a constant therefore we obtain the two approximate solutions to the mixed parity nonlinear oscillator one is expressed in terms of the exact amplitude in the negative direction the other in terms of the approximate amplitude these solutions are more accurate than the second approximate solution obtained by the lindstedt poincare method for large amplitudes

a generalized harmonic function perturbation method for determining limit cycles and homoclinic orbits of helmholtz duffing oscillator abstract in this paper a novel description of periodic solution and homoclinic orbit of undamped helmholtz duffing oscillator is proposed via nonlinear time transformation based on this novel description a generalized harmonic function perturbation method is presented to determine the limit cycles and homoclinic orbits of helmholtz duffing oscillator with nonlinear damping the amplitude of limit cycle and critical value of the homoclinic bifurcation parameter can be also predicted to illustrate the accuracy of the present method the solutions obtained in this paper are compared with those of runge kutta method which shows the method proposed in this paper is effective and feasible

on r l o we prove there exists a somewhat simpler description of l and its fuzzy topology than the one given by hutton 2 and a considerably simpler description of than the one given by rodabaugh 4 as a result of this simplification we are able to give a direct proof of the continuity of an open problem in 4

on fuzzy pseudo normed vector spaces abstract we provide a method for introducing fuzzy pseudo metric topologies on sets and fuzzy pseudo normed topologies on vector spaces over r or c which will be fuzzy linear topologies we define fuzzy pseudo metrics for pairs of crisp points and fuzzy pseudo norms for crisp points as fuzzy real numbers 0 as defined by hutton we define the associated fuzzy open balls with crisp points for their centres and fuzzy real numbers 0 for their radii these form a basis for the associated fuzzy pseudo metric topology the axioms governing fuzzy pseudo metric and fuzzy pseudo norm are straightforward extensions for the corresponding axioms in the crisp case the formulation conforms with zadeh s extension principle we show that katsaras concept of fuzzy seminorm fuzzy sets and systems 12 1984 143 154 is equivalent to ours in the sense that both concepts will result in same fuzzy linear topologies

on fuzzy neighborhood rings abstract we introduce the concept of fuzzy neighborhood rings we give necessary and sufficient conditions for a ring structure and a fuzzy neighborhood system to be compatible we also give necessary and sufficient conditions for a prefilter to be a neighborhood prefilter of 0 in a fuzzy neighborhood rings the notion of bounded fuzzy set is introduced in fuzzy neighborhood rings and some results in this context are provided

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

criterion for the convergence of the solution of the riccati differential equation the optimal control problem for a linear system with a quadratic cost function leads to the matrix riccati differential equation the convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix a necessary and sufficient condition for convergence is derived for stabilizable systems even if the output in the cost function is not detectable an algorithm is developed to determine the limiting value of the solution which is one of the symmetric positive semidefinite solutions of the algebraic riccati equation examples for convergence and nonconvergence are given a discussion is also included of the convergence properties of the solution of the riccati differential equation to any real symmetric not necessarily positive semidefinite solution of the algebraic riccati equation

analysis of a beam column system under varying axial forces of elliptic type the exact solution of lame s equation abstract we investigate the dynamical response of a beam column system with hinged ends subjected to an axial pulsating force of elliptic type it is shown that the resulting equation is of the form d 2 y 1 d 2 y 1 d 1 d 2 cn 2 k 2 0 which is the well known lame equation higher transcendental functions bateman manuscript project edited by mcgraw hill new york vol 3 in this paper we obtain the general exact solution of this equation that reveals stable behavior of the beam column system if the assigned initial conditions are of the form y1 0 y10 and y 1 0 0 it is also found that at a certain value of the modulus of the elliptic force the lateral vibrational frequency is independent of the material properties of the beam column system

an elliptic perturbation method for certain strongly non linear oscillators abstract an elliptic perturbation method is presented for calculating periodic solutions of strongly non linear oscillators of the formx c1x c3x3 f x x in which the jacobian elliptic functions are employed instead of usual circular functions in the conventional perturbation procedure three types of duffing van der pol equation withf x x 1 x2 xare studied in detail the present method can give a second approximate solution and therefore its accuracy is much higher than that of other methods in which the solutions are only first approximations

a new method for evaluating the log likelihood gradient the hessian and the fisher information matrix for linear dynamic systems a method is presented for evaluating the log likelihood gradient score the hessian and the fisher information matrix of the parameters of linear dynamic stochastic systems the method incorporates the optimal kalman smoothing equations and is therefore ideal for simultaneous state estimation and parameter identification the result can be used for efficient implementation of gradient based algorithms for maximum likelihood identification of the unknown system parameters and for assessing the mean square estimation accuracy

exact solution of the quadratic mixed parity helmholtz duffing oscillator abstract in this paper the exact solution of the quadratic mixed parity helmholtz duffing oscillator is derived by using jacobi elliptic functions it is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind at the end of the paper we examine the stability of the system and determine the regions for which periodic and unbounded motions take place

solution of a quadratic nonlinear oscillator by the method of harmonic balance abstract this paper deals with the quadratic nonlinear oscillator qno x x e x 2 0 this oscillator is compared with the nonlinear oscillators x x e x x 0 which are solved by the method of harmonic balance therefore we obtain the first approximate solution to the qno this solution is more accurate than the second approximate solution obtained by the lindstedt poincare method for qnos it has been pointed out that the angular frequency for x 0 should be different from the angular frequency for x 0 if solutions of higher degree of accuracy are required

explicit solutions to the linear optimal estimation problem of discrete time invariant linear processes the linear optimal estimation problem of linear discrete time invariant processes is considered using a mixed z and time domain approach explicit expressions am obtained in the case where the measurement record is infinitely long for the constant kalman gain and the minimum error covariance matrix of the filtered estimate in terms of the zero structure of the measurement power spectrum density matrix these expressions are simple to obtain both conceptually and computationally and they provide a geometric insight to the structure of the solution they are also applied to the filtering problems with finite number of measurements and to the optimal discrete smoothing problems where similar explicit results are obtained

errata complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines abstract we construct a fuzzy multiplication for the case l a chain in the l fuzzy real line r l which is jointly fuzzy continuous and which is the unique piecewise consistent extension of ordinary multiplication to r l distributing over the fuzzy addition constructed by the author in 47 we also specify the fuzzy integers and give an extensive analysis of fuzzy contractions and expansions these results the constructions of this paper and the results of 47 are used to prove r l and the stratified r c l are nontopologically isomorphic complete fuzzy topological hyperfields if l is a chain with l 3 we can now completely answer for l a chain the following question of 27 in the affirmative can a set of fuzzy numbers its fuzzy topology and its fuzzy addition and multiplication be defined such that these operations are jointly fuzzy continuous extensions of the usual addition and multiplication of r and can we specify the fuzzy integers these results in combination with recent papers of several authors would seem to answer in the affirmative the following philosophical question cf 34 does fuzzy topology have deep specific canonical examples

continuity of fuzzy multiplication in the n euclidean space abstract the n euclidean space m i is a fuzzy normed vector space whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers the fuzzy multiplication on the fuzzy real line easily extends to a multiplication on m i we show that under that multiplication m i is a fuzzy normed algebra in consequence fuzzy multiplication is shown to be continuous in the n euclidean topology on m i

compact probabilistic metrics on bounded closed intervals of distribution functions a menger space a special type of probabilistic metric spaces is said to be compact if its strong uniformity is compact we construct in a natural way menger t metrics on the set of distribution functions one for each copula t a special type of continuous t norms we show that on each bounded closed interval of distribution functions the strong uniformities of all our spaces are induced by the modified levy metric hence they are compact we establish an alternative description and a number of good properties of our menger spaces which may render them well behaved examples for workers in the field

fuzzy inner product spaces abstract the n euclidean space m i is a fuzzy pseudo normed algebra whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers we introduce axioms for a fuzzy inner product on a unitary m i module with values in m i we establish the fuzzy schwarz inequality for fuzzy inner products we introduce the cartesian product of fuzzy inner product spaces and investigate the relations between their underlying topologies we conclude the research by supplying two interesting examples of fuzzy inner product spaces

completeness of the standard fuzzy real line for fns abstract we consider a special class of fuzzy metric neighbourhood spaces where the fuzzy distance is induced by a fuzzy pseudo norm this class is contained in lowen s well known category fns of fuzzy neighbourhood spaces we define the fuzzy limit of a sequence a fuzzy cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces we prove that the set of all nonnegative fuzzy real numbers r i under the n euclidean metric the standard fuzzy real line for fns is complete but that the smallest fuzzy neighbourhood real algebra m i including r i is not complete we also characterize the fuzzy cauchy sequences in m i in terms of their corresponding real sequences of cuts

harmonic balance and iteration calculations of periodic solutions to y y 1 0 we use two methods harmonic balance and iteration to calculate analytical approximations to the periodic solutions of a nonlinear singular oscillator

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

application of the elliptic balance method to a nonlinear singular oscillator abstract new analytical approximate solutions based on jacobi elliptic functions are formulated to compute the angular frequency value of a nonlinear singular oscillator it is shown that when our derived analytical angular frequency solutions are compared to the exact angular frequency value the third order elliptic balance solution provides the best estimate to the exact value since the percentage error value is about 0 0059 which is 216 times lower than the best estimate percentage error value reported in the literature

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

exact solution of a quadratic nonlinear oscillator abstract the exact solution of a quadratic nonlinear oscillator qno is presented by using the jacobian elliptic sine function the constants in the solution are expressed in terms of the initial conditions the formulas for the period of oscillation t and amplitude b for x 0 are also given this exact solution can serve as a benchmark solution for various approximate solutions it also can be utilized for educational purpose

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism

fixed points in fuzzy metric spaces abstract in this note the well known fixed point theorems of banach and edelstein are extended to fuzzy metric spaces in the settle of kramosil and michalek