a new method for solving interval and fuzzy equations linear case a new approach to the solution of interval and fuzzy equations based on the generalized procedure of interval extension called interval extended zero method is proposed the central for the proposed approach is the treatment of the interval zero as an interval centered around zero it is shown that such proposition is not of heuristic nature but is the direct consequence of the interval subtraction operation some methodological problems concerned with this definition of interval zero are discussed it is shown that the resulting solution of interval linear equations based on the proposed method may be naturally treated as a fuzzy number an important advantage of a new method is that it substantially decreases the excess width effect on the other hand we show that it can be used as a reliable practical tool for solving linear interval and fuzzy equations as well as the systems of them the features of the method are illustrated by the example of the solution of the well known leontief input output problem in the interval setting

a family of fifth order runge kutta pairs the construction of a runge kutta pair of order 5 4 with the minimal number of stages requires the solution of a nonlinear system of 25 order conditions in 27 unknowns we define a new family of pairs which includes pairs using 6 function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step this is achieved by making use of kutta s simplifying assumption on the original system of the order conditions i e that all the internal nodes of a method contributing to the estimation of the endpoint solution provide at these nodes cost free second order approximations to the true solution of any differential equation in both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters optimal runge kutta pairs with respect to minimized truncation error coefficients maximal phase lag order and various stability characteristics are presented these pairs were selected under the assumption that they are used in local extrapolation mode the propagated solution of a problem is the one provided by the fifth order formula of the pair numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients instead of some other existing pairs

a new approach to normalization of interval and fuzzy weights a new approach to normalization of interval and fuzzy weights based on the so called interval extended zero method is proposed the three desirable intuitively obvious properties of the normalization procedure are defined the main of them is based on the assumption that the sum of normalized interval or fuzzy weights should be an interval or a fuzzy value centered around 1 with a minimal width the advantages of a new approach are illustrated with the use of numerical examples it is shown that a new approach performs better than known methods for normalization of interval and fuzzy weights as it provides the results with the properties which are close to the desirable ones

surface signatures an orientation independent free form surface representation scheme for the purpose of objects registration and matching this paper introduces anew free form surface representation scheme for the purpose of fast and accurate registration and matching accurate registration of surfaces is a common task in computer vision the proposed representation scheme captures the surface curvature information seen from certain points and produces images called surface signatures at these points matching signatures of different surfaces enables the recovery of the transformation parameters between these surfaces we propose using template matching to compare the signature images to enable partial matching another criterion the overlap ratio is used this representation scheme can be used as a global representation of the surface as well as a local one and performs near real time registration we show that the signature representation can be used to recover scaling transformation as well as matching objects in 3d scenes in the presence of clutter and occlusion applications presented include free form object matching multimodal medical volumes registration and dental teeth reconstruction from intraoral images

parametric representation of fuzzy numbers and application to fuzzy calculus we present several models to obtain simple parametric representations of the fuzzy numbers or intervals based on the use of piecewise monotonic functions of different forms the representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers we use the standard cuts setting but also the membership functions are obtained immediately and can be used directly to obtain error controlled approximations of the fuzzy calculus in terms of a finite set of parameters the general setting is the hermite type interpolation where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces and of the parameters by which the results are obtained the representations are designed to model the lower and the upper extremal values of each cut compact intervals of the fuzzy numbers and are able to produce almost any possible configuration differentiable continuous or with a finite number of discontinuity points by using parametric monotonic functions of different types we show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems including the numerical solution of fuzzy differential equations the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics the proposed model is called the lower upper representation and we denote the associated fuzzy numbers or intervals as lu fuzzy

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

neural networks with multidimensional transfer functions we present a new type of neural network nn where the data for the input layer are the value x spl epsiv r the vector y spl epsiv r sup m associated to an initial value problem ivp with y x f y x and a steplength h then the stages of a runge kutta rk method with trainable coefficients are used as hidden layers for the integration of the ivp using f as transfer function we take as output two estimations y y spl circ of ivp at the point x h training the rk method at some test problems and counting the cost of the method under the coefficients used we may achieve coefficients that help the method to perform better at a wider class of problems

explicit runge kutta pairs with one more derivative evaluation than the minimum existing classes of 4 5 5 6 and 6 7 pairs use the minimum number of derivative evaluations per step this ensures that the cost of each step is minimized however the cost per unit step is not necessarily minimized because it may be possible to take larger steps if an extra evaluation is used the change in efficiency when one extra evaluation per step is used is investigated for classes of 4 5 5 6 and 6 7 pairs with the minimum number of evaluations the aim is to show that using an extra evaluation does not necessarily increase the cost per unit step and that in some cases the cost may decrease the paper begins with the derivation of the new classes then the new and existing classes are compared using standard functions of the free parameters following this numerical comparisons of representative pairs from the new and existing classes are presented the new 4 5 and 5 6 pairs are found to be more efficient and just as reliable as the existing 4 5 and 5 6 pairs for the 6 7

a reconsideration of some embedded runge kutta formulae abstract the rk5 4 and rk6 5 embedded runge kutta formulae are reconsidered with regard to enlarging regions of absolute stability while retaining satisfactory truncation error norms results from standard tests for the above pairs are presented in comparison with an efficient rk8 7 embedded formula

a general family of explicit runge kutta pairs of orders 6 5 explicit runge kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations so far several runge kutta pairs of order 6 5 have appeared in the literature these pairs use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients depending on the use of the fsal first stage as last device these equations form a set of necessary and sufficient conditions that a 6 5 runge kutta pair must satisfy the solution of the latter is achieved by employing various types of simplifying assumptions in this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions using only these simplifying assumptions we define a new family of 6 5 runge kutta pairs its main characteristic which is also a property that no other known family shares is that all of its nodes except the last one which eq

a generalized borgonovo s importance measure for fuzzy input uncertainty in this work the way in which fuzzy uncertainty in a model s output is apportioned to fuzzy uncertainty in model inputs is studied through a sensitivity analysis here an optimization technique is employed to obtain the membership functions of the fuzzy structural response for which a global sensitivity indicator is introduced to measure the influence of fuzzy input uncertainty on fuzzy output uncertainty the global sensitivity indicator is the important measure of the fuzzy input uncertainty which extends borgonovo s measure in this study the mathematical properties of the important measure of the fuzzy input uncertainty are discussed and proved the results of numerical examples and engineering examples show that the proposed importance measure can effectively describe the effect of fuzzy input uncertainty on fuzzy structural response when the sensitivity indicator is larger the basic fuzzy valued variable becomes more important the sensitivity indicators of the fuzzy structural response can give an essential importance sequence of all the basic fuzzy valued variables and identify key contributing fuzzy valued variables the sensitivity indicators can provide the availability guidance to reduce the number of basic variables and optimize the fuzzy response model

importance analysis for model with mixed uncertainties in engineering sparse or imprecise data often leads to epistemic uncertainty about the distribution parameters of the aleatory input variables separating and estimating the individual contributions of the aleatory variability and epistemic parameter uncertainties to the uncertainty in the model output can assist in resource allocation for data collection as natural variability is irreducible whereas parameter uncertainty is reducible for structural systems involving aleatory inputs with epistemic distribution parameters a new kind of importance measure is proposed to distinguish and quantify the individual contributions of these two kinds of uncertainties in which probability distribution is used to describe the aleatory uncertainty and fuzzy membership function is employed to represent the epistemic distribution parameter the mathematical properties of the proposed importance measures are discussed and proved the defined importance measures are easy to apprehend and can evaluate the contributions of the aleatory variability and epistemic parameter uncertainties even when the information of the epistemic parameters is very sparse thus they can measure the importance of the two kinds of uncertainties more reasonably for efficiently estimating the proposed importance measures a monotonicity analysis is conducted for model function and the probability transformation process the results indicate that the extreme values of model function at each membership level can generally be obtained by combining the bounds of membership intervals of the fuzzy distribution parameters based on the monotonicity analysis an efficient algorithm is then formulated to compute the proposed importance measures several examples demonstrate the rationality and effectiveness of the proposed importance measures and the efficiency of the presented algorithm

importance analysis on the failure probability of the fuzzy and random system and its state dependent parameter solution for structure systems with fuzzy input uncertainty as well as random one the effects of the two kinds of uncertainties on the failure probability of the structure are studied and an importance analysis model is established to quantitatively evaluate these effects based on the fact that the fuzziness of the output is determined by that of the input the importance measure is defined to evaluate the effect of the fuzzy input variable for the random input variable the established model analyzes its importance from two aspects 1 its effect on the most plausible value of the failure probability 2 its effect on the imprecision of the failure probability in the process of computing the importance measures the conditional failure probability and unconditional one need to be evaluated which is time demanding for practical engineering problems for efficiently performing importance analysis in the presence of the fuzzy input variables and the random ones the importance sampling is method combined with the state dependent parameter sdp method is presented several examples show that the established importance analysis model can reflect the effects of the two kinds of input variables on the safety of the structure comprehensively and reasonably and the presented solution can improve computational efficiency considerably with acceptable precision

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

explicit runge kutta methods for starting integration of lane emden problem abstract traditionally when constructing explicit runge kutta methods we demand the satisfaction of the trivial simplifying assumption thus f 1 f x 0 y 0 is always used as the first stage of these methods when applied to the initial value problem ivp y x f x y y x 0 y 0 here we examine the case with f 1 f x 0 c 1 h y 0 h the step and c1 0 we derive the order conditions for arbitrary order and construct a 5th order method at the standard cost of six stages per step this method is found to outperform other classical runge kutta pairs with orders 5 4 when applied to problems with singularity at the beginning e g lane emden problem

global error estimation with runge kutta triples abstract the applications of runge kutta rk interpolation to global error estimation using the zadunaisky and related techniques are considered it is shown that the pseudo problem can be based on dense output values within any one step and reliable global error estimates can be obtained at the integration mesh points by using special rk formulae some special formulae of orders 2 6 are presented together with numerical results

on the reliability of the influence measure in the transformation method of fuzzy arithmetic the transformation method has been proposed as a practical tool for the simulation and the analysis of systems with uncertain parameters using fuzzy arithmetic a major advantage of this method among others is the fact that in the analysis part of the method the degree of influence for each uncertain parameter can be computed with a rather simple formula however a thorough examination of how accurate this measure of the degree of influence really is has not yet been performed in the paper the influence measure provided by the transformation method is compared to a rather classical analytical approach based on differential calculus furthermore an additional measure is introduced which can serve as an indicator for the effectiveness and the reliability of the influence measure calculated by the transformation method

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection

a new uncertainty analysis for the transformation method in this paper a new uncertainty analysis for the transformation method tm is proposed as a practical implementation of fuzzy arithmetic the tm is a convenient tool for the simulation and analysis of systems with uncertain parameters that are expressed by fuzzy numbers the proposed uncertainty analysis and the sensitivity analysis of the tm complete each other in providing some quantification of the relationship between the uncertainties of the system input and the system output the computation of gain factors is proposed which allows the estimation of the absolute and relative measures of uncertainty these measures allow the quantification of the influence of the uncertainty of the input on the uncertainty of the output