improving order and efficiency composition with a modified newton s method in this paper a zero finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented the key idea in deriving this procedure is to compose a given iterative method with a modified newton s method that introduces just one evaluation of the function to carry out this procedure some classical methods with different orders of convergence are used to obtain new methods that can be generalized in banach spaces

brief sliding mode control for two time scale systems stability issues the paper presents some results on global exponential stability of linear time invariant systems with different time scales the full system is decomposed in two reduced ones by means of singular perturbations and a smooth approximation of a variable structure control is synthesized for each of them using reaching law approach the global closed loop stability is then proved for the whole system using lyapunov methods and a particular state space decomposition moreover a literal form for e a parameter which measures the minimum separation between time scales is derived

modified chebyshev halley methods with sixth order convergence abstract in this paper we present a family of modified chebyshev halley methods the new methods add one evaluation of the function at the point iterated by chebyshev halley methods analysis of convergence shows that these methods have sixth order convergence numerical results show that the new methods can be of practical interest

wild pfister forms over henselian fields k theory and conic division algebras the epicenter of this paper concerns pfister quadratic forms over a field f with a henselian discrete valuation all characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but f does not we also prove results about round quadratic forms composition algebras generalizations of composition algebras we call conic algebras and central simple associative symbol algebras finally we give relationships between these objects and kato s filtration on the milnor k groups of f

improvements of the efficiency of some three step iterative like newton methods an improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative the third and fourth order of known two step like newton methods have been improved and the efficiency has also been increased

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

symmetric bilinear forms quadratic forms and milnor k theory in characteristic two is bijective for any field f of characteristic two and for any n 0 here k f is the k group of milnor defined in 11 and i is the unique maximal ideal of the witt ring w f of non degenerate symmetric bilinear forms over f on the other hand sah studied quadratic forms in characteristic two in 15 let f be a field of characteristic two wq f the witt group of non degenerate quadratic forms over f and define i wq f n o by the canonical w f module structure on wq f then he proved that wq f iwq f is isomorphic to the group of quadratic cyclic extensions of f and iwq f i2wq f is isomorphic to the subgroup br f 2 of the brauer group br f of f consisting of elements annihilated by 2 in this paper we prove the above milnor s conjecture and describe the precise structures of w f and wq f generalizing the result of sah for any field f of characteristic p 0 let be the n th exterior power over f of the absolute differential module n t g21 let d 2 1 be the f z f fp image of the exterior derivation d f2 t f2 and let v n v resp h 1 f be the kernel resp cokernel of the homomorphism

semilocal convergence of the secant method under mild convergence conditions of differentiability abstract in this work we obtain a semilocal convergence result for the secant method in banach spaces under mild convergence conditions we consider a condition for divided differences which generalizes those usual ones i e lipschitz continuous and holder continuous conditions also we obtain a result for uniqueness of solutions

variable structure control of synchronous generator singularly perturbed analysis synchronous generators have a natural different time scale dynamics that is why for modelling and control design in such systems the methods of singular perturbations are widely used in this paper the possibilities of sliding mode control design for synchronous generators are analysed with this aim the concept of singular perturbation is revised in order to use it for relay control systems the obtained results are used for sliding mode control of synchronous generator

quadratic forms and pfister neighbors in characteristic 2 we study pfister neighbors and their characterization over fields of characteristic 2 where we include the case of singular forms we give a somewhat simplified proof of a theorem of fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a pfister form in terms of the hyperbolicity of this form over the function field of a form which is dominated by q from this we derive an analogue in characteristic 2 of a result by knebusch saying that in characteristic 2 a form is a pfister neighbor if its anisotropic part over its own function field is defined over the base field our result includes certain cases of singular forms but we also give examples which show that knebusch s result generally fails in characteristic 2 for singular forms as an application we characterize certain forms of height 1 in the sense of knebusch whose quasi linear parts are of small dimension we also develop some of the basics of a theory of totally singular quadratic forms this is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper

a two step steffensen s method under modified convergence conditions a simplification of a third order iterative method is proposed the main advantage of this method is that it does not need to evaluate neither any frechet derivative nor any bilinear operator a semilocal convergence theorem in banach spaces under modified kantorovich conditions is analyzed a local convergence analysis is also performed finally some numerical results are presented

a family of fifth order iterations composed of newton and third order methods abstract in this paper we present a family of fifth order methods formed by the composition of newton and third order methods for solving non linear equations per iteration the new methods require two evaluations of the function one of its first derivative and one of its second derivative analysis of efficiency shows that the new methods can compete with newton s method and the classical third order methods numerical results show that the methods have definite practical utility

shells of twisted flag varieties and the rost invariant we introduce two new general methods to compute the chow motives of twisted flag varieties and settle a 20 year old conjecture of markus rost about the rost invariant for groups of type e 7

convergence and numerical analysis of a family of two step steffensen s methods we provide sufficient conditions for the semilocal convergence of a family of two step steffensen s iterative methods on a banach space the main advantage of this family is that it does not need to evaluate neither any frechet derivative nor any bilinear operator but having a high speed of convergence some numerical experiments are also presented

on totally decomposable algebras with involution in characteristic two abstract a necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution in terms of its frobenius subalgebras is given it is also proved that a bilinear pfister form recently introduced by a dolphin can classify totally decomposable central simple algebras of orthogonal type

springer s theorem for tame quadratic forms over henselian fields a quadratic form over a henselian valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension the witt group of tame quadratic forms is shown to be canonically isomorphic to the witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation hence also isomorphic to a direct sum of copies of the witt group of the residue field indexed by the value group modulo 2

variable structure control for singularly perturbed linear continuous systems with matched disturbances a variable structure system can be studied using the singular perturbation theory the discontinuous control that leads to a finite time reaching of the sliding surface creates fast time transients analogous to the stable boundary layer dynamics of a singularly perturbed system as the sliding mode is attained the slow time dynamics prevails just as that of a singularly perturbed system after the boundary layer dynamics fades away in this technical note the problem of sliding mode control for singularly perturbed systems in the presence of matched bounded external disturbances is investigated a composite sliding surface is constructed from solutions of algebraic lyapunov equations which are derived from both the fast and the slow subsystems the resultant sliding motion ensures lyapunov stability with disturbance rejection two proposed schemes that ensure the asymptotic stability of the system are presented the effectiveness of the proposed methods is demonstrated in a numerical example of a magnetic tape control system

third order iterative methods under kantorovich conditions we study a class of third order iterative methods for nonlinear equations on banach spaces a characterization of the convergence under kantorovich type conditions and optimal estimates of the error are found though in general these methods are not very extended due to their computational costs we will show some examples in which they are competitive and even cheaper than other simpler methods we center our analysis in both analytic and computational aspects

sliding mode control for singularly perturbed systems this paper addresses the design of sliding mode controllers for singularly perturbed systems the full order system is separated into slow and fast subsystems and a sliding mode controller is designed for each subsystem a composite of these controls is then implemented on the full order system it is shown that if specific reaching conditions are satisfied for the reduced order models and an additional condition is met then the reaching conditions are also satisfied for the full order model furthermore sufficient conditions are derived which guarantee that the approximation error between the controlled reduced order response and the controlled full order model response is small a numerical example is given which successfully demonstrates the techniques developed in this paper

the convergence ball of the secant method under holder continuous divided differences under the hypothesis that nonlinear operators have holder continuous divided differences of order one an estimate of the radius of the convergence ball of the secant method is obtained error estimate is also established

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

a class of quasi newton generalized steffensen methods on banach spaces we consider a class of generalized steffensen iterations procedure for solving nonlinear equations on banach spaces without any derivative we establish the convergence under the kantarovich ostrowski s conditions the majorizing sequence will be a newton s type sequence thus the convergence can have better properties finally a numerical comparation with the classical methods is presented