modeling identification and control of a class of nonlinear systems in this paper we propose a new fuzzy hyperbolic model for a class of complex systems which is difficult to model the fuzzy hyperbolic model is a nonlinear model in nature and can be easily derived from a set of fuzzy rules it can also be seen as a feedforward neural network model and so we can identify the model parameters by bp algorithm we prove that the stable controller can be designed based on linear system theory two methods of designing the controller for the fuzzy hyperbolic model are proposed the results of simulation support the effectiveness of the model and the control scheme

genus 4 trigonal reduction of the benney equations it was shown by gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 that n parameter reductions of the benney equations correspond to particular n parameter families of conformal maps in recent papers baldwin and gibbons 2003 j phys a math gen 36 8393 417 baldwin and gibbons 2004 j phys a math gen 37 5341 54 the present authors have constructed examples of such reductions where the mappings take the upper half p plane to a polygonal slit domain in the plane in those cases the mapping function was expressed in terms of the derivatives of kleinian functions of hyperelliptic curves restricted to the one dimensional stratum 1 of the divisor this was done using an extension of the method given in enolskii et al 2003 j nonlinear sci 13 157 extended to a genus 3 curve enolski v z and gibbons j addition theorems on the strata of the theta divisor of genus three hyperelliptic curves in preparation here we use similar ideas but now applied to a trigonal curve of genus 4 fundamental to this approach is a family of differential relations which satisfies on the divisor again it is shown that the mapping function is expressible in terms of quotients of derivatives of on the divisor 1 one significant by product is an expansion of the leading terms of the taylor series of for the given family of 3 5 curves to the best of the authors knowledge this is new

wave solutions of evolution equations and hamiltonian flows on nonlinear subvarieties of generalized jacobians the algebraic geometric approach is extended to study evolution equations associated with the energy dependent schr dinger operators having potentials with poles in the spectral parameter in connection with hamiltonian flows on nonlinear subvarieties of jacobi varieties the general approach is demonstrated by using new parametrizations for constructing quasi periodic solutions of the shallow water and dym type equations in terms of theta functions a qualitative description of real valued solutions is provided

rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

parallel solution of symmetric positive definite systems with hyperbolic rotations abstract an algorithm based on hyperbolic rotations is presented for the solution of linear systems of equations ax b with symmetric positive definite coefficient matrix a forward elimination and backsubstitution are replaced by matrix vector multiplications rendering the method amenable to implementation on a variety of parallel and vector machines this method can be simplified and formulated without square roots if a is also toeplitz a systolic vlsi architecture implementing the resulting recurrence equations is more efficient than previously proposed pipelined toeplitz system solvers the hardware count becomes independent of the matrix size if its inverse is banded

decay rates of inverses of banded m matrices that are near to toeplitz matrices the decaying behavior of inverses of positive definite band matrices is analysed for m matrices that are in some sense close to toeplitz matrices estimates based on the factorization are derived that are better than existing ones in particular for nonsymmetric matrices some examples are given

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

higher genus hyperelliptic reductions of the benney equations it was shown by gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 that n parameter reductions of the benney equations correspond to n parameter families of conformal maps here we consider a specific set of these the hyperelliptic reductions the mapping function for this is calculated explicitly by inverting a second kind abelian integral on the stratum 1 of the jacobi variety of a genus g g 3 hyperelliptic curve this is done using a method based on the result of jorgenson 1992 isr j math 77 273

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

toeplitz matrices with banded inverses abstract given a toeplitz matrix t with banded inverse i e t 1 ij 0 for j i p we show that the elements of t can be expressed in terms of the roots of a polynomial then using properties we have previously established we generalize this result appropriately to allow singular t and show that the converse also holds finally we give a sufficient condition for the decay of the elements of t as one moves away from the diagonal

relationship between the prime form and the sigma function for some cyclic r s curves in this paper we study some cyclic r s curves x given by we give an expression for the prime form where p q x in terms of the sigma function for some such curves specifically any hyperelliptic curve r s 2 2g 1 as well as the cyclic trigonal curve r s 3 4 where r is a certain multi index of differentials here u1 and v1 are respectively the first components of u w p and v w q which are given by the abel map where g is the genus of x these explicit formulae are useful in applications for instance to the problem of constructing classes of schwarz christoffel maps to slit domains

a genus six cyclic tetragonal reduction of the benney equations a reduction of benney s equations is constructed corresponding to schwartz christoffel maps associated with a family of genus six cyclic tetragonal curves the mapping function a second kind abelian integral on the associated riemann surface is constructed explicitly as a rational expression in derivatives of the kleinian function of the curve

on axiomatic characterisations of crisp approximation operators abstract in rough set theory one can find the result that the upper approximation defines a closure operator on the power set of the universe fixed in the paper presented first we have solved the problem under which additional assumptions a given closure operator can be represented by an upper approximation within the concepts of rough set theory second we have solved this problem within modal logic for a diamond operator represented by an arbitrary binary relation on the universe finally we have generalized these problems to lower approximations and modal box operators to rough approximations of fuzzy sets rough fuzzy sets to fuzzy approximations of crisp sets fuzzy rough sets and fuzzy approximations of fuzzy sets fuzzy diamond and fuzzy box

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

the vector nonlinear schrodinger hierarchy abstract an article published recently by one of the authors jne presents closed form solutions for zero curvature representations of the vector nonlinear schrodinger hierarchy several of these results are confirmed computationally but left unproven in this article we begin by providing strict algebraic proofs of these results the forms of the hierarchy s associated spectral curves are investigated and proven to have only odd genus without the introduction of integration constants we admit specific non zero constants of integration to the hierarchy and show that even genus curves may be introduced in a very straightforward manner thereby accessing the full family of finite gap solutions to the vector nonlinear schrodinger equation finally an original construction of the infinite set of conserved densities of the hierarchy is given

hamiltonian structures of reductions of the benney system we show how to construct the hamiltonian structures of any reduction of the benney chain dispersionless kp the construction follows the scheme suggested by ferapontov leading in general to nonlocal hamiltonian structures in some special cases these reduce to local structures all the geometric objects which define the poisson bracket the metric connection and riemann curvature are obtained explicitly in terms of the n parameter family of conformal maps associated with the reduction

integrable discretization of hodograph type systems hyperelliptic integrals and whitham equations based on the well established theory of discrete conjugate nets in discrete differential geometry we propose and examine discrete analogues of important objects and notions in the theory of semi h

abelian functions associated with a cyclic tetragonal curve of genus six we develop the theory of abelian functions defined using a tetragonal curve of genus six discussing in detail the cyclic curve y 4 x 5 4x 4 3x 3 2x 2 1x 0 we construct abelian functions using the multivariate function associated with the curve generalizing the theory of the weierstrass function we demonstrate that such functions can give a solution to the kp equation outlining how a general class of solutions could be generated using a wider class of curves we also present the associated partial differential equations satisfied by the functions the solution of the jacobi inversion problem a power series expansion for u and a new addition formula

genus n algebraic reductions of the benney hierarchy within a schottky model by exploiting a new theoretical connection between reductions of the benney hierarchy and the dirichlet problem for laplace s equation the solution to a spectral problem associated with n parameter algebraic reductions of the benney hierarchy is found explicitly the solutions can be written in terms of the modified green s function associated with reflectionally symmetric n connected planar domains whose holes are all centred on the symmetry axis explicit formulae for the modified green s function in a canonical class of circular domains are constructed using a schottky model of the schottky double of these domains uniformizations of the spectral problem associated with two different types of reductions then follow from these formulae

hyperelliptic reduction of the benney moment equations we consider n parameter reductions of the benney moment equations these were shown in gibbons and tsarev 1996 phys lett a 211 19 1999 phys lett a 258 263 to correspond to n parameter families of conformal maps and to satisfy a particular system of pde a specific known example of this the n 2 elliptic reduction l yu and j gibbons 2000 inverse probl 16 605 is described we then consider an analogous reduction for a genus 2 hyperelliptic curve n 3 the mapping function is given by the inversion of a second kind abelian integral on the divisor this is found explicitly following a method given by enolskii et al 2003 j nonlinear sci 13 157

characterisation and properties of r toeplitz matrices abstract an r toeplitz rt matrix can be regarded as a block toeplitz matrix with order some multiple of r from which a bottom border and a right hand border have been removed it is shown that some known properties of toeplitz matrices extend to this new class in particular a necessary and sufficient condition is given in terms of a linear matrix equation for a nonsingular matrix to have rt form from which it follows that the elements of the inverse are completely determined by those in its first and last r rows and columns in addition some results are presented for cases when the inverse of an rt matrix has certain banded forms

two new operators in rough set theory with applications to fuzzy sets in this paper two new operators are introduced for the rough set theory using them two inequalities well known in the rough set theory can now be modified to become equalities with this change no information will be lost in the new expressions hence many properties in rough set theory can be improved and in particular the union the intersection and the complement operations can be redefined based on the two equalities furthermore the collection of rough sets of an approximation space forms a boolean algebra under these new operators finally roughness properties of fuzzy sets are analyzed using the new operations

a roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

rough fuzzy functions in classification this paper generalizes the concept of rough membership functions in pattern classification tasks to rough fuzzy membership functions and rough fuzzy ownership functions unlike the rough membership value of a pattern which is sensitive only towards the rough uncertainty associated with the pattern the rough fuzzy membership or ownership value of the pattern signifies the rough uncertainty as well as the fuzzy uncertainty associated with the pattern in this paper various set theoretic properties of the rough fuzzy functions are exploited to characterize the concept of rough fuzzy sets these properties are also used to measure the rough fuzzy uncertainty associated with the given output class finally a few possible applications of the rough fuzzy functions are mentioned

explicit and asymptotic formulas for ldmt factorization of banded toeplitz matrices abstract we study the ldmt factorization of banded toeplitz matrices we show that under fairly general conditions the l m and d factors exhibit exponential convergence along diagonals we also give explicit formulas for the limiting values our method of proof involves a generalization to matrices of the theory of continued fractions of quadratic irrationals