a remark on regularity criterion for the dissipative quasi geostrophic equations abstract this paper concerns with a regularity criterion of solutions to the 2d dissipative quasi geostrophic equations based on a logarithmic sobolev inequality in besov spaces the absence of singularities of in 0 t is derived for a solution on the interval 0 t satisfying the condition l r 0 t b p 0 for 2 p r 4 p this is an extension of earlier regularity results in the serrin s type space l r 0 t l p

a multiple valued reed muller transform for incompletely specified functions this paper considers the reed muller transform for incompletely specified multiple valued logic functions which is obtained as the finite field polynomial representation a new algorithm for dealing with single variable functions is presented it is applicable to finite fields of small sizes two to four which is of interest because only these fields are readily implementable with today s mvl technology it is shown how such or any similar single variable algorithm can be used to obtain a fast n variable reed muller transform based on this transform a heuristic scheme is derived for dealing with incompletely specified functions it has better computational properties than other methods and achieves the best results when applied to functions with a large number of unspecified points

brezis gallouet wainger type inequalities and blow up criteria for navier stokes equations in unbounded domains we shall find the weakest norm that satisfies the brezis gallouet wainger type inequality under some conditions as an application of the brezis gallouet wainger type inequality we shall establish beale kato majda type blow up criteria of smooth solutions to the 3 d navier stokes equations in unbounded domains

an extension of the beale kato majda criterion for the euler equations the beale kato majda criterion asserts that smooth solutions to the euler equations remain bounded past t as long as is finite ohgr being the vorticity we show how to replace this by a weaker statement on where j is a frequency localization around

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3d nse the first part of the paper provides spatio temporal localization of a family of analytic regularity classes for the 3d nse obtained by beirao da veiga space time integrability of the gradient of the velocity on mathbb r 3 times 0 t which is out of the range of the sobolev embedding theorem reduction to the classical foias ladyzhenskaya prodi serrin space time integrability conditions on the velocity as well as the localization of the beale kato majda regularity criterion time integrability of the l norm of the vorticity the second part introduces a family of local scaling invariant hybrid geometric analytic classes in which coherence of the vorticity direction serves as a weight in the local spatio temporal integrability of the vorticity magnitude

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

a method for modulo 2 minimization a minimization approach is presented for a modulo 2 expansion of a switching function this method uses mixed polarity representation for the variables a local optimization procedure is presented which requires modest computation and is easily expanded the procedure applied to the 32 des select functions of six variables is reported

extension criterion via partial components of vorticity on strong solutions to the navier stokes equations in higher dimensions abstract we consider the extension criterion of strong solutions to the navier stokes equations in r n it is proved that among n n 1 2 components of the vorticity n 2 components are negligible for the criterion whether the time local solutions can be extended beyond the critical time our result may be regarded as generalization to the higher dimensional case of chae choe 4 in the 3d case which showed that only two components in l q 3 2 q of the vorticity contribute to such an extension criterion furthermore the critical case q originally treated by kato ponce 8 in r n is also generalized in such a way that n 2 components of vortex matrix are redundant for the extension criterion

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

regularity criterion for a bona colin lannes system abstract we prove a regularity criterion of the strong solutions for a bona colin lannes system in besov space as a byproduct we show the existence of globally smooth solutions for a symmetric bona colin lannes system

navier stokes equations with vorticity in besov spaces of negative regular indices abstract this paper studies the cauchy problem for the three dimensional navier stokes equations and shows that the condition u l 2 2 r 0 t b r 0 r 2 ensures the regularity of the solution on 0 t this improves and extends many previous results

existence theorem and blow up criterion of the strong solutions to the two fluid mhd equation in r3 abstract we first give the local well posedness of strong solutions to the cauchy problem of the 3d two fluid mhd equations and then study the blow up criterion of the strong solutions by means of the fourier frequency localization and bony s paraproduct decomposition it is proved that the strong solution u b can be extended after t t if either u l t q b p 0 with 2 q 3 p 1 and b l t 1 b 0 or j l t q b p 0 with 2 q 3 p 2 where t u denotes the vorticity of the velocity and j b stands for the current density

components reduction regularity results for the navier stokes equations in general dimensions abstract we consider the cauchy problem of the navier stokes equations in arbitrary dimensions and establish several new components reduction regularity criteria

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

on the euler equations in the critical triebel lizorkin spaces in this paper we study the initial value problem of the incompressible euler equations in n for initial data belonging to the critical triebel lizorkin spaces i e v0 fn 11 q q 1 we prove the blow up criterion of solutions in fn 11 q for n 2 3 for n 2 in particular we prove global well posedness of the euler equations in f31 q q 1 for the proof of these results we establish a sharp moser type inequality as well as a commutator type estimate in these spaces the key methods are the littlewood paley decomposition and the paradifferential calculus by j m bony

on the youla kucera parametrization for nonlinear systems abstract we consider the problem of stabilizing a nonlinear plant through the use of a possibly unstable pre compensator and a stable feedback compensator and parameterizing a class of such stabilizers in terms of a bibo stable map q in the linear case by means of superposition the pre and feedback compensator can be combined in such a way that the bibo map q occurs in just one feedback loop within the controller feeding back output residuals prediction errors to the control inputs the main contribution of this paper is to develop a nonlinear generalization of this property building on earlier work in forming the youla kucera parametrization for nonlinear systems we show the equivalence of the class of all bounded input stabilizing nonlinear pre and feedback compensators to a class of possibly unstable feedback controllers in which a map q s is present in only the one feedback loop we then show that necessary and sufficient condition to achieve stability of the system is that q s be bibo stable one advantage of the new formulation is that differential boundedness assumptions do not involve the parametrization q s in any way just as the linear versions of our results have applications in the areas of optimal control and adaptive control of linear systems it is conjectured that the present results will underlie more general results for adaptive control and nonlinear systems

internally stable nonlinear systems with disturbances a parameterization this paper deals with the control of a nonlinear system whose output is subjected to an additive disturbance the main result is a simple parameterization of the set of all system responses that can be obtained through internally stable control of the given system the parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system the class of achievable responses is determined by the numerator of a right coprime fraction representation of the system being controlled

generalized partially mixed polarity reed muller expansion and its fast computation generalized partially mixed polarity reed muller gpmprm expansion a canonical subfamily of exclusive sum of products esop is presented an efficient algorithm in two dimensional data flow is proposed for computation of the gpmprm forms mcnc benchmark experimental results show that the minimal gpmprm forms of these functions on the average have similar number of terms to their sum of products sop counterparts while there are many functions for which the gpmprm circuits are much smaller

two regularity criteria for the 3d mhd equations this work establishes two regularity criteria for the 3d incompressible mhd equations the first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction

coprime fractional representations and stability of non linear feedback systems fractional representation methods for finite dimensional linear time invariant fdlti systems have been developed in an algebraic framework of principal ideal domains the concept of coprime fractional representations has been generalized to discrete time non linear systems by hammer 1985 1987 by viewing the bezout identity as the basis of the definition of coprimeness in this paper we view the role of right coprime fractional representation in representing the set of all stable input output pairs of a possibly unstable non linear system this notion of coprimeness is shown to be equivalent to that of fdlti systems using coprime fractional representations of non linear systems we obtain a characterization of the stability of non linear feedback systems this characterization is similar to that obtained in the case of fdlti systems we also obtain a parameterization of all non linear and time varying stabilizing controllers for linear possibly time varying plants which has the same form as t

regularity criteria for a lagrangian averaged magnetohydrodynamic α model abstract in this paper we study the n dimensional lagrangian averaged magnetohydrodynamic lamhd model with 5 n 7 throughout the whole space various regularity criteria are established for both the cases with and those without viscosity

on the deformations of the incompressible euler equations we consider systems of deformed system of equations which are obtained by some transformations from the system of incompressible euler equations these have similar properties to the original euler equations including the scaling invariance for one form of deformed system we prove that finite time blow up actually occurs for generic initial data while for the other form of the deformed system we prove the global in time regularity for smooth initial data moreover using the explicit functional relations between the solutions of those deformed systems and that of the original euler system we derive the condition of finite time blow up of the euler system in terms of solutions of one of its deformed systems as another application of those relations we deduce a lower estimate of the possible blow up time of the 3d euler equations

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

operator based robust nonlinear control system design for mimo nonlinear plants with unknown coupling effects in this article operator based robust nonlinear control system design for multi input multi output mimo plants with unknown coupling effects is considered that is by using operator based robust nonlinear control design coupling effects existing in the mimo nonlinear plants can be decoupled based on a feedback design and robust right coprime factorisation approach the coupling effects caused by controllers and plant outputs can be stabilised by using definition of lipschitz operator and contraction mapping theorem and output tracking performance can be realised by a tracking design scheme finally a simulation example about temperature control process of 3 input 3 output aluminum plate is given to support the theoretical analysis

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

a new regularity criterion of the 2d mhd equations abstract in this paper we establish a new regularity criterion for the two dimensional incompressible generalized magnetohydrodynamics equations

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

global well posedness of incompressible flow in porous media with critical diffusion in besov spaces abstract in this paper we study the model of heat transfer in a porous medium with a critical diffusion we obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in besov spaces b p 1 3 p r 3 with 1 p by the method of modulus of continuity and fourier localization technique

parametrization of nonlinear stabilizing controllers the observer controller configuration the paper uses the author s 1996 previous results to develop the parametrization of nonlinear stabilizing observer controller compensators of a given nonlinear system the configuration described permits a simple parametrization of stabilizing controllers using coprime factorisation techniques

on regularity criterion for the dissipative quasi geostrophic equations abstract in this paper we consider the 2d dissipative quasi geostrophic equations and study the regularity criterion of the solutions by means of a commutator estimate based on frequency localization and bony s paraproduct decomposition we obtain a regularity criterion l r 0 t b p 0 for 2 p r 2 p which is an obvious extension of the result in b dong z chen a remark on regularity criterion for the dissipative quasi geostrophic equations j math anal appl 329 2007 1212 1217 where their restrictive condition on p is 4 p

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

a remark on the blow up criterion of strong solutions to the navier stokes equations abstract in this note we prove that b 1 norm of the vorticity controls the blow up phenomena of strong solutions to the navier stokes equations our criterion is a limiting case in yuan zhang 11 and an improvement of giga s result 3

local well posedness for the homogeneous euler equations abstract we introduce triebel lizorkin lorentz function spaces based on the lorentz l p q spaces instead of the standard l p spaces and prove a local in time unique existence and a blow up criterion of solutions in those spaces for the euler equations of inviscid incompressible fluid in r n n 2 as a corollary we obtain global existence of solutions to the 2 d euler equations in the triebel lizorkin lorentz space for the proof we establish the beale kato majda type logarithmic inequality and commutator estimates in our spaces the key methods of proof used are the littlewood paley decomposition and the paradifferential calculus by j m bony

coprimeness properties of nonlinear fractional system realizations in this paper the relationship between the bezout and the set theoretic approaches to left coprimeness is studied it is shown that left coprimeness in the set theoretic sense implies left coprimeness in the bezout sense in addition to these results we investigate whether some properties for linear left coprime realizations carry over to the nonlinear case for example we examine the relations between two left coprime realizations of the same system

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

remarks on gagliardo nirenberg type inequality with critical sobolev space and bmo we consider the generalized gagliardo nirenberg inequality in mathbb r n in the homogeneous sobolev space dot h s r mathbb r n with the critical differential order s n r which describes the embedding such as l p mathbb r n cap dot h n r r mathbb r n subset l q mathbb r n for all q with p q n r which may be regarded as a generalization of the brezis gallouet wainger inequality

regularity criterion of axisymmetric weak solutions to the 3d navier stokes equations abstract we consider the regularity of axisymmetric weak solutions to the navier stokes equations in r 3 let u be an axisymmetric weak solution in r 3 0 t w curl u and w be the azimuthal component of w in the cylindrical coordinates chae lee d chae j lee on the regularity of axisymmetric solutions of the navier stokes equations math z 239 2002 645 671 proved the regularity of weak solutions under the condition w l q 0 t l r with 3 2 r 2 q 3 r 2 we deal with the marginal case r which they excluded it is proved that u becomes a regular solution if w l 1 0 t b 0

uniformly local l p estimate for 2 d vorticity equation and its application to euler equations with initial vorticity in bmo we shall prove the global existence theorem for the 2 dimensional euler equations in open image in new window with the initial vorticity in bmo containing functions which do not decay at infinity and have logarithmic singularities

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

on the weak strong uniqueness of koch tataruʼs solution for the navier stokes equations abstract we prove the weak strong uniqueness between koch tataru s solution and leray s weak solution for three dimensional incompressible navier stokes equations

a beale kato majda blow up criterion for a hydrodynamic system modeling vesicle and fluid interactions abstract in this paper we establish an analog of the beale kato majda type criterion for singularities of smooth solutions of a hydrodynamic system modeling vesicle and fluid interactions the result shows that the maximum norm of the vorticity alone controls the breakdown of smooth solutions

beale kato madja type criteria of smooth solutions to 3d hall mhd flows in this article we study the initial value problem for the impressible hall mhd flows in three space dimensions several beale kato madja type blow up criteria of smooth solutions in term of the vorticity in homogeneous besov space are established

global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity abstract we prove the global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity

remarks on the blowup criteria for oldroyd models abstract we provide a new method to prove and improve the chemin masmoudi criterion for viscoelastic systems of oldroyd type in j y chemin n masmoudi about lifespan of regular solutions of equations related to viscoelastic fluids siam j math anal 33 1 2001 84 112 in two space dimensions our method is much easier than the one based on the well known losing a priori estimate and is expected to be easily adopted to other problems involving the losing a priori estimate

regularity criterion for solutions to the navier stokes equations in the whole 3d space based on two vorticity components abstract we prove among others the following regularity criterion for the solutions to the navier stokes equations if u is a global weak solution satisfying the energy inequality and u then u is regular on 0 t t 0 if two components of belong to the space l q 0 t b 3 p for p 3 and 2 q 3 p 2 this result is an improvement of the results presented by chae and choe 1999 7 or zhang and chen 2005 38 our method of the proof uses a suitable application of the bony decomposition and can also be used for the proofs of some other kin criteria one such example is presented in appendix

global strong solution to the 2d nonhomogeneous incompressible mhd system abstract in this paper we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible mhd system as long as the initial data satisfies some compatibility condition as a corollary the global existence of strong solution with vacuum to the 2d nonhomogeneous incompressible navier stokes equations is also established our main result improves all the previous results where the initial density need to be strictly positive the key idea is to use some critical sobolev inequality of logarithmic type which is originally due to brezis and wainger 1980 7

strichartz estimates for the euler equations in the rotational framework abstract we consider the initial value problems of the incompressible euler equations in the rotational framework we obtain the optimal range of the strichartz estimate for the linear group associated with the coriolis forces as an application we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high

on the complexity of mod 2l sum pla s consideration is given to the realization of logic functions by using plas with an exclusive or exor array where a function is represented by mod 2 exor sum of products esops and both true and complemented variables are used the authors propose a new pla structure using an exor array they derive upper bounds on the number of products of this type of pla that are useful for estimating the size of a pla as well as for assessing the minimality of the solutions obtained by heuristic esop minimization algorithms computer simulation using randomly generated functions shows that plas with the exor array require on the average fewer products than conventional plas for symmetric functions the authors conjecture that the plas with an exor array require at most as many products as the conventional plas the proposed plas can be made easily testable by adding a small amount of hardware

stabilization of nonlinear systems based on a generalized be zout identity the relationship between the bezout and input output approaches to right coprimeness is investigated we introduce a generalized bezout identity and analyze its relationship to right coprimeness in terms of the generalized bezout identity a right factorization of a nonlinear plant is coprime according to the input output definition if and only if it satisfies the generalized bezout identity therefore the stability of an observer controller configuration is a necessary and sufficient condition for a nonlinear plant to have a right coprime factorization finally we present generalized bezout maps for some nonlinear plants with a state space representation

on blow up criteria of smooth solutions to the 3 d euler equations in a bounded domain abstract we show that a smooth solution of the 3 d euler equations in a bounded domain breaks down if and only if a certain norm of vorticity blows up at the same time here this norm is weaker than bmo norm

operator based nonlinear feedback control design using robust right coprime factorization in this note robust stabilization and tracking performance of operator based nonlinear feedback control systems are studied by using robust right coprime factorization specifically a new condition of robust right coprime factorization of nonlinear systems with unknown bounded perturbations is derived using the new condition a broader class of nonlinear plants can be controlled robustly when the spaces of the nonlinear plant output and the reference input are different a space change filter is designed and in this case this note considers tracking controller design using the exponential iteration theorem

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

logarithmically improved blow up criteria for a phase field navier stokes vesicle fluid interaction model a b s t r a c t in this paper we study a hydrodynamical system modeling the deformation of the vesicle membrane under external incompressible viscous flow fields the system is in the eulerian formulation and is governed by the coupling of the incompressible navier stokes equations with a phase field equation in the three dimensional case we establish two logarithmically improved blow up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous besov spaces

operator based robust control design for nonlinear plants with perturbation in this article operator based robust control design for nonlinear plants with perturbation is considered by using robust right coprime factorisation approach in detail the property of unimodular operator is studied and a control design structure is proposed based on the proposed design scheme the designed system is robustly stable also output tracking performance can be realised simultaneously finally the effectiveness of the proposed design scheme is demonstrated by a simulation example

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

regularity criteria for the navier stokes landau lifshitz system abstract in this paper we study regularity criteria for the navier stokes landau lifshitz system using delicate estimates the regularity criteria for smooth solution of navier stokes landau lifshitz system in besov spaces and the multiplier spaces are obtained the navier stokes landau lifshitz system is coupled system of the navier stokes equation and landau lifshitz system our results generalize the related results for navier stokes equation and landau lifshitz system to our system

local well posedness and regularity criterion for the density dependent incompressible euler equations in this paper local well posedness for the density dependent incompressible euler equations is established in besov spaces we also obtain a blow up criterion for the corresponding solution

boolean matrix transforms for the minimization of modulo 2 canonical expansions fast transforms for computing modulo 2 ring sum canonical expansions of a boolean function are described using kronecker products of elementary boolean matrices these transforms unify and generalize existing ones in the literature previous algorithms which employ such transforms for finding a minimal fixed polarity expansion fpe are reviewed analyzed and extended to a minimal fixed basis expansion fbe

the class of stabilizing nonlinear plant controller pairs in this paper a general approach is taken to yield a characterization of the class of stable plant controller pairs which is a generalization of the youla parameterization for linear systems this is based on the idea of representing the input output pairs of the plant and controller as elements of the kernel of some related operator which is denoted the kernel representation of the system it is demonstrated that in some sense the kernel representation is a generalization of the left coprime factorization of a general nonlinear system results giving one method of deriving a kernel representation for a nonlinear plant with a general state space description are presented