poincare friedrichs type constants for operators involving grad curl and div theory and numerical experiments abstract we give some theoretical as well as computational results on laplace and maxwell constants i e on the smallest constants c n 0 arising in estimates of the form u l 2 c 0 grad u l 2 e l 2 c 1 curl e l 2 h l 2 c 2 div h l 2 besides the classical de rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of hilbert complexes we consider mixed boundary conditions and bounded lipschitz domains of arbitrary topology our numerical aspects are presented by examples for the de rham complex in 2d and 3d which not only confirm our theoretical findings but also indicate some interesting conjectures

optimal controllers for finite wordlength implementation when a controller is implemented by a digital computer with a d and d a conversion numerical errors can severely affect the performance of the control system there exist realizations of a given controller transfer function exhibiting arbitrarily large effects from computational errors assuming sufficient excitation of the system the problem of designing an optimal controller in the presence of both external disturbances and internal roundoff errors is solved the results reduce to the standard lqg controller when infinite precision computation is used for finite precision however the separation principle does not hold a penalty is also added to the cost function to penalize the sum of the wordlengths used to compute the fractional part of each state variable of the controller this sum can be used to represent the lower bound on computer memory needed for controller synthesis it measures controller complexity and is minimized penalized here

on a selfadjoint realization of curl in exterior domains the first order vectorial operator curl occurs in many areas of applications e g in electrodynamics fluid mechanics hydrodynamics magnetohydrodynamics and elasticity theory it is the purpose of this note to discuss a particular instance of curl where a domain can be given to curl so that it becomes a selfadjoint operator the particular domain of selfadjointness under consideration is singled out by its relevance in applications to establish this operator we initially consider curl as an operator curl c read restricted to in l2 defined by curl c c l2 l2 0 1

comments on robust fragile or optimal with reply the issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by keel bhattacharyya in the above mentioned paper ibid vol 42 1997 the commenter points out that the controller fragility problem described in the paper is 1 well known in the literature 2 due to the poor choice of examples 3 can easily be fixed by changing the optimization criteria and 4 can be fixed by changing the implementation in reply keel bhattacharyya disagree with him on all counts and explain their argument

well posedness for a general class of differential inclusions abstract we consider an abstract class of differential inclusions which covers differential algebraic and non autonomous problems as well as problems with delay under weak assumptions on the operators involved we prove the well posedness of those differential inclusions in a pure hilbert space setting moreover we study the causality of the associated solution operator the theory is illustrated by an application to a semistatic quasilinear variant of maxwell s equations

an elementary proof for a compact imbedding result in generalized electromagnetic theory propriete de plongement compact et certaines de ses consequences on etudie la validite du plongement suivant r 9 g e q 1 d q g l 2 q loc g

rearrangeability of 2 protectn 1 stage shuffle exchange networks rearrangeable networks can realize each and every permutation in one pass through the network shuffle exchange networks provide an efficient interconnection scheme for implementing various types of parallel processes whether 2n stage shuffle exchange networks with n 2n inputs outputs are rearrangeable has remained an open question for approximately three decades this question has been answered affirmatively in this paper an important corollary of the main result is the proof that two passes through an omega network are sufficient and necessary to implement any permutation in obtaining the main results of this paper frames that look like grids with horizontal links of different lengths are shown to be remarkable tools for identifying and characterizing the binary matrix representations of permutations

noise reduction strategies for digital filters error spectrum shaping versus the optimal linear state space formulation the noise reduction performance of error spectrum shaping ess structures and of the optimal linear state space lss structure are compared for second order digital filter sections it is shown that optimal direct form 1 and direct form 2 ess realizations have a higher signal to noise ratio than the optimal lss structure in practice suboptimal ess structures with simple hardware implementations are of greater interest several of these implementations are considered and optimal values for the ess coefficients in these structures are derived for filters with zeros at z 1 it is shown that some of the simple ess structures can outperform the optimal lss structure for elliptic filters it is shown that several of the suboptimal ess structures perform poorly but that others still perform well

on the maxwell and friedrichs poincare constants in nd we prove that for bounded and convex domains in arbitrary dimensions the maxwell constants are bounded from below and above by friedrichs and poincare s constants respectively especially the second positive maxwell eigenvalues in nd are bounded from below by the square root of the second neumann laplace eigenvalue

real interpolation of spaces of differential forms in this paper we study interpolation of hilbert spaces of differential forms using the real method of interpolation we show that the scale of fractional order sobolev spaces of differential l forms in hs with exterior derivative in hs can be obtained by real interpolation our proof heavily relies on the recent discovery of smoothed poincare lifting for differential forms m costabel and a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains math z 265 2 297 320 2010 they enable the construction of universal extension operators for sobolev spaces of differential forms which in turns pave the way for a fourier transform based proof of equivalences of k functionals

poincare meets korn via maxwell extending korn s first inequality to incompatible tensor fields for a bounded domain r3 r3 with lipschitz boundary and some relatively open lipschitz subset t t of we prove the existence of some c 0c 0 such that equation 0 1 c t l2 r3 3 symt l2 r3 3 curlt l2 r3 3 holds for all tensor fields in h curl h curl i e for all square integrable tensor fields t r3 3t r3 3 with square integrable generalized rotation curlt r3 3 having vanishing restricted tangential trace on t t if t t 0 1 still holds at least for simply connected and for all tensor fields t h curl t h curl which are l2 l2 perpendicular to so 3 so 3 i e to all skew symmetric constant tensors here both operations curl and tangential trace are to be understood row wise for compatible tensor fields t vt v 0 1 reduces to a non standard variant of the well known korn s first inequality in r3r3 namely c v l2 r3 3 sym v l2 r3 3 for all vector fields v h1 r3 v h1 r3 for which vn vn n 1 3n 1 3 are normal at t t on the other hand identifying vector fields v h1 r3 v h1 r3 having the proper boundary conditions with skew symmetric tensor fields t 0 1 turns to poincare s inequality since 2c v l2 r3 c t l2 r3 3 curlt l2 r3 3 2 v l2 r3 therefore 0 1 may be viewed as a natural common generalization of korn s first and poincare s inequality from another point of view 0 1 states that one can omit compatibility of the tensor field t at the expense of measuring the deviation from compatibility through curl t decisive tools for this unexpected estimate are the classical korn s first inequality helmholtz decompositions for mixed boundary conditions and the maxwell estimate

balanced realizations via gradient flow techniques abstract the task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations by viewing such algebraic tasks in the context of calculus they are amenable to analog computational solutions or parallel processing machines perhaps even neural networks the convergence rates of the differential equations is exponential and consequentially convergence is rapid and numerical stability properties are attractive

reconstruction from projections based on detection and estimation of objects parts i and ii performance analysis and robustness analysis the problem of reconstructing a multidimensional field from noisy limited projection measurements is approached using an object based stochastic field model objects within a cross section are characterized by a finite dimensional set of parameters which are estimated directly from limited noisy projection measurements using maximum likelihood estimation in part i the computational structure and performance of the ml estimation procedure are investigated for the problem of locating a single object in a deterministic background simulations are also presented in part ii the issue of robustness to modeling errors is addressed

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

optimal realizations of floating point implemented digital controllers with finite word length considerations the closed loop stability issue of finite precision realizations is investigated for digital controllers implemented in floating point arithmetic unlike the existing methods which only address the effect of the mantissa bits in floating point implementation to the sensitivity of closed loop stability the sensitivity of closed loop stability is analyzed with respect to both the mantissa and exponent bits of floating point implementation a computationally tractable fwl closed loop stability measure is then defined and the method of computing the value of this measure is given the optimal controller realization problem is posed as searching for a floating point realization that maximizes the proposed fwl closed loop stability measure and a numerical optimization technique is adopted to solve for the resulting optimization problem simulation results show that the proposed design procedure yields computationally efficient controller realizations with enhanced fwl closed loop stability performance

analysis of coefficient quantization errors in state space digital filters in this paper we examine the sensitivity of state space digital filters to coefficient quantization errors a cost function using an l p norm criteria is used to measure the deviation in frequency response of the filter the cost function is used to indicate the connection between roundoff noise and filter sensitivity to coefficient quantization errors then based on this cost function it is shown that low roundoff noise state space digital filters filters in balanced coordinates and filters based on polynomials orthogonal on the unit circle have low sensitivity to coefficient quantization errors furthermore the sensitivity of the filters is shown to depend on the second order modes thereby exhibiting certain robustness and immunity to filter bandwidth

finite wordlength controller realisations using the specialised implicit form a specialised implicit state space representation is introduced to deal with finite wordlength effects in controller implementations this specialised implicit form provides a macroscopic description of the algorithm to be implemented so it constitutes a unifying framework allowing to encompass various implementation forms such as the operator the direct form ii transposed observer based and many other realisations usually considered separately in the literature different measures quantifying the finite wordlength effects on the overall closed loop behaviour are defined in this new context they concern both stability and performance the gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise analysis the problem of determining a realisation with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods the approach is illustrated with an example

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

rearrangeability of bit permutation networks in this paper we introduce the concept of routing grid as a tool for analyzing realizability of permutations on bit permutation networks bpns we extend a result by linial and tarsi which characterizes permutations realizable on shuffle exchange networks to any bpns a necessary condition for a bpn to he rearrangeable is given and the rearrangeability of two families of bpns are explored finally we present a method which may help to tackle one kind of balanced matrix problems whose solution implies an answer to benes conjecture hopefully our treatment brings some new insight into the problem of permutation routing

on a first order hyperbolic system including bean s model for superconductors with displacement current abstract this paper deals with a class of first order systems with generally multivalued damping terms including bean s model for superconductors with displacement current the main goal is to prove the convergence of solutions to stationary states as t

small roundoff noise realization of fixed point digital filters and controllers a method for finding digital realizations of discrete time transfer functions by fixed point arithmetic is presented the obtained realizations attain very small roundoff noise intensity this intensity is significantly smaller than the intensities that have been achieved so far the method applies many 0 1 coefficients in the realization it therefore requires a small number of multiplications and additions and it is significantly faster than many other realization methods a detailed comparison is made between this method and other existing methods for digital realization two different cases are investigated a butterworth filtering realization and the realization of an lqg linear quadratic gaussian optimal controller

optimizing stability bounds of finite precision pid controller structures this paper investigates a recently derived lower bound stability measure for sampled data controller structures subject to finite wordlength fwl constraints the optimal realization of the digital pid controller with fwl considerations is formulated as a nonlinear optimization problem and an efficient strategy based on adaptive simulated annealing asa is adopted to solve this complex optimization problem a numerical example of optimizing the finite precision pid controller structure for a simulated steel rolling mill system is presented to illustrate the effectiveness of the proposed strategy

brief paper an efficient controller structure with minimum roundoff noise gain based on a polynomial operator approach a new sparse controller structure is derived which is actually an improved version of the recently proposed structure li g 2004 a polynomial operator based dfiit structure for iir filters ieee transactions on circuits and systems ii 51 147 151 the performance of the proposed structure is analyzed by deriving the corresponding expression of roundoff noise gain and the problem of finding optimized sparse structures is solved efficiently with a genetic algorithm ga a numerical example is given which shows that the newly developed structure can achieve much better performance than some well known structures and particularly outperforms the traditional optimal fully parametrized realization greatly in terms of reducing roundoff noise and implementation complexity

generalised modal realisation as a practical and efficient tool for fwl implementation finite word length fwl effects have been a critical issue in digital filter implementation for almost four decades although some optimisations may be attempted to get an optimal realisation with regards to a particular effect for instance the parametric sensitivity or the round off noise gain the purpose of this article is to propose an effective one i e taking into account all the aspects based on the specialised implicit form a new effective and sparse structure named modal realisation is proposed this realisation meets simultaneously accuracy low sensitivity round off noise gain and overflow risk few and flexible computational efforts with a good readability thanks to sparsity and simplicity no tricky optimisation is required to obtain it as well two numerical examples are included to illustrate the modal realisation s interest

a new routing algorithm for a class of rearrangeable networks this paper presents a routing algorithm for a class of multistage interconnection networks specifically the concatenation of two omega networks which has 2 log sub 2 n stages is treated it is shown that this kind of asymmetric omega omega network can be converted into a symmetric omega sup 1spl times omega network or a symmetric omega spl times omega sup 1 network however they have butterfly connections between the two center stages a general algorithm is developed which routes a class of symmetric networks the algorithm routes the network from center stages to outer stages at both the input and the output sides simultaneously the algorithm presented is simpler and more flexible than the well known looping algorithm in that it can be applied adaptively according to the structure of the network it can be applied to routing the omega based networks regardless of the center stage connection patterns i e straight skewed straight simple butterfly or skewed butterfly as long as the networks are symmetric the sufficient conditions for proper routing are shown and proved in addition an example is shown to demonstrate the algorithm

roundoff noise analysis of two efficient digital filter structures in this correspondence two efficient structures are analyzed for the implementation scheme of rounding before multiplication rbm the first one is the direct form ii transposed structure in spl rho operator spl rho dfiit recently proposed in the expression of roundoff noise gain is derived based on which a revised spl rho dfiit structure denoted as spl rho rdfiit is obtained it is shown that the spl rho rdfiit structure having the same implementation complexity yields a smaller roundoff noise gain than the spl rho dfiit the roundoff noise gain for the spl rho rdfiit structure with error feedbacks is also derived each structure can be optimized with regard to the free parameters it possesses with a genetic algorithm numerical examples are presented which show that the optimized structures are very competitive because they can even outperform the traditional optimal state space realization in terms of the roundoff noise performance as well as the implementation simplicity

roundoff noise minimization using delta operator realizations digital filters the authors examine the advantages of using delta operator state space realizations rather than shift operator realizations of transfer functions in terms of minimizing roundoff noise gain they give several conditions under which the optimal roundoff noise gain for delta operator realizations is smaller than the optimal gain for shift operator realizations they then illustrate that even sparse and hence nonoptimal delta operator realizations can have smaller roundoff noise gain than the optimal shift operator realizations

on a structural observation in generalized electromagnetic theory are satisfied equations 0 2 describe the electromagnetic properties of the medium where 0 1 is considered if this medium is confined to a certain region g of r3 then the electromagnetic properties of its complent cc are usually represented via boundary conditions on dg we shall deal here with the boundary condition appearing if cc is assumed to be a perfect conductor i e nxe t o on c7g 13 0 0 3

asymptotic behavior of electromagnetic waves in nonlinear dielectric media this paper is concerned with maxwell s equations including a nonlinear dielectric polarization and a generally nonlinear law for the electric current the large time behavior of the electromagnetic field is investigated in the case of a bounded spatial domain as well as in the exterior domain case

optimum realizations of sampled data controllers for fwl sensitivity minimization the problem of an optimal finite word length state space realization of a digital controller is investigated the closed loop to be considered consists of a continuous time plant a discrete time controller a sampler a zero order hold and an antialiasing filter an effective algorithm is proposed to find the optimal sampled data controller realization minimizing the sensitivity of the closed loop performance with respect to coefficient errors in the state variable matrices of the controller realization in order to get a tractable problem a two step procedure is to be used very fast sampling at a multiple of the sampling frequency followed by blocking or lifting to obtain a single rate system the procedure allows consideration of the system s intersample behaviour

on asymptotic equipartition of energy abstract a particular class of selfadjoint 2 2 operator matrices a of the form a 0 b b 0 acting on a direct sum hilbert space h h 1 h 2 h i i 1 2 hilbert spaces is considered the square of the norm i e the energy of a solution of the corresponding initial value problem is shown to be distributed asymptotically evenly to its first and second component as t if the initial values are in the absolute continuous subspace the result extends and simplifies proofs of results obtained previously in this area a list of examples covered by the general results is included

an improved closed loop stability related measure for finite precision digital controller realizations the pole sensitivity approach is employed to investigate the stability issue of the discrete time control system where a digital controller implemented with finite word length fwl is used a stability related measure is derived which is more accurate in estimating the closed loop stability robustness of an fwl implemented controller than some existing measures for the pole sensitivity analysis this improved stability measure thus provides a better criterion to find the optimal realizations for a generic controller structure that includes output feedback and observer based controllers a numerical example is used to verify the theoretical analysis and to illustrate the design procedure

on the correctness of inside out routing algorithm recently a new routing algorithm called inside out routing algorithm was proposed for routing an arbitrary permutation in the omega based 2log sub 2 n stage networks this paper discusses the problems of the inside out routing algorithm and shows that the suggested condition for proper routing in the omega omega network is insufficient an extended necessary and sufficient condition for proper routing in the omega omega network is also suggested however it is unknown if any permutation can be successfully routed by a heuristic algorithm which follows the condition thus the rearrangeability of the omega omega network still remains an open problem

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

a new approach to the discretization of continuous time controllers a new method for discretizing continuous time controllers is derived using principles of controller approximation it focuses on the closed loop use of the discrete time controller the resulting approximation criterion is a measure for stability of the control system provides an upper bound on the sampling time for which stability can be guaranteed and because it is based on continuous time controller approximation it indicates the cost of discretization in terms of performance degradation the discrete time controller is obtained through minimization of this criterion which can be performed with standard software used in h sub infinity controller design

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

on electric and magnetic problems for vector fields in anisotropic nonhomogeneous media r 3 2 initiated by saranen 131 in the above n is the outward drawn unit normal to the boundary and a denotes the exterior product according to the simple models for static magnetic fields resp electric fields which are governed by 0 1 resp 0 2 we call 0 1 the magnetic type problem and 0 2 the electric type problem considering bounded smooth domains a c r3 we discussed in 131 by means of an appropriate hilbert space method the solvability and the representation of the solutions for both problems 0 1 and 0 2 such a new approach was necessary to cover the general nonhomogeneous cases where v and e are matrix valued functions here our aim is twofold first we will now include exterior domains in our consideration this will be achieved by using certain weighted spaces which slightly restrict the behavior of the fields at infinity to find the scalar potentials we have to solve the dirichlet and neumann problems in exterior domains consequently in order to define the vector potentials we solve appropriate auxiliary second order boundary problems for vector fields the other aspect of this paper is to point out that the dimension of the space of solutions for 0 1 or 0 2 with homogeneous right side is 254 0022 247x 83 010254 22 03 00 0

on the structure of digital controllers with finite word length consideration it is well known that a stable discrete time control system may become unstable when the digital controller is actually implemented with a digital control processor due to the finite word length effects and that the stability behavior of the system depends on the structure of the digital controller due to the difficulties in adopting the existing stability robustness measures a new measure is derived which is tractable and hence enables us to handle the digital controller structure problem easily the optimal controller structure problem is formulated by maximizing this measure an analytical solution is presented an algorithm is developed for finding sparser structures

frames a simple characterization of permutations realized by frequently used networks rearrangeable multistage networks such as the benes network realize any permutation yet their routing algorithms are not cost effective on the other hand there exist inexpensive routing algorithms for nonrearrangeable networks but no simple technique exists to characterize all the permutations realized on these networks this paper introduces the concept of frame and shows how it can be used to characterize all the permutations realized on various multistage interconnection networks they include subnetworks of baseline benes and cascaded baseline and shuffle exchange networks

a search algorithm for a class of optimal finite precision controller realization problems with saddle points with game theory we review the optimal digital controller realization problems that maximize a finite word length fwl closed loop stability measure for a large class of these optimal fwl controller realization problems which have saddle points a minimax based search algorithm is derived for finding a global optimal solution the algorithm consists of two stages in the first stage the closed form of a transformation set is constructed which contains global optimal solutions in the second stage a subgradient approach searches this transformation set to obtain a global optimal solution this algorithm does not suffer from the usual drawbacks associated with using direct numerical optimization methods to tackle these fwl realization problems furthermore for a small class of optimal fwl controller realization problems which have no saddle point the proposed algorithm also provides useful information to help solve them

low sensitivity h filter design for linear delta operator systems with sampling time jitter this article is concerned with the problem of designing h filters for a class of linear discrete time systems with low sensitivity to sampling time jitter via delta operator approach delta domain model is used to avoid the inherent numerical ill condition resulting from the use of the standard shift domain model at high sampling rates based on projection lemma in combination with the descriptor system approach often used to solve problems related to delay a novel bounded real lemma with three slack variables for delta operator systems is presented a sensitivity approach based on this novel lemma is proposed to mitigate the effects of sampling time jitter on system performance then the problem of designing a low sensitivity filter can be reduced to a convex optimisation problem an important consideration in the design of correlation filters is the optimal trade off between the standard h criterion and the sensitivity of the transfer function with respect to sampling time jitter finally a numer

topologies of combined 2 log n 1 stage interconnection networks a combined 2 log n 1 stage interconnection network denoted by spl delta spl oplus spl delta is constructed by concatenating two omega equivalent networks spl delta and spl delta with the rightmost stage of spl delta and the leftmost stage of spl delta overlapped benes network and the 2 log n 1 stage shuffle exchange network are two examples of such networks although these two networks have received intensive studies the research on the topology of entire class of spl delta spl oplus spl delta networks is very limited so far in this paper we study the topological structure of spl delta spl oplus spl delta networks and propose an algorithm for determining topological equivalence between two given spl delta spl oplus spl delta networks we also present a simplified spl omega equivalence checking algorithm as a supporting result

the essential spectrum of the nonself adjoint maxwell operator in a bounded domain 1 introductionin this work we study the spectrum of the maxwell operator in theself adjoint case it is well known that the spectrum consists of oneeigenvalue with infinite multiplicity in zero and other eigenvalues havingfinite multiplicities however the spectrum of the nonself adjoint operatorwas not studied much though the nonself adjoint perturbation rises from avery common phenomena namely from the conductivity in this work weshow that the maxwell system with conductivity in a bounded domain hasquite a large essential spectrum the essential spectrum contains aninterval or another nondiscrete set let v be a

a unified closed loop stability measure for finite precision digital controller realizations implemented in different representation schemes a computationally tractable unified finite word length closed loop stability measure is derived which is applicable to fixed point floating point and block floating point representation schemes both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure for each arithmetic scheme the optimal controller realization problem is defined and a numerical optimization approach is adopted to solve it numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes