Output feedback stabilizability and stabilization algorithms for 2D systems

Alternative methods are proposed for test of output feedback stabilizability and construction of a stable closed-loop polynomial for 2D systems. By the proposed methods, the problems can be generally reduced to the 1D case and solved by using 1D algorithms or Gröbner basis approaches. Another feature of the methods is that their extension to certain specialnD (n>2) cases can be easily obtained.Moreover, the Rabinowitsch trick, a technique ever used in showing the well-known Hilbert's Nullstellensatz, is generalized in some sense to the case of modules over polynomial ring. These results eventually lead to a new solution algorithm for the 2D polynomial matrix equationD(z, w)X(z, w)+N(z, w)Y(z, w)=V(z, w) withV(z, w) stable, which arises in the 2D feedback design problem. This algorithm shows that the equation can be effectively solved by transforming it to an equivalent Bezout equation so that the Gröbner basis approach for polynomial modules can be directly applied.Notation R the field of real numbers - C the field of complex numbers - R[z, w] commutative ring of 2D polynomials inz andw with coefficients inR - M(R[z, w]) set of matrices with appropriate dimensions with entries inR[z, w] - R[z, w] ⁿ module of orderedn-tuples inR[z, w] - R[z, w] ^{n ×m} set ofn ×m matrices with entries inR[z, w] - closed unit disc inC, i.e., {z C| |z| 1} - ² closed unit bidisc, i.e., {(z, w) C ²| |z| 1, |w| 1} - A ^T transpose of matrixA     

On the distribution of characteristics in bijective mappings

Differential cryptanalysis is a method of attacking iterated mappings based on differences known as characteristics. The probability of a given characteristic is derived from the XOR tables associated with the iterated mapping. If is a mapping : Z ₂ ^m , then for each , X, Y Z ₂ ^m the XOR table for gives the number of input pairs of difference X=X+X for which gp(X)+(X)=Y.The complexity of a differential attack depends upon two properties of the XOR tables: the density of zero entries in the table, and the size of the largest entry in the table. In this paper we present the first results on the expected values of these properties for a general class of mappings . We prove that if : Z ₂ ^m Z ₂ ^m is a bijective mapping, then the expected size of the largest entry in the XOR table for is bounded by 2m, while the fraction of the XOR table that is zero approaches e ^–1/2=0.60653. We are then able to demonstrate that there are easily constructed classes of iterated mappings for which the probability of a differential-like attack succeeding is very small.The author is presently employed by the Distributed System Technology Center, Brisbane, Australia.     

A Fast Algorithm to Map Functions Forward

Mapping functions forward is required in image warping and other signal processing applications. The problem is described as follows: specify an integer d 1, a compact domain D R ^d, lattices L ₁,L ₂ R ^d, and a deformation function F : D R ^d that is continuously differentiable and maps D one-to-one onto F(D). Corresponding to a function J : F(D) R, define the function I = J F. The forward mapping problem consists of estimating values of J on L ₂ F(D), from the values of I and F on L ₁ D. Forward mapping is difficult, because it involves approximation from scattered data (values of I F ^-1 on the set F(L ₁ $#x22C2; D)), whereas backward mapping (computing I from J) is much easier because it involves approximation from regular data (values ofJ on L ₂ D). We develop a fast algorithm that approximates J by an orthonormal expansion, using scaling functions related to Daubechies wavelet bases. Two techniques for approximating the expansion coefficients are described and numerical results for a one dimensional problem are used to illustrate the second technique. In contrast to conventional scattered data interpolation algorithms, the complexity of our algorithm is linear in the number of samples.     

An auxiliary theorem for stability analysis in the presence of interval-valued parameters

A theorem which helps to determine the boundary of the image of an interval inR ⁿ under a differentiable mapping,F:R ⁿ C, is expounded. Examples illustrating the application of the theorem are given.     

Symmetry relations in multidimensional Fourier transform pairs

A relation between the types of symmetries that exist in signal and Fourier transform domain representations is derived for continuous as well as discrete domain signals. The symmetry is expressed by a set of parameters, and the relations derived in this paper will help to find the parameters of a symmetry in the signal or transform domain resulting from a given symmetry in the transform or signal domain respectively. A duality among the relations governing the conversion of the parameters of symmetry in the two domains is also brought to light. The application of the relations is illustrated by a number of two-dimensional examples.Notation R the set of real numbers - R ^m R × R × ... × R m-dimensional real vector space - continuous domain real vector - L {¦ – _i , i = 1,2,..., m} - m-dimensional frequency vector - W { – _i ,i=1,2,..., m} - m-dimensional normalized frequency vector - _P {¦ – _i , i=1,2,...,m} - g(ol) g (₁,₂,..., _m ) continuous domain signal - () ( _{1 2},..., _m )=G (j ₁,j ₂,..., j _m ) Fourier transform ofg (ol) - (A,b,,,) parameters ofT- symmetry - N the set of integers - N ^m N × N × ... × N m-dimensional integer vector spacem-dimensional lattice - h(n) h (n ₁,.,n _m ) discrete domain signal - H() Fourier transform ofh (n) - v ₁,v ₂,..., v_m m sample-direction and interval vectors - V (v ₁ v ₂ ...v _m ) sampling basis matrix - [x]^* complex conjugate ofx - detA determinant ofA - X {x¦ – x _i , i=1,2,..., m} - A ^–t [A ^–1] ^t ,t stands for transpose This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A-7739 to M. N. S. Swamy and in part by Tennessee Technological University under its Faculty Research support program to P. K. Rajan.     

Estimation of Energy Aliasing Error for Nonbandlimited Signals

In this paper, we investigate the problem of approximating a given (not necessarily bandlimited) signal, x(t), by a (bandlimited) interpolation or sampling series of the form:

Permissively structured transfinite electrical networks

Transfinite electrical networks of ranks larger than 1 have previously been defined by arbitrarily joining together various infinite extremities through transfinite nodes that are independent of the networks' resistance values. Thus, some or all of those transfinite nodes may remain ineffective in transmitting current through infinity. In this paper, transfinite nodes are defined in terms of the paths that permit currents to reach infinity. This is accomplished by defining a suitable metricd ^v on the node setN _S ^v of eachv-sectionS ^v, av-section being a maximal subnetwork whose nodes are connected by two-ended paths of ranks no larger thanv. Upon taking the completion ofN _S ^v under that metricd ^v, we identify those extremities (now calledv-terminals) that are accessible to current flows. These are used to define transfinite nodes that combine such extremities. The construction is recursive and is carried out through all the natural number ranks, and then through the first arrow rank and the first limit-ordinal rank . The recursion can be carried still further. All this provides a more natural development of transfinite networks and indeed simplifies the theory of electrical behavior for such networks.This work was partially supported by the National Science Foundation under Grant MIP-9423732.     

Universal approximation capability of EBF neural networks with arbitrary activation functions

The purpose of this paper is to investigate the approximation capability of elliptic basis function (EBF) neural networks. The main results are: (1) A necessary and sufficient condition for a functionS (R ¹) to be qualified as an activation function in the hidden layer of an EBF neural network is given. (2) Every nonzero functionG L ²(R ⁿ ) is qualified to be an activation function in the elliptic neural network to approximate any function in L²(Rⁿ). (3) As applications, we give new proofs of the theorems concerning the approximation capability of affine basis function (ABF) neural networks and generalized radial basis function (GRBF) neural networks (including radial basis function neural networks) with arbitrary activation functions. In particular, we solve the problem of the approximation capability of sigma-pi neural networks.Work was supported in part by CNSF, the Shanghai Science Foundations and Doctoral Program of Education Commissions of China.     

A VLSI decoder for a new type of constellations adapted to the Rayleigh Fading Channel

Diversity is the key solution to obtain efficient channel coding in wireless communications, where the signal is subject to fading (Rayleigh Fading Channel). For high spectral efficiency, the best solutions used nowadays are based on QAM constellations of 1-order diversity, associated with a binary code or a trellis coded modulation to increase the overall diversity. It has been shown that a new class of d-dimensional non-QAM constellations, named -constellations, can bring a d-order diversity without the addition of redundancy. Combined with classical coding techniques, -constellations are very efficient. However, the decoding algorithm is far more complicated for -constellations than for QAM-constellations. A sub-optimal algorithm that allows the decoding of -constellations is proposed. An example of an application for a 4 bits/Hz/s spectral efficiency with a 4-D -constellation is given. The VLSI architecture of the decoder is described. The implementation leads to 72 K gates, a binary rate of 32 Mbits/s and a BER of 10^-3 for a SNR of 14 dB.     

Ad hoc networks beyond unit disk graphs

In this paper, we study an algorithmic model for wireless ad hoc and sensor networks that aims to be sufficiently close to reality as to represent practical realworld networks while at the same time being concise enough to promote strong theoretical results. The quasi unit disk graph model contains all edges shorter than a parameter d between 0 and 1 and no edges longer than 1. We show that—in comparison to the cost known for unit disk graphs—the complexity results of geographic routing in this model contain the additional factor 1/d ². We prove that in quasi unit disk graphs flooding is an asymptotically message-optimal routing technique, we provide a geographic routing algorithm being most efficient in dense networks, and we show that classic geographic routing is possible with the same asymptotic performance guarantees as for unit disk graphs if .

A doubling procedure for constructing minimal broadcast networks

Broadcast is the task of disseminating a message from any node to all the other nodes in a network. A minimal broadcast network (mbn) withn nodes is a communication network in which a message originated at any node can be broadcasted in [log₂ n] time units. An optimal broadcast network (obn) is an mbn with minimum number of edges. No method is known for constructing an obn with an arbitrary number of nodes. In this paper, a new method called the doubling procedure is presented to construct mbn's with 2n and 2n–1 nodes when an obn or a good mbn withn nodes is known. The new construction method is based on the concepts of center node and center node set of an mbn. An algorithm is proposed to find a center node set of a given mbn. It is shown that an obn with 2n nodes can be constructed based on a known obn withn nodes for alln 9,n=15, 31 and 63,n=2 ^m –1 andn=2 ^m ,mZ ⁺, by applying the doubling procedure. This method also generates the best mbn's for some values of [n64.     

Factorizations in the elementary Abelian p-group and their cryptographic significance

Let G be a finite group and let A _i 1 i s, be subsets of G where ¦A _i ¦ 2, 1 i s and s 2. We say that (A₁, A₂,..., A₃) is a factorization of G if and only if for each g G there is exactly one way to express g = a ₁ a ₁ a ₂··· a ₃, where a _j A _i , 1 i s.The problem of finding factorizations of this type was first introduced by Hajos [3] in 1941. Since then a number of papers have appeared on the subject. More recently, Magliveras [6] has applied factorization of permutation groups to cryptography to obtain a private-key cryptosystem. Factorizations in the elementary abelian p-group were exploited (but not explicitly stated in these terms) by Webb [13] to produce a public-key cryptosystem conceptually similar to cryptosystems based on the knapsack problem.Using the result that certain types of factorizations in the elementary abelian p-group are necessarily transversal (a term introduced by Magliveras), this paper shows that the public-key system proposed by Webb is insecure.     

Uniformity of Optical Constants in Amorphous Ta2O5 Thin Films as Measured by Spectroscopic Ellipsometry

Amorphous Ta₂O₅ films are deposited on silicon substrates by reactive sputtering. The uniformity of refractive index and absorbance is evaluated by spectroscopic ellipsometry over the range 1.9–4.9 eV. The respective surface variations of the thickness d and the refractive index n are measured by scanning ellipsometry at a wavelength of 632.8 nm. It is established that n and d are uniform over a film to within n/n = 2.4% and d/d = 1%. The optical energy gap is estimated at 4.20 ± 0.05 eV. It is noted that the optical properties of the films indicate stoichiometric composition. It is found by AFM examination that the films have an almost atomically smooth surface.     

BIBO Stability of D-Dimensional Filters

Consider the class of d-dimensional causal filters characterized by a d-variate rational function analytic on the polydisk . The BIBO stability of such filters has been the subject of much research over the past two decades. In this paper we analyze the BIBO stability of such functions and prove necessary and sufficient conditions for BIBO stability of a d-dimensional filter. In particular, we prove if a d-variate filter H(z) analytic on has a Fourier expansion that converges uniformly on the closure of , then H(z) is BIBO stable. This result proves a long standing conjecture set forth by Bose in [3].     

Temperature dependent current–voltage (I–V) characteristics of Au/n-Si (1 1 1) Schottky barrier diodes with PVA(Ni,Zn-doped) interfacial layer

Current–voltage (I–V) characteristics of Au/PVA/n-Si (1 1 1) Schottky barrier diodes (SBDs) have been investigated in the temperature range 80–400 K. Here, polyvinyl alcohol (PVA) has been used as interfacial layer between metal and semiconductor layers. The zero-bias barrier height (Φ_B0) and ideality factor (n) determined from the forward bias I–V characteristics were found strongly dependent on temperature. The forward bias semi-logarithmic I–V curves for different temperatures have an almost common cross-point at a certain bias voltage. The values of Φ_B0 increase with the increasing temperature whereas those of n decrease. Therefore, we have attempted to draw Φ_B0 vs. q/2kT plot in order to obtain evidence of a Gaussian distribution (GD) of the barrier heights (BHs). The mean value of BH and standard deviation (σ₀) were found to be 0.974 eV and 0.101 V from this plot, respectively. Thus, the slope and intercept of modified vs. q/kT plot give the values of and Richardson constant (A^?) as 0.966 eV and 118.75 A/cm²K², respectively, without using the temperature coefficient of the BH. This value of A^* 118.75 A/cm²K² is very close to the theoretical value of 120 A/cm²K² for n-type Si. Hence, it has been concluded that the temperature dependence of the forward I–V characteristics of Au/PVA/n-Si (1 1 1) SBDs can be successfully explained on the basis of the Thermionic Emission (TE) theory with a GD of the BHs at Au/n-Si interface.     

A method for robust stabilization

LetP(y,M) be a family of polynomials, depending on a controller parameter vector y in ⁿ⁺¹, defined as follows: a family of interval matrices, and y in ⁿ⁺¹, set Given an initial stabilizing controller y⁰, this paper provides a simple method to robustify y⁰, i.e., to obtain a controller parameter vector y¹ which is more robust than y⁰. Furthermore we define the stability factort for a given robustly stable controller y, which serves as a measure of the robustness of y.     

The prime and generalized nullspaces of right regular pencils

The classical notion of the -generalized nullspace, defined on a matrixA R ^n×n,where is an eigenvalue, is extended to the case of ordered pairs of matrices(F, G), F, G R ^m×nwhere the associated pencilsF – G is right regular. It is shown that for every C {} generalized eigenvalue of (F, G), an ascending nested sequence of spaces {P ⁱ ,i=1, 2,...} and a descending nested sequence of spaces {ie495-02 i=1, 2,...} are defined from the -Toeplitz matrices of (F, G); the first sequence has a maximal elementM ^* , the -generalized nullspace of (F, G), which is the element of the sequence corresponding to the index , the -index of annihilation of (F, G), whereas the second sequence has the first elementP ^* as its maximal element, the -prime space of (F, G). The geometric properties of the {M ⁱ ,i=1, 2,..., and {P ⁱ ,i=1, 2,...sets, as well as their interrelations are investigated and are shown to be intimately related to the existence of nested basis matrices of the nullspaces of the -Toeplitz matrices of (F, G). These nested basis matrices characterize completely the geometry ofM ^* and provide a systematic procedure for the selection of maximal length linearly independent vector chains characterizing the-Segre characteristic of (F, G).     

Advanced modeling of silicon oxidation

The reaction-diffusion (i.e. the linear-parabolic) mechanism, widely in use for modeling the thermal oxidation of silicon, can be rebutted on the following issues:• There is a poor fit of the linear-parabolic law or of its derivative X² +

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X =

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(t + τ) or

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dt//dx = 2X +

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with dry oxidation data.• The experimental P_H2O dependence of

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/

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and

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contradicts the basic assumptions of the reaction-diffusion model.• The model fails to explain all technically important phenomena, such as: the nature of the fixed oxide charge, Q_f, the orientation dependence, the cross-over effect, the cleaning effects, the weak spots, the bird's beak and other 2D and 3D effects.Thermal oxidation of silicon is much better described by the extended Jorgensen model, i.e. the classical ionic-transport model modified to include non-linear conduction.• This concept leads to an excellently fitting power-parabolic growth law: X² + AX^2-α = Bt or B dt/dX = 2X + (2 - α)AX^1-α which holds for the growth data for dry and wet oxidation, including the initial part of the dry growth curves.• It gives the correct P_H2O dependence of A and B. It fully accounts for the generation of fixed oxide charge while it easily explains the Deal annealing triangle.• The model has remarkable potential to explain the orientation and cross-over effects, the cleaning effect, the weak spots, bird's beak and other 2D and 3D effects.     

A new algorithm to compute the discrete inverse radon transform

The aim of this paper is to compute the discrete inverse Radon transform over ⁿ . The Radon transform is a function with domainS ^n–1×. It is shown that under different measure this function can be defined with domain ⁿ . In this case one can compute the discrete inverse Radon transform in the Cartesian coordinate system without interpolating from polar to Cartesian coordinates or using the backprojection operator.     

Integral expressions for the numerical evaluation of product form expressions over irregular multidimensional integer state spaces

We consider stochastic systems defined over irregular, multidimensional, integer spaces that have a product form steady state distribution. Examples of such systems include closed and BCMP type of queuing networks, polymerization and genetic models. In these models the system state is a vector of integers, n=[n ₁,...,n _M ] and the steady state solution has product form of the type (n)= _i=1 ^M f _i (n _i ). To obtain useful statistics from such product form solutions, (n) has to be summed over some subset of the space over which it is defined. We consider situations when these subsets are defined by a set of equalities and inequalities with integer coefficients, as is most often the case and provide integral expressions to obtain these sums. Typically, a brute force technique to obtain the sum is computationally very expensive. Algorithmic solutions are available for only specific forms of f _i (n _i ) and shapes of the state space. In this paper we derive general integral expressions for arbitrary state spaces and arbitrary f _i (n _i ). The expressions that we derive here become especially useful if the generating functions f _i (n _i ) can be expressed as a ratio of polynomials in which case, exact closed form expressions can be obtained for the sums. We demonstrate the wide applicability of the integral expressions that we derive here through three examples in which we model finite highway cellular systems, copy networks in multicast packet switches and a BCMP queuing network modeling a multiuser computer system.