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We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. Numerical results are shown for optimal control problems subject to a nonlinear heat equation.

Optimal Sobolev norms under volume preserving affine transformations are considered. It turns out that this minimal transform is equivalent to the ( p , 2 ) Fisher information matrix defined by Lutwak, Lv, Yang, and Zhang. Furthermore, some analytic inequalities regarding to the L p affine and L p sine energies for the optimal function are investigated.

In this paper, a novel Full Reference method is proposed for image quality assessment, using the combination of two separate metrics to measure the perceptually distinct impact of detail losses and of spurious details. To this purpose, the gradient of the impaired image is locally decomposed as a predicted version of the original gradient, plus a gradient residual. It is assumed that the detail attenuation identifies the detail loss, whereas the gradient residuals describe the spurious details. It turns out that the perceptual impact of detail losses is roughly linear with the loss of the positional Fisher information, while the perceptual impact of the spurious details is roughly proportional to a logarithmic measure of the signal to residual ratio. The affine combination of these two metrics forms a new index strongly correlated with the empirical Differential Mean Opinion Score (DMOS) for a significant class of image impairments, as verified for three independent popular databases. The method allowed alignment and merging of DMOS data coming from these different databases to a common DMOS scale by affine transformations. Unexpectedly, the DMOS scale setting is possible by the analysis of a single image affected by additive noise., Comment: 15 pages, 9 figures. Copyright notice: The paper has been accepted for publication on the IEEE Trans. on Image Processing on 19/09/2017 and the copyright has been transferred to the IEEE

Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.

In this paper, we deal with uniform- and arbitrary-time controllability problem of affine control systems on several Lie group. For this class of control systems, we establish uniform-time controllability properties on simply connected semisimple Lie groups. We also show that if the system is controllable, then an affine control system on compact Lie groups is controllable at uniform time. Finally, we show that arbitrary-time controllability of affine control systems on Carnot groups is established by relating to their associated bilinear part.

Formulations of multidimensional scaling (MDS) in dually flat spaces are proposed. First the space supposed in the classical MDS is extended to a tangent space around a specific point in a dually flat space. We see that Riemannian metric of the tangent point plays a key role in the extension. Next, in order to remove the restriction of symmetry in dissimilarities, the affine connection is incorporated. We pay attention to the fact that it is an affine connection term that causes an asymmetry in dissimilarities in infinitesimal space. To mitigate the difficulty in treating the affine connection term, an approximation is shown and we can see the effect of the affine connection term to modify the effective Riemannian metric. Finally a numerical example is shown.

This paper develops a novel stability analysis and robust controller design method for affine fuzzy systems. The emphasis of the paper is to present more relaxed stability conditions based on nonquadratic fuzzy Lyapunov function and affine parallel distributed compensation. At first, diffeomorphic transformations are used to treat more general class of nonlinear systems in a unified manner. Then, by introducing slack matrices, the Lyapunov matrices are decoupled from the feedback gain matrices and controller affine terms which lead to eliminate the structural constraints of Lyapunov matrices and consequently reduces the conservativeness of the proposed approach. Because of the bias terms, the stabilization conditions are obtained in terms of bilinear matrix inequalities. A nonsingular state transformation together with using the S-procedure and also slack variables lead to derive the stabilization conditions in the formulation of linear matrix inequalities which can be solved by convex optimization techniques. Moreover, H ∞ controller is used to reject the disturbances. Finally, the merit and applicability of the proposed approach are demonstrated via comparative numerical and industrial case studies.

We present a numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach constructs a partition of the state space, called a triangulation, and then computes values at the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous and Piecewise Affine (CPA) function. Verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Numerical examples are presented demonstrating different aspects of the proposed method.

We consider affine Markov processes taking values in convex cones. In particular, we characterize all affine processes taking values in irreducible symmetric cones in terms of certain Levy–Khintchine triplets. This is the natural, coordinate-free formulation of the theory of Wishart processes on positive semidefinite matrices, as put forward by Bru (J Theor Probab 4(4):725–751, 1991) and Cuchiero et al. (Ann Appl Probab 21(2):397–463, 2011), in the more general context of symmetric cones, which also allows for simpler, alternative proofs.

The goal of this chapter is to solve a concentration of energy problem associated with the Special Affine Fourier Transformation (SAFT). Since an explicit, closed form solution seems to be elusive, we will solve the problem numerically. The problem can be reduced to finding the largest eigenvalues and their associated eigenfunctions of two-dimensional integral equations. The numerical solutions are obtained by using the Gaussian quadrature method in two dimensions. We use also the Gaussian quadrature method in two dimensions to solve concentration of energy problems in other cases, including a problem for kernels of convolution type, and to compute the so-called generalized prolate spheroidal wave functions (GPSWF).

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering.. In this work, the notion of the quarternary generalized multiresol- ution structure (TGMS) of subspace is proposed. The characteristics of quarternary affine pseudoframes for subspaces is investigated. Construction of a GMS of Paley-Wiener subspace of is studied. The pyramid decomposition scheme is obtained based on such a TGMS and a sufficient condition for its existence is provided. A constructive method for affine frames of based on a TGMS is established.

The existence of affine-periodic solutions is studied. These types of solutions may be periodic, harmonic or even quasi-periodic. Mainly, via the topological degree theory, a general existence theorem is proved, which asserts the existence of affine-periodic solutions, extending some classical results. The theorem is applied to establish the Lyapunov function type theorem and the invariant region principle relative to affine-periodic solutions.

In geospatial applications, the datum transformation has been necessarily employed to transform the geospatial outcomes from the data-collection system to the user-interested system. Its key is to compute the transformation parameters that describe the geometric relation between two datum systems. The ordinary least-squares based transformation parameter estimation needs the iterative computations unless the initial values of parameters are approximate enough, which is usually time-consuming. Particularly with the development of (near) real-time data collection techniques, such iterative datum transformation method cannot meet the real-time applications. In this paper, we study the noniterative method in terms of the multivariate least-squares theory with two-dimensional empirical affine transformation as a case study. We address the noniterative transformation for the partially and fully error-affected affine models, respectively. The study indicates that the noniterative solution exists when the...

This note proposes a method to analyze uniform asymptotic stability and uniform ultimate boundedness of uncertain piecewise affine systems whose dynamics are only defined in a bounded and possibly non-invariant set X of states. The approach relies on introducing fake dynamics outside X and on synthesizing a piecewise affine and possibly discontinuous Lyapunov function via linear programming. The existence of such a function proves stability properties of the original system and allows the determination of a region of attraction contained in X. The procedure is particularly useful in practical applications for analyzing the stability of piecewise affine control systems that are only defined over a bounded subset X of the state space, and to determine whether for a given set of initial conditions the trajectories of the state vector remain within the domain X.

We study improper affine spheres with some admissible singularities, called improper affine maps, and associated to the unimodular Hessian equation. In particular, we characterize when a curve of R 3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails. Also, we consider improper affine maps with isolated singularities and show some similarities and differences between the Hessian +1 equation and the Hessian −1 equation. As a consequence, we construct global examples with the desired singularities.

In this paper, we derive transform formulae for linear functionals of affine processes and their bridges whose state space is the set of positive semidefinite d × d matrices. Particularly, we investigate the relationship between such transforms and certain integral equations. Our findings extend and unify the well known results of Cuchiero et al. (2011) [5] and Pitman and Yor (1982) [19] , who analysed affine processes on positive semidefinite matrices and transforms of linear functionals of squared Bessel processes, respectively. We are, then, able to derive analytic expressions for Laplace transforms of some functionals of Wishart bridges.

Phase stability analysis is a crucial step in the determination of multiphase equilibrium. This analysis by the tangent plane distance (TPD) minimization is a well-known technique, as well as the difficulties in providing guarantees that the global minimum has been found. On this regard, interval methods are powerful tools since they provide such guarantees. In this work, an interval Newton method plus generalized bisection, based on a modified affine arithmetic, is used to reliably find all possible stationary points of the TPD function. Additionally, an improved convergence test is suggested as well as a special treatment for mole fraction weighted averages. Several mixtures with up to 5 components, including LLE island type ternary systems, were studied. Both activity coefficient models and cubic equations of state were considered. For all the cases tested, the proposed modified affine arithmetic method was superior to other interval-based methods.

Affine transformation that allows the axis-specific rotations and scalars to capture the more transformation details has been extensively applied in a variety of geospatial fields. In tradition, the computation of affine parameters and the transformation of non-common points are individually implemented, in which the coordinate errors only of the target system are taken into account although the coordinates in both target and source systems are inevitably contaminated by random errors. In this article, we propose the seamless affine error-in-variables EIV transformation model that computes the affine parameters and transforms the non-common points simultaneously, importantly taking into account the errors of all coordinates in both datum systems. Since the errors in coefficient matrix are involved, the seamless affine EIV model is nonlinear. We then derive its least squares iterative solution based on the Euler–Lagrange minimization method. As a case study, we apply the proposed seamless affine EIV model to the map rectification. The transformation accuracy is improved by up to 40%, compared with the traditional affine method. Naturally, the presented seamless affine EIV model can be applied to any application where the transformation estimation of points fields in the different systems is involved, for instance, the geodetic datum transformation, the remote sensing image matching, and the LiDAR point registration.

We give a discrete geometric (differential-free) proof of the theorem underlying the solution of the well known Fermat–Torricelli problem, referring to the unique point having minimal distance sum to a given finite set of non-collinear points in d-dimensional space. Further on, we extend this problem to the case that one of the given points is replaced by an affine flat, and we give also a partial result for the case where all given points are replaced by affine flats (of various dimensions), with illustrative applications of these theorems.

The main contribution of this paper is to present a novel robust observer-based controller design method for discrete-time piecewise affine systems with norm-bounded uncertainties. The key ideas are to construct a piecewise-quadratic Lyapunov function to guarantee the stability of the closed-loop systems, approximate polytopic operating regions by ellipsoids, and use the singular value decomposition technique to treat the constraint of matrix equality. It is shown that the suggested control method can be formulated as linear matrix inequalities that are numerically feasible with commercially available software. A numerical example is also given to verify the proposed approach.

In this paper, global input-to-state stability (ISS) for discrete-time piecewise affine systems with time-delay are considered. Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhin and Lyapunov-Krasovskii methods are used. The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown, respectively.

Scale Invariant Feature Transform (SIFT) is a method to detect and match invariant feature points on images, and is robust against contrast, rotation, and scale changes. However, SIFT cannot find many correct matching points between affine transformed images because this method employs Gaussian function for scale parameter which specifies a circle area on image planes. In this paper, we propose a method to use Bi-dimensional Empirical Mode Decomposition (BEMD) for keypoint detection, where a given image is decomposed into Intrinsic Mode Functions (IMFs). Our method also employs Affine Moment Invariants (AMIs) instead of SIFT's feature values. As a result, the proposed method detects more matching points than SIFT in a steep affine transformed image.

This paper studies output statistics of an adaptive line enhancer that is based on an affine combination of two NLMS adaptive filters. Combination of adaptive filters is a new interesting way of improving the performance of adaptive algorithms. The structure consists of two adaptive filters that adapt on the same input signal, one with a large and the other one with a small step size. Such a combination is capable of achieving fast initial convergence and small steady state error at the same time. In this paper we investigate the second order statistics of the output signal of adaptive line enhancer based on the combination in steady state. The result is given in terms of the parameters of the adaptive combination, input process statistics, and the optimal Wiener filter weights for the problem at hands.

The notion of complex equiaffine manifolds is an affine generalization of Calabi–Yau manifolds. Similarly as in the Riemannian case the minimality of affine Lagrangian submanifolds in complex equiaffine spaces can be studied via calibrations, phase functions and variational formulas.

This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques.

Problem statement: In this study, a new descriptor for shape retrieval under affine transformations had been proposed; the affine length parameterization was used in this approach. Approach: The zero crossing of affine curvature was used to extract a descriptor of shapes in database. For each two successive zero crossing points we extract two parameters: the first parameter was a maximal surface area of the triangle constructed by these points and a point between them. The second parameter is the average of affine curvature of the arc constructed by these two points. Results: The method was also evaluated objectively through a three classified databases which are a subset of the MCD database with a vast variety of shapes. The obtained results and the comparison with geometric moment and Fourier descriptors indicated a promising performance of our method. Conclusion: In this study, we examined the performance of the proposed descriptor under affine transformations. It was observed that the performance of the proposed method is promising even under severe deformations caused by shear.

In this paper we present a general formalism for the establishment and mean-square performance analysis of the family of selective partial update affine projection (SPU-AP), selective regressor affine projection (SR-AP), and selective partial update subband adaptive filter (SPU-SAF) algorithms. This analysis is based on energy conservation arguments and does not need to assume a Gaussian or white distribution for the regressors. We demonstrate through simulations that the results are useful in predicting the performance of these adaptive filter algorithms.

Two modifications of Newton’s method to accelerate the convergence of the nth root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p∈N,p≥2. We consider affine combinations of the two modified pth-order methods which lead to a family of methods of order p with arbitrarily small asymptotic constants. Moreover the methods are of order p+1 for some specific values of a parameter. Then we consider affine combinations of the three methods of order p+1 to get methods of order p+1 again with arbitrarily small asymptotic constants. The methods can be of order p+2 with arbitrarily small asymptotic constants, and also of order p+3 for some specific values of the parameters of the affine combination. It is shown that infinitely many pth-order methods exist for the nth root computation of a strictly positive real number for any p≥3.

This work, proposes a robust digital watermarking scheme based on feature points as a defense against illegal user attacks, such as, rotating, scaling and translation. First of all, a Gaussian filter is first adopted to discover the invariant feature points. The local maximum and the local minimum values are then calculated from those feature points. The Affine Invariant Region (AIR) is then divided by combination the values: local maximum, local minimum values and the center of mass. The AIR region is then transformed using affine transform to a block image, called the normalized image, in which the size of the block image is predefined. The watermarks are then embedded into the normalized image. Experimental results indicate that the proposed method can still extract the correct watermark, even after a geometric attack. The proposed method has a lower false positive rate and higher accuracy than Lu's [12] that making it a robust blind watermarking scheme.

In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM.

Let $S_1$ and $S_2$ be two affine semigroups, and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.

Several pattern-matching techniques have focused on affine invariant pattern matching, mainly because rotation, scale, translation, and shear are common image transformations. In some situations, other transformations may be modeled as a small deformation on top of an affine transformation. This work presents an algorithm which aims at improving existing Fourier Transform (FT)-based pattern matching techniques in such a situation. The pattern is first decomposed into non-overlapping concentric circular rings, which are centered in middle of the pattern. Then, the FT of each ring is computed. Essentially, adding the individual complex-valued FTs provides the overall FT of the pattern. Past techniques used the overall FT to identify the parameters of the affine transformation between two patterns. In this work, it is assumed that the rings may be rotated with respect to each other, thus, parameters of transformations beyond the affine ones can be computed. The proposed method determines this variable angle of rotation starting from the FT of the outermost ring and moving inwards to the FT of the innermost ring. The variable angle of rotation provides information about the directional properties of a pattern. Two methods are investigated, namely a dynamic programming algorithm and a greedy algorithm, in order to determine the variable angle of rotation. The intuition behind this approach is that since the rings are not necessarily aligned in the same manner for different patterns, their ring FTs may also be rotated with respect to each other. Simulations demonstrate the effectiveness of the proposed technique.

In this chapter, I outline the GATSM framework, explain how GATSMs may be estimated from yield curve data, and use estimation results to detail the practical pitfalls of applying them in ZLB environments. The outline first serves to establish the present standard for term structure modeling, given that GATSMs are used extensively by researchers and practitioners, as already discussed in chapter 2. I also use GATSMs to represent the shadow term structure in the shadow/ZLB-GATSMs in the remainder of the book, so it is important to establish the precise notation and estimation methods for GATSMs that will carry over to those shadow/ZLB-GATSMs. Related to that point, I can then present the shadow/ZLB-GATSM framework and its estimation in chapter 4 as a relatively tractable modification to the GATSM class.

This paper deals with the robust H ∞ filter design problem for a class of nonlinear stochastic systems which are modeled by uncertain Takagi-Sugeno (T-S) fuzzy affine Ito stochastic models affected by a multidimensional Wiener process. The objective is to design an admissible full-order filter so that the resulting stochastic filtering error system is mean-square asymptotically stable with a prescribed disturbance attenuation level in an H ∞ sense. It is assumed that the plant premise variables are not necessarily measurable so that the filter implementation with state space partition may be not synchronized with the state trajectories of the plant. Based on piecewise quadratic Lyapunov functions, some new results are proposed for the filtering design of T-S fuzzy affine Ito stochastic systems. The filter gain of each region can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, two simulation examples are provided to illustrate the effectiveness of the proposed filtering design approach.

This paper proposes a new image smoothing method using a metric tensor for affine invariant scale space. In the field of image processing and recognition, Gaussian filtering is a common procedure for image smoothing. For example, scale space construction based on Gaussian filtering is sometimes used as a preprocessing of various image processing tasks. However Gaussian filtering is not affine invariant. This paper proposes a new method for image smoothing that is invariant under such affine transformation that does not change the area of any region in the image. It is shown that a scale space representation can be constructed collaterally with the image smoothing. Experimental results show that the proposed method is almost never affected by affine transformation different from usual Gaussian filtering. In the proposed method, processing results are expected to be not affected much by variation of the viewpoint.

This paper investigates necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz (or Schur) stability of a polytope of matrices. A systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function. Necessity is asymptotically attained through a relaxation based on a generalization of Polya’s Theorem to the case of matrix valued functions. Numerical experiments illustrate the results.

In this paper, illumination-affine invariant methods are presented based on affine moment normalization techniques, Zernike moments, and multiband correlation functions. The methods are suitable for the illumination invariant recognition of 3D color texture. Complex valued moments (i.e., Zernike moments) and affine moment normalization are used in the derivation of illumination affine invariants where the real valued affine moment invariants fail to provide affine invariants that are independent of illumination changes. Three different moment normalization methods have been used, two of which are based on affine moment normalization technique and the third is based on reducing the affine transformation to a Euclidian transform. It is shown that for a change of illumination and orientation, the affinely normalized Zernike moment matrices are related by a linear transform. Experimental results are obtained in two tests: the first is used with textures of outdoor scenes while the second is performed on the well-known CUReT texture database. Both tests show high recognition efficiency of the proposed recognition methods.

Affine projection algorithms are useful adaptive filters whose main purpose is to speed the convergence of LMS-type filters. Most analytical results on affine projection algorithms assume special regression models or Gaussian regression data. The available analysis also treat different affine projection filters separately. This paper provides a unified treatment of the mean-square error, tracking, and transient performances of a family of affine projection algorithms. The treatment relies on energy conservation arguments and does not restrict the regressors to specific models or to a Gaussian distribution. Simulation results illustrate the analysis and the derived performance expressions.

In this paper we present various algorithms both for stability and performance analysis of discrete-time Piece-Wise Affine (PWA) systems. For stability, different classes of Lyapunov functions are considered and it is shown how to compute them through Linear Matrix Inequalities that take into account the switching structure of the systems. We also show that the continuity of the Lyapunov function is not required in the discrete-time case. Moreover, the tradeoff between the degree of conservativeness and computational requirements is discussed. Finally, by using arguments from the dissipativity theory for nonlinear systems, we generalize our approach to solve $H_infty$ and generalized $H_2$ analysis problems.

Previous research [Appl. Opt.52, A290 (2013)] has revealed that Fourier analysis of three-dimensional affine transformation theory can be used to improve the computation speed of the traditional polygon-based method. In this paper, we continue our research and propose an improved full analytical polygon-based method developed upon this theory. Vertex vectors of primitive and arbitrary triangles and the pseudo-inverse matrix were used to obtain an affine transformation matrix representing the spatial relationship between the two triangles. With this relationship and the primitive spectrum, we analytically obtained the spectrum of the arbitrary triangle. This algorithm discards low-level angular dependent computations. In order to add diffusive reflection to each arbitrary surface, we also propose a whole matrix computation approach that takes advantage of the affine transformation matrix and uses matrix multiplication to calculate shifting parameters of similar sub-polygons. The proposed method improves hologram computation speed for the conventional full analytical approach. Optical experimental results are demonstrated which prove that the proposed method can effectively reconstruct three-dimensional scenes.

We present an affine multifactor Gaussian term structure model, where we let factors be principal components of yields. Taking forward the affine-yield idea presented in Duffie and Kan (1996) we derive general expressions for the coefficients of a multidimensional affine Gaussian stochastic process given an exogenous matrix of loadings of selected yields on the state variables.

We prove that any countably piecewise affine homeomorphism from an open set of ℝ2 can be approximated, together with its inverse, by diffeomorphisms in the W 1,p and the L ∞ norms.

The stabilization problems are considered for program motion of affine systems. To solve these problems we transform affine system into canonical form in a neighborhood of program trajectory. The stabilising control is designed by the method of feedback linearization. The closed loop system is linear one and we can fix the Lyapunov function. Then we consider saturated feedback control and corresponding closed loop system. To estimate the stabilisation domain of program trajectory under saturated control we use the fixed Lyapunov function for linear system.

Frequently, affine recurrence equations can be scheduled more efficiently by quadratic scheduling functions than by linear scheduling functions. In this paper, the problem of finding optimal quadratic schedules for affine recurrence equations is formulated as a convex nonsmooth programming problem. In particular, sufficient constraints for causality are used generalizing Lamport's condition. In this way, the presented problem formulation becomes independent of the problem size. The research tool AQUAD is described implementing this problem formulation. Several nontrivial examples demonstrate that AQUAD can be effectively used to calculate quadratic schedules for affine recurrence equations. Finally, it is shown how array processors can be synthesized from affine recurrence equations which are scheduled by quadratic functions with a singular Hessian matrix.

A wide variety of problems in system and control theory can be formulated or reformulated as convex optimization problems involving linear matrix inequalities (LMIs), that is, constraints requiring an affine combination of symmetric matrices to be positive semidefinite. For a few very special cases, there are analytical solutions to these problems, but in general LMI problems can be solved numerically in a very efficient way. Thus, the reduction of a control problem to an optimization problem based on LMIs constitutes, in a sense, a solution to the original problem. The objective of this article is to provide a tutorial on the application of optimization based on LMIs to robust control problems. In the first part of the article, we provide a brief introduction to optimization based on LMIs. In the second part, we describe a specific example, that of the robust stability and performance analysis of uncertain systems, using LMI optimization.

The relations to curvatures of a Riemannian map to be affine are investigated. Also some splitting theorems related to Riemannian maps and curvatures are obtained.

We find a representative set of left cells of the affine Weyl groupWaof typeC4as well as its left cell graphs by applying an algorithm. This algorithm was designed in my previous paper (Tôhoku J. Math.46,1994, 105–124). It is reformulated and improved in a more efficient form here. These representatives of left cells are presented as the vertices of so-called essential graphs so that the generalized τ-invariants of left cells ofWaare actually described explicitly, the latter almost characterize the left cells ofWa. Some comments and conjectures are proposed on cells of affine Weyl groups, mostly for the case of typeCl,l≥2.