System identification in nonlinear structural seismic dynamics

This paper deals with the development of efficient identification methods for the determination of parameters associated with the nonlinear response of structural systems subject to seismic conditions. The equation of motion of the system is given by $\text{M}\text{u}\text{?}\text{+ h(/.u, u, a) = p(t)}$, where the external force vector $\text{p}$ and the mass matrix $\text{M}$ are assumed to be known. Also given is an observation vector $\text{w}$, consisting of some or all of the components of $\text{u}$, the vector of the nodal displacements. The vector function $\text{h}$, denoting the restoring forces of the mechanical system, is parametrically given in terms of a constant vector $\text{a}$. Three different methods are presented for the determination of the parameters governing the vector function $\text{h}$. The first one is a direct approach requiring the unknown coefficients to appear linearly in the model equation. The remaining two are based on methods of control and optimization theory, and are exempt from the limitations of the direct method. The relative merits of each approach are discussed and extensive numerical experimentation is presented. Only numerical results for one degree of freedom systems are reported in this paper.     

Characterizations for functional dependency and Boyce-Codd normal form families

A functional dependency (fd) family was recently defined [20] as the set of all instances satisfying some set of functional dependencies. A Boyce-Codd normal form, abbreviated BCNF, family is defined here as an fd-family specified by some BCNF set of functional dependencies. The purpose of this paper is to present set-theoretic/algebraic characterizations relating to both types of families.Two characterizations of $\text{F}$($\text{I}$), the smallest fd-family containing the family $\text{I}$ of instances, are established. The first involves the notion of agreement, a concept related to that of a closed set of attributes. The second describes $\text{F}$($\text{I}$) as the smallest family of instances containing $\text{I}$ and closed under four specific operations on instances. Companion results are also given for BCNF- families.The remaining results concern characterizations involving the well-known operations of projection, join and union. Two characterizations for when the projection of an fd-family is again an fd-family are given. Several corollaries are obtained, including the effective decidability of whether a projection of an fd-family is an fd-family. The problem for BCNF-families disappears since it is shown that the projection of a BCNF-family is always a BCNF-family. Analogous to results for fd-families presented in [20], characterizations of when the join and union of BCNF-families are BCNF-families are given. Finally, the collections of all fd-families and all BCNF-families are characterized in terms of inverse projection operations and intersection.     

Job sequencing with continuous machine operation

This paper presents the methods for minimizing the total processing time of the $\text{n-}\text{job}$, $\text{m-}\text{machine}$ sequencing problem. To commence working it must continue processing consecutive jobs without having to wait between 2 jobs. The problem can be formulated as a linear programming problem which requires a large number of variables and constraints. However, when the same job sequence is to be processed on each machine, a branch-and-bound procedure for solving the problem is proposed. The 2 methods are used to solve a sample problem and the results are given.     

Errors in reduction methods

A mathematical basis is given for comparing the relative merits of various techniques used to reduce the order of large linear and non-linear dynamics problems during their numerical integration. In such techniques as Guyan-Irons, path derivatives, selected eigenvectors, Ritz vectors, etc., the nth order initial value problem of $\text{[}\text{/.y}\text{=}\text{f(y)}$ for t > 0, y(0) given] is typically reduced to the mth order (m ? n) problem of $\text{z}\text{?}\text{= g(z)}$ for t > 0, z(0) given] by the transformation $\text{y}\text{=}\text{Pz}$ where P changes from technique to technique. This paper gives an explicit approximate expression for the reduction error e_i in terms of P and the Jacobian of f. It is shown that: (a) reduction techniques are more accurate when the time rate of change of the response y is relatively small; (b) the change in response between two successive stations contributes to the errors at future stations after the change in response is transformed by a filtering matrix H, defined in terms of P; (c) the error committed at a station propagates to future stations by a mixing and scaling matrix G, defined in terms of P, Jacobian of f, and time increment h. The paper discusses the conditions under which the reduction errors may be minimized and gives guidelines for selecting the reduction basis vector, i.e. the columns of P.     

Equilibrium strategies in dynamic games with multi-levels of hierarchy

This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, $\text{J}$1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of $\text{J}$1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, $\text{J}$2), which forces the remaining player to jointly minimize the cost function of $\text{J}$2 under the declared equilibrium strategy of $\text{J}$1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.     

Estimating the quality of computational algorithms — Part 1

Computational algorithms for approximate calculation of the mapping $\text{A:X → Y}$ on some element $\text{f}$ of the compactum $\text{X}$ are considered, where $\text{A}$ is a one-to-one continuous mapping. Topology on $\text{X}$ and $\text{Y}$ is set by imbedding them into some Banach spaces. The relation between $\text{?(N; X)}$ and $\text{?(N);Y}$ plays the main role in the problems concerning the construction of such computational algorithms. Here $\text{?(N);X)}$ is a function reciprocal to the ?-entropy of the compactum $\text{X}$. The computational algorithm $\text{U}$ is characterized first of all by the volume of the input data $\text{N}$ and that accuracy ? with which elements $\text{g}$ on the output are defined. If $\text{N}$ is given (but under condition that a table of any $\text{f}$ is admissible), then inf $\text{? = ?}{}_1\text{(N)}$ exists.For the given concrete algorithm the quantity $\text{?/?}{}_1\text{(N)}$ characterizes its qualities and is one of the most important characteristics of the algorithm. In this paper some of the simplest current algorithms are analyzed, and their quality is determined as well. A number of recommendations relative to the questions of the construction of computational algorithms is proposed.     

Improving loops and vanishing variables in long transportation problems

In this paper the following two results are presented: (1)A method which determines the optimal values of certain variables during the iterative solution process. The closer the current primal feasible solution is to the optimal solution, the greater the number of variables which may be determined. (2) For each current feasible solution (X_ij) of the given m × n transportation problem A, a feasible solution ($\text{X}\text{?}{}_{\mathrm{ij}}$) of an auxiliary m × m(m ?1) transportation problem $\text{A}\text{?}$ is constructed. Problem $\text{A}\text{?}$ is shown to be equivalent to an m(m ? 1) × m(m ? 1) assignment problem with two admissible cells per column. The optimally of (X_ij) is shown to imply the optimality of ($\text{X}\text{?}{}_{\mathrm{ij}}$) and conversely. The best “improving loops” (including the improving loops used in MODI) of $\text{A}\text{?}$ are shown to be the best “improving loops” of A as well.     

The complexity of monadic recursion schemes: exponential time bounds

We study the computational complexity of decision problems for the class $\text{M}$ of monadic recursion schemes. By the “executability problem” for a class 't of monadic recursion schemes, we mean the problem of determining whether a given defined function symbol of a given scheme in .'t can be called during at least one computation. The executability problem for a class $\text{I}$ of very simple monadic recursion schemes is shown to require deterministic exponential time. Using arguments about executability problems and about the class $\text{I}$, a number of decision problems for. $\text{M}$ and for several of. $\text{M}$'s subclasses are shown to require deterministic exponential time. Deterministic exponential time upper bounds are also presented for several of these decision problems.     

Reflectance of leaf components

The reflectance of leaf components was evaluated over the 370 to 1100 nm wavelength interval. Kodak high speed, black-and-white infrared photographs at 850 nm showed that: (1) Leaf epidermises of Elodea (Anacharis canandensis, Planch.) and Lemna L, diffused incoming infrared light; (2) infrared light was reflected from surfaces inside leaves of Rhoeo discolor Hance, through stomatal apertures; and (3) crystals and chloroplasts in the expressed sap of Zebrina pendula Schnizl. reflected infrared light. Scans (370 to 1 100 nm) showed that reflectance from complex cell walls of Agave americana L. compared with that of the adjacent cytoplasm was significantly greater ($\text{p=0.01}$) than the reflectance of simpler cell walls of Heliconia humile L. compared with that of the adjacent cytoplasm. The reflectance of Vicia faba L. nuclei was larger (significant, $\text{p=0.10}$) than that of adjacent cell areas. Results show that refractive index discontinuities in leaves cause the reflectance of near-infrared light.     

Surface drawing made simple

A new approach to plotting $\text{z=f(x,y)}$ from a table of $\text{z}$ over a regular x?y grid is explained in comparison with other hidden-line methods. Its application to drawing crosshatched and simple outline pictures of surfaces is described.     

Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves

Solving boundary value problems in plane elastostatics with the aid of integral equations the boundary of the elastic body usually represents the domain of definition of the sought solution as well as of the given boundary values. However, it is also possible to define the solution and the boundary values on two spatially separated curves $\text{S}\text{?}$ and S. In this paper the numerical properties of both methods are investigated and compared and in particular the influence of the distance between $\text{S}\text{?}$ and S is estimated. For these purposes the eigenvalues and eigenfunctions of the operators in the integral equations are determined.     

An algorithm for finding the intersection lines between two B-spline surfaces

A divide-and-conquer algorithm is presented for finding all the intersection lines between two B-spline surfaces. Each surface is organized as an $\text{n-}\text{branch}$ tree. For each intersection line, an initial point is detected after a depth-first search along one tree, ie the host tree. Extrapolation methods are then used to trace the entire length of the line, thus the line appears naturally in a continuous form. Efficiency of the algorithm is achieved by the employment of an adaptive division strategy and by the careful choice of the representation basis of the patches on both surfaces.     

Optimal input design for an AR model with output constraints

The output power constraint problem of optimal input design for parameter estimation for an autoregressive model is considered. A simple method to obtain an optimal design by solving two sets of $\text{p}\text{2-}\text{linear}$ simultaneous equations and a polynomial equation of $\text{p}\text{2}\text{th}$ order is proposed and two nontrivial examples are given to illustrate this methodology.     

Response time sensitivity measurement for computer systems and general closed queuing networks

A method is described for measuring the performance sensitivity variable $\text{S = (}\text{?T}\text{?μ}\text{) (}\text{μ}\text{T}\text{)}$ when T is a mean response time and μ is the processing rate of one device in a computer system, or one server in a closed or partly open queuing network. The derivative is with respect to a hypothetical change in μ, in a system operating on the identical job sequence. The measurement method is basically an algorithm for reducing operating data, and can be applied to statistically nonstationary systems, for which repeated experiments are difficult to interpret. No previous work on these lines is known to the author. The paper introduces the derivative $\text{?T}\text{?μ}$, shows it exists and describes an algorithm to obtain it; demonstrates it on simulated data; describes and tests an approximation, and briefly investigates the sampling error of S. The measured values show insignificant bias and are close to expected theoretical values computed from analytic queuing models.     

Solution errors in finite element analysis

The development of the finite element method so far indicates that it is a discretization technique especially suited for positive definite, self-adjoint, elliptic systems, or systems with such components. The application of the method leads to the discretized equations in the form of $\text{u}\text{?}\text{=}\text{f(u)}$, where u lists the response of the discretized system at n preselected points called nodes. Instead of explicit expressions, vector function f and its Jacobian f,_u are available only numerically for a numerically given u. The solution of $\text{u}\text{?}\text{=}\text{f(u)}$ is usually a digital computer. Due to finiteness of the computer wordlength, the numerical solution $\text{u}{}_{\mathrm{c}}$ is in general different from u. Let $\text{u}\text{(}\text{x}\text{, t)}$ denote the actual response of the system in continuum at points corresponding to those of u. In the literature. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}$ is called the discretization errors, $\text{u}\text{-}\text{u}{}_{\mathrm{c}}$ the round-off errors, and the s is. $\text{u}\text{(}\text{x}\text{, t)-}\text{u}{}_{\mathrm{c}}$ is called the solution errors. In this paper, a state-of-the-art survey is given on the identification, growth, relative magnitudes, estimation, and control of the components of the solution errors.     

Analysis of the detectability of faults by random test patterns in a special class of NAND networks

In this paper we first define the structure of a specific subclass of NAND networks, called $\text{C}$ networks. We then investigate the problem of detecting stuck-at faults in $\text{C}$ networks by applying a sequence of random input patterns. Analytic results are derived for: (1) the average number of faults detected by a random input, and (2) the average number of faults detected by a sequence of d random inputs. These analytic results compare quite favorably to experimental results obtained.     

A dynamic programming branch and bound algorithm for pure integer programming

Several algorithms for linear integer programming have been proposed in which one or more of the conditions are relaxed to produce a solvable problem form. Reimposing these relaxed conditions can lead to a branch and bound process. In this paper an alternative relaxation is proposed in which the integer conditions are maintained but the feasibility conditions are relaxed in a special way. Reimposing feasibility by sequentially setting variables creates the branch and bound process. In special cases, the algorithm reverts to the normal form of dynamic programming.The algorithm is applicable to both linear and non-linear pure integer problems. It has been programmed to solve pure $\text{0}\text{1}$ problems and results of computational experience with some linear problems previously used in the literature are given.     

On the stability of convolution feedback systems with dynamical feedback

This paper considers distributed n-inputn-output convolution feedback systems characterized by $\text{y = G}{}_1\text{1e}$, $\text{z = G}{}_2\text{1y}$ and e = u ? z, where the forward path transfer function $\text{G}\text{?}{}_1$ and the feedback path transfer function $\text{G}\text{?}{}_2$ both contain a real single unstable pole at different locations. Theorem 1 gives necessary and sufficient conditions for both input-error and input-output stability. In addition to usual conditions that guarantee input-error stability a new condition is found which results in the fact that input-error stability will guarantee input-output stability. These conditions require to investigate only the open-loop characteristics. A basic device is the consideration of the residues of different transfer functions at the open-loop unstable poles. An example is given.     

Efficient decomposition of separable algebras

We present new, efficient algorithms for computations on separable matrix algebras over infinite fields. We provide a probabilistic method of the Monte Carlo type to find a generator for the center of a given algebra $\text{A}\text{⊆}\text{F}{}^{\mathrm{m\times m}}$ over an infinite field $\text{F}$. The number of operations used is within a logarithmic factor of the cost of solving m×m systems of linear equations. A Las Vegas algorithm is also provided under the assumption that a basis and set of generators for the given algebra are available. These new techniques yield a partial factorization of the minimal polynomial of the generator that is computed, which may reduce the cost of computing simple components of the algebra in some cases.