Indeterminate Problem Research Articles

An algorithm to allow humerus translation in the indeterminate problem of shoulder abduction

The shoulder is one of the most complex joints of the human body, mainly because of its large range of motion, but also because of its active muscular stabilisation. Actually, the numerous stabilizing muscles and degrees of freedom yield indeterminate biomechanical models. To solve this indeterminate, most models use reverse dynamics with a simplified ball-socket joint, preventing therefore the natural humerus translation. In this paper, an algorithm was specifically developed to solve the indeterminate problem by a feedback control of muscle activation, allowing the natural humerus translation. Abduction was considered in the scapular plane, accounting for the three deltoid parts and the rotator cuff muscles. The major aim of this study was to validate the numerical algorithm, which was here restricted to two-dimensions in order to compare the numerical solution to a known algebraic one. This comparison gave a relative error below 0.1%. The joint reaction force was comparable to other models and the humerus translation was in agreement with in vivo or in vitro studies.

Self-Adjoint Difference Operators and Symmetric Al-Salam–Chihara Polynomials

The symmetric Al-Salam–Chihara polynomials for q > 1 are associated with an indeterminate moment problem. There is a self-adjoint second-order difference operator on l2(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christiansen and Ismail using a different method, and we give an explicit orthogonal basis for the corresponding weighted l2-space. In particular, the orthocomplement of the polynomials is described explicitly. Taking a limit we obtain all the N-extremal solutions to the q-1-Hermite moment problem, a result originally obtained by Ismail and Masson in a different way. Some applications of the results are discussed.

Indeterminacy of reverse engineering of Gene Regulatory Networks: the curse of gene elasticity.

BackgroundGene Regulatory Networks (GRNs) have become a major focus of interest in recent years. A number of reverse engineering approaches have been developed to help uncover the regulatory networks giving rise to the observed gene expression profiles. However, this is an overspecified problem due to the fact that more than one genotype (network wiring) can give rise to the same phenotype. We refer to this phenomenon as “gene elasticity.” In this work, we study the effect of this particular problem on the pure, data-driven inference of gene regulatory networks.MethodologyWe simulated a four-gene network in order to produce “data” (protein levels) that we use in lieu of real experimental data. We then optimized the network connections between the four genes with a view to obtain the original network that gave rise to the data. We did this for two different cases: one in which only the network connections were optimized and the other in which both the network connections as well as the kinetic parameters (given as reaction probabilities in our case) were estimated. We observed that multiple genotypes gave rise to very similar protein levels. Statistical experimentation indicates that it is impossible to differentiate between the different networks on the basis of both equilibrium as well as dynamic data.ConclusionsWe show explicitly that reverse engineering of GRNs from pure expression data is an indeterminate problem. Our results suggest the unsuitability of an inferential, purely data-driven approach for the reverse engineering transcriptional networks in the case of gene regulatory networks displaying a certain level of complexity.

Optimal weight design problem of elastic structure by GA

Abstract In this paper, we formulate an optimal weight design (OWD) problem of elastic frame structure for a constrained relative story displacement as a statically indeterminate structure problem and solve it directly by keeping the constraints based on an improved genetic algorithm (iGA). We discuss the efficiency between the traditional optimum criteria methods and the proposed method.

An indeterminate rational moment problem and Carathéodory functions

Let { α n } n = 1 ∞ be a sequence of points in the open unit disk in the complex plane and let B 0 = 1 and B n ( z ) = ∏ k = 0 n α k ¯ | α k | α k - z 1 - α k ¯ z , n = 1 , 2 , … , ( α k ¯ / | α k | = - 1 when α k = 0 ). We put L = span { B n : n = 0 , 1 , 2 , … } and we consider the following “moment” problem: Given a positive-definite Hermitian inner product 〈 · , · 〉 in L , find all positive Borel measures ν on [ - π , π ) such that 〈 f , g 〉 = ∫ - π π f ( e i θ ) g ( e i θ ) ¯ d ν ( θ ) for f , g ∈ L . We assume that this moment problem is indeterminate. Under some additional condition on the α n we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞ .

Statically indeterminate contacts in axially loaded wire strand

This paper is concerned with the analysis of simple wire strands made from one layer of circular wires helically wrapped around a circular straight core wire. The undeformed configuration is the special case in which each of the surrounding helical wires touches its two adjacent neighbouring helical wires as well as touching the core wire. This forms a statically indeterminate contact problem. A simple finite element model is developed to analyse this situation. Numerical results show that contacts can occur simultaneously at all possible contact points when the strand is subjected to extension with both ends fixed against rotation.

Power transmission for gastrointestinal microsystems using inductive coupling

Much research on the development of a robotic capsule and microrobot for the diagnosis of gastrointestinal diseases has been carried out. The powering of these microsystems is becoming very important as the implementation of such systems is becoming more feasible by advances in relevant technologies. A wireless power transmission system employing two coils, based on inductive coupling, is discussed in this note. The coupling coefficient between the two coils was measured on axial, lateral and pitch misalignment. The power transmission efficiency was deduced and optimized by selecting proper tuning capacitors and transmission frequency. A serial resonant circuit (SRC) on a receiving side is more adaptive by comparison with parallel resonant circuit (PRC) topology. A multi-receiving coil structure was proposed and examined in an attempt to resolve the orientation indetermination problem of a receiving coil. Experiments show that a dc power level up to 170 mW, with an efficiency of 1.3%, could be steadily received when the receiving coil was placed at the center of the transmitting coil, which can meet the requirement for electric power of some microsystems.

A Family of Heat Functions as Solutions of Indeterminate Moment Problems

We construct a family of functions satisfying the heat equation and show how they can be used to generate solutions to indeterminate moment problems. The following cases are considered: log-normal, generalized Stieltjes-Wigert, andq-Laguerre.

“Organized crime” and “organized crime”: Indeterminate problems of definition

Despite decades of effort, the search for a universal definition of organized crime has eluded both academics, criminal justice agencies, as well as international bodies. More than two decades ago, a content analysis of such definitional efforts by this writer (Hagan, 1983) noted that, while many writers, including those of textbooks, failed to supply explicit definitions of organized crime, some consensus was apparent. These earlier findings are explored and compared with updated content analyses of American criminology and criminal justice textbooks and organized crime textbooks. Also discussed are definitions offered by criminal justice agencies and those by international organizations. A distinction is made between “Organized Crime” groups and “organized crime,” activities by groups that are organized.

Self-adjoint difference operators and classical solutions to the Stieltjes–Wigert moment problem

The Stieltjes–Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted L 2 -spaces. Under some additional assumptions these measures are exactly the solutions to the q-Pearson equation. In the case of discrete and absolutely continuous measures the difference operator is essentially self-adjoint, and the corresponding spectral decomposition is given explicitly. In particular, we find an orthogonal set of q-Bessel functions complementing the Stieltjes–Wigert polynomials to an orthogonal basis for L 2 ( μ ) when μ is a discrete orthogonality measure solving the q-Pearson equation. To obtain the spectral decomposition of the difference operator in case of an absolutely continuous orthogonality measure we use the results from the discrete case combined with direct integral techniques.

Inference in the indeterminate parameters problem

We face an indeterminate parameters problem when there are two sets of parameters, x and g, say, such that the null hypothesis H0:x=x0 makes the likelihood independent of g. A consequence of indeterminacy is the singularity of the information matrix. For this problem the standard results, such as the asymptotic chi-squared distribution of the Wald test statistic, are generally false. In the paper we propose an estimator of the parameters of interest, x, so that a Wald-type test statistic can be used for testing H0. Such an estimator is obtained through the maximization of a modified (penalized) log-likelihood function. We show that a solution to the (penalized) likelihood equation is consistent and asymptotically normally distributed with variance-covariance matrix approximated by the Moore-Penrose pseudoinverse of the information matrix. These properties allow one to construct a Wald-type test statistic useful for inferential purposes.

NONLINEAR AND WAVE THEORY CONTRIBUTIONS OF T. BROOKE BENJAMIN (1929–1995)

Brooke Benjamin's original theories of fluid mechanical phenomena changed our basic understanding of cavitation bubbles, surface and internal waves, gravity currents, instabilities of shear flow over flexible surfaces, and swirling flows. For some types of finite-amplitude wave phenomena, he generated integral constraints and derived new partial differential equations; by establishing their general properties he showed how they have wide application. He developed a complementary approach based on functional analysis that was quite new to fluid mechanics. He demonstrated methods for deriving, without detailed calculation, the essential features of nonlinear and indeterminate flow problems that are otherwise intractable.

Some Cubic Birth and Death Processes and their Related Orthogonal Polynomials

The orthogonal polynomials with recurrence relation (λ,n +μn-z)Fn(z) = μn+1Fn+1(z)+λn-1Fn-1(z) with two kinds of cubic transition rates λn and μn, corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.

Preface

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna theory. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.

From Asymptotics to Spectral Measures: Determinate Versus Indeterminate Moment Problems

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna parametrization. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of the Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.

Orthogonal polynomials and analytic functions associated to positive definite matrices

For a positive definite infinite matrix A, we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation A n of size n × n . For the particular case of A being a Hankel or a Hankel block matrix, our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems, respectively.

Determination of a New Set of Lipophilicity Fragmental Contributions Highlights the Problem of Indetermination Due to Two Phenomena: Fragmental Induction and Fragmental Isofrequency

The principle of determining lipophilicity contribution values fi of atomic fragments assumes the additive character of the logarithm of the partition coefficient of a molecule to be log P=Σ aifi, where ai is a numerical factor indicating the frequency of a given atomic fragment (i) in the structure i.e. the number of times this atomic fragment appears in the molecule. However, two phenomena induce unresolved fi values due to the generation of identical equations. We propose to name these phenomena “fragmental isofrequency” and “fragmental induction”.

Mixed forward and inverse solutions in movement biomechanics

There are many more muscles crossing any given joint than there are degrees of freedom prescribing joint movement. One consequence of this arrangement is that muscle forces cannot be determined uniquely. This is the indeterminate problem in biomechanics, and virtually all attempts to solve it are based on either the inverse- or the forward-dynamics method in mechanics. The inverse method uses measurements of body motions and external forces to determine the net muscle torques acting about each joint. The forward approach uses net muscle torques (or muscle forces) as inputs to the model equations to calculate the corresponding body motions. Although both methods have their strong points, there are instances when neither is suitable if applied alone. One possibility is to combine the two methods, thereby exploiting the advantages of each. An example is given in which forward dynamics is used to calculate muscle forces in normal walking, and knee-ligament forces are then found by applying the inverse approach.

On the prediction of passive contact forces of workpiece-fixture systems

The prediction of passive forces in a frictional workpiece-fixture system is an important problem, since the contact forces have a strong influence on clamp design and on workpiece accuracy during machining. This paper presents a general method for the computation of passive contact forces. Firstly, an indeterminate system of static equilibrium is defined, in which the passive, frictional contact forces cannot be determined arbitrarily as in an actively controlled robotic multifinger grasp. Then, a locally elastic contact model is used to describe the non-linear coupling between the contact forces and elastic deformations at the contact point. This model captures the essence of the passive contact. Further, a set of ‘compatibility’ equations is given so that the relationship can be developed between the elastic deformations at all contacts and the displacement of the workpiece. Finally, combining the force equilibrium, the locally elastic contact model and the ‘compatibility’ conditions, the passive force computation problem is transformed into a determinate system of non-linear equations governing all of the elastic deformations at all of the passive contacts. By solving the resulting non-linear equations, all passive contact forces can be accurately predicted in the frictional workpiece-fixture system. This method is illustrated with example cases. The method presented here may also have an application to other passive, indeterminate problems such as power grasps in robotics.

Evaluation of Probabilistic Point Estimate Methods in Uncertainty Analysis for Environmental Engineering Applications

This paper begins with a comprehensive review of various point estimate methods, with an emphasis on their differences and similarities in theory and application. The Rosenblueth probabilistic point estimate method is a computationally straightforward technique for the uncertainty analysis of engineering problems. It is capable of estimating a statistical moment of any order of a model output involving several stochastic variables that are correlated or uncorrelated, symmetric, or asymmetric. However, in multivariate problems with more than two stochastic variables involved, the Rosenblueth method is not able to provide a unique solution, rather an approximate solution to indeterminate problems. This is attributed to the fact that the number of unknowns to be solved is larger than the number of governing equations provided. An improved modified Rosenblueth point estimate method is proposed to circumvent the drawback of the nonunique solution of the original Rosenblueth method and to increase the computational efficiency in modeling. One example application on the particle terminal velocity computation is presented for illustration. A quantitative performance index is introduced to assess the performance of various point estimate methods. It is concluded in this study that the modified Rosenblueth method has a comparable performance to the Rosenblueth method and yet resolves the nonuniqueness problem in solutions.