Moment Constraints Research Articles

Muscle forces predicted using optimization methods are coordinate system dependent

Optimization methods are widely used to predict in vivo muscle forces in musculoskeletal joints. Moment equilibrium at the joint center (usually chosen as the origin of the joint coordinate system) has been used as a constraint condition for optimization procedures and the joint reaction moments were assumed zero. This study, through the use of a three-dimensional elbow model, investigated the effect of coordinate system origin (joint center) location on muscle forces predicted using a nonlinear static optimization method. The results demonstrated that moving the origin of the coordinate system medially and laterally along the flexion–extension axis caused dramatic variations in the predicted muscle forces. For example, moving the origin of the coordinate system from a position 5 mm medial to 5 mm lateral of the geometric elbow center caused the predicted biceps force to vary from 12% to 46% and the brachialis force to vary from 80% to 34% of the total muscle loading. The joint reaction force reduced by 24% with this medial to lateral variation of the coordinate system origin location. This data revealed that the muscle forces predicted using the optimization method are sensitive to the coordinate system origin location due to the zero joint reaction moment assumption in the moment constraint condition. For accurate prediction of muscle load distributions using optimization methods, it is necessary to determine the accurate coordinate system origin location where the condition of a zero joint reaction moment is satisfied.

Entropic approximation in kinetic theory

Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys. 83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global existence theorem for such an approximation and derive as a by-product a necessary and sufficient condition for the so-called problem of moment realizability . Applications of the above result are given in kinetic theory: first in the context of Levermore's approach and second to design generalized BGK models for Maxwellian molecules.

Information-Theoretic Distribution Tests with Application to Symmetry and Normality

We derive distribution free tests based on the Maximum Entropy densities to test the null hypotheses of symmetry and normality. The proposed tests are derived from maximizing the difierential entropy subject to moment constraints. By exploiting the equivalence between Maximum Entropy and Maximum Likelihood estimates of the exponential families, we can use the conventional Likelihood Ratio, Wald and Lagrange Multiplier testing principles in the maximum entropy framework. Monte Carlo evidence suggests that they have desirable small sample properties, when compared with the standard parametric tests used in the literature, such as the standardized skewness coe‐cient test for symmetry or the Jarque and Bera (1980) test for normality. We apply the proposed symmetry tests to test the nominal wage rigidity hypothesis of wage determination process.

Aerodynamic Shape Design of Rotor Airfoils Undergoing Unsteady Motion

A new design method is suggested that can handle the dynamic response of an airfoil subjected to unsteady motion and flow conditions. A new objective function and constraints are defined to represent the overall characteristics of the rotor airfoil over a single cycle. A weighting function is also introduced to enable a designer to determine the specific azimuthal location at which the performance is improved intensively. A numerical analysis code that can handle the oscillatory pitching motion of an airfoil and the inflow Mach number variation is developed. The analyses during the design process are limited to the cases without dynamic stall because the numerical analysis code cannot predict the dynamic stall phenomena with sufficient accuracy. Response surface methodology is exclusively employed to construct and optimize the response surface models of the objective functions and constraints. The advantages of the method are demonstrated by a comprehensive comparison with a static three-point design method. The dynamic method can yield a better airfoil for the following reasons: 1) The static performance is not always proportional to the dynamic performance. 2) The moment constraint used in the static three-point design is too tight to allow enough variation of the design variables. 3) The objective function of the static three-point design may be implicitly overweighted at a certain design point so that the overall aerodynamic performance is miscalculated, unlike in the present method. The variations of the aerodynamic performance with weighting functions are also discussed.

Limiting distributions for minimum relative entropy calibration

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.

Limiting distributions for minimum relative entropy calibration

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.

THE CONSTRAINTS ON CP VIOLATING PHASES IN MODELS WITH A DYNAMICAL GLUINO PHASE

We analyze the electric dipole moment and the Higgs mass constraints on the supersymmetric model which offers dynamical solutions to the μ and strong CP problems. The trilinear coupling phases, and tan β-|μ| are strongly correlated, particularly in the low-tan β regime. Certain values of the phases of the trilinear couplings are forbidden, whereas the CP violating phase from the chargino sector is imprisoned to lie near a CP conserving point, by the Higgs mass and electric dipole moment constraints.

A Computationally Efficient Multivariate Maximum-Entropy Density Estimation (MEDE) Technique

Density estimation is the process of taking a set of multivariate data and finding an estimate for the probability density function (pdf) that produced it. One approach for obtaining an accurate estimate of the true density f(x) is to use the polynomial-moment method with Boltzmann-Shannon entropy. Although rigorous mathematically, the method is difficult to implement in practice because the solution involves a large set of simultaneous nonlinear integral equations, one for each moment or joint moment constraint. Solutions available in the literature are generally not easily applicable to multivariate data, nor computationally efficient. In this paper, we take the functional form that was developed in this problem and apply pointwise estimates of the pdf as constraints. These pointwise estimates are transformed into basis coefficients for a set of Legendre polynomials. The procedure is mathematically similar to the multidimensional Fourier transform, although with different basis functions. We apply this technique, called the maximum-entropy density estimation (MEDE) technique, to a series of multivariate datasets.

Time-optimal multidimensional gradient waveform design for rapid imaging.

Magnetic resonance imaging (MRI) is limited in many cases by long scan times and low spatial resolution. Recent advances in gradient systems hardware allow very rapid imaging sequences, such as steady-state free precession (SSFP), which has repetition times (TRs) of 2-5 ms. The design of these rapid sequences demands time-optimal preparatory gradient waveforms to provide maximum readout duty-cycle, and preserve spatial resolution and SNR while keeping TRs low. Time-optimal gradient waveforms can be synthesized analytically for certain simple cases. However, certain problems, such as time-optimal 2D and 3D gradient design with moment constraints, either may not have a solution or must be solved numerically. We show that time-optimal gradient design is a convex-optimization problem, for which very efficient solution methods exist. These methods can be applied to solve gradient design problems for oblique gradient design, spiral imaging, and flow-encoding using either a constant slew rate or the more exact voltage-limited gradient models. Ultimately, these methods provide a time-optimal solution to many 2D and 3D gradient design problems in a sufficiently short time for interactive imaging.

Impact of CP phases on a light sbottom and gluino sector

We study a scenario in which light bottom squarks and light gluinos with masses in the range 2–5.5 GeV and 12–16 GeV, respectively, can coexist in the MSSM, without being in conflict with flavor-conserving low-energy observables. We find that in such a scenario, the anomalous magnetic moment of a muon could be as large as 10−9, if the theory conserves CP. However, if the theory violates CP, we conclude that not both, the gluino and bottom squark, can be light at the same time, after the neutron electric dipole moment constraint on Weinberg's 3-gluon operator has been taken into account.

Direct detection of dark matter in the minimal supersymmetric standard model with non-universal Higgs boson masses

We calculate dark matter scattering rates in the minimal supersymmetric extension of the standard model (MSSM), allowing the soft supersymmetry-breaking masses of the Higgs multiplets, ${m}_{1,2},$ to be nonuniversal (NUHM). Compared with the constrained MSSM (CMSSM) in which ${m}_{1,2}$ are required to be equal to the soft supersymmetry-breaking masses ${m}_{0}$ of the squark and slepton masses, we find that the elastic scattering cross sections may be up to two orders of magnitude larger than values in the CMSSM for similar lightest supersymmetric particle masses. We find the following preferred ranges for the spin-independent cross section, ${10}^{\ensuremath{-}6}\mathrm{pb}\ensuremath{\gtrsim}{\ensuremath{\sigma}}_{\mathrm{SI}}\ensuremath{\gtrsim}{10}^{\ensuremath{-}10}\mathrm{pb},$ and, for the spin-dependent cross section, ${10}^{\ensuremath{-}3}\mathrm{pb}\ensuremath{\gtrsim}{\ensuremath{\sigma}}_{\mathrm{SD}},$ with the lower bound on ${\ensuremath{\sigma}}_{\mathrm{SI}}$ dependent on using the putative constraint from the muon anomalous magnetic moment. We stress the importance of incorporating accelerator and dark matter constraints in restricting the NUHM parameter space, and also of requiring that no undesirable vacuum appear below the grand unified theory (GUT) scale. In particular, values of the spin-independent cross section another order of magnitude larger would appear to be allowed, for small $\mathrm{tan}\ensuremath{\beta},$ if the GUT vacuum stability requirement were relaxed, and much lower cross-section values would be permitted if the muon anomalous magnetic moment constraint were dropped.

Calculation of maximum entropy densities with application to income distribution

The maximum entropy approach is a flexible and powerful tool for density approximation. This paper proposes a sequential updating method to calculate the maximum entropy density subject to known moment constraints. Instead of imposing the moment constraints simultaneously, the sequential updating method incorporates the moment constraints into the calculation from lower to higher moments and updates the density estimates sequentially. The proposed method is employed to approximate the size distribution of U.S. family income. Empirical evidence demonstrates the efficiency of this method.

NP-Optimal Kernels for Nonparametric Sequential Detection Rules

An attractive nonparametric method to detect change-points sequentially is to apply control charts based on kernel smoothers. Recently, the strong convergence of the associated normed delay associated with such a sequential stopping rule has been studied under sequences of out-of-control models. Kernel smoothers employ a kernel function to downweight past data. Since kernel functions with values in the unit interval are sufficient for that task, we study the problem to optimize the asymptotic normed delay over a class of kernels ensuring that restriction and certain additional moment constraints. We apply the key theorem to discuss several important examples where explicit solutions exist to illustrate that the results are applicable.

NP-Optimal Kernels for Nonparametric Sequential Detection Rules

An attractive nonparametric method to detect change-points sequentially is to apply control charts based on kernel smoothers. Recently, the strong convergence of the associated normed delay associated with such a sequential stopping rule has been studied under sequences of out-of-control models. Kernel smoothers employ a kernel function to downweight past data. Since kernel functions with values in the unit interval are sufficient for that task, we study the problem to optimize the asymptotic normed delay over a class of kernels ensuring that restriction and certain additional moment constraints. We apply the key theorem to discuss several important examples where explicit solutions exist to illustrate that the results are applicable.

Calculation of Maximum Entropy Densities with Application to Income Distribtuions

This paper shows that there exits a unique maximum entropy density for any finite sample when arithmetic sample moments are used as side conditions. A sequential updating method to calculate the maxent entropy density subject to known moment constraints is proposed. Instead of imposing the moment constraints simultaneously, the sequential updating method incorporates the moment constraints into the calculation from lower to higher moments and updates the density estimates sequentially. The proposed method is employed to approximate the size distribution of U.S. family income. Numerical experiments and empirical evidence demonstrate the efficiency of this method.

CP-violating chargino contributions to the Higgs coupling to photon pairs in the decoupling regime of the Higgs sector

In most supersymmetric theories, charginos $\tilde{\chi}^\pm_{1,2}$ belong to the class of the lightest supersymmetric particles and the couplings of Higgs bosons to charginos are in general complex so that the CP--violating chargino contributions to the loop--induced coupling of the lightest Higgs boson to photon pairs can be sizable even in the decoupling limit of large pseudoscalar mass $m_A$ with only the lightest Higgs boson kinematically accessible at future high energy colliders. We introduce a specific benchmark scenario of CP violation consistent with the electric dipole moment constraints and with a commonly accepted baryogenesis mechanism in the minimal supersymmetric Standard Model. Based on the benchmark scenario of CP violation, we demonstrate that the fusion of the lightest Higgs boson in linearly polarized photon--photon collisions can allow us to confirm the existence of the CP--violating chargino contributions {\it even in the decoupling regime of the Higgs sector} for nearly degenerate SU(2) gaugino and higgsino mass parameters of about the electroweak scale.

Constrained mixture distributions

The problem of estimating the weights associated with mixture distributions subject to several constraints, such as the percentile and/or moment constraints, is analyzed using the general theory of polyhedral convex cones and systems of inequalities. We address three problems associated with constrained mixture distributions: (a) Compatibility: a set of inequalities is obtained to check whether or not any given set of constraints lead to a feasible solution for the weights, (b) Feasible solutions: a general expression for building feasible solutions for the weights associated with the given constraints is obtained, and (c) Equivalence: the set of all feasible weights is obtained. In addition, the problem is shown to lead to a new mixture distribution, without extra constraints. This new mixture distribution can then be easily used for the statistical analysis (e.g. estimation and hypotheses testing) instead of the original mixture distribution with extra constraints. The proposed methods are illustrated by numerical examples.

Shape coexistence in even-even superheavy nuclei

The structures of the even-even superheavy nuclei with proton number $Z=100--114$ are systematically investigated using self-consistent relativistic mean-field theory. The calculated binding energies are in good agreement with all available experimental data. The experimental alpha decay energies and lifetimes of the newly discovered superheavy nuclei are also reasonably reproduced by the model. Large scale calculations with a quadrupole moment constraint clearly show the variation of energy with the quadrupole deformation parameter. It reliably demonstrates that there is shape coexistence in superheavy nuclei. In some cases the configuration with superdeformation may be the ground state of superheavy nuclei near $Z=114$ and $N=174.$

Nuclear anapole moments

Nuclear anapole moments are parity-odd, time-reversal-even E1 moments of the electromagnetic current operator. Although the existence of this moment was recognized theoretically soon after the discovery of parity nonconservation (PNC), its experimental isolation was achieved only recently, when a new level of precision was reached in a measurement of the hyperfine dependence of atomic PNC in 133Cs. An important anapole moment bound in 205Tl also exists. In this paper, we present the details of the first calculation of these anapole moments in the framework commonly used in other studies of hadronic PNC, a meson exchange potential that includes long-range pion exchange and enough degrees of freedom to describe the five independent $S-P$ amplitudes induced by short-range interactions. The resulting contributions of pi-, rho-, and omega-exchange to the single-nucleon anapole moment, to parity admixtures in the nuclear ground state, and to PNC exchange currents are evaluated, using configuration-mixed shell-model wave functions. The experimental anapole moment constraints on the PNC meson-nucleon coupling constants are derived and compared with those from other tests of the hadronic weak interaction. While the bounds obtained from the anapole moment results are consistent with the broad ``reasonable ranges'' defined by theory, they are not in good agreement with the constraints from the other experiments. We explore possible explanations for the discrepancy and comment on the potential importance of new experiments.

Connections Between Entropic and Linear Projections in Asset Pricing Estimation

The concept of entropy has a long and distinguished history in the physical sciences and engineering, in fields ranging from thermodynamics to image processing. Each of these applications employs a probability distribution that solves a relative entropy projection problem, i.e. an optimization problem with an entropy objective, subject to linear (e.g. moment) constraints. This paper develops the relationship between relative entropy projection approaches and the better-known linear projection approaches to problems of estimation and performance diagnostics for stochastic discount factor models in asset pricing. Frequentist interpretations of relative entropy, enabled by large deviations theory, are used to unify the interpretation of the seemingly disparate procedures.