Explicit Function Representations Research Articles

Multi-modal molecular determinants of clinically relevant osteoporosis subtypes

Due to a rapidly aging global population, osteoporosis and the associated risk of bone fractures have become a wide-spread public health problem. However, osteoporosis is very heterogeneous, and the existing standard diagnostic measure is not sufficient to accurately identify all patients at risk of osteoporotic fractures and to guide therapy. Here, we constructed the first prospective multi-omics atlas of the largest osteoporosis cohort to date (longitudinal data from 366 participants at three time points), and also implemented an explainable data-intensive analysis framework (DLSF: Deep Latent Space Fusion) for an omnigenic model based on a multi-modal approach that can capture the multi-modal molecular signatures (M3S) as explicit functional representations of hidden genotypes. Accordingly, through DLSF, we identified two subtypes of the osteoporosis population in Chinese individuals with corresponding molecular phenotypes, i.e., clinical intervention relevant subtypes (CISs), in which bone mineral density benefits response to calcium supplements in 2-year follow-up samples. Many snpGenes associated with these molecular phenotypes reveal diverse candidate biological mechanisms underlying osteoporosis, with xQTL preferences of osteoporosis and its subtypes indicating an omnigenic effect on different biological domains. Finally, these two subtypes were found to have different relevance to prior fracture and different fracture risk according to 4-year follow-up data. Thus, in clinical application, M3S could help us further develop improved diagnostic and treatment strategies for osteoporosis and identify a new composite index for fracture prediction, which were remarkably validated in an independent cohort (166 participants).

Optimization of circuitry arrangements for heat exchangers using derivative-free optimization

Optimization of the refrigerant circuitry can improve a heat exchanger's performance. Design engineers currently choose the refrigerant circuitry according to their experience and heat exchanger simulations. However, the design of an optimized refrigerant circuitry is difficult. The number of refrigerant circuitry candidates is enormous. Therefore, exhaustive search algorithms cannot be used and intelligent techniques must be developed to explore the solution space efficiently. In this paper, we formulate refrigerant circuitry design as a binary constrained optimization problem. We use CoilDesigner, a simulation and design tool of air to refrigerant heat exchangers, in order to simulate the performance of different refrigerant circuitry designs. We treat CoilDesigner as a black-box system since the exact relationship of the objective function with the decision variables is not explicit. Derivative-free optimization (DFO) algorithms are suitable for solving this black-box model since they do not require explicit functional representations of the objective function and the constraints. The aim of this paper is twofold. First, we compare four mixed-integer constrained DFO solvers and one box-bounded DFO solver and evaluate their ability to solve a difficult industrially relevant problem. Second, we demonstrate that the proposed formulation is suitable for optimizing the circuitry configuration of heat exchangers. We apply the DFO solvers to 17 heat exchanger design problems. Results show that TOMLAB/glcDirect and TOMLAB/glcSolve can find optimal or near-optimal refrigerant circuitry designs after a relatively small number of circuit simulations.

Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner's work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh's six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner's work conjugate integral and theJ-integral in these cases is obtained by introducing a complementary stress-displacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.

Matrix product approach for the asymmetric random average process

We consider the asymmetric random average process which is a one-dimensional stochastic lattice model with nearest neighbour interaction but continuous and unbounded state variables. First, the explicit functional representations, so-called beta densities, of all local interactions leading to steady states of product measure form are rigorously derived. This also completes an outstanding proof given in a previous publication. Then, we present an alternative solution for the processes with factorized stationary states by using a matrix product ansatz. Due to continuous state variables we obtain a matrix algebra in form of a functional equation which can be solved exactly.

Bueckner's work conjugate integrals and weight functions for a crack in anisotropic solids

The Bueckner work conjugate integrals are studied for cracks in anisotropic elastic solids. The difficulties in separating Lekhnitskii's two complex arguments involved in the integrals are overcome and explicit functional representations of the integrals are given for several typical cases. It is found that the pseudoorthogonal property of the eigenfunction expansion forms presented previously for isotropic cases, isotropic bimaterials, and orthotropic cases, are proved to be also valid in the present case of anisotropic material. Finally, Some useful path-independent integrals and weight functions are proposed.

Investigation of EEF properties for a crack in a plane orthotropic elastic solid with purely imaginary characteristic roots

The properties of the eigenfunction expansion form for a semi-infinite crack in a plane orthotropic elastic body with purely imaginary characteristic roots are studied in detail. The difficulties in performing Bueckner's integral with two complex arguments are overcome and the expected explicit function representations for the integral are derived. It is found that the pseudo-orthogonal property of EEF found in the isotropic case is proved to be valid in the orthotropic case with purely imaginary characteristic roots. Some useful path-independent integrals and weight functions are derived.

Asymptotic expansions for constant-composition dew-bubble curves near the critical locus

Explicit functional representations are developed for constant-composition dew and bubble curves near critical according to the modified Leung-Griffiths theory. The pressure and temperature incrementsΔP=P−Pc andΔT= T−Tc, where c denotes critical, are linearly transformed to new variablesΔP′ andΔT′. In the transformed space, the coexistence curves are no longer double-valued and can be expressed as a nonanalytic expansion, where the coefficients are functions of the critical properties and their derivatives. A similar asymptotic expansion is developed forΔT in terms of the density incrementΔρ=ρ−ρc. In the approximation that the critical exponentsα=0 andβ=1/3, the critical point in temperature-density space is shown to be a point of maximum concave upward curvature, rather than an inflection point as previously conjectured.

Minimum-fuel aircraft transition trajectories

Sequential quadratic programming is used to solve several minimum-fuel flight path optimization problems for a point-mass aircraft flying in a vertical plane. Each optimal control problem involves two control functions: a “steering” control and a “thrust” control. Range is the independent variable. Four dynamic models are examined. The usual point-mass model requires four states (speed, flight path angle, altitude, and mass) and angle of attack as the steering control. An intermediate model with three states (speed, altitude, and mass) neglects the flight path angle dynamics and uses flight path angle as the steering control. An energy-state approximation and a point-mass model including engine dynamics are also treated. Explicit function representations are used to model the aircraft's thrust, lift, and drag characteristics. Numerical solutions are presented for several fuel-optimal transition flight paths between specified states. A comparison of results for the various dynamic models is also given.

On Implicit Forms of Multiproduct‐Multifactor Production Functions

A quest for generality in economic analysis appears to underlie the long tradition of the use of implicit functions. However, insufficient attention is given to weaknesses and limitations of implicit functions as they are traditionally used (e.g., Cohen and Cyert, Hadar, Henderson and Quandt, Malinvaud, Naylor, Naylor and Vernon, and Samuelson, among others). Careless use of implicit functions can result in faulty analysis. In particular, clarification of these limitations is needed, especially as they pertain to the study of multiproduct-multifactor (MP-MF) production processes.' This paper is a rigorous examination of certain issues regarding implicit functions and MP-MF production processes. Because any relation that explicitly determines one argument as a function of the remaining arguments can be written in implicit function form, our results also apply to explicit function representations of MP-MF production. The functions examined here are continuously differentiable and can be used as production feasibility constraints in a Lagrangian or Kuhn-Tucker analysis of the MP-MF firm. The theory of MP-MF production is relevant in the conceptualization of resource allocation problems throughout agriculture. For example, few important real-world analyses of agricultural production or marketing operations are single product in nature. In fact, in his extensive literature review on production efficiency in agricultural processing and marketing, French states multiple outputs are effectively the rule rather than the exception. Similarly, MP-MF theory is prominent in the study of environmental and natural resource problems involving technological externalities. Despite the importance of the theory, it appears that work to date is a collection of special cases of a potentially more general theory. It is our hope that this paper will stimulate a reexamination of MP-MF production theory.