The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations

This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals—called constrained expressions—and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.

Analyzing Stress as a Multivariate Phenotype

The stress phenotype is multivariate. Recent advances have broadened our understanding beyond characterizing the stress response in a single dimension. Simultaneously, the toolbox available to ecophysiologists has expanded greatly in recent years, allowing the measurement of multiple biomarkers from an individual at a single point in time. Yet these advances—in our conceptual understanding and available methodologies—have not yet been combined in a unifying multivariate statistical framework. Here, we offer a brief review of the multivariate stress phenotype and describe a general statistical approach for analysis using nonparametric multivariate analysis of variance with residual randomization in permutation procedures (RRPP) implemented using the “RRPP” package in R. We also provide an example illustrating the novel insights that can be gained from a holistic multivariate approach to stress and provide a tutorial for how we analyzed these data, including annotated R code and a guide to interpretation of outputs (Online Appendix 1). We hope that this statistical methodology will provide a quantitative framework facilitating the unification of our theoretical understanding and empirical observations into a quantitative, multivariate theory of stress.

The Multivariate Theory of Connections

This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g ( x , y ) , satisfy all constraints no matter what the g ( x , y ) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations.

Target Damage Assessment Based on the Detection Information

With the wide application of all kinds of precision guided munitions, the rhythm of combat become faster, the environment of battle field become more complicated. It brings a great challenge to the battle damage assessment. the basic principle of battle damage assessment based on detection information is analyzed. Pretreatment and reliability analysis of battle field information and target function damage are elaborated. Single function damage analysis and target damage assessment of the Aircraft-borne guidance ammunition technical area are realized, based on target function damage analysis and fuzzy multivariate theory analysis. With the widely application of all kinds of precision guided munitions, the cost of combat grow rapidly，the rhythm of combat become faster, the environment of battle field become complicated， the damage effect to the target of the evaluation has brought an unprecedented challenges. At present , our army damage effect evaluation of target research has just started, there is no perfect theory system and mature evaluation method, so, target damage effect evaluation model and the method has important practical significance, on account of the scene to collect information on target is the important basis of damage effect evaluation, therefore, how to use all kinds of target damage information damage effect evaluation decision target efficiently collecting from battlefield is the primary problem studying damage effect evaluation.

Multiple Skill Deficits in Depression

This study involves the description of the presence or absence of several theoretically relevant skill deficits in 35 cases of depression. Until recently, most research and theory about depression has been based on the assumption that this is a homogeneous disorder with one determinant common to all incidences. Theories proposed by Beck (1967), Lewinsohn (1974), Seligman (Abramson, Seligman & Teasdale, 1978), and Rehm (1977) are examples of unitary theories. It is argued that a comprehensive account of depression must involve a multivariate theory such as that proposed by Staats and Heiby (1985). The case study involves a preliminary uncontrolled test of the hypothesis that there exist several skills deficits that may determine depression. The findings suggest support for the possibility that the following skill deficits deserve further investigation for their role in depression aetiology and treatment: logical errors including misattributions, poor self-control skills, poor social skills in terms of excessive aggression and deficit assertiveness, and inadequate hedonic capacity in terms of deficit positive interests and excessive negative interests (anhedonia). Suggestions for future research are offered.