Covariance Theorem Research Articles

An Improved Detection Statistic for Systems with Unsteady Trend

The object of this paper is to address data-driven fault detection design for systems with unsteady trend, which shows cyclicity, monotonicity and non-zero mean. Firstly, mean theorem and covariance theorem are proposed and proved. The former is the mean property of projection matrix, and the latter is the recursive formula for covariance matrix of regression residual. Secondly, an improved fault detection statistic, called Least Square T2 (LST2), is proposed. It can partly solve the detection problem for systems with unsteady trend. The improvement can also partly cope with the limitations of the traditional multivariate detection methods, such as Principal Component Analysis (PCA). Thirdly, based on the two theorems, a recursive algorithm and a moving window algorithm of LST2 are given, thus both time and space complexity are greatly reduced for online detection. The effectiveness of the presented detection statistic is evaluated with an application of monitoring satellite attitude control system. The case study result shows that the false alarm rate of LST2 is much lower than that of T2 based on PCA, while LST2 is more sensitive to fault.

Planck Scale Potential Associated with Particles

As the particles originating from point-like entities are associated with infinite self energies, a postulate, that the scalar-potential associated with particles are bounded by a Planck scale potential is introduced. By defining the self energy of a particle, equivalences between charge-energy and mass-energy are obtained. The electromagnetic energy-momentum equation, de-Broglie’s electromagnetic wave-length and frequency for a charge particle in motion are presented resolving the “4/3” discrepancy. The non-covariance nature of the present classical electrodynamics is discussed and how the proposed postulate makes it a fully covariant theorem with the rest of the classical electrodynamics is presented. A way electromagnetic energy-momentum equation could potentially resolve the stability-problem of a charge particle is discussed and thereby a theoretical explanation to electron’s spin is presented.

Planck Scale Potential Associated with Particles

As the particles originating from point-like entities are associated with infinite self energies, a postulate, that the scalar-potential associated with particles are bounded by a Planck scale potential is introduced. By defining the self energy of a particle, equivalences between charge-energy and mass-energy are obtained. The electromagnetic energy-momentum equation, de-Broglie’s electromagnetic wave-length and frequency for a charge particle in motion are presented resolving the “4/3” discrepancy. The non-covariance nature of the present classical electrodynamics is discussed and how the proposed postulate makes it a fully covariant theorem with the rest of the classical electrodynamics is presented. A way electromagnetic energy-momentum equation could potentially resolve the stability-problem of a charge particle is discussed and thereby a theoretical explanation to electron’s spin is presented.

Causal Categories: Relativistically Interacting Processes

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.

Scaring Them into Learning!? Using a Snake Screen to Enhance the Knowledge Transfer Effectiveness of a Web Interface

ABSTRACTIt seems that surprise events have the potential to turn short‐term memories into long‐term memories, an unusual phenomenon that may have limited but interesting applications in learning tasks. This surprise‐enhanced cognition phenomenon is theoretically modeled based on the notion that many human mental traits have evolved through natural selection; a mathematical analysis building on Price's covariance theorem is employed in this modeling effort. Additionally, the phenomenon is discussed in the context of an online learning task, based on a study involving 186 student participants. A simulated threat was incorporated into a human–computer interface with the goal of increasing the interface's knowledge‐transfer effectiveness. The participants were asked to review Web‐based learning modules and subsequently take a test on what they had learned. Data from six learning modules in two experimental conditions were contrasted. In the treatment condition, a Web‐based screen with a snake in attack position was used to surprise the participants; the snake screen was absent in the control condition. As predicted, the participants in the treatment condition did significantly better in the test for the modules immediately before and after the snake screen than the participants in the control condition. These findings are extrapolated to classroom applications in general. Ethical considerations are also discussed.

THE PRICE EQUATION, FISHER'S FUNDAMENTAL THEOREM, KIN SELECTION, AND CAUSAL ANALYSIS.

A general framework is presented to unify diverse models of natural selection. This framework is based on the Price Equation, with two additional steps. First, characters are described by their multiple regression on a set of predictor variables. The most common predictors in genetics are alleles and their interactions, but any predictor may be used. The second step is to describe fitness by multiple regression on characters. Once again, characters may be chosen arbitrarily. This expanded Price Equation provides an exact description of total evolutionary change under all conditions, and for all systems of inheritance and selection. The model is first used for a new proof of Fisher's fundamental theorem of natural selection. The relations are then made clear among Fisher's theorem, Robertson's covariance theorem for quantitative genetics, the Lande-Arnold model for the causal analysis of natural selection, and Hamilton's rule for kin selection. Each of these models is a partial analysis of total evolutionary change. The Price Equation extends each model to an exact, total analysis of evolutionary change for any system of inheritance and selection. This exact analysis is used to develop an expanded Hamilton's rule for total change. The expanded rule clarifies the distinction between two types of kin selection coefficients. The first measures components of selection caused by correlated phenotypes of social partners. The second measures components of heritability via transmission by direct and indirect components of fitness.

Kinship and covariance

Price's (1970) covariance theorem can be used to derive an expression for gene frequency change in kin selection models in which the fitness effect of an act is independent of the genotype of the recipient. This expression defines a coefficient of relatedness which subsumes r (Wright, 1922), b (Hamilton, 1972), ρ (Orlove & Wood, 1978), and R (Michod & Hamilton, 1980). The new coefficient extends the domain of Hamilton's rule to models in which the average gene frequency of actors differs from that of recipients.