Probability Thermodynamics and Probability Quantum Field

In this paper, we introduce probability thermodynamics and probability quantum fields. By probability we mean that there is an unknown operator, physical or nonphysical, whose eigenvalues obey a certain statistical distribution. Eigenvalue spectra define spectral functions. Various thermodynamic quantities in thermodynamics and effective actions in quantum field theory are all spectral functions. In the scheme, eigenvalues obey a probability distribution, so a probability distribution determines a family of spectral functions in thermodynamics and in quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. There are two types of spectra: lower bounded spectra, corresponding to the probability distribution with nonnegative random variables, and the lower unbounded spectra, corresponding to probability distributions with negative random variables. For lower unbounded spectra, we use the generalized definition of spectral functions. In some cases, we encounter divergences. We remove the divergence by a renormalization procedure. Moreover, in virtue of spectral theory in physics, we generalize some concepts in probability theory. For example, the moment generating function in probability theory does not always exist. We redefine the moment generating function as the generalized heat kernel, which makes the concept definable when the definition in probability theory fails. As examples, we construct examples corresponding to some probability distributions. Thermodynamic quantities, vacuum amplitudes, one-loop effective actions, and vacuum energies for various probability distributions are presented

Algorithm and experiments of six-dimensional force/torque dynamic measurements based on a Stewart platform

AbstractStewart platform (SP) is a promising choice for large component alignment, and interactive force measurements are a novel and significant approach for high-precision assemblies. The designed position and orientation (P&O) adjusting platform, based on an SP for force/torque-driven (F/T-driven) alignment, can dynamically measure interactive forces. This paper presents an analytical algorithm of measuring six-dimensional F/T based on the screw theory for accurate determination of external forces during alignment. Dynamic gravity deviations were taken into consideration and a compensation model was developed. The P&O number was optimized as well. Given the specific appearance of repeated six-dimensional F/T measurements, an approximate cone shape was used for spatial precision analysis. The magnitudes and directions of measured F/Ts can be evaluated by a set of standards, in terms of accuracy and repeatability. Experiments were also performed using a known applied load, and the proposed analytical algorithm was able to accurately predict the F/T. A comparison between precision analysis experiments with or without assembly fixtures was performed. Experimental results show that the measurement accuracy varies under different P&O sets and higher loads lead to poorer accuracy of dynamic gravity compensation. In addition, the preferable operation range has been discussed for high-precision assemblies with smaller deviations

Optimal Nonparametric Inference on Network Effects with Dependent Edges

Testing network effects in weighted directed networks is a foundational problem in econometrics, sociology, and psychology. Yet, the prevalent edge dependency poses a significant methodological challenge. Most existing methods are model-based and come with stringent assumptions, limiting their applicability. In response, we introduce a novel, fully nonparametric framework that requires only minimal regularity assumptions. While inspired by recent developments in $U$-statistic literature (arXiv:1712.00771, arXiv:2004.06615), our approach notably broadens their scopes. Specifically, we identified and carefully addressed the challenge of indeterminate degeneracy in the test statistics $-$ a problem that aforementioned tools do not handle. We established Berry-Esseen type bound for the accuracy of type-I error rate control. Using original analysis, we also proved the minimax optimality of our test's power. Simulations underscore the superiority of our method in computation speed, accuracy, and numerical robustness compared to competing methods. We also applied our method to the U.S. faculty hiring network data and discovered intriguing findings.Comment: 29 pages, 3 figure