High-temperature cluster expansion for classical and quantum spin lattice systems with multi-body interactions

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum -- and, in particular, classical -- interactions. Our approach is based on the use of ``decoupling parameters", advocated by Park [34], which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a $\beta$-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small $\beta$. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams [2]. As an application we show that our lower bound of the radius of $\beta$-analyticity is larger than Park's for quantum systems two-body interactions

Rates of Approximation by ReLU Shallow Neural Networks

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the H\"older space $W_\infty^r([-1, 1]^d)$ with rates $O((\log m)^{\frac{1}{2} +d}m^{-\frac{r}{d}\frac{d+2}{d+4}})$ when $r<d/2 +2$. Such rates are very close to the optimal one $O(m^{-\frac{r}{d}})$ in the sense that $\frac{d+2}{d+4}$ is close to $1$, when the dimension $d$ is large

Finite Element Analysis of Low-Velocity Impact Damage on Stiffened Composite Panels

To understand the factors which affect impact damage on composite structures, particularly the effects of impact position and ribs. In this paper, a finite element model (FEM) of low-velocity impact damage on the composite structure was established via the nonlinear finite element method, combined with the user-defined materials subroutine (VUMAT) of the ABAQUS software. The structural elements chosen for the investigation comprised a series of stiffened composite panels, representative of real aircraft structure. By impacting the panels at different positions relative to the ribs, the effect of relative position of ribs was found out. Then the simulation results and the experiments data were compared. Finally, the factors which affect impact damage on the structures were discussed. The paper was helpful for the design of stiffened composite structures

The Role of Human Capital: Evidence From Patent Generation

Firms exhibit persistence in innovation output. This paper focuses on the role played by individual inventors. Compared to firm organizational capital, human capital embedded in inventors explains a majority of the variation in innovation performance but much less in innovation style. Inventors contribute more when they are better networked, in firms with higher inventor mobility, and in industries in which innovation is more difficult. Additional tests suggest that our main findings are unlikely driven by inventors’ endogenous moving. This paper highlights the importance of human capital in enhancing firm innovation and sheds new light on the theory of the firm

Do Individuals or Firms Matter More? The Case of Patent Generation

This paper studies the relative importance of individual inventors’ human capital and firms’ organizational capital in promoting a firm’s innovation output. We decompose the variation in innovation output into inventor- and firm-specific components. Inventors’ human capital is about 13 times as important as firms’ organizational capital in explaining a firm’s innovation performance in terms of patent counts and citations, while inventors’ human capital is only about the same as important when explaining the firm’s innovation styles in terms of patent exploratory and exploitive scores. In the cross section, inventors contribute more to innovation output when they are better networked, in firms with higher inventor mobility, in industries in which innovation is more difficult to achieve, and in publicly traded firms. Additional tests suggest that our main findings continue to hold after accounting for inventors’ endogenous moving. This paper highlights the importance of individual inventors in enhancing firm innovation and sheds new light on the theory of the firm

Free energy expansions for renormalized systems for colloids

We consider a binary system of small and large spheres of finite size in a continuous medium interacting via a non-negative potential. We work in the canonical ensemble and compute upper and lower bound for the free energy at finite and infinite volume by first integrating over the small spheres and then treating the effective system of the large ones which now interact via a multi-body potential. By exploiting the underlying structure of the original binary system we prove the convergence of the cluster expansion for the latter system and obtain a sufficient condition which involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods directly to the binary system). Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction

Understanding Gravitational Form Factors with the Weizs\"acker-Williams Method

Understanding the internal structure of nucleons and nuclei has been a topic of enduring interest in high-energy physics. Gravitational form factors (GFFs) provide an important portal for us to probe the energy-momentum/mass distribution of nucleons and nuclei. This letter presents the study of the photon and gluon momentum GFFs, also known as the A-GFFs, of relativistic hadrons using the Weizs\"acker-Williams method. To begin, we express the photon A-GFFs in terms of charge form factors and discuss the corresponding photon radius. Furthermore, an integral relation between the gluon A-GFF and the Laplacian of dipole scattering amplitude is derived in the small-$x$ framework, and it allows us to unravel the gluon energy momentum distribution inside hadrons through measurements at the upcoming Electron-Ion Collider. In addition, we generalize the analysis to study the A-GFF of nuclei and propose employing the nuclear gluon mean square radius, together with the charge distribution, to constrain the neutron distribution for large nuclei. This work provides an interesting perspective into the fundamental structure of high-energy hadrons.Comment: 8 pages, 2 figures; Minor text revision