Large Exponent Research Articles

Complex hyperbolic Kleinian groups of large critical exponents

Complex hyperbolic Kleinian groups of large critical exponents

A Modified Initial Mass Function of the First Stars with Explodability Theory under Different Enrichment Scenarios

The most metal-poor stars record the earliest metal enrichment triggered by Population III stars. By comparing observed abundance patterns with theoretical yields of metal-free stars, physical properties of their first star progenitors can be inferred, including zero-age main-sequence mass and explosion energy. In this work, the initial mass distribution of the first stars is obtained from the largest analysis to date of 406 very metal-poor stars with the newest LAMOST/Subaru high-resolution spectroscopic observations. However, the mass distribution fails to be consistent with the Salpeter initial mass function, which is also reported by previous studies. Here, we modify the standard power-law function with explodability theory. The mass distribution of Population III stars could be well explained by ensuring that the initial metal enrichment originates from successful supernova explosions. Based on the modified power-law function, we suggest an extremely top-heavy or nearly flat initial mass function with a large exponent for the explosion energy. This indicates that supernova explodability should be considered in the earliest metal enrichment process in the Universe.

Explaining neural scaling laws

The population loss of trained deep neural networks often follows precise power-law scaling relations with either the size of the training dataset or the number of parameters in the network. We propose a theory that explains the origins of and connects these scaling laws. We identify variance-limited and resolution-limited scaling behavior for both dataset and model size, for a total of four scaling regimes. The variance-limited scaling follows simply from the existence of a well-behaved infinite data or infinite width limit, while the resolution-limited regime can be explained by positing that models are effectively resolving a smooth data manifold. In the large width limit, this can be equivalently obtained from the spectrum of certain kernels, and we present evidence that large width and large dataset resolution-limited scaling exponents are related by a duality. We exhibit all four scaling regimes in the controlled setting of large random feature and pretrained models and test the predictions empirically on a range of standard architectures and datasets. We also observe several empirical relationships between datasets and scaling exponents under modifications of task and architecture aspect ratio. Our work provides a taxonomy for classifying different scaling regimes, underscores that there can be different mechanisms driving improvements in loss, and lends insight into the microscopic origin and relationships between scaling exponents.

Continuous data assimilation for the 3D and higher-dimensional Navier–Stokes equations with higher-order fractional diffusion

We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the 3D Navier–Stokes equations modified to have finer-order fractional diffusion. The fractional diffusion case is of particular interest, as it is known to be globally well-posed for sufficiently large diffusion exponent α. In this work, we prove that the assimilation equations are globally well-posed, and we demonstrate that the solutions produced by the AOT algorithm exhibit exponential convergence in time to the reference solution, given a sufficiently high spatial resolution of observations and a sufficiently large nudging parameter. We note that the results hold in arbitrary spatial dimensions d where d≥2, so long as α≥12+d4. Though the cases d>3 are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof.

Influence of Mechanical Properties of Tube on Forming Limits and Deformation Characteristics in an Expansion Drawing Process

Abstract This paper describes an efficient method of manufacturing thin-walled metal tubes. The proposed method is an expansion drawing and it consists of two stages, flaring and drawing, during which high thickness reduction is achieved because the tube being processed is subjected to tensile stress in both the circumferential and axial directions. The optimal plug half-angle for various materials is investigated using a finite element analysis and experiments, revealing the plug shape with the highest expansion limit and the highest thickness reduction ratio. To investigate the deformation characteristics of different materials during the expansion drawing process, the strain in tubes formed using different materials was measured experimentally using a square grid on the outer surface of the tubes. In addition, a finite element analysis using hypothetical material properties revealed that the work-hardening exponent and friction coefficient affect tube deformation. The results of this work explain the influences of material properties on experimentally observed phenomena. Materials with a large work-hardening exponent are less prone to thinning during forming and have a greater forming limit. On the other hand, the larger the friction coefficient between the plug and the mother tube, the larger the deformation in the axial direction was, and then the thickness reduction increased due to biaxial tension.

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We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the W-1,p′(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{-1,p'}(\\Omega )$$\\end{document} norm with exponents p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p> 2$$\\end{document}. The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents p≫2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\gg 2$$\\end{document}. Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.

Scaling relationships between human leg muscle architectural properties and body size.

A skeletal muscle's peak force production and excursion are based on its architectural properties that are, in turn, determined by its mass, muscle fiber length and physiological cross-sectional area (PCSA). In the classic interspecific study of mammalian muscle scaling, it was demonstrated that muscle mass scales positively allometrically with body mass whereas fiber length scales isometrically with body mass, indicating that larger mammals have stronger leg muscles than they would if they were geometrically similar to smaller ones. Although this relationship is highly significant across species, there has never been a detailed intraspecific architectural scaling study. We have thus created a large dataset of 896 muscles across 34 human lower extremities (18 females and 16 males) with a size range including approximately 90% and 70% of the United States population height and mass, respectively, across the range 36-103 years. Our purpose was to quantify the scaling relationships between human muscle architectural properties and body size. We found that human muscles depart greatly from isometric scaling because muscle mass scales with body mass1.3 (larger exponent than isometric scaling of 1.0) and muscle fiber length scales with negative allometry with body mass0.1 (smaller exponent than isometric scaling of 0.33). Based on the known relationship between architecture and function, these results suggest that human muscles place a premium on muscle force production (mass and PCSA) at the expense of muscle excursion (fiber length) with increasing body size, which has implications for understanding human muscle design as well as biomechanical modeling.

New sign-changing solutions for the 2D Lane-Emden problem with large exponents

New sign-changing solutions for the 2D Lane-Emden problem with large exponents

A unified explanation for variation of the grain growth exponent based on grain boundary migration kinetics

Based on molecular dynamics simulation on grain boundary migration, the large grain growth exponent extensively observed in high-purity nanocrystalline materials is proved to be ascribed to specific boundary migration kinetics determined. It is also found that temperature, topology of polycrystal and grain boundary type all affect the migration kinetics, which therefore causes the exponent to vary. The findings obtained in this work are capable of rationalizing various grain growth phenomena. In addition, triple junction drag against boundary migration has been evaluated and discussed.

A double-phase eigenvalue problem with large exponents

Abstract In the present article, we consider a double-phase eigenvalue problem with large exponents. Let λ ( p n , q n ) 1 {\lambda }_{\left({p}_{n},{q}_{n})}^{1} be the first eigenvalues and u n {u}_{n} be the first eigenfunctions, normalized by ‖ u n ‖ ℋ n = 1 \Vert {u}_{n}{\Vert }_{{{\mathcal{ {\mathcal H} }}}_{n}}=1 . Under some assumptions on the exponents p n {p}_{n} and q n {q}_{n} , we show that λ ( p n , q n ) 1 {\lambda }_{\left({p}_{n},{q}_{n})}^{1} converges to Λ ∞ {\Lambda }_{\infty } and u n {u}_{n} converges to u ∞ {u}_{\infty } uniformly in the space C α ( Ω ) {C}^{\alpha }\left(\Omega ) , and u ∞ {u}_{\infty } is a nontrivial viscosity solution to a Dirichlet ∞ \infty -Laplacian problem.

Relaxed Kačanov Scheme for the \(\boldsymbol{p}\)-Laplacian with Large Exponent

Relaxed Kačanov Scheme for the \(\boldsymbol{p}\)-Laplacian with Large Exponent

Characterization of the neuronal and network dynamics of liquid state machines

Reservoir computing (RC) is a relatively new machine-learning framework that uses an abstract neural network model, called reservoir. The reservoir forms a complex system with high dimensionality, nonlinearity, and intrinsic memory effect due to recurrent connections among individual neurons. RC manifests a best-in-class performance in processing information generated by complex dynamical systems, yet little is known about its microscopic/macroscopic dynamics underlying the computational capability. Here, we characterize the neuronal and network dynamics of liquid state machines (LSMs) using numerical simulations and Modified National Institute of Standards and Technology (MNIST) database classification tasks. The computational performance of LSMs largely depends on a dynamic range of neuronal avalanches whereby the avalanche patterns are determined by the neuron and network models. A larger dynamic range leads to higher performance—the MNIST classification accuracy is highest when the avalanche sizes follow a slowly decaying power-law distribution with an exponent of ∼1.5, followed by the power-law statistics with a larger exponent and the mixture of power-law/log-normal distributions. Network-theoretic analysis suggests that the formation of large functional clusters and the promotion of dynamic transitions between large and small clusters may contribute to the scale-invariant nature. This study provides new insight into our understanding of the computational principles of RC concerning the actions of the individual neurons and the system-level collective behavior.

Dynamics of spatiotemporal soliton solutions to a generalized nonlinear Schrödinger equation with inhomogeneous coefficients

In this paper, we study the dynamics of spatiotemporal soliton solutions to a generalized nonlinear Schrödinger equation with inhomogeneous coefficients in analytical and numerical forms. The characteristics and physical applications of the soliton solutions are investigated under different modulations of the coefficients depicting diffraction, potential, nonlinearity, source, and gain. We find that structures of these soliton solutions can be effectively manipulated by appropriately selecting the source term. Besides, the impact of gate effect on soliton propagation is investigated by employing the hyper-Gaussian and rectangle functions as diffraction coefficients. The results show that (i) the hyper-Gaussian modulation can be replaced by the rectangle modulation for large exponent and (ii) the solitons undergo a broadening in width and a sudden decrease in amplitude when passing through the gate functions. Our findings may have potential applications for the investigation of soliton control in the fields of nonlinear optics and Bose–Einstein condensations.

Ultra Rate‐Dependent Pressure Sensitive Adhesives Enabled by Soft Elasticity of Liquid Crystal Elastomers

AbstractThe fabrication of pressure sensitive adhesives (PSAs) using liquid crystal elastomers (LCEs), which are known for their excellent dissipation properties, is explored in this work. The adhesive properties of the PSAs are evaluated using the 180° peeling test at various conditions. The performance of the LCE adhesives is found to show significant rate and temperature dependence. When the adhesion energy is plotted against the rate, LCE shows an anomalously large power law exponent (n ≈ 1.17) compared to existing PSAs (n ≈ 0.1–0.6). The unusual rate sensitivity is hypothesized to originate from the synergy of soft elasticity and non‐linear viscoelasticity. The adhesive properties at various rates and temperatures are correlated to the results from dynamic mechanical analysis. Moreover, the large strain stiffening behavior of LCE under uniaxial tension reveals the distinctive advantages offered by LCE as adhesives. Time‐temperature superposition is used to obtain a master curve of adhesion energy that spans rates beyond typical experimental limits. The extreme rate dependence and the large strain stiffening of LCE yield a new category of adhesives that possess special properties, such as reversible adhesion and impact resistance, unlike traditional adhesives.

Chemo-hydro-mechanical effects of CO2 injection into a permeable limestone

CO2 storage in carbonate aquifers is a way to mitigate atmospheric CO2 emissions, but it leads to acid-producing reactions that may induce alterations in the rock pore structure and hydromechanical properties. To gain a better understanding of these changes in rock properties, percolation experiments under constant flow-rate conditions with (i) CO2-saturated water (PCO2 = 100 bar and T = 60 °C) and (ii) HCl solutions (P = 1 bar and T = 60 °C) were performed on centimetric cores of the grain-supported Pont du Gard Limestone (permeability about 10−14–10−13 m2). Effluent chemistry analyses, X-ray imaging, and measurements of the hydromechanical properties of intact and altered specimens were used to quantify acid-induced changes in the two acid-rock systems. Experimental results show that the rock dissolution patterns highly depend on the acid type and pore space heterogeneity. Under the flow conditions of these experiments, complete HCl dissociation (strong acid) and rapid HCl-rock reaction result in a compact dissolution pattern that only affects the hydromechanical properties of the core inlet. Conversely, partial dissociation of H2CO3 (weak acid) produces chemical reactions all along the core. Initial structural heterogeneities localize chemical reactions along preferential flow pathways, causing instabilities in the reaction front and wormhole formation, which markedly enhances permeability. A powerlaw relationship with a very large exponent (as high as 24) accounts for the variation in permeability as a function of porosity. Altered cores render both a significant attenuation in mechanical rock properties and ultrasonic velocities, which is reproduced using a Differential Effective Medium (DEM) homogenization approach. The high-pressure percolation experiments of this study using CO2-saturated water represent an extreme scenario of CO2-brine-rock interactions in carbonate reservoirs, which needs to be considered to improve the prediction and monitoring of the storage performance.

Information diversity and anomalous scaling in asymmetric social contagion process on low-dimensional static networks.

To understand how competition affects the diversity of information, we study the social contagion model introduced by Halvorsen-Pedersen-Sneppen (HPS) [G. S. Halvorsen, B. N. Pedersen, and K. Sneppen, Phys. Rev. E 103, 022303 (2021)2470-004510.1103/PhysRevE.103.022303] on one-dimensional (1D) and two-dimensional (2D) static networks. By mapping the information value to the height of the interface, we find that the width W(N,t) does not satisfy the well-known Family-Vicsek finite-size scaling ansatz. From the numerical simulations, we find that the dynamic exponent z should be modified for the HPS model. For 1D static networks, the numerical results show that the information landscape is always rough with an anomalously large growth exponent, β. Based on the analytic derivation of W(N,t), we show that the constant small number of influencers created for unit time and the recruitment of new followers are two processes responsible for the anomalous values of β and z. Furthermore, we also find that the information landscape on 2D static networks undergoes a roughening transition, and the metastable state emerges only in the vicinity of the transition threshold.

On upper bounds for the first ℓ²-Betti number

This article presents a method for proving upper bounds for the first ℓ 2 \ell ^2 -Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first ℓ 2 \ell ^2 -Betti number. Our approach extends to generalizations of ℓ 2 \ell ^2 -Betti numbers that are defined using characters. We illustrate this flexibility by generalizing results of Peterson-Thom on q-normal subgroups to this setting.

Emergent XY* transition driven by symmetry fractionalization and anyon condensation

Anyons in a topologically ordered phase can carry fractional quantum numbers with respect to the symmetry group of the considered system, one example being the fractional charge of the quasiparticles and quasiholes in the fractional quantum Hall effect. When such symmetry-fractionalized anyons condense, the resulting phase must spontaneously break the symmetry and display a local order parameter. In this paper, we study the phase diagram and anyon condensation transitions of a \mathbb{Z}_2ℤ2 topological order perturbed by Ising interactions in the Toric Code. The interplay between the global (``onsite’’) Ising (\mathbb{Z}_2ℤ2) symmetry and the lattice space group symmetries results in a non-trivial symmetry fractionalization class for the anyons, and is shown to lead to two characteristically different confined, symmetry-broken phases. To understand the anyon condensation transitions, we use the recently introduced critical torus energy spectrum technique to identify a line of emergent 2+1D XY* transitions ending at a fine-tuned (Ising^22)* critical point. We provide numerical evidence for the occurrence of two symmetry breaking patterns predicted by the specific symmetry fractionalization class of the condensed anyons in the explored phase diagram. In combination with large-scale quantum Monte Carlo simulations we measure unusually large critical exponents \etaη for the scaling of the correlation function at the continuous anyon condensation transitions, and we further identify lines of (weakly) first order transitions in the phase diagram. As an important additional result, we discuss the phase diagram of a resulting 2+1D Ashkin-Teller model, where we demonstrate that torus spectroscopy is capable of identifying emergent XY/O(2) critical behaviour, thereby solving some longstanding open questions in the domain of the 3D Ashkin-Teller models. To establish the generality of our results, we propose a field theoretical description capturing the transition from a \mathbb{Z}_2ℤ2 topological order to either \mathbb{Z}_2ℤ2 symmetry broken phase, which is in excellent agreement with the numerical results.

Homogenization process for a semi-diffusive coupled inclusion system

<p style='text-indent:20px;'>This work concerns the study of homogenization for a semi-diffusive system of inclusions with large diffusion and variable exponents. We show that under certain conditions the asymptotic dynamics is determined by an ordinary differential coupled inclusion system.</p>

Design and Analysis of RSA and Paillier Homomorphic Cryptosystems Using PSO-Based Evolutionary Computation

Non-lattice based homomorphic encryption schemes usually involve huge modular exponentiation operations. Thus, improving the efficiency of modular exponentiation for large exponents is a real-world issue. Modular exponentiation can be computed by a series of modular multiplications. However, performing a series of modular multiplications is computationally expensive. This paper proposes hardware /software co-design for efficient modular exponentiation. This article first explores the use of particle swarm optimization (PSO) for the modular exponentiation in software, and present an efficient hardware co-design utilizing FPGA. Our findings reveal that the suggested PSO approach surpasses all other deterministic and non-deterministic approaches already in use. Further, we also demonstrate a comprehensive analysis for the optimal performance and parameter selection of our proposed PSO approach. Finally, we implement homomorphic encryption schemes, such as RSA and Paillier, using our PSO-based hardware /software co-design. Our approach gains resource savings for 1024-bit as follows: RSA encryption/decryption - 60.7% (area) and 65.3% (DSP); Paillier encryption - 46.3% (area) and 40% (DSP) and Paillier decryption - 73.7% (area) and 66.6% (DSP). We have obtained area-time improvements of 1024-bit as follows: RSA encryption/decryption - 2.7x; Paillier encryption - 2x and Paillier decryption - 4.6x using Xilinx Virtex-7 FPGAs.