Exponential Decay Research Articles

Dynamics for wave equations connected in parallel with nonlinear localized damping

Abstract This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

Quantum Fisher kernel for mitigating the vanishing similarity issue

Quantum kernel (QK) methods exploit quantum computers to calculate QKs for the use of kernel-based learning models. Despite a potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; the exponential decay of the expectation value and the variance of the QK deteriorates implementation feasibility and trainability of the model with the increase of the number of qubits. This implies the need to design QKs alternative to the fidelity-based one. In this work, we propose a new class of QKs called the quantum Fisher kernels (QFKs) that take into account the geometric structure of the data source. We analytically and numerically demonstrate that the QFK can avoid the issue when shallow alternating layered ansatzes are used. In addition, the Fourier analysis numerically elucidates that the QFK can have the expressivity comparable to the fidelity-based QK. Moreover, we demonstrate synthetic classification tasks where QFK outperforms the fidelity-based QK in performance due to the absence of vanishing similarity. These results indicate that QFK paves the way for practical applications of quantum machine learning toward possible quantum advantages.

Band-limited wavelets beyond Gevrey regularity

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert [Formula: see text]-function.

Real-Time Simulation of Ultrafast Electronic Dynamics of Nanoscale Systems Involving an Organic Molecule and a Nanoparticle Dimer.

Understanding and predicting the behavior of nanomaterials composed of plasmons interacting with quantum emitters at ultrafast timescales is crucial for the better manipulation of light at the nanoscale and advancing technologies like ultrafast communication and computing. Here we perform a simulation of the "real-time" electronic dynamics of a coupled molecule-metal nanoparticle dimer interacting with an ultrashort resonant laser pulse by combining the real-time time-dependent density functional theory (RT-TDDFT) approach with the time-domain frequency-dependent fluctuating charge (TD-ωFQ) model, an atomistic electromagnetic (AEM) model for the dynamic plasmonic response of nanoparticles. It is shown that the induced dipoles evolve from an exponential decay pattern to a beat pattern with an increase in coupling strength, which is altered by changing the molecular orientation relative to the dimer axis. It is further shown that in the strong coupling regime, both the excited molecule and the plasmon relax rapidly due to the molecule-plasmon interaction, and the efficient coherent energy exchange between the interacting molecule and plasmon modes occurs on a femtosecond (fs) timescale. This work provides guidance on manipulating light-matter interaction and studying molecular plasmonics at extremely fast timescales.

Transient internal wave excitation of resonant modes in a density staircase

The density of the ocean generally increases continuously with depth as a consequence of variations in salinity and temperature. In some regions, however, the density profile of the ocean adopts a (double diffusive) staircase structure in which successive layers of uniform density fluid are separated by rapid density jumps. Previous work has theoretically examined the transmission and reflection of periodic internal (gravity) waves incident upon a density staircase. This predicted the existence of transmission spikes (global modes) for certain combinations of frequency and horizontal wave number in which the incident waves transmit perfectly across a density staircase. It was hypothesized that the transmission spikes occur when the incident waves resonate with natural modes of disturbances in the staircase. Here we derive theory to investigate the interactions between incident internal waves and modes. We demonstrate a close correspondence between the frequency for incident waves at a transmission spike and the real part of the frequency of modes at the same horizontal wave number. However, the frequency of the corresponding modes has a negative imaginary part corresponding to exponential decay of the modes in time. We perform numerical simulations to examine the impact of this near-resonant coupling when a vertically localized, quasimonochromatic internal wave packet interacts with a density staircase. In a range of simulations with fixed incident wave frequency and varying horizontal wave number, the measured transmission coefficient does not exhibit transmission spikes, but decreases monotonically with increasing horizontal wave number about the critical wave number separating strong and weak transmission. We show this occurs because the incident wave excites modes that then slowly transmit energy above and below the staircase at a rate consistent with the predicted decay rate of the modes. This rate is slower for staircases with more steps with the decay time increasing as the cube of the number of steps. Published by the American Physical Society 2024

Exact minimal effective amounts of three 1D continuous functions and their use in Anderson transitions

A conceptual quantity—the minimal effective amount of a quantum state ϕ(rj) in d-dimensional systems, defined by N∗=∑j=1Nmin{N|ϕ(rj)|2,1} , is newly proposed, where system sizes N=Ld . The effective dimension d IR can be calculated by N∗=h∗(L)LdIR , where h∗(L) does not change faster than any nonzero power. However, the nature of h∗(L) is unknown priori in any given model, but is at the same time very important for its numerical analysis. Hence, analytical results can provide insights on h∗(L) in more complex situations. In this paper, we get exact results of 1D continuous sine functions, exponential decay functions and power-law decay functions. They are used to distinguish extended and localized phases in the 1D uniform potential model, Anderson model and HMP (hopping rates modulated by a power-law function) model.

Micromechanical property evolution and damage mechanism of coal subjected to ScCO2 treatment

Carbon dioxide (CO2) injection into deep coal seams holds considerable significance in both carbon mitigation and enhanced recovery of clean coalbed methane (CBM). The CO2-coal interaction leads to changes in the physicochemical properties, potentially impacting the stability of the target sequestration coal reservoir. This study applied the nanoindentation, scanning probe microscopy (SPM), X-ray diffraction (XRD), and electron probe X-ray Micro-Analyzer (EPMA) techniques to probe the micromechanical property change and damage mechanism of coal subjected to ScCO2 injection. The result shows that the load-displacement behavior of the tested coal changes with continuous ScCO2 treatment, and the mechanical strength significantly weakens after 4 days of ScCO2 treatment. The nanoindentation test indicates that the peak and creep displacements of tested coal increased by 168.79 % and 1046.32 % respectively with 4-day ScCO2 treatment, inferring that the coal deformation increases and changes from elastic to plastic after ScCO2 injection. As the ScCO2 treatment time increases, Young's modulus and hardness of coal exhibit an exponential decay trend and rapidly decrease by 77.77 %∼89.76 % and 68.37 %∼81.63 % respectively after the initial 3-day treatment, followed by a slow decrease with the ScCO2 duration prolonging. The XRD and EPMA results show that the carbonate minerals reacted preferentially with ScCO2, and silicate minerals experienced gradual dissolution, while the amorphous carbon fluctuated almost unchanged in the tested coal. Carbonate minerals in tested coal have decreased by 70.68 % within the first 1-day ScCO2 treatment, making the most significant contribution to the initial rapid weakening of coal mechanics. The mineral distribution is closely related to the mechanical anisotropy of coal, which is confirmed by the phenomenon that the coal anisotropy gradually weakens with the mineral reaction process. It is inferred that the mineral component and distribution of coal seams are important indicators for evaluating the intensity of physical and chemical reactions in ScCO2-injected reservoirs, which directly determines the mechanical damage behavior of sequestration reservoirs. This research provides basic support for site selection and safety assessment of carbon sequestration in deep coal seams.

Improved calculation method for corresponding characteristics of foundation pit excavation on the diaphragm wall

The excavation of the pit causes displacement of the surrounding soil, and the excessive deformation causes damage to the existing support structure, which in turn affects the safety of the pit. Reasonable calculation of structural deformation and internal force is crucial for design and construction. Most of the existing theoretical methods simplify the diaphragm wall(DW) as an Euler-Bernoulli beam acting on the Winkler foundation, consider beam-soil interaction, and simplify the soil as isotropic and continuous. However, the shear effects due to the differential deformation of the structure, the unloading stresses acting on the structure due to foundation excavation, and the discontinuous nature of the multilayered soil are neglected. In this paper, an improved analysis method is proposed based on the elastic foundation beam theory. The DW is simplified as a Timoshenko beam and the foundation is simplified as a Vlasov two-parameter model, and a proposed model considering the shear effect of the DW and the interaction of adjacent springs is established, and the proposed method for the deformation and internal force of the DW is obtained by the finite-difference method. The correctness and applicability of the proposed method are verified by numerical simulation and field monitoring data. The effects of equivalent bending stiffness, equivalent shear stiffness, soil elastic modulus, and excavation depth on the deformation and internal force of the DW were further analyzed. The results show that the proposed method can accurately solve the deformation and internal force of the DW, and the maximum errors between the proposed method and the numerical simulation results are only 4.5 % and 1.3 %, respectively. The equivalent bending stiffness of the DW and the elastic modulus of the soil have more significant effects on the horizontal deformation and internal force. The excavation depth is more sensitive to the deformation of the DW, and there is an exponential decay trend between the two. When the equivalent shear and bending stiffnesses reach 6.8×107 kN·m2 and 2.9×107 kN/m, the effect on the horizontal deformation is no longer obvious. The proposed method in this paper can accurately calculate the internal force and deformation of the DW.

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ \epsilon -approximate inverse Cholesky factors of the kernel matrices with complexity O ( N log d ⁡ ( N / ϵ ) ) O(N\log ^d(N/\epsilon )) in space and O ( N log 2 d ⁡ ( N / ϵ ) ) O(N\log ^{2d}(N/\epsilon )) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

Investigation on uncoordinated deformation and failure mechanism and damage modeling of rock mass with weak interlayer zone

The weak interlayer zone (WIZ) is a high-risk geotechnical system with poor mechanical properties, strong ductility, loose structure, and wide distribution. To comprehensively investigate the uncoordinated deformation characteristics and failure mechanism of rock mass with WIZ, a series of uniaxial loading tests using digital image correlation (DIC) technique, and numerical simulation tests were carried out. Besides, the damage constitutive model of rock mass with WIZ was established, then the damage evolution law was also analyzed. Results show that a high strain localized zone appears in the WIZ from the beginning of loading, which then develops towards the contact surface between the WIZ and the rock mass. As loading proceeds, the WIZ exhibits layered spalling and ultimately forms through-cracks after the initiation of micro-cracks, and vertical tensile cracks to appear in the rock. causing vertical tensile cracks to appear in the rock. With the increase of the WIZ thickness and dip angle, the uniaxial compressive strength and deformation modulus of specimen show a negative exponential decay. When rock mass with WIZ collapse, the bonding of WIZ particles fails and the number of tensile-shear cracks gradually increases, eventually resulting in the plastic squeezing-out of WIZ. Moreover, the particles of the rock masses with smaller WIZ dip angles are mainly displaced vertically, and the failure is mainly caused by tensile splitting. The significant lateral displacement of particles in the rock mass with larger WIZ dip angles leads to the destruction of the rock mass mainly through shear slip, in particular a pronounced sliding down along the WIZ. Furthermore, the damage constitutive model of the rock mass with WIZ was established by distinguishing the total damage of the rock mass with WIZ into the damage of the WIZ and the damage of the rock mass. The model prediction results show that the total damage D increases with the increase of WIZ thickness and dip angle, and develops faster in the early stage and slower in the later stage. The research can provide effective basis and feasible ideas for the uncoordinated deformation and failure evolution, macro–micro failure mechanism, as well as the constitutive model of rock mass with WIZ in underground cavern under high geostress.

Serum measures of docosahexaenoic acid (DHA) synthesis underestimates whole body DHA synthesis in male and female mice

Females have higher docosahexaenoic acid (DHA) levels than males, proposed to be a result of higher DHA synthesis rates from α-linolenic acid (ALA). However, DHA synthesis rates are reported to be low, and have not been directly compared between sexes. Here, we apply a new compound specific isotope analysis model to determine n-3 PUFA synthesis rates in male and female mice and assess its potential translation to human populations. Male and female C57BL/6N mice were allocated to one of three 12-week dietary interventions with added ALA, eicosapentaenoic acid (EPA) or DHA. The diets included low carbon-13 (δ13C)-n-3 PUFA for four weeks, followed by high δ13C-n-3 PUFA for eight weeks (n=4 per diet, time point, sex). Following the diet switch, blood and tissues were collected at multiple time points, and fatty acid levels and δ13C were determined and fit to one-phase exponential decay modeling. Hepatic DHA synthesis rates were not different (P>.05) between sexes. However, n-3 docosapentaenoic acid (DPAn-3) synthesis from dietary EPA was 66% higher (P<.05) in males compared to females, suggesting higher synthesis downstream of DPAn-3 in females. Estimates of percent conversion of dietary ALA to serum DHA was 0.2%, in line with previous rodent and human estimates, but severely underestimates percent dietary ALA conversion to whole body DHA of 9.5%. Taken together, our data indicates that reports of low human DHA synthesis rates may be inaccurate, with synthesis being much higher than previously believed. Future animal studies and translation of this model to humans are needed for greater understanding of n-3 PUFA synthesis and metabolism, and whether the higher-than-expected ALA-derived DHA can offset dietary DHA recommendations set by health agencies.

Topological edge states in reconfigurable multi-stable mechanical metamaterials

Multi-stable mechanical structures find cutting-edge applications across various domains due to their reconfigurability, which offers innovative possibilities for engineering and technology advancements. This study explores the emergence of topological states in a one-dimensional chain-like multi-stable mechanical metamaterial composed of bistable units through a combination of mechanical and optical experiments. Drawing inspiration from the SSH (Su-Schrieffer-Heeger) model in condensed matter physics, we leverage the unique mechanical properties of the reconfigurable ligament-oscillator metamaterial to engineer a system with coexisting topological phases. Based on the one-dimensional periodic bistable chain, there is an exponential decay diffusion of elastic energy from both end boundaries towards the interior of the body. Experimental characterizations demonstrate the existence of stable topological phases within the reconfigurable multi-stable mechanical metamaterial. The findings underscore the potential of reconfigurable mechanical metamaterials as versatile platforms for flexibly exploring and manipulating topological phenomena, with applications ranging from impact resistance to energy harvesting and information processing.

Thin twigs decompose faster than thick ones under stagnant and flowing water: a double exponential decay model parameterization

Thin twigs decompose faster than thick ones under stagnant and flowing water: a double exponential decay model parameterization

An Examination of Learning Using Fourier Analysis of Mathematical Models of Consciousness

The concept of consciousness remains quite possibly the ultimate mystery for humanity. It is somehow linked to the ability to learn and experience the external environment. The process of learning is of utmost importance to any species and yet remains largely not understood. While many experts from varying fields throughout history have attempted to quantify and measure learning, all failed to capture the process with a mathematical explanation. The purpose of this paper is to present three different models of consciousness and their related Fourier transform and evaluate the ability of each to capture some of the behavior that is understood about the conscious experience. There is some evidence that the experience of consciousness as a function of time can be measured in an expression of phase. With this hypothesis, the technique of Fourier transform becomes a useful tool to examine a mathematical model of consciousness that can be transformed from the phase plane to the linearization of time. This paper summarizes some current research on consciousness and learning and perception of time. This paper then presents three different models of consciousness represented as a pulse function, Dirac Delta function, and exponential decay function and examines each model utilizing Fourier transform. Finally, this paper concludes that the intersection between psychology, neuroscience and cognitive science can be made by utilizing analysis tools in the field of applied mathematics.

Neural-network quantum-state study of the long-range antiferromagnetic Ising chain.

We investigate quantum phase transitions in the transverse field Ising chain with algebraically decaying long-range (LR) antiferromagnetic interactions using the variational Monte Carlo method with the restricted Boltzmann machine employed as a trial wave function ansatz. First, we measure the critical exponents and the central charge through the finite-size scaling analysis, verifying the contrasting observations in the previous tensor network studies. The correlation function exponent and the central charge deviate from the short-range (SR) Ising values at a small decay exponent α_{LR}, while the other critical exponents examined are very close to the SR Ising exponents regardless of α_{LR} examined. However, in the further test of the critical Binder ratio, we find that the universal ratio of the SR limit does not hold for α_{LR}<2, implying a deviation in the criticality. On the other hand, we find evidence of the conformal invariance breakdown in the conformal field theory (CFT) test of the correlation function. The deviation from the CFT description becomes more pronounced as α_{LR} decreases, although a precise breakdown threshold is yet to be determined.

A second gradient theory of thermoviscoelasticity

This paper is concerned with a rate-type theory of thermoviscoelasticity in which the second gradient of the displacement and the second temperature gradient are added to the classical set of independent constitutive variables. Viscoelasticity and related phenomena are of great importance in the study of biological materials. An adequate modeling of rubber-like materials and of biological soft tissues requires the use of the theory of viscoelasticity. Introduction of the concept of thermal displacement and the theory of multipolar continua allows us to show that Green-Naghdi thermomechanics can be used to derive a second gradient theory. The basic equations of the theory are established and the boundary conditions associated to nonsimple materials are investigated. The stress tensor and hyperstress tensor are shown to depend on the first and second temperature gradients. For rigid heat conductors we find that the temperature satisfies a fourth order equation. The boundary-initial-value problems are formulated. A uniqueness result in the dynamic theory of thermoviscoelastic materials is presented. We establish an existence result and prove the analyticity of the solutions. As a consequence, the exponential decay of the solutions and their impossibility of localization are obtained.

Experimental investigation of the behavior of UHPCFST under repeated axial tension

A ultra-high performance concrete filled steel tube (UHPCFST) is a composite structural component that extends the performance of both steel and concrete. It is a promising component to be used in a diagrid structure to further reduce self-weight. Compared with the research on compressive performance of UHPCFST, there is a lack of knowledge on the mechanical behavior of UHPCFST under axial tension. This paper fills this knowledge gap by carrying out experiments on UHPCFST subjected to monotonic and repeated tension. The test parameters are steel tube thickness and volume fraction of ultra-high performance concrete (UHPC). The failure modes, load-strain curves, tensile strength and tensile stiffness are studied in detail. Stiffness degradation is also studied. The test results show that: (1) under axial tension load, a UHPCFST typically experiences fracture failure of the outer steel tube, followed by section fracture of the UHPC, and notable deformation before a final ductile failure; (2) tensile strength increases with the increase of the thickness of the steel tube, while it is less obvious in a UHPCFST with a higher steel ratio; (3) the force-strain curve of a UHPCFST under monotonical axial tension is close to that of the UHPCFST under repeated axial tension, suggesting that the accumulated damage during unloading and reloading is limited. An exponential decay formula is proposed to predict stiffness degradation observed in the repeated axial tensile tests. It is found that the design codes from Europe, USA, and China underestimate tensile strength and stiffness of UHPCFST. Finally, a three-phase empirical model is proposed for the load-strain curve of UHPCFST under tension.

Nonmesonic Quantum Many-Body Scars in a 1D Lattice Gauge Theory.

We investigate the meson excitations (particle-antiparticle bound states) in quantum many-body scars of a 1D Z_{2} lattice gauge theory coupled to a dynamical spin-1/2 chain as a matter field. By introducing a string representation of the physical Hilbert space, we express a scar state |Ψ_{n,l}⟩ as a superposition of all string bases with an identical string number n and a total length l. For the small-l scar state |Ψ_{n,l}⟩, the gauge-invariant spin exchange correlation function of the matter field hosts an exponential decay as the distance increases, indicating the existence of stable mesons. However, for large l, the correlation function exhibits a power-law decay, signaling the emergence of nonmesonic excitations. Furthermore, we show that this mesonic-nonmesonic crossover can be detected by the quench dynamics, starting from two low-entangled initial states, respectively, which are experimentally feasible in quantum simulators. Our results expand the physics of quantum many-body scars in lattice gauge theories and reveal that the nonmesonic state can also manifest ergodicity breaking.

Stringy evidence for a universal pattern at infinite distance

Infinite distance limits in the moduli space of a quantum gravity theory are characterized by having infinite towers of states becoming light, as dictated by the Distance Conjecture in the Swampland program. These towers imply a drastic breakdown in the perturbative regimes of the effective field theory at a quantum gravity cut-off scale known as the species scale. In this work, we find a universal pattern satisfied in all known infinite distance limits of string theory compactifications, which relates the variation in field space of the mass of the tower and the species scale: ∇→mm·∇→ΛspΛsp=1d−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\\overrightarrow{\ abla}m}{m}\\cdotp \\frac{\\overrightarrow{\ abla}{\\Lambda}_{\ extrm{sp}}}{\\Lambda_{\ extrm{sp}}}=\\frac{1}{d-2} $$\\end{document} in d spacetime dimensions. This implies a more precise definition of the Distance conjecture and sharp bounds for the exponential decay rates. We provide plethora of evidence in string theory in diverse dimensions and with different levels of supersymmetry, and identify some sufficient conditions that allow the pattern to hold from a bottom-up perspective.

A new method for estimating megacity NOx emissions and lifetimes from satellite observations

Abstract. We present a new method for estimating NOx emissions and effective lifetimes from large cities based on NO2 measurements from the TROPOspheric Monitoring Instrument (TROPOMI) (PAL dataset, May 2018–November 2021). As in previous studies, the estimate is based on the downwind plume evolution for different wind directions separately. The novelty of the presented approach lies in the simultaneous fit of downwind patterns for opposing wind directions, which makes the method far more robust (i.e., less prone to local minima with nonphysically high or low lifetimes) than a single exponential decay fit. In addition, the new method does not require the assumption of a city being a “point source” but also derives the spatial distribution of emissions. The method was successfully applied to 100 cities worldwide on a seasonal scale. Fitted emissions generally agree reasonably with the Emissions Database for Global Atmospheric Research (EDGAR) v6 (R=0.72) and are on average 14 % lower, while estimated uncertainties are still rather large (≈ 30 %–50 %). Lifetimes were found to be rather short (2.44 ± 0.68 h) and show no distinct dependency on season or latitude, which might be a consequence of discarding observations at high solar zenith angles (&gt;65°). The main limitations of this and similar methods are the underlying assumptions of steady state (meaning constant emissions, wind fields and chemical conditions) within about 100 km downwind from a city, which is probably a simplification that is too strong in order to reach higher accuracies.