Quadratic Scaling Research Articles

Low-frequency hybridized excess vibrations of two-dimensional glasses

One hallmark of glasses is the existence of excess vibrational modes at low frequencies ω beyond Debye’s prediction. Numerous studies suggest that understanding low-frequency excess vibrations could help gain insight into the anomalous mechanical and thermodynamic properties of glasses. However, there is still intensive debate as to the frequency dependence of the population of low-frequency excess vibrations. In particular, excess modes could hybridize with phonon-like modes and the density of hybridized excess modes has been reported to follow D exc(ω) ∼ ω 2 in 2D glasses with an inverse power law potential. Yet, the universality of the quadratic scaling remains unknown, since recent work suggested that interaction potentials could influence the scaling of the vibrational spectrum. Here, we extend the universality of the quadratic scaling for hybridized excess modes in 2D to glasses with potentials ranging from the purely repulsive soft-core interaction to the hard-core one with both repulsion and attraction as well as to glasses with significant differences in density or interparticle repulsion. Moreover, we observe that the number of hybridized excess modes exhibits a decrease in glasses with higher density or steeper interparticle repulsion, which is accompanied by a suppression of the strength of the sound attenuation. Our results indicate that the density bears some resemblance to the repulsive steepness of the interaction in influencing low-frequency properties.

Recovering Marcus Theory Rates and Beyond without the Need for Decoherence Corrections: The Mapping Approach to Surface Hopping.

It is well-known that fewest-switches surface hopping (FSSH) fails to correctly capture the quadratic scaling of rate constants with diabatic coupling in the weak-coupling limit, as expected from Fermi's golden rule and Marcus theory. To address this deficiency, the most widely used approach is to introduce a "decoherence correction", which removes the inconsistency between the wave function coefficients and the active state. Here we investigate the behavior of a new nonadiabatic trajectory method, called the mapping approach to surface hopping (MASH), on systems that exhibit an incoherent rate behavior. Unlike FSSH, MASH hops between active surfaces deterministically and can never have an inconsistency between the wave function coefficients and the active state. We show that MASH not only can describe rates for intermediate and strong diabatic coupling but also can accurately reproduce the results of Marcus theory in the golden-rule limit, without the need for a decoherence correction. MASH is therefore a significant improvement over FSSH in the simulation of nonadiabatic reactions.

Assessing the functional vulnerability of woody plant communities within a large scale tropical rainforest dynamics plot.

Tropical forests are characterized by intricate mosaics of species-rich and structurally complex forest communities. Evaluating the functional vulnerability of distinct community patches is of significant importance in establishing conservation priorities within tropical forests. However, previous assessments of functional vulnerability in tropical forests have often focused solely on isolated factors or individual disturbance events, with limited consideration for a broad spectrum of disturbances and the responses of diverse species. We assessed the functional vulnerability of woody plant communities in a 60-ha dynamic plot within a tropical montane rainforest by conducting in silico simulations of a wide range disturbances. These simulations combined plant functional traits and community properties, including the distribution of functional redundancy across the entire trait space, the distribution of abundance across species, and the relationship between species trait distinctiveness and species abundance. We also investigated the spatial distribution patterns of functional vulnerability and their scale effects, and employed a spatial autoregressive model to examine the relationships between both biotic and abiotic factors and functional vulnerability at different scales. The functional vulnerability of tropical montane rainforest woody plant communities was generally high (the functional vulnerability of observed communities was very close to that of the most vulnerable virtual community, with a value of 72.41% on average at the 20m×20m quadrat scale), and they exhibited significant spatial heterogeneity. Functional vulnerability decreased with increasing spatial scale and the influence of both biotic and abiotic factors on functional vulnerability was regulated by spatial scale, with soil properties playing a dominant role. Our study provides new specific insights into the comprehensive assessment of functional vulnerability in the tropical rainforest. We highlighted that functional vulnerabilities of woody plant communities and their sensitivity to environmental factors varied significantly within and across spatial scales in the tropical rainforest landscape. Preserving and maintaining the functionality of tropical ecosystems should take into consideration the variations in functional vulnerability among different plant communities and their sensitivity to environmental factors.

Efficient Multimodal Transformer With Dual-Level Feature Restoration for Robust Multimodal Sentiment Analysis

With the proliferation of user-generated online videos, Multimodal Sentiment Analysis (MSA) has attracted increasing attention recently. Despite significant progress, there are still two major challenges on the way towards robust MSA: 1) inefficiency when modeling cross-modal interactions in unaligned multimodal data; and 2) vulnerability to random modality feature missing which typically occurs in realistic settings. In this paper, we propose a generic and unified framework to address them, named Efficient Multimodal Transformer with Dual-Level Feature Restoration (EMT-DLFR). Concretely, EMT employs utterance-level representations from each modality as the global multimodal context to interact with local unimodal features and mutually promote each other. It not only avoids the quadratic scaling cost of previous local-local cross-modal interaction methods but also leads to better performance. To improve model robustness in the incomplete modality setting, on the one hand, DLFR performs low-level feature reconstruction to implicitly encourage the model to learn semantic information from incomplete data. On the other hand, it innovatively regards complete and incomplete data as two different views of one sample and utilizes siamese representation learning to explicitly attract their high-level representations. Comprehensive experiments on three popular datasets demonstrate that our method achieves superior performance in both complete and incomplete modality settings.

Investigating the phase behaviour of binary suspensions of cellulose nanocrystals and montmorillonite with nonlinear rheology, SAXS and polarized optical microscopy

Using microscopy imaging techniques, X-ray scattering, and nonlinear rheological measurements, we have examined microstructural changes, and phase behavior of cellulose nanocrystals (CNC)/montmorillonite (MMT) composite colloidal systems. Fourier-transform (FT) rheology-stress decomposition methods, Lissajous-Bowditch plots, and other techniques are used to both qualitatively and quantitatively determine the various phases that the suspensions exhibit. In the MAOS (medium amplitude oscillatory shear) region, multiple scaling regions, as well as quadratic scaling regions are obtained. The dependence of the normalized third relative intensity (I3/1) values on angular frequency (ω) also provided new insights into how nonlinear parameters depend on the various CNC/MMT phases. These findings allow us to differentiate various phases exhibited by the composite system, including the isotropic, biphasic, and chiral nematic liquid crystalline (LC) phases. The CNC/MMT nanocomposite system exhibits a variety of liquid crystalline phases, which may contribute to improved mechanical properties for the system, including increased mixture strength, stiffness, and toughness because of the particle alignment in the LC phase. Additionally, the field of optics and photonics may find use for this nanocomposite system in the fabrication of sensors, polarizers, and optical films; energy storage applications where the aligned structures facilitate ion transport; and the formulation of paints and coatings where the strain stiffening and shear thinning behavior of the CNC/MMT composite system are critical.

Anomalous scaling law of strength and toughness in polymers with strong interfacial secondary bonds

In materials design, the increase of toughness usually leads to the decrease of strength, which is a long-standing challenge. Polymeric materials with strong interfacial secondary bonds have recently emerged as promising solutions due to their remarkable capabilities to increase toughness and strength simultaneously. Nevertheless, a quantitative understanding on how toughness increases with strength is unavailable, posing challenges for deeper understanding and on-demand design for stronger and tougher advanced materials. In this work, we reveal a quadratic scaling law of simultaneously increased strength and toughness that unifies multiscale evidences from theories, computations and experiments across multiple material systems, where the deformation mechanisms are dominated by breaking and reforming of interfacial secondary bonds. Our results highlight the significance of interfacial secondary bonding combined with topology design to enable a wide range of strong and tough advanced materials.

Decomposition of the drag force in steady and oscillatory flow through a hexagonal sphere pack

We investigate steady and oscillatory flow through a hexagonal close-packed arrangement of spheres in the framework of the volume-averaged momentum equation. We quantify the friction and pressure drag based on a direct numerical simulation dataset. Using the pressure decomposition of Graham (J. Fluid Mech., vol. 881, 2019), the pressure drag can be further split up into an accelerative, a viscous and a convective contribution. For the accelerative pressure, a closed-form expression can be given in terms of the potential flow solution. We investigate the contributions of the different drag components to the volume-averaged momentum budget and their Reynolds number scaling. For steady flow, we find that the friction and viscous pressure drag are proportional to $Re$ at low Reynolds numbers and scale with $Re^{1.4}$ for high Reynolds numbers. This is close to the steady laminar boundary layer scaling. For the convective pressure drag, we find a cubic scaling at low and a quadratic scaling at high Reynolds numbers. The Reynolds stresses have a minor contribution to the momentum budget. For oscillatory flow at low and medium Womersley numbers, the amplitudes of the drag components are similar to the steady cases at the same Reynolds number. At high Womersley numbers, the drag components behave quite differently and the friction and viscous pressure drag are relatively insignificant. The drag components are not in phase with the forcing and the superficial velocity; the phase lag increases with the Womersley number. This suggests that new models beyond the current quasisteady approaches need to be developed.

Multi-scale biomass estimation of an alpine desert shrub based on relative cover: A case study of Asterothamnus centraliasiaticus in the Qaidam Desert on the Qinghai–Tibet Plateau

Biomass estimates and the allocation characteristics of shrubs are essential indicators for studying the structure, function, and material and energy flows of alpine desert ecosystems. However, conventional destructive biomass estimation methods are unsuitable for large-scale application in alpine desert areas that are characterized by sparse vegetation and comprise extremely fragile ecosystems. In this study, we propose a multi-scale method for estimating the biomass of alpine desert shrubs based on relative cover and verified its accuracy, feasibility, and efficiency using data obtained from 100 individual plants and 27 quadrat plots of Asterothamnus centraliasiaticus, an important species growing in the Qaidam Desert on the Qinghai–Tibet Plateau. Having gained this information, we developed models to estimate the new branch (NBB), aboveground (AGB), and belowground (BGB) biomasses of A. centraliasiaticus at individual, quadrat, and regional scales. The modelling results revealed that an allometric relationship model (power law function) could be used to accurately represent the synergistic relationship between plant morphological variables and biomass increases at all assessed scales. Moreover, we established that at the individual scale, estimation models (NBB = 0.3015 × C1.0610 and AGB = 1.0046 × C1.4239) incorporating canopy area (C) as a parameter can better reflect the cumulative characteristics of the new branch and aboveground biomasses of A. centraliasiaticus, respectively, whereas canopy volume (CH, resembles a cylinder) is the most appropriate parameter for the model (BGB = 0.3262 × CH0.8822) used to estimate belowground biomass. In addition, at the sample scale, the total canopy cover (TC) of A. centraliasiaticus in the sample was established to be an appropriate parameter for estimating total biomass (TNBB = 0.0042 × TC1.4647, TAGB = 0.0033 × TC1.6308, and TBGB = 0.0110 × TC1.2554). Furthermore, at the regional scale, we found that incorporating relative canopy cover (RC) in models was effective for estimating the biomass density of A. centraliasiaticus (DNBB = 0.0012 × RC1.4647, DAGB = 0.0014 × RC1.6308, and DBGB = 0.0020 × RC1.2554) and can be combined with regional area estimates to determine the total biomass of A. centraliasiaticus at any spatial scale. On the basis of our findings, we established that the biomass estimation model for alpine desert shrubs, incorporating the parameter relative cover, can effectively solve the longstanding problem of scale conversion and facilitates the rapid and accurate estimation of the biomass components of alpine desert shrubs. In future studies, biomass could be efficiently estimated by combining models based on the relative cover data of shrub communities in relevant areas obtained using aerial photography and remote sensing. The results of this study have important implications for estimating the biomass of alpine desert shrubs, and will accordingly make a valuable contribution to the management of alpine desert ecosystems.

Efficient parallelization of quantum basis state shift

Basis state shift is central to many quantum algorithms, most notably the quantum walk. Efficient implementations are of major importance for achieving a quantum speedup for computational applications. We optimize the state shift algorithm by incorporating the shift in different directions in parallel. This provides a significant reduction in the depth of the quantum circuit in comparison to the currently known methods, giving a linear scaling in the number of gates versus working qubits in contrast to the quadratic scaling of the state-of-the-art method based on the quantum Fourier transform. For a one-dimensional array of size 2 n for n > 4, we derive the total number of two-qubit CX gates in the parallel circuit, using a total of qubits including an ancilla register for the decomposition of multi-controlled gates. We focus on the one-dimensional and periodic shift, but note that the method can be extended to more complex cases.

Causal explicit algorithm for heat conduction in a plasma

Hyperbolic heat conduction extends standard Spitzer-Harm heat conduction by including a term proportional to the time derivative of the heat flux. The new term arises from a kinetic derivation of the heat flux that includes higher order corrections. We present a causal explicit numerical algorithm for solving the nonlinear hyperbolic heat conduction equation in an unmagnetized plasma. The maximum stable timestep for the causal explicit algorithm scales linearly with the cell size, owing to the hyperbolic nature of the problem. This is in contrast to the quadratic scaling of the maximum stable timestep with the cell size for the parabolic forward time centered space algorithm. The favorable scaling of the timestep with the cell size enables a practical explicit implementation of heat conduction in high-performance massively parallel plasma codes. In particular, we have implemented the causal explicit algorithm in the laser plasma interaction code pF3D. We verify the CE algorithm and analyze its convergence rate by simulating a harmonic mode, which has an analytic solution within the context of the HHC model. We also compare simulations using the CE algorithm to those using the forward time centered space algorithm on a pair of test problems: evolution in time of a Gaussian temperature perturbation in a uniform plasma and heat transport in the presence of inverse bremsstrahlung heating by a Gaussian laser speckle.

Disorder-induced effects in high-harmonic generation process in fullerene molecules

The objective of this article is to investigate the profound nonlinear optical response exhibited by inversion symmetric fullerene molecules under the influence of different types of disorders described by the Anderson model. Our aim is to elucidate the localization effects on the spectra of high harmonic generation in such molecules. We show that the disorder-induced effects are imprinted onto molecules’ high-harmonic spectrum. Specifically, we observe a presence of strong even-order harmonic signals already for relatively small disorders. The odd-order harmonics intrinsic for disorder-free systems are generally robust to minor disorders. Both diagonal and off-diagonal disorders lift the degeneracy of states, opening up new channels for interband transitions, leading to the enhancement of the high-harmonic emission. The second harmonic signal has a special behavior depending on the disorder strength. Specifically in the case of diagonal disorder, the second harmonic intensity exhibits a quadratic scaling with the disorder strength, which enables the usage of the harmonic spectrum as a tool in measuring the type and the strength of a disorder.

Improved order of magnitude estimate of rotational effects on HAWTs

This work proposes an improved order of magnitude estimate methodology of rotational effects on wind turbines, by applying the partial pressure field concept to the case of rotating wind turbine blades, which allows isolating acceleration that rises as a result of the rotation of the blades; then an order of magnitude scaling is performed to estimate the partial pressure due to rotation for both fully attached flow and massively separated flow. In the case of fully attached flow, it was found that rotational effects are negligible; while in the case of massively separated flow, rotational effects were found to be important. The pressure reduction was found to scale linearly with the aspect ratio inside the separated boundary layer, while it scales quadratically with the aspect ratio in the outer inviscid flow. A new estimate of the Rossby number is proposed, which is found to be the inverse of the pressure reduction coefficient estimate in the outer inviscid flow. Then, a semi-empirical model of the pressure reduction is proposed based on the order of magnitude estimate in the outer inviscid flow. The proposed semi-empirical model is then compared against the reference data of three wind turbines, namely, MEXICO, NREL phase VI, and AVATAR 10 MW rotors; along with the already known centrifugal pumping model. The results showed that the centrifugal pumping alone, inside the separated boundary layer, is not sufficient to produce the large force augmentation observed in experimental data. In contrast, the improved semi-empirical model provided a good agreement with experimental data, thanks to the quadratic scaling of the aspect ratio. And, the new estimate of the Rossby number proved to be advantageous over the existing Rossby number estimate in terms of the dependency on the angle of attack. Furthermore, the tangential and drag force coefficients were found to be strongly dependent on the airfoil type and its corresponding pressure shape. In contrast, the normal and lift force coefficients were found to be less dependent on the airfoil type. The proposed semi-empirical model is still strongly empirical, future work needs to account for the complex effect of airfoil type and its consequent interaction with the outer inviscid flow through the boundary layer.

Quantum Metrology Using Time-Frequency as Quantum Continuous Variables: Resources, Sub-Shot-Noise Precision and Phase Space Representation.

We study the role of the electromagnetic field's frequency on the precision limits of time measurements from a quantum perspective, using single photons as a paradigmatic system. We demonstrate that a quantum enhancement of precision is possible only when combining both intensity and spectral resources and, in particular, that spectral correlations enable a quadratic scaling of precision with the number of probes. We identify the general mathematical structure of nonphysical states that achieve the Heisenberg limit and show how a finite spectral variance may cause a quantum-to-classical-like transition in precision scaling for pure states similar to the one observed for noisy systems. Finally, we provide a clear and consistent geometrical time-frequency phase space interpretation of our results, identifying what should be considered as spectral classical resources.

Event-Triggered Control Strategy for 2-DoF Helicopter System Under DoS Attacks

This paper focuses on the event-triggered strategy design of a two-degree-of-freedom (DoF) helicopter system (HS) with denial-of-service (DoS) attacks, which has essential applications in traffic information security and military information security. First, the attack characteristics of DoS are introduced from the cloud servers and converters at the physical layer. Then, hackers launching a DoS attack on HS also are considered. Second, a new first-order weight sampling (FOWS) control method is developed, fully considering sampling intervals resource consumption. Meanwhile, novel looped-functionals are constructed, which contain more sampling time information and sampling interval weight information. In addition, quadratic scaling is used to process the integral terms to obtain a tighter upper limit, which helps optimize the two control algorithms (TCAs). Furthermore, TCAs are obtained using the linear convex combination method (LCCM). Then, a new event-triggered controller under DoS attacks is designed to ensure the 2-DoF HS is asymptotically stable. Finally, the feasibility and superiority of our designed approach are verified through an experimental simulation.

Parabolic avalanche scaling in the synchronization of cortical cell assemblies

Neurons in the cerebral cortex fire coincident action potentials during ongoing activity and in response to sensory inputs. These synchronized cell assemblies are fundamental to cortex function, yet basic dynamical aspects of their size and duration are largely unknown. Using 2-photon imaging of neurons in the superficial cortex of awake mice, we show that synchronized cell assemblies organize as scale-invariant avalanches that quadratically grow with duration. The quadratic avalanche scaling was only found for correlated neurons, required temporal coarse-graining to compensate for spatial subsampling of the imaged cortex, and suggested cortical dynamics to be critical as demonstrated in simulations of balanced E/I-networks. The corresponding time course of an inverted parabola with exponent of χ = 2 described cortical avalanches of coincident firing for up to 5 s duration over an area of 1 mm2. These parabolic avalanches maximized temporal complexity in the ongoing activity of prefrontal and somatosensory cortex and in visual responses of primary visual cortex. Our results identify a scale-invariant temporal order in the synchronization of highly diverse cortical cell assemblies in the form of parabolic avalanches.

Hydrodynamic clogging of micro-particles in planar channels under electrostatic forces

Hydrodynamic clogging in planar channels is studied via direct numerical simulation for the first time, utilising a novel numerical test cell and stochastic methodology with special focus on the influence of electrostatic forces. Electrostatic physics is incorporated into an existing coupled lattice Boltzmann-discrete element method framework, which is verified rigorously. First, the dynamics of the problem is governed by the Stokes number, $St$ . At low $St$ , the clogging probability, $P$ , increases with $St$ due to increasing collision frequency. At high $St$ , however, $P$ decreases with $St$ due to quadratic scaling of hydrodynamic force acting on arches. Under electrostatic forces, clogging is well represented by the wall adhesion number, $Ad_w$ . For $Ad_w \lesssim 4$ , the mechanical dependence on $St$ is exhibited, while for $4 < Ad_w < 20$ , there is a transition to high $P$ as sliding along, and attachment to, the channel surface occurs increasingly. For $Ad_w \gtrsim 20$ , clogging occurs with $P > 0.95$ . Particle agglomeration, however, can also decrease $P$ due to diminished interaction with channel walls. Distinct parametric regions of clogging are also observed in relation to the channel width, while a critical width $w/d^*=2.6$ is reported, which increases to $w/d^*=4$ with strong electrostatic surface attachment. The number of particles that form stable arches across a planar channel is determined to be $n=\left \lceil {w/d}\right \rceil + 1$ . Finally, sensitivity to the Coulomb friction coefficient is determined in favour of calibrating numerical parameters to bulk system behaviour. The greatest sensitivities occur in situations where the arch stability is lowest, while clogging becomes independent of friction for strong wall adhesion.

Performance Boost of a Collective Qutrit Refrigerator

A single qutrit with transitions selectively driven by weakly-coupled reservoirs can implement one of the world's smallest refrigerators. We analyze the performance of $N$ such fridges that are collectively coupled to the reservoirs. We observe a quantum boost, manifest in a quadratic scaling of the steady-state cooling current with $N$. As $N$ grows further, the scaling reduces to linear, since the transitions responsible for the quantum boost become energetically unfavorable. Fine-tuned inter-qutrit interactions may be used to maintain the quantum boost for all $N$ and also for not-perfectly collective scenarios.

Scalable In-Memory Clustered Annealer With Temporal Noise of Charge Trap Transistor for Large Scale Travelling Salesman Problems

Combinatorial optimization problems (COP) are ubiquitous and practical in our daily life, but many of them are NP-hard and intractable to solve using conventional computing. Ising model-based annealer has gained increasing attention due to its efficiency in finding fast and approximate solutions. However, such solvers are suffering from scalability problem due to quadratically increasing spins, and there are few hardware designs proposed to support large-scale COPs. In this paper, we propose scalable in-memory annealers to solve the large-scale travelling salesman problem (TSP) with crossbar arrays of FinFET based charge trap transistors. The intrinsic temporal noise of the drain current caused by trapping/detrapping is used to realize the annealing process. We apply two kinds of hardware implementations to map Ising model: 1) Hopfield neural network (HNN) based design, converting from a graph network to a regular neural network to utilize crossbar memory arrays; 2) index-based design, using index to record the order to avoid the quadratic scaling up of spins to represent all permutations. For both implementations, hierarchical clustering algorithms are adopted to enable the solving of large-scale TSP and overcome the scalability challenge. We further speed up the system convergence by updating non-adjacent clusters in parallel with equal size k-means clustering. At last, three hardware designs (HNN with conventional and equal-size clustering, index method with conventional clustering) are tested with up to 1060-city problems and benchmarked by the system-level simulation to show their trade-offs. They all show benefits on fast convergence and ultra-low energy compared with state-of-the-art annealers, and present advantages on accuracy, chip area, or latency/energy consumption, respectively.

Squeezing and Overcoming the Heisenberg Scaling with Spin-Orbit Coupled Quantum Gases.

We predict that exploiting spin-orbit coupling in a harmonically trapped spinor quantum gas can lead to scaling of the optimal measurement precision beyond the Heisenberg scaling. We show that quadratic scaling with the number of atoms can be facilitated via squeezed center-of-mass excitations of the atomic motion using 1D spin-orbit coupled fermions or strongly interacting bosons (Tonks-Girardeau gas). Based on predictions derived from analytic calculations of the corresponding quantum Fisher information, we then introduce a protocol which overcomes the Heisenberg scaling (and limit) with the help of a tailored excited and entangled many-body state of a noninteracting Bose-Einstein condensate. We identify corresponding optimal measurements and argue that even finite temperature as a source of decoherence is, in principle, rather favorable for the obtainable precision scaling.

Nonlinearities in Black Hole Ringdowns.

The gravitational wave strain emitted by a perturbed black hole (BH) ringing down is typically modeled analytically using first-order BH perturbation theory. In this Letter, we show that second-order effects are necessary for modeling ringdowns from BH merger simulations. Focusing on the strain's (ℓ,m)=(4,4) angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic (4,4) mode's amplitude exhibits quadratic scaling with the fundamental (2,2) mode-its parent mode. The nonlinear mode's amplitude is comparable to or even larger than that of the linear (4,4) mode. Therefore, correctly modeling the ringdown of higher harmonics-improving mode mismatches by up to 2 orders of magnitude-requires the inclusion of nonlinear effects.