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Unlike one-dimensional polymers, the theoretical framework on the behaviors of two-dimensional (2D) polymers is far from completeness. In this study, we model single-layer flexible 2D polymers of different sizes and examine their scaling behaviors in solution, represented by Rg ∼ Lν, where Rg is the radius of gyration and L is the side length of a 2D polymer. We find that the scaling exponent ν is 0.96 for a good solvent and 0.64 for under poor solvent condition. Interestingly, we observe a previously unnoticed phenomenon: under intermediate solvent conditions, the 2D polymer folds to maintain a flat structure, and as L becomes larger, multiple folded structures emerge. We introduce a shape parameter Q to diagram the relationship of folded structures with the polymer size and solvent condition. Theoretically, we explain the folding transitions by the competition between bending and solvophobic free energies. [ABSTRACT FROM AUTHOR]

We present results from the molecular dynamics simulation of surface-directed spinodal decomposition in binary fluid mixtures (A + B) with off-critical compositions. The aim is to elucidate the role of composition ratio in the early time wetting kinetics under the influence of long-range surface potential. In our simulations, the attractive part of surface potential varies as V(z) = −ϵa/zn, with ϵa being the surface-potential strength. The surface prefers the "A" species to form the wetting layer. Its thickness [R1(t)] for the majority wetting (number of A-type particles [NA] > number of B-type particles [NB]) grows as a power-law with an exponent of 1/(n + 2). This is consistent with the early time kinetics in the form of potential-dependent growth present in the Puri–Binder model. However, for minority wetting (NA < NB), the growth exponent in R1(t) is less than 1/(n + 2). Furthermore, on decreasing the field strength ϵa, we recover 1/(n + 2) for a minority wetting case. We provide phenomenological arguments to explain the early time wetting kinetics for both cases. [ABSTRACT FROM AUTHOR]

In this paper, we report on the electrical impedance measurement of human endothelial cellular networks and show the existence of emergent power law behavior in its admittance. In particular, we find that the admittance scales with the frequency ω as ωα, with the exponent that varies with the degree of the disruption caused by the inflammation in the endothelial cellular network. We demonstrate that the power law of the measured electrical admittance can be understood by a simple percolation model of a large R–C (resistor–capacitor) network, which allows us to relate quantitatively and the intensity of inflammation. Our results suggest that the electrical properties of heterogeneous biomaterials, like living tissues, behave as a complex microstructural network. [ABSTRACT FROM AUTHOR]

Sizable glass formers feature numerous unique properties and potential applications, but many questions regarding their glass transition dynamics have not been resolved yet. Here, we have analyzed structural relaxation times measured as a function of temperature and pressure in combination with the equation of state obtained from pressure–volume–temperature measurements. Despite evidence from previous dielectric studies indicating a remarkable sensitivity of supercooled dynamics to compression, and contrary to intuition, our results demonstrated the proof for the almost equivalent importance of thermal energy and free volume fluctuations in controlling reorientation dynamics of sizable molecules. The found scaling exponent γ = 3.0 and Ev/Ep ratio of 0.6 were typical for glass-forming materials with relaxation dynamics determined by both effects with a minor advantage of thermal fluctuations involvement. It shows that the high values of key parameters characterizing the sensitivity of the glass transition dynamics to pressure changes, i.e., activation volume ΔV and dTg/dP, are not a valid premise for a remarkable contribution of volume to glass transition dynamics. [ABSTRACT FROM AUTHOR]

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by n, the algorithm uses O(n1.43) field operations, breaking through the 3/2 barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require O(n1.63) field operations in general, and n3/2+o(1) field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in n5/3+o(1) operations, while previous ones run in O(n2) operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation. [ABSTRACT FROM AUTHOR]

The correlation properties of a random system of densely packed disks, obeying a power-law size distribution, are analyzed in reciprocal space in the thermodynamic limit. This limit assumes that the total number of disks increases infinitely, while the mean density of the disk centers and the range of the size distribution are kept constant. We investigate the structure factor dependence on momentum transfer across various number of disks and extrapolate these findings to the thermodynamic limit. The fractal power-law decay of the structure factor is recovered in reciprocal space within the fractal range, which corresponds to the range of the size distribution in real space. The fractal exponent coincides with the exponent of the power-law size distribution as was shown previously by the authors of the work of Cherny et al. [J. Chem. Phys. 158(4), 044114 (2023)]. The dependence of the structure factor on density is examined. As is found, the power-law exponent remains unchanged but the fractal range shrinks when the packing fraction decreases. Additionally, the finite-size effects are studied at extremely low momenta of the order of the inverse system size. We show that the structure factor is parabolic in this region and calculate the prefactor analytically. The obtained results reveal fractal-like properties of the packing and can be used to analyze small-angle scattering from such systems. [ABSTRACT FROM AUTHOR]

Minima of the log-multiphase variational integral w ↦ ∫ Ω [ | D w | log ⁡ (1 + | D w |) + a (x) | D w | q + b (x) | D w | s ] d x , have locally Hölder continuous gradient under sharp quantitative bounds linking the growth powers (q , s) to the Hölder exponents of the modulating coefficients a (⋅) and b (⋅) respectively. [ABSTRACT FROM AUTHOR]

At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann) steady state. Here we consider scale-invariant power-law potentials V (x) ∼ | x | m , where m > 0, and employ the optimal fluctuation method (OFM) to determine the large- | x | tails of the steady-state probability distribution P (x) of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large- | x | tails of ln ⁡ P (x) up to a dimensionless factor α (H , m) , where 0 < H < 1 is the Hurst exponent. We determine α (H , m) analytically (i) in the limits of H → 0 and H → 1, and (ii) for m = 2 and arbitrary H, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact P (x) and autocovariance. The form of the tails of P (x) yields exact conditions, in terms of H and m, for the particle confinement in the potential. For H ≠ 1 / 2 , the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for m = 2. To compute the optimal paths and the factor α (H , m) for arbitrary permissible H and m, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which accounts analytically for an intrinsic cusp singularity of the optimal paths for H < 1 / 2 . [ABSTRACT FROM AUTHOR]

We present a comprehensive study of nearly associative algebras. Our research shows that these algebras are power associative Lie-admissible for which the related Lie algebra is solvable, and are also Jordan-admissible. Furthermore, we describe the features of finite-dimensional nearly associative algebras that are nilpotent. In addition, we define the concepts of radical and semisimplicity for this type of algebras. We give a characterization of semisimple nearly associative algebras and prove the Wedderburn Principal Theorem for them. Lastly, we deal with quadratic nearly associative algebras. We provide a characterization and then an inductive description of them through the process of double extension, enriching our understanding of the structural intricacies of this class of algebras. [ABSTRACT FROM AUTHOR]

Let S be a Cohen-Macaulay ring which is local or standard graded over a field, and let I be an unmixed ideal that is generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal J ⊆ I (m) equals the m -th symbolic power I (m) of I. Second, we provide a saturation-type formula to compute I (m) and employ it to deduce a theoretical criterion for when I (m) = I m. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of I (m). Along the way, we prove a generalized version of a conjecture raised by Eisenbud and Mazur about ann S (I (m) / I m) , and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals. [ABSTRACT FROM AUTHOR]

The aim of this paper is the study of double‐phase problems with variable exponent. Using the Clark's theorem and the symmetric mountain pass lemma, we prove the existence of infinitely many small solutions and infinitely many large solutions, respectively. [ABSTRACT FROM AUTHOR]

In addition to the usual slope and location parameters included in a regular two‐parameter logistic model (2PL), the logistic positive exponent (LPE) model incorporates an item parameter that leads to asymmetric item characteristic curves, which have recently been shown to be useful in some contexts. Although this model has been used in some empirical studies, an identifiability analysis (i.e., checking the (un)identified status of a model and searching for identifiablity restrictions to make an unidentified model identified) has not yet been established. In this paper, we formalize the unidentified status of a large class of fixed‐effects item response theory models that includes the LPE model and related versions of it. In addition, we conduct an identifiability analysis of a particular version of the LPE model that is based on the fixed‐effects one‐parameter logistic model (1PL), which we call the 1PL‐LPE model. The main result indicates that the 1PL‐LPE model is not identifiable. Ways to make the 1PL‐LPE useful in practice and how different strategies for identifiability analyses may affect other versions of the model are also discussed. [ABSTRACT FROM AUTHOR]

In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920's. We study the set of singular vectors on an affine subspace of \mathbb {R}^n. We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamical stability of spiky steady states to a chemotaxis-consumption model with signal-dependent motilities on the half space, where no-flux boundary condition for the bacteria and inhomogeneous Dirichlet boundary condition for the nutrient are prescribed. The motilities of bacteria depend on the concentration of nutrient in the form of power law, which means that they are degenerate if the power-law exponent is positive and are singular if the exponent is negative at the zero state of nutrient. We show that in both cases the spiky patterns admitted by the model are nonlinearly stable under appropriate perturbations. We introduce a unified approach to dealing with both degeneracy and singularity in the diffusivity for this model. The approach is based on a combination of weighted energy estimates and weighted elliptic estimates, where various weights are artfully constructed to capture the complex behaviors of solutions near the zero state. Numerical simulations are also carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of nontrivial solutions of the following problem { − Δ p u + (− Δ p) s u = λ | u | q − 2 u + | u | p ⁎ − 2 u in Ω , u = 0 on R N ∖ Ω , where Ω ⊂ R N is a bounded open set with smooth boundary, dimension N ≥ 2 , parameter λ > 0 , exponents 0 < s < 1 < p < N , while q ∈ (1 , p ⁎) with p ⁎ = N p N − p. The problem is driven by an operator of mixed order obtained by the sum of the classical p -Laplacian and of the fractional p -Laplacian. We analyze three different scenarios depending on exponent q. For this, we combine variational methods with some topological techniques, such as the Krasnoselskii genus and the Lusternik–Schnirelman category theories. [ABSTRACT FROM AUTHOR]

At temporal light modulation (TLM) frequencies between 80 Hz and 20 000 Hz observers may perceive a series of repeated images called the phantom array effect (PAE) when they move their eyes in large saccades across a modulating light source or across a scene lit by the modulating light source. To date, there is no well-established measure for quantifying PAE visibility, but there is growing awareness of the need for one among design professionals and sensitive populations. This paper documents a new measure, the phantom array visibility measure (PAVM), which is based on the results of recent human factors experiments. The measure follows the mathematical underpinning used by the flicker visibility measure and the stroboscopic visibility measure, where the time-domain TLM waveform is converted into its Fourier frequency components; each component is evaluated through a threshold curve of modulation depth, then summed through an equation employing a Minkowski exponent. This scales the PAVM so that a value of 1 indicates a waveform at a threshold visibility in the conditions of the underlying experiment. [ABSTRACT FROM AUTHOR]

Prime Power RSA is a variant of the RSA scheme due to Takagi with modulus N = p r q for r ⩾ 2 , where p , q are of the same bit-size. In this paper, we concentrate on one type of Prime Power RSA which assumes e · d ≡ 1 mod p r − 1 (p − 1) (q − 1) . Two new attacks on this type of Prime Power RSA are presented when given two pairs of public and private exponents, namely, (e 1 , d 1) and (e 2 , d 2) with the same modulus N. Suppose that d 1 < N β 1 , d 2 < N β 2 . In 2015, Zheng and Hu showed that when β 1 β 2 < (r − 1) 3 / (r + 1) 3 , N may be factored in probabilistic polynomial time. The first attack of this paper shows that one can obtain the factorization of N in probabilistic polynomial time, provided that β 1 β 2 < r / (r + 1) 3 . Later, in the second attack, we improve both the first attack and the attack of Zheng and Hu, and show that the condition β 1 β 2 < r (r − 1) 2 / (r + 1) 3 already suffices to break the Prime Power RSA. By introducing multiple parameters, our lattice constructions take full advantage of known information, and obtain the best known attack. Specifically, we make full use of the information that p r is a divisor of N, while the attack of Zheng and Hu only assumes that p r − 1 is a divisor of N. As a consequence, this method implies a better lattice construction, and thus improves the previous attack. The experiments which reach a better upper bound than before also verify it. Our approaches are based on Coppersmith's method for finding small roots of bivariate modular polynomial equations. [ABSTRACT FROM AUTHOR]

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro‐geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums. [ABSTRACT FROM AUTHOR]

We prove that a large family of pairs of graphs satisfy a polynomial dependence in asymmetric graph removal lemmas. In particular, we give an unexpected answer to a question of Gishboliner, Shapira and Wigderson by showing that for every t⩾4$t \geqslant 4$, there are Kt$K_t$‐abundant graphs of chromatic number t$t$. Using similar methods, we also extend work of Ruzsa by proving that a set A⊂{1,⋯,N}$\mathcal {A}\subset \lbrace 1,\dots,N \rbrace$ which avoids solutions with distinct integers to an equation of genus at least two has size O(N)$\mathcal {O}(\sqrt {N})$. The best previous bound was N1−o(1)$N^{1 - o(1)}$ and the exponent of 1/2$1/2$ is best possible in such a result. Finally, we investigate the relationship between polynomial dependencies in asymmetric removal lemmas and the problem of avoiding integer solutions to equations. The results suggest a potentially deep correspondence. Many open questions remain. [ABSTRACT FROM AUTHOR]

The performance of multivariate calibration models ŷ = f(x) for the prediction of a numerical property y from a set of x‐variables depends on the type of scaling of the x‐variables. Common scaling methods are autoscaling (dividing the centered x by its standard deviation s) and Pareto scaling (dividing the centered x by sP with P = 0.5). The adjusted Pareto scaling presented here varies the exponent P between 0 (no scaling) and 1 (autoscaling) with the aim of obtaining an optimum prediction performance for ŷ. Related scaling methods based on the variable spread are range scaling and vast scaling; while level scaling is based on the location (central value) of the variable. These scaling methods and robust versions are compared for models created by partial least‐squares (PLS) regression. The applied strategy repeated double cross validation (rdCV) evaluates the model performance for test set objects and considers its variability. Results with three data sets from chemistry show: (a) the efficacy of the different scaling methods depends on the data structure; (b) optimization of the Pareto exponent P is recommended; (c) range scaling or vast scaling may be better than adjusted Pareto scaling; (d) in general a heuristic search for the best scaling method is advisable. Overall, the consideration of different variants of scaling allow for a flexible adjustment of the variable contributions to the calibration model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic plate system with a boundary frictional damping having a variable exponent $ m(x) $ m (x). We investigate the competition between the two types of damping and their effects on the energy decay rates. Using the multiplier method, we establish explicit decay results depending on both m and the relaxation function g. Our results address both the optimality and generality and improve earlier related results in the literature. [ABSTRACT FROM AUTHOR]

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n × n Boolean matrices of the form O (n 2. 5 7 5) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the classical computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time Õ (n 2 + λ / 2) = Õ (n 2. 4 3 4) , where λ satisfies the equation ω (1 , λ , 1) = 1 + 1. 5 λ and ω (1 , λ , 1) is the exponent of the multiplication of an n × n λ matrix by an n λ × n matrix. Next, we consider a relaxed version of the MW problem (in the classical model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most ℓ in time Õ ((n / ℓ) n ω (1 , log n ℓ , 1)). Then, by reducing the relaxed problem to the so called k -witness problem, we provide an algorithm that reports for each non-zero entry C [ i , j ] of the product matrix C a witness of rank O (⌈ W C (i , j) / k ⌉) , where W C (i , j) is the number of witnesses for C [ i , j ] , with high probability. The algorithm runs in Õ (n ω k 0. 4 6 5 3 + n 2 + o (1) k) time, where ω = ω (1 , 1 , 1). [ABSTRACT FROM AUTHOR]

To study the effects of long-range temporal correlations on kinetic roughening of the molecular beam epitaxy (MBE) growth systems in both (1 + 1) - and (2 + 1) -dimensions, we adopt fast fractional Gaussian noise (FFGN) technique to generate temporally correlated noise to the continuum growth equations including Mullins–Herring (MH) and Villain–Lai–Das Sarma (VLDS), and the typical discrete growth models including Das Sarma–Tamborenea (DT) and Wolf–Villain (WV) with slight modifications. Extensive numerical simulations on these continuum and discrete growth systems are performed in the presence of long-range temporal correlations, and the scaling exponents are obtained correspondingly. We find that these correlated growth systems exhibit high dependence on the temporal correlation exponent within the large temporal correlated regimes, and there exist non-trivial scaling properties in the correlated DT and WV models. Our results also show that the scaling exponents in these linear and nonlinear MBE growth equations are in good agreement with the theoretical predictions. Furthermore, the saturated surface morphologies are compared qualitatively through simulating numerically these continuum and discrete growth systems in the presence of long-range temporal correlations. Generally, as the temporal correlation exponent increases, the surface heights of these correlated discrete and continuum growth systems exhibit evident increasing trends. Likewise, with the temporal correlation exponent increasing, the surface morphologies of the modified DT and WV models undergo a gradual transition from self-affine hills to sharp peaks, while the growing surfaces of the correlated MH and VLDS equations gradually become relatively smooth. [ABSTRACT FROM AUTHOR]

We study the spectral properties of the black hole X-ray transient binary 4U 1630–472 during its 2022 and 2023 outbursts with Insight-HXMT observations. We find that the outbursts are in peculiar soft states. The effect of the hardening factor on the disk temperature is taken into account by the kerrbb model, and the flux and temperature of the disk are found to follow F ∝ T eff 3.92 ± 0.13 and F ∝ T eff 4.91 ± 1.00 for the two outbursts, respectively. The flux–temperature relation is roughly consistent with a standard disk. By fitting with a p -free model, the p -value, the exponent of the radial dependence of the disk temperature, is found to have an anticorrelation with disk temperature. Combining a joint diagnostic analysis with a diagram of the relation between the nonthermal fraction and luminosity, we find a possible scenario that the disk evolves from 2022 to 2023 toward a slim one with a decreasing radiation efficiency, where such an evolution may depend on the fraction of the nonthermal emission in the high soft state. [ABSTRACT FROM AUTHOR]

The weak commutativity group χ (G) is generated by two isomorphic groups G and G φ subject to the relations [ g , g φ ] = 1 for all g ∈ G . We establish a finiteness criterion for the subgroups of χ (G) in terms of the set T χ (G) = { [ g 1 , g 2 φ ] | g i ∈ G } . We also obtain sufficient conditions under which some specific (local) classes of groups are invariant under the operator χ. Moreover, we prove that if G is a locally finite group with exp (G) = n , then χ (G) is locally finite and has finite n-bounded exponent. Communicated by Mark Lewis [ABSTRACT FROM AUTHOR]

This article deals with the results of technological tests of pressability (Erichsen's - IE index and capping - limit drawing ratio (LDR)). It also presents on which material parameter (ductility, strain hardening exponent, coefficient of normal anisotropy...) should be given more emphasis when choosing a steel sheet, if stretching stress (uniaxial tension, biaxial tension) dominates during the production of stampings, or if dominates the "pull-pressure" technique (pulling out from under the flange). [ABSTRACT FROM AUTHOR]

Experiments have revealed that biological cells exhibit a universal power-law rheology, but the underlying mechanisms remain elusive. Here, we present a minimal model to explain the power-law creep of cell cortex, which is abstracted as chains of crosslinkers with random binding energies. Using this model, we show that when both the load and chain length are small, the logarithm of both the strain and time scales with the fraction of unbound crosslinkers, leading to power-law creep with a constant exponent, as observed in many experiments. Increasing the load alters the latter relationship between time and unbinding fraction, and thus, increases the power-law exponent, explaining the stress-induced nonlinearity in some experiments. Increasing the chain length alters this relationship as well, and as a result, the exponent grows proportionally with the chain length, explaining the crosslinker-density-induced nonlinearity in other experiments. This work provides a mesoscopic explanation for the linear and nonlinear power-law creep of cell cortex and may serve as a basis for understanding the cytoskeletal mechanics. [ABSTRACT FROM AUTHOR]

Clouds are one of the largest sources of uncertainty in climate predictions. Global km‐scale models need to simulate clouds and precipitation accurately to predict future climates. To isolate issues in their representation of clouds, models need to be thoroughly evaluated with observations. Here, we introduce multifractal analysis as a method for evaluating km‐scale simulations. We apply it to outgoing longwave radiation fields to investigate structural differences between observed and simulated anvil clouds. We compute fractal parameters which compactly characterize the scaling behavior of clouds and can be compared across simulations and observations. We use this method to evaluate the nextGEMS ICON simulations via comparison with observations from the geostationary satellite GOES‐16. We find that multifractal scaling exponents in the ICON model are significantly lower than in observations. We conclude that too much variability is contained in the small scales (<100km) $(< 100\ \mathrm{k}\mathrm{m})$ leading to less organized convection and smaller, isolated anvils. Plain Language Summary: In this paper, we present a new approach to evaluating state‐of‐the‐art high‐resolution climate models. We use a type of analysis that captures how a field like outgoing radiation varies between two points in space; it is called multifractal analysis. We apply multifractal analysis to snapshots of climate model simulations and satellite observations, and compare the results to evaluate the model. In contrast to traditional evaluation approaches, our method focuses on the evaluation of the spatio‐temporal structure of cloud fields, exploiting previously untapped information content. Hence, it can take into account the fine details in time and space that high‐resolution climate models provide. We use our method to evaluate the ICON atmospheric model. We find that the simulations does not contain enough large clusters of clouds, as found in big thunderstorms, but instead clouds are randomly distributed in space: the simulated clouds are not organized enough. Key Points: Quantifiable, structural evaluation metrics such as multifractal analysis should be used to evaluate and improve km‐scale modelsMultifractal analysis finds that deep convection in the ICON model is not organized enough leading to smaller fractal parametersThe model's bias toward smaller fractal parameters can be attributed to clouds simulated over the ocean [ABSTRACT FROM AUTHOR]

Abstract.In this article, we introduce a new concept of integrability called extended residually h-integrability with exponent r, which deals with weighted sums of arrays under sublinear expectation. The classical copy of this concept extends the residually h-integrability given by Wang, Hu, and Volodin (2018). On this basis, we obtain three forms of weak laws of large numbers for weighted sums of arrays of rowwise negatively dependent or rowwise independent random variables, respectively, under sublinear expectation. As corollaries, we get some weak laws of large numbers for arrays of random variables with normalizing factor kn under sublinear expectation. [ABSTRACT FROM AUTHOR]

Let ϕ(z)$\phi (z)$ be a non‐isotrivial rational function in one‐variable with coefficients in F¯p(t)$\overline{\mathbb {F}}_p(t)$ and assume that γ∈P1(F¯p(t))$\gamma \in \mathbb {P}^1(\overline{\mathbb {F}}_p(t))$ is not a post‐critical point for ϕ$\phi$. Then we prove that the diophantine approximation exponent of elements of ϕ−m(γ)$\phi ^{-m}(\gamma)$ are eventually bounded above by ⌈deg(ϕ)m/2⌉+1$\lceil \deg (\phi)^m/2\rceil +1$. To do this, we mix diophantine techniques in characteristic p$p$ with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we take any wandering point a∈P1(F¯p(t))$a\in \mathbb {P}^1(\overline{\mathbb {F}}_p(t))$ and write ϕn(a)=an/bn$\phi ^n(a)=a_n/b_n$ for some coprime polynomials an,bn∈F¯p[t]$a_n,b_n\in \overline{\mathbb {F}}_p[t]$, then we prove that 12⩽lim infn→∞deg(an)deg(bn)⩽lim supn→∞deg(an)deg(bn)⩽2,$$\begin{equation*} \frac{1}{2}\leqslant \liminf _{n\rightarrow \infty } \frac{\deg (a_n)}{\deg (b_n)} \leqslant \limsup _{n\rightarrow \infty } \frac{\deg (a_n)}{\deg (b_n)}\leqslant 2, \end{equation*}$$whenever 0 and ∞$\infty$ are both not post‐critical points for ϕ$\phi$. In characteristic p$p$, the Thue–Siegel–Dyson–Roth theorem is false, and so our proof requires different techniques than those used by Silverman. [ABSTRACT FROM AUTHOR]

AbstractRecent theoretical studies have shown that heavy-tails can emerge in stochastic optimization due to ‘multiplicative noise’, even under surprisingly simple settings, such as linear regression with Gaussian data. While these studies have uncovered several interesting phenomena, they consider conventional stochastic optimization problems, which exclude decentralized settings that naturally arise in modern machine learning applications. In this paper, we study the emergence of heavy-tails in decentralized stochastic gradient descent (DE-SGD), and investigate the effect of decentralization on the tail behavior. We first show that, when the loss function at each computational node is twice continuously differentiable and strongly convex outside a compact region, the law of the DE-SGD iterates converges to a distribution with polynomially decaying (heavy) tails. To have a more explicit control on the tail exponent, we then consider the case where the loss at each node is a quadratic function, and show that the tail-index can be estimated as a function of the step-size, batch-size, and the topological properties of the network of the computational nodes. Then, we provide theoretical and empirical results showing that DE-SGD has heavier tails than centralized SGD. We also compare DE-SGD to disconnected SGD where nodes distribute the data but do not communicate. Our theory uncovers an interesting interplay between the tails and the network structure: we identify two regimes of parameters (stepsize and network size), where DE-SGD can have lighter or heavier tails than disconnected SGD depending on the regime. Finally, to support our theoretical results, we provide numerical experiments conducted on linear regression and neural networks that suggest the heavier tails is correlated with better generalization in the decentralized setting. [ABSTRACT FROM AUTHOR]

The phenomenon of adversarial examples is one of the most attractive topics in machine learning research these days. These are particular cases that are able to mislead neural networks, with critical consequences. For this reason, different approaches are considered to tackle the problem. On the one side, defense mechanisms, such as AutoEncoder-based methods, are able to learn from the distribution of adversarial perturbations to detect them. On the other side, chaos theory and Lyapunov exponents (LEs) have also been shown to be useful to characterize them. This work proposes the combination of both domains. The proposed method employs these exponents to add more information to the loss function that is used during an AutoEncoder training process. As a result, this method achieves a general improvement in adversarial examples detection performance for a wide variety of attack methods. [ABSTRACT FROM AUTHOR]

The crystallization kinetics behaviors of PVA (poly (vinyl alcohol))/TP (tea polyphenol) composites were studied. Results showed that the addition of TP component in PVA formed composites would reduce the crystallinity (%), leading to the formation of a lower Avrami exponent, n. Under the fixed cooling rate condition, the n values of Avrami and α values of Mo are both higher for the above composites with TP ingredient at 10–20%. Subsequently, the influence of the hydrogen bonding (H-bonding) in PVA/TP composites, pure PVA and non-hydrogen bonding (non-H-bonding) in such composites were further quantitatively assessed by analyzing the H-bonding factor, F%, in relation to different crystallization kinetic parameters including Avrami exponent n, Zt, α and F(T). Results showed that the relationship of Avrami exponent n and α were presented as n = -a1 + b1FPVA/TPH-bonding-c1F2PVA/TPH-bonding and α = -a5 + b5FPVA/TPH-bonding-c5F2PVA/TPH-bonding. The Zt was described as the functions like Zt = a2-b2e(FPVA/TPH-bonding−c2)/d, Zt = -a3 + b3FPVAH-bonding-C3F2PVAH-bonding, and Zt = a4(Fnon-H-bonding-b4)c4, respectively. The F(T) was presented as F(T) = a6-b6FPVA/TPH-bonding+c6F2PVA/TPH-bonding, where a, b and c are positive constants. [ABSTRACT FROM AUTHOR]

The phenomenon of droplet impingement on solid surfaces is prevalent in various natural and industrial contexts. Research on impact dynamics on conical surfaces keeps emerging, with superhydrophobic cones receiving more attention than hydrophilic ones. This study systematically investigates water droplet impact dynamics on both hydrophilic and superhydrophobic cones using a two-phase numerical solver under different Weber numbers (We) and cone angles (φ). Three distinct phases are identified in the We– φ map to describe the different outcomes on each type of cones. Generally, deposition occurs ultimately on hydrophilic cones, whereas rebounding is observed on superhydrophobic ones. The maximum spreading area β Amax on hydrophilic cones depends only slightly on φ but consistently increases with We, following a scaling law of We0.5 at higher We. In contrast, on superhydrophobic cones, β Amax increases significantly with both We and φ , and the exponent in the scaling laws of β Amax with respect to We increases notably as φ increases. Three characteristic times are defined to describe important motion states on both types of cones. Corresponding scaling laws for each time with We are established. Two theoretical models are developed to predict the maximum spreading position for droplets on hydrophilic cones and the rebound position on superhydrophobic cones, respectively. Gravitational potential energy is included in the energy budget for both models, and an auxiliary viscous dissipation due to spontaneous spreading is accounted for the hydrophilic case. Satisfactory agreement between the theoretical and numerical results is achieved. [ABSTRACT FROM AUTHOR]

The statistical characteristics of bubble plumes generated by inclined liquid jets remain inadequately understood, particularly regarding their spatial evolution. This study employs an optical fiber probe for sequential multi-point measurements to investigate the evolution of bubble plumes produced by shallow-angle inclined jets impacting both quiescent fluid and crossflow. Measurements of air volume fractions reveal that crossflow significantly attenuates the jet's penetration capability. Under both conditions, the bubble size distributions exhibit power-law behavior, with variations in the exponents observed at different spatial locations. In quiescent water, a power-law exponent of −2 is observed within the jet impact region, where the strong shear effect of the oblique jet introduces more large bubbles and extends their survival time. This exponent gradually decreases with increasing distance from the jet's influence. In the presence of crossflow, the bubble size spectrum closely resembles that observed under breaking waves. This similarity arises because the jet induces intense surface overturning and fragmentation, becoming the primary source of bubble entrainment. Multi-point measurements further indicate that bubble fragmentation dominates near the free surface, while bubble coalescence becomes more prevalent at greater depths. The results for the bubble velocity distribution indicate that, under both conditions, the distribution of high-speed bubbles follows a power law with an exponent of −6. However, the physical mechanisms driving this behavior remain unclear. [ABSTRACT FROM AUTHOR]

We report psychophysical experiments and time series analyses to investigate sensorimotor tapping strength fluctuations in human periodic tapping with and without a metronome. The power spectral density of tapping strength fluctuations typically decays in an inverse power law (1 / f β -noise) associated with long-range correlations, i.e., with a slow power-law decay of tapping strength autocorrelations and scale-free behavior. The power-law exponents β are scattered around β = 1 ranging from 0.67 to 1.8. A log-linear representation of the power spectral densities reveals rhythmic peaks at frequencies f = 0.25 (and f = 0.5) and a tendency to slightly accentuate every fourth (and second) stroke when subjects try to synchronize their tapping with a metronome. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka–Łojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence. [ABSTRACT FROM AUTHOR]

This study employs multifractal detrended fluctuation analysis to investigate the impact of fuel cell introduction in the Korean electricity market via the lens of multifractal scaling behavior. Using multifractal analysis, the research delineates discrepancies between peak and off-peak hours, accounting for the daily cyclicity of the electricity market, and proposes a crossover point detection method based on the Chow test. Furthermore, the impacts of fuel cell introduction are evidenced through various methods that encompass multifractal spectra and market efficiency. The findings initially indicate a higher degree of multifractality during off-peak hours relative to peak hours. In particular, the crossover points emerged solely during off-peak hours, unveiling short- and long-term dynamics predicated on a near-annual cycle. Additionally, the average Hurst exponent for the short-term was 0.542, while the average for the long-term was 0.098, representing a notable discrepancy. The introduction of fuel cells attenuated the heterogeneity in the scaling behavior, which is potentially attributable to the decreased volatility in both the supply and demand spectra. Remarkably, after the introduction of fuel cells, there was a discernible decrease in the influence of long-range correlation within multifractality, and the market exhibited an increased propensity toward random-walk behavior. This phenomenon was also detected in the market deficiency measure, from an average of 0.536, prior to the introduction, to an average of 0.267, following the introduction, signifying an improvement in market efficiency. This implies that the introduction of fuel cells into the market engendered increased supply stability and a consistent increase in demand, mitigating volatility on both the supply and demand sides, thus increasing market efficiency. [ABSTRACT FROM AUTHOR]

This study explores the application of neural networks for anomaly detection in time series data exhibiting fractal properties, with a particular focus on changes in the Hurst exponent. The objective is to investigate whether changes in fractal properties can be identified by transitioning from the analysis of the original time series to the analysis of the sequence of Hurst exponent estimates. To this end, we employ an LSTM autoencoder neural network, demonstrating its effectiveness in detecting anomalies within synthetic fractal time series and real EEG signals by identifying deviations in the sequence of estimates. Whittle's method was utilized for the precise estimation of the Hurst exponent, thereby enhancing the model's ability to differentiate between normal and anomalous data. The findings underscore the potential of machine learning techniques for robust anomaly detection in complex datasets. [ABSTRACT FROM AUTHOR]

This paper investigates the (semi)group action of |$\textrm{SL}_{n}({\mathbb R})$| on |${\mathbb P}({\mathbb R}^{n})$|⁠ , a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32 ], they obtain a dimension formula for the stationary measures on |${\mathbb P}({\mathbb R}^{3})$|⁠. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in |$\textrm{SL}_{3}({\mathbb R})$| and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to |$1.5$|⁠. [ABSTRACT FROM AUTHOR]

We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent p ⁢ (⋅) -Laplacian as a forward model with the interior density data σ ⁢ | ∇ ⁡ u | q , and find out that single measurement specifies the boundary conductivity when p - q ≥ 1 , and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available. [ABSTRACT FROM AUTHOR]

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p into L q and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 푝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 푝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 푝-Laplacian with sub-homogeneous right-hand side, as the exponent 푝 diverges to ∞. The case of first eigenfunctions of the 푝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of weak solutions of the p (x) -Laplacian problems that are subject to Dirichlet boundary conditions, employing variational techniques as our primary analytical framework. [ABSTRACT FROM AUTHOR]

In this paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. The critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions. [ABSTRACT FROM AUTHOR]

In this present paper, we are first of all interested in some continuity and compactness results for the space ψ-fractional 핊 풜 α , β ; ψ ⁢ ( Λ ) {\mathbb{S}^{\alpha,\beta;\psi}_{\mathcal{A}}(\Lambda)} . In this sense, we investigate the existence of at least two solutions with constant signs using truncation arguments and comparison methods of a new class of fractional differential equations with m ⁢ ( ξ ) {m(\xi)} -Laplacian with double phase. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the initial vertex (Adam) in a Barabási–Albert tree process (T (n) : n ≥ 1) at large times. More precisely, given ε > 0 , one wants to output a subset P ε (n) of vertices of T (n) so that the initial vertex belongs to P ε (n) with probability at least 1 - ε when n is large. It has been shown by Bubeck, Devroye and Lugosi, refined later by Banerjee and Huang, that one needs to output at least ε - 1 + o (1) and at most ε - 2 + o (1) vertices to succeed. We prove that the exponent in the lower bound is sharp and the key idea is that Adam is either a "large degree" vertex or is a neighbor of a "large degree" vertex (Eve). [ABSTRACT FROM AUTHOR]

Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth–death processes with n distinct types as a birth–death model of EEX in a growing population. Each type-i cell divides independently (i) → (i) + (i) or mutates (i) → (i + 1) at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-n appears, which cell type can only divide or die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general n-type critical birth–death process. We find that the mass function of the number of cells of type-k has algebraic and stationary tail (size) - 1 - χ k , with χ k = 2 1 - k , for k = 2 , ⋯ , n , in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability (time) - ξ k . We present applications of the results for studying extinction due to intolerable mutation rates in biological populations. [ABSTRACT FROM AUTHOR]

We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix p weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this theory by generalizing the matrix p condition to the variable exponent setting. We prove boundedness of the convolution operator f ↦ ϕ ∗ f for ϕ ∈ C c ∞ (Ω) , and show that the approximate identity defined using ϕ converges in matrix weighted, variable Lebesgue spaces L p (⋅) (W , Ω) for W in matrix p (⋅) . Our approach to this problem is through averaging operators; these results are of interest in their own right. As an application of our work, we prove a version of the classical H = W theorem for matrix weighted, variable exponent Sobolev spaces. [ABSTRACT FROM AUTHOR]

Let ℤ n be the ring of integers modulo n. Let Ct, Em, and F r , s respectively denote the cyclic group of order t, the elementary abelian 2-group of order 2 m , and the abelian group of exponent 4 with order 2 r 4 s . In this article, we find the structure and generators of the unit group V (ℤ n C 2). We also solve the normal complement problem in V (ℤ n C 2) . Additionally, we provide a normal complement of Em in V (ℤ 2 n E m). At the end, we determine the structure of V (ℤ p n F r , s) for an odd prime p and establish that F r , s does not have a normal complement in V (ℤ p n F r , s) . [ABSTRACT FROM AUTHOR]

In this article we prove generalizations of the well-known Mertens's formula. We also define some arithmetic functions related to the exponents of the primes in the prime factorization of the positive integers and study its average. Finally, we study square-free numbers, s-full numbers and perfect powers not exceeding x which can be represented as sums of two squares. [ABSTRACT FROM AUTHOR]