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In this paper, we studied the stability of the Cauchy problem for a class of first-order linear quasi-periodically forced PDEs on the m-dimensional torus: {∂tu + (ξ + f(x, ωt, ξ)) · ∂xu = 0, u(x, 0) = u0(x), where ξ ∈ ℝm, x ∈ Tm, ω ∈ ℝd, in the case of multidimensional Liouvillean forced frequency. We proved that for each compact set O ∈ ℝm there exists a Cantor subset Oγ of O with positive Lebesgue measure such that if ξ ∈ Oγ, then for a perturbation f being small in some analytic Sobolev norm, there exists a bounded and invertible quasi-periodic family of linear operator Ψ(ωt), such that the above PDEs are reduced by the transformation v := Ψ(ωt)−1 [u] into the following PDE: ∂tv + (ξ + m∞(ωt)) · ∂xv = 0, provided that the forced frequency ω ∈ ℝd possesses finite uniform Diophantine exponent, which allows Liouvillean frequency. The reducibility can immediately cause the stability of the above Cauchy problem, that is, the analytic Sobolev norms of the Cauchy problem are controlled uniformly in time. The proof is based on a finite dimensional Kolmogorov-Arnold-Moser (KAM) theory for quasi-periodically forced linear vector fields with multidimensional Liouvillean forced frequency. As we know, the results on Liouvillean frequency existing in the literature deal with two-dimensional frequency and exploit the theory of continued fractions to control the small divisor problem. The results in this paper partially extend the analysis to higher-dimensional frequency and impose a weak nonresonance condition, i.e., the forced frequency ω possesses finite uniform Diophantine exponent. Our result can be regarded as a generalization of analytic cases in the work [R. Feola, F. Giuliani, R. Montalto and M, Procesi, Reducibility of first order linear operators on tori via Moser’s theorem, J. Funct. Anal., 2019] from Diophantine frequency to Liouvillean frequency. [ABSTRACT FROM AUTHOR]

We consider the anisotropic integral functional of the calculus of variations ... where ci≥0 and 2 ≤pi≤pi+1 for every i = 1 ,...n-1. We exhibit a minimizer functional, for an opportune choice of the exponents pi, which turns out to be bounded everywhere and Lipschitz continuous (or even of class C¹) in an opportune subset of Ω. [ABSTRACT FROM AUTHOR]

The present paper explores the spectral cocycle defined in [9] in the special case of bijective substitutions on two letters, the most prominent example being the Thue-Morse substitution. We derive an explicit subexponential behaviour of the deviations from the expected exponential behavior. Moreover, these sharp bounds will be exploited to prove that the top Lyapunov exponent is greater or equal to the top exponent of the subshift topological factors after a renormalization. In order to obtain such results for the substitutive subshift factors, we define a special kind of sum, which is a multiple version of the twisted Birkhoff sum. For the particular case of the Thue-Morse substitution, we derive that the exponent is zero, we give an explicit subexponential bound for the twisted Birkhoff sums and we do the same for subshift topological factors. [ABSTRACT FROM AUTHOR]

Based on the variational methods and critical-point theory, we are concerned with the existence results for a second-order impulsive boundary value problem involving an ordinary differential equation with p(x)-Laplacian operator, and Neumann conditions. [ABSTRACT FROM AUTHOR]

This article is dedicated to studying a class of frictional contact problems involving the p(x)-Laplacian-like operator, on a bounded domain Ω ⊆ R². Using an abstract Lagrange multiplier technique and the Schauder fixed point theorem we establish the existence of a weak solution. Furthermore, we also obtain the uniqueness of the solution assuming that the datum f1 satisfies a suitable monotonicity condition. The results here extend earlier theorems due to Cojocaru- Matei to the quasilinear case, with semilinearity f1. [ABSTRACT FROM AUTHOR]

In this paper, we study about order and hyper-order of growth of nontrivial solutions of f' + A(z)f' + B(z)f = 0, where A(z) and B(z) are entire functions having some restrictions. These restrictions involve notions of Yang’s inequality, Borel exceptional value, deficient value and accumulation ray. [ABSTRACT FROM AUTHOR]

We give an improved lower bound for the average of the Erdős–Hooley function Δ(n)$\Delta (n)$, namely ∑n⩽xΔ(n)≫εx(loglogx)1+η−ε$\sum _{n\leqslant x} \Delta (n) \gg _\varepsilon x(\log \log x)^{1+\eta -\varepsilon }$ for all x⩾100$x\geqslant 100$ and any fixed ε$\varepsilon$, where η=0.3533227⋯$\eta = 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of ≫xloglogx$\gg x \log \log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of x(loglogx)11/4$x (\log \log x)^{11/4}$ of the second and third authors. [ABSTRACT FROM AUTHOR]

In this article, we systematically study the high-order stability for a stochastic reaction-diffusion equation under random fluctuation. We obtain the convergence of solutions and upper semi-continuity of random attractors for the fluctuation equations in the initial space L2(ℝN), without an additional assumption on the nonlinearity. By means of a splitting method, we technically prove that the Lp∩H1-difference of solutions is controlled by the L2-difference of initial values near the initial time, where p , p > 2 , is the dissipation exponent of the nonlinearity. This along with the L2-stability is sufficient for us to establish the Lp∩H1-stability of random attractors. By utilizing some iteration arguments, we show that the solution is uniformly bounded on a bounded interval near the initial time in Lδ(ℝN) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established. [ABSTRACT FROM AUTHOR]

Multifractality is a concept that extends locally the usual ideas of fractality in a system. Nevertheless, the multifractal approaches used lack a multifractal dimension tied to an entropy index like the Shannon index. This paper introduces a generalized Shannon index (GSI) and demonstrates its application in understanding system fluctuations. To this end, traditional multifractality approaches are explained. Then, using the temporal Theil scaling and the diffusive trajectory algorithm, the GSI and its partition function are defined. Next, the multifractal exponent of the GSI is derived from the partition function, establishing a connection between the temporal Theil scaling exponent and the generalized Hurst exponent. Finally, this relationship is verified in a fractional Brownian motion and applied to financial time series. In fact, this leads us to proposing an approximation called local fractional Brownian motion approximation, where multifractal systems are viewed as a local superposition of distinct fractional Brownian motions with varying monofractal exponents. Also, we furnish an algorithm for identifying the optimal q-th moment of the probability distribution associated with an empirical time series to enhance the accuracy of generalized Hurst exponent estimation. [ABSTRACT FROM AUTHOR]

This work uses multispectral measurements of vertically resolved aerosol extinction coefficients from the Stratospheric Aerosol and Gas Experiment (SAGE) III on the International Space Station (ISS) to demonstrate how the use of the Ångström exponent for interpolation of aerosol data between two different wavelengths creates a bias. An empirical relationship is derived between the magnitude of this bias and the Ångström exponent at several different SAGE wavelengths. This relationship can thus be used as a correction factor for other studies, such as multi-instrument intercomparisons or merging, that wish to convert aerosol data from one wavelength to another using the Ångström exponent and is applicable to all stratospheric non-cloud aerosol except highly aged particles that are evaporating at altitudes above the Junge layer. [ABSTRACT FROM AUTHOR]

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function V E of a subset E of C n , also called a pluricomplex Green function or global extremal function of E, equals the logarithm of the Siciak function Φ E if E is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on E, and the Siciak function is the upper envelope of m-th roots of the modulus of polynomials p in P m (C n) of degree ≤ m such that | p | ≤ 1 on E. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space P m (C n) is replaced by P m S (C n) consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of R + n with 0 ∈ S . It states that if q is an admissible weight on a closed set E in C n then V E , q S = log Φ E , q S on C ∗ n if and only if the rational points in S form a dense subset of S. [ABSTRACT FROM AUTHOR]

We study the unique solvability of the Cauchy problem for a mixed-type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst exponents . We prove a theorem on the existence and uniqueness of strong solutions of these stochastic differential equations. [ABSTRACT FROM AUTHOR]

We present the results of numerical simulation and experimental studies of the propagation of fle-xural elastic waves in a notched metal bar with a rectangular cross section that approximates the acoustic black hole effect. The sample is a notched bar; the depth of notches increases according to a power law with an exponent of 4/3. The experimental results and the results of numerical simulation confirm that such bars slow the propagation velocity of an elastic wave towards the end of the bar. It is demonstrated that flexural waves in such structures exhibit dispersion and their amplitude at the end of the bar for some eigenfrequencies is higher than that in a solid bar. The eigenmode shapes of a solid and notched bar are compared together with the distribution of the flexural wave amplitude along the bars. The frequency dependence of the flexural wave length is studied during wave propagation towards the end of the notched bar. [ABSTRACT FROM AUTHOR]

A universal number (Unum) is a number representation format that can reduce the memory contention issues in multicore processors and parallel computing systems by optimizing the bit storage in the arithmetic operations. Given the safety-critical nature of applications of Unum format, there is a dire need to rigorously assess the correctness of the Unum based arithmetic operations. Unums are of three types, namely, Unum-I, Unum-II and Unum-III (commonly known as Posits). In this paper, we provide a higher-order-logic formalization of Unum-III (posits). In particular, we formally model a posit format (binary encoding of a posit), which is comprised of the sign, exponent, regime and fraction bits, using the HOL Light theorem prover. In order to prove the correctness of a posit format, we formally verify various properties regarding conversions of a real number to a posit and a posit to a real number and the scaling factors of the regime, exponent and fraction bits of a posit using HOL Light. [ABSTRACT FROM AUTHOR]

We consider Campanato spaces with exponents λ,p$\lambda, p$ on domains of class C0,γ$C^{0,\gamma }$ in the N‐dimensional Euclidean space endowed with a natural anisotropic metric depending on γ$\gamma$. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces can be extended to the whole of the Euclidean space without deterioration of the exponents λ,p$\lambda, p$. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the weight under which the Haar system is a basis in a weighted Lebesgue space with variable exponent. [ABSTRACT FROM AUTHOR]

We prove the parametric version of Forstnerič splitting lemma for certain families of weakly pseudoconvex domains allowing regularity and stability of Kohn's solution to $ \bar {\partial } $ ∂ ¯ -problem in Sobolev spaces with exponent 2. [ABSTRACT FROM AUTHOR]

We define and study a notion of minimal exponent for a local complete intersection subscheme 푍 of a smooth complex algebraic variety 푋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 푉-filtration associated to 푍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ (O X) , where 푟 is the codimension of 푍 in 푋. We also study its relation to the Bernstein–Sato polynomial of 푍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 푉-filtration associated to any ideal (f 1 , ... , f r) in terms of the microlocal 푉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i . [ABSTRACT FROM AUTHOR]

We consider an eigenvalue problem driven by the anisotropic (p, q)-Laplacian and with a Carathéodory reaction which is (p(z) − 1)-sublinear as x → + ∞. We look for positive solutions. We prove an existence, nonexistence and multiplicity theorem which is global in the parameter λ > 0, that is, we prove a bifurcation-type theorem which describes in an exact way the changes in the set of positive solutions as the parameter λ varies on ℝ̊+ = (0, + ∞). [ABSTRACT FROM AUTHOR]

This article focuses on studying the existence of solutions for the following indefinite Choquard equation in R N : where N ≥ 2 . Here, α is a positive constant satisfying 0 < α < N , and I α represents the Riesz potential of order α . The potential V(x) is nonperiodic and changes sign, leading to inf σ (- Δ + V) < 0 , where σ (- Δ + V) denotes the spectrum of the operator - Δ + V , which the problem become indefinite. The exponent p is chosen from the range given by 0.1 N - 2 N + α < 1 p < 1 2 < N N + α. In this work, we establish the existence of nontrivial solutions by employing significant outcomes from spectral theory, along with a version of the linking theorem presented by [13], and the interaction between translated solutions of the problem at infinity. [ABSTRACT FROM AUTHOR]

The stress singularity is determined using matched asymptotics for complete flow of a White–Metzner (WM) fluid around a re-entrant corner. The model is considered in the absence of a solvent viscosity, with power-law forms for the relaxation time and polymer viscosity. In this form, the model shares the same stress singularity as the upper convected Maxwell (UCM) model, but its wall boundary layers may be thinner or thicker than those for UCM depending upon the relative difference in the power-law exponents. If the exponent for the relaxation time is greater than that for the polymer viscosity, the boundary layer is narrower, whilst it is thicker if the polymer viscosity exponent exceeds that of the relaxation time. When the exponents are the same, the WM boundary layer thickness is the same size as that for UCM. A self-similar solution is derived for the stress and velocity fields and matched to both upstream and downstream boundary layers. Restrictions on the sizes of the power-law exponents are also given for validity of this solution. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results. [ABSTRACT FROM AUTHOR]

We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that ∑ n ≤ N e (α n 4) ≪ ε , α N 5 / 6 + ε for any ε > 0 and any quadratic irrational α ∈ R − Q. Classically one would have had the exponent 7 / 8 + ε for such α. In contrast to the author's earlier work [2] on cubic Weyl sums (which was conditional on the abc -conjecture), we show that the van der Corput AB -steps are sufficient for the quartic case, rather than the B A A B -process needed for the cubic sum. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of distributional solutions in the anisotropic Sobolev space ... with variable exponents and zero boundary, for a class of variable exponents anisotropic nonlinear elliptic equations having a compound N nonlinearity G(x, u) = ... on the right-hand side, such that f is in the variable exponents anisotropic Lebesgue space ... [ABSTRACT FROM AUTHOR]

In this study, we establish some new characterizations for a class of anisotropic Herz spaces in which all exponents are considered as variables. We also provide a description of these spaces based on bloc decomposition. As an application, we investigate the boundedness of certain sublinear operators within these function spaces. [ABSTRACT FROM AUTHOR]

Assume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P -generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $ , into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X , there exists a discontinuous homomorphism of $C(X)$ , the algebra of continuous real-valued functions on X. [ABSTRACT FROM AUTHOR]

One of the main constituents of any reservoir characterization is an accurate forecast of water saturation, which is highly dependent upon the cementation exponent. Even though there have been a lot of studies, the most common correlations depend on total porosity. This means that they do not work as well in heterogeneous carbonate reservoirs, especially tight formations with total porosities less than 10%. This study aims to develop accurate and universal models for estimating the cementation exponent in deep and tight carbonate pore systems located in West Asia. Two heuristic algorithms, including the radial basis function neural network optimized by ant colony optimization (RBFNN-ACO) and gene expression programming (GEP), were employed to calculate the cementation exponent. To do this, we prepared a databank incorporating cementation exponents, total porosity, and various pore types. Then, the databank is classified into the test subset (for model prediction checking) and the train subset (for model construction). The reliability of the new recommended models is inspected by applying several statistical quality measures associated with graphical analyses. So, the consequences of the modeling disclose the large precision of the above-mentioned RBFNN-ACO, GEP Model-I, and GEP Model-II by average absolute percentage relative deviations (AAPRD%) of 6.28%, 6.39%, and 7.45%, respectively. Based on the outliers analysis, nearly 95% of the databank and model estimations are, respectively, valid and reliable. Additionally, the three input variables, including moldic porosity (with a + 70% impact value), non-fabric-selective dissolution (connected) porosity (with a -30% impact value), and interparticle porosity (with a -23% impact value), exhibit the main affecting parameters on the cementation exponent. Comparing current results with traditional literature correlations demonstrates the supremacy of the RBFNN-ACO model (AAPRD = 6.28% and root mean squared error (RMSE) = 0.17) over the examined literature correlations such as Borai's equation (AAPRD = 12.30% and RMSE = 0.41). In addition, RBFNN-ACO can give better results than Borai's Eqn. for tight (porosity less than 10%) and deep carbonate samples. [ABSTRACT FROM AUTHOR]

In this article, we study the cyclicity problem of elliptic curves |$E/\mathbb{Q}$| modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for elliptic curves E , which also presents a generalization of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg and Wu. [ABSTRACT FROM AUTHOR]

This paper examines the asymmetric multifractality and efficiency in four DeFi assets (BAT-Basic Attention Token, LINK-Chainlink, MKR-Maker, and SNX-Synthetix) using the asymmetric MF-DFA approach and Hurst exponents. The results show different multifractality during downward and upward trends. Furthermore, the asymmetric multifractality intensifies with scale rises. Before the COVID-19 pandemic, BAT, Maker, and Link are more inefficient under upward trend and SNX under downward trend. Conversely, the four DeFi assets are more inefficient during downward trend. Link asset is the most inefficient market before and during COVID-19. These results have important implications for retail investors and funds mangers. [ABSTRACT FROM AUTHOR]

The correct selection of the value of p is a complex and iterative procedure that requires experience in the interpretation of the obtained interpolated maps. Inverse Distance Weighting is a method applied to the porosities of the K and L hydrocarbon reservoirs discovered in the Neogene (Lower Pontian) subsurface sandstones in northern Croatia (Pannonian Basin System). They represent small and large data samples. Also, a standard statistical analysis of the data was made, followed by a qualitative–quantitative analysis of the maps, based on the selection of different values for the power distance exponent (p-value) for the K and L reservoir maps. According to the qualitative analysis, for a small data set, the p-value could be set at 1 or 2, giving the optimal result, while for a large data set, a p value of 3 or 4 could be applied. For quantitative analysis, in the case of a small data set, p = 2 is recommended, resulting in a root mean square error value of 0.03458, a mean absolute error of 0.02013 and a median absolute deviation of 0.00546. In contrast, a p-value of 3 or 4 is selected as appropriate for a large data set, with root mean square errors of 0.02435 and 0.02437, mean square errors of 0.01582 and 0.01509 and median absolute deviations 0.00896 and 0.00444. Eventually for a small data set, it is recommended to use a p-value of 2, and for a large data set, a p-value of 3 or 4. [ABSTRACT FROM AUTHOR]

We consider the generalized Choquard equation of the type - Δ Q + Q = I (| Q | p ) | Q | p - 2 Q , for 3 ≤ n ≤ 5 , with Q ∈ H rad 1 (R n) , where the operator I is the classical Riesz potential defined by I (f) (x) = (- Δ) - 1 f (x) and the exponent p ∈ (2 , 1 + 4 / (n - 2)) is energy subcritical. We consider Weinstein-type functional restricted to rays passing through the ground state. The corresponding real valued function of the path parameter has an appropriate analytic extension. We use the properties of this analytic extension in order to show local uniqueness of ground state solutions. The uniqueness of the ground state solutions for the case p = 2 , i.e. for the case of Hartree–Choquard, is well known. The main difficulty for the case p > 2 is connected with a possible lack of control on the L p norm of the ground states as well on the lack of Sturm's comparison argument. [ABSTRACT FROM AUTHOR]

Network modeling has been explored extensively by means of theoretical analysis as well as numerical simulations for Network Reconstruction (NR). The network reconstruction problem requires the estimation of the power-law exponent (γ) of a given input network. Thus, the effectiveness of the NR solution depends on the accuracy of the calculation of γ. In this article, we re-examine the degree distribution-based estimation of γ, which is not very accurate due to approximations. We propose X-distribution, which is more accurate than degree distribution. Various state-of-the-art network models, including CPM, NRM, RefOrCite2, BA, CDPAM, and DMS, are considered for simulation purposes, and simulated results support the proposed claim. Further, we apply X-distribution over several real-world networks to calculate their power-law exponents, which differ from those calculated using respective degree distributions. It is observed that X-distributions exhibit more linearity (straight line) on the log-log scale than degree distributions. Thus, X-distribution is more suitable for the evaluation of power-law exponent using linear fitting (on the log-log scale). The MATLAB implementation of power-law exponent (γ) calculation using X-distribution for different network models and the real-world datasets used in our experiments are available at https://github.com/Aikta-Arya/X-distribution-Retraceable-Power-Law-Exponent-of-Complex-Networks.git. [ABSTRACT FROM AUTHOR]

This document is an erratum for a paper titled "Unusual dynamics of tetrahedral liquids caused by the competition between dynamic heterogeneity and structural heterogeneity" published in the Journal of Chemical Physics. The erratum points out several typographical errors in Figure 5 of the original paper, including incorrect labels and missing information. The corrections provided in the erratum do not impact the results or conclusions of the original paper. The erratum also clarifies a reference to a figure in the Results and Discussions section. [Extracted from the article]

In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd exponents. In this short note we prove almost sure polynomialfree unisolvence in such instances, by a deeper analysis of the interpolation matrix determinant and fundamental properties of analytic functions. [ABSTRACT FROM AUTHOR]

An empirical multi-parameter equation of state in terms of the reduced Helmholtz energy is presented for the Mie (λr-6) fluid with a repulsive exponent λr from 11 to 13. The equation is fitted to an extensive dataset from molecular dynamics simulation as well as the second and third thermal virial coefficients. It is comprehensively compared with the SAFT-VR model and is a more accurate description of the considered fluid class. The equation is valid for reduced temperatures T/Tc from 0.55 to 4.5 and for reduced pressures of up to p/pc = 265. A good extrapolation behavior and the occurrence of a single Maxwell loop down to the vicinity of the triple point temperature are realized. [ABSTRACT FROM AUTHOR]

In the PISA 2021 framework, Computational Thinking (CT) is described as a detailed mathematical solution to the problem to be solved. However, CT-based learning still needs to be widely applied in Indonesia. This study aims to describe the CT ability of students in grade IX of junior high school based on CT indicators on the material of signed numbers. The data collection techniques in this study were test questions and interviews. Students who obtained high categories with scores above 45.76 were six students with a percentage of 21%, students who received medium categories with scores between 11.94 and 45.76 were 19 students with a rate of 66%, and students who obtained low categories with scores below 11.94 were four people with a percentage of 13%. The results of the study state that as many as 39% of students can decompose the problems given, 17% of students can recognize patterns in the problem, 24% of students can sort out the information in the situation or abstract, and 26% of students can solve problems well according to algorithm indicators.

Physical kinetic roughening processes are well-known to exhibit universal scaling of observables that fluctuate in space and time. Are there analogous dynamic scaling laws that are unique to the chemical reaction mechanisms available synthetically and occurring naturally? Here, we formulate an approach to the dynamic scaling of stochastic fluctuations in thermodynamic observables at and away from equilibrium. Both analytical expressions and numerical simulations confirm our dynamic scaling ansatz with associated scaling exponents, function, and law. A survey of common chemical mechanisms reveals classes that organize according to the molecularity of the reactions involved, the nature of the reaction vessel and external reservoirs, (non)equilibrium conditions, and the extent of autocatalysis in the reaction network. Varying experimental parameters, such as temperature, can cause coupled reactions capable of chemical feedback to transition between these classes. While path observables, such as the dynamical activity, have scaling exponents that are time-independent, the variance in the entropy production and flow can have time-dependent scaling exponents and self-averaging properties as a result of temporal correlations that emerge during thermodynamically irreversible processes. Altogether, these results establish dynamic universality classes in the nonequilibrium fluctuations of thermodynamic observables for well-mixed chemical reactions. [ABSTRACT FROM AUTHOR]

Spatial confinement of a polymer chain results in a reduction of conformational entropy. For confinement of a flexible N-mer chain in a planar slit or cylindrical pore (confining dimension D), a blob model analysis predicts the asymptotic scaling behavior ΔF/N ∼ D−γ with γ ≈ 1.70, where ΔF is the free energy increase due to confinement. Here, we extend this scaling analysis to include the variation of local monomer density upon confinement giving ΔF/N ∼ D−γ(1 − h(N, D)), where the correction-to-scaling term has the form h ∼ Dy/NΔ with exponents y = 3 − γ ≈ 1.30 and Δ = 3/γ − 1 ≈ 0.76. To test these scaling predictions, we carry out Wang–Landau simulations of confined and unconfined tangent-hard-sphere chains (bead diameter σ) in the presence of a square-well trapping potential. The fully trapped chain provides a common reference state, allowing for an absolute determination of the confinement free energy. Our simulation results for 32 ≤ N ≤ 1024 and 3 ≤ D/σ ≤ 14 are well-described by the extended scaling relation giving exponents of γ = 1.69(1), y = 1.25(2), and Δ = 0.75(6). [ABSTRACT FROM AUTHOR]

In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using an integral inequality due to Komornik, the general decay outcome is obtained in the case b=0$$ b&#x0003D;0 $$. [ABSTRACT FROM AUTHOR]

We establish a new Harnack inequality for nonnegative solutions to the p(x)-Laplace equation with two-phase exponent p(x) taking two constant values p and q in the case where the phase interface is a hyperplane. [ABSTRACT FROM AUTHOR]

This work studies the inhomogeneous Schrödinger coupled system with potential iðtuj - Hvuj = ±|x| -τ (∑m k=1 ajk|uk||p ) |uj|p-2 uj, 1 ≤ j ≤ m. The wave function components read uj: R × R3 → C for 1 ≤ j ≤ m. The inhomogeneous singular term exponent is τ > 0. The source term is energy sub-critical: 1 < p < 3 - τ. Moreover, in order to avoid a singularity of the term |uj| p-2, one assumes that p ≥ 2. The linear Schrödinger operator reads Hv = -Δ + V, where V is a potential satisfying some assumptions which imply that the dispersive and Strichartz estimates hold. The purpose is two-fold. First, we develop a local well-posedness theory in the energy space [H¹v]m:= [{ƒεL²(R3), √{ƒεL²(R³)}]m. Second, we present a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground-state threshold in the inter-critical focusing regime. The scattering is obtained by using the new approach of Dodson-Murphy which is based on Tao's scattering criteria and Morawetz estimates. The novelty here is the presence of the potential V. The challenge is to deal with three technical problems: a coupled nonlinearity, an inhomogeneous singular term |·| -τ, and the presence of the potential V. This note naturally extends previous works by the authors about the above problem for V = 0. [ABSTRACT FROM AUTHOR]

A long-standing problem in meteor science has been the persistent presence of bias in the measured value of atmospheric scale heights obtained from radio meteor echoes. A common practice of fitting a linear function for bias correction follows the assumption that the systematic bias is devoid of seasonal asymmetry. This would be true if the mass and the velocity distribution of meteoroids remain invariant, both spatially and temporally. But so far no such convincing evidence has been published. On the contrary, fundamental arguments suggest that a universal mass function of radio meteoroids is counterintuitive. This parameter cannot remain invariant due to the intrinsic variability in the meteor response function resulting from the Earth's motion on the plane of ecliptic. In this paper, we show that an inverse relation exists between the width of meteor height distribution, expressed in unit of atmospheric scale height, and the exponent of the mass function. The overall mean of this exponent for the Sodankylä radar is |$1.91 \pm 0.02$|⁠ , modulated by a seasonal variation from the mean of the order of |$\sim \pm 0.1$|⁠. The stated inverse relation is applied to correct for the effect of mass distribution on the height distribution. Allowing for variable mass correction effectively removes the non-linear bias in the measured scale heights in the meteor ionization region. [ABSTRACT FROM AUTHOR]

The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its scaling theory is controlled by a single parameter, the spectral dimension ds, which plays the role of the relevant parameter on complex geometries. The graph under consideration allows us to tune the value of the spectral dimension continuously also to noninteger values and to find the universal exponents as continuous functions of the dimension. By means of extensive numerical simulations, we probe the scaling exponents of a simple instance of O (N) symmetric models on the LRDG showing quantitative agreement with the theoretical prediction of universal scaling in real dimensions. Universality of critical behaviour of O(N) field theories on regular homogeneous lattices is established, but open questions remain for more complex lattices. Bighin et al. study universality on a non-homogeneous graph showing that its scaling theory is controlled by a single parameter, the spectral dimension. [ABSTRACT FROM AUTHOR]

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term. [ABSTRACT FROM AUTHOR]

The paper introduces and explores the notions of ergodicity index and ergodicity exponent of a set of stochastic matrices. For the ergodicity exponent a sharp upper bound is obtained. A particular case of this bound is the well-known Paz bound. Also a connection between the ergodicity index and the Protasov–Voynov imprimitivity index is established. [ABSTRACT FROM AUTHOR]

In this paper, we give direct theorems on approximation by Haar and Walsh polynomials in variable exponent grand Lebesgue space. Also the degree of approximation by Borel, Abel-Poisson, Riesz-Zygmund and Euler linear means of Haar-Fourier series are treated in the above-cited space. [ABSTRACT FROM AUTHOR]

Considering the solutions of a class of noncooperative Kirchhoff-type p (x) -Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the (P S) c * condition using another method, diverging from the approaches cited in the previous literature. [ABSTRACT FROM AUTHOR]

We show that sequences of the form |$\alpha n^{\theta } \pmod {1}$| with |$\alpha> 0$| and |$0 < \theta < \tfrac {43}{117} = \tfrac {1}{3} + 0.0341 \ldots $| have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for |$\alpha> 0$| and |$0 < \theta < \tfrac {14}{41} = \tfrac {1}{3} + 0.0081 \ldots $|⁠. We reduce the problem of establishing Poissonian pair correlation to a counting problem using a form of amplification and the Bombieri–Iwaniec double large sieve. The counting problem is then resolved non-optimally by appealing to the bounds of Robert–Sargos and (Fouvry–Iwaniec–)Cao–Zhai. The exponent |$\theta = \tfrac {2}{5}$| is the limit of our approach. [ABSTRACT FROM AUTHOR]

The greater the persistence in a financial time series, the more predictable it becomes, allowing for the development of more effective investment strategies. Desirable attributes for financial portfolios include persistence, smoothness, long memory, and higher auto-correlation. We argue that these properties can be achieved by adjusting the composition weights of the portfolio. Considering the fractal nature of typical financial time series, the fractal dimension emerges as a natural metric to gauge the smoothness of the portfolio trajectory. Specifically, the Hurst exponent is designed for measuring the persistence of time series. In this paper, we introduce an optimization method inspired by the Hurst exponent and signal processing to mitigate the irregularities in the portfolio trajectory. We illustrate the effectiveness of this approach using real data from an S &P100 dataset. [ABSTRACT FROM AUTHOR]