Set Of Exponents Research Articles

Approximation of $$L^\infty $$ functionals with generalized Orlicz norms

AbstractThe aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. We generalize results proven by Bertazzoni, Harjulehto and Hästö in Journ. of Math. Anal. and Appl. (2024) for integral type energies (in generalized Orlicz spaces), considering milder convexity assumptions. $$\Gamma $$ Γ -convergence results and related representation theorems in terms of $$L^\infty $$ L ∞ functionals are proven. The convexity hypotheses are completely removed in the variable exponent setting, thus extending the results in Eleuteri-Prinari in Nonlinear Anal. Real. World Appl. (2021) and Prinari-Zappale in JOTA (2020).

Multilinear rough singular integral operators

AbstractWe study ‐linear homogeneous rough singular integral operators associated with integrable functions on with mean value zero. We prove boundedness for from to when and in the largest possible open set of exponents when and . This set can be described by a convex polyhedron in .

Ramanujan type congruences for quotients of Klein forms

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p>19. Canonical bases for modular forms of level 5≤p≤19 are constructed by summing weight one Hecke Eisenstein series of levels 5≤p≤19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ1(p) and Γ(p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ1(p) as a direct sum of subspaces of modular forms for Γ(p) of the form qr/pZ[[q]]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

POLYNOMIALS WITH A RESTRICTED RANGE AND CURVES WITH MANY POINTS

In this paper, we introduce novel properties of restricted range polynomials compared to those developed by L. Rédei in ([1]). We showcase a new method for determining their exponent set and leverage this to construct curves over finite fields with a multitude of rational points.

Fractal Analysis of S&P 500 Sector Indexes

In this study multifractal properties of S&P 500 sector indexes are investigated with Multifractal Detrended Fluctuation Analysis (MF-DFA). The MF-DFA is a signal processing technique that is used to describe the multifractal properties of a time series data. It is an extension of Detrended Fluctuation Analysis (DFA), which is a widely utilized method for estimating the scaling behavior of a time series. Main idea behind MF-DFA is to decompose a time series into multiple scales using a coarse-graining procedure, and then to estimate the scaling behavior of each scale using DFA. This gives a set of scaling exponents that describe the multifractal features of the time series. Our MF-DFA results indicates the presence of multifractality in all S&P 500 sector indexes. Since these indexes are multifractal, we can conclude that they possess properties such as scaling variability, nonlinear dynamics, self-similarity, long-range dependence, multiscale correlations and nonstationary.

Continuously varying critical exponents in long-range quantum spin ladders

We investigate the quantum-critical behavior between the rung-singlet phase with hidden string order and the Néel phase with broken SU(2)-symmetry in quantum spin ladders with algebraically decaying unfrustrated long-range Heisenberg interactions. To this end, we determine high-order series expansions of energies and observables in the thermodynamic limit about the isolated rung-dimer limit. This is achieved by extending the method of perturbative continuous unitary transformations (pCUT) to long-range Heisenberg interactions and to the calculation of generic observables. The quantum-critical breakdown of the rung-singlet phase then allows us to determine the critical phase transition line and the entire set of critical exponents as a function of the decay exponent of the long-range interaction. We demonstrate long-range mean-field behavior as well as a non-trivial regime of continuously varying critical exponents implying the absence of deconfined criticality contrary to a recent suggestion in the literature.

Special transition and extraordinary phase on the surface of a two-dimensional quantum Heisenberg antiferromagnet

Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4)ys=0.86(4) and \eta_{\parallel}=-0.33(1)η∥=−0.33(1), which are distinct from the special transition of the classical O(3) model and indicate a new surface universality class of the 3D O(3) Wilson-Fisher theory.

Modified diffusive epidemic process on Apollonian networks.

We present an analysis of an epidemic spreading process on an Apollonian network that can describe an epidemic spreading in a non-sedentary population. We studied the modified diffusive epidemic process using the Monte Carlo method by computational analysis. Our model may be helpful for modeling systems closer to reality consisting of two classes of individuals: susceptible (A) and infected (B). The individuals can diffuse in a network according to constant diffusion rates [Formula: see text] and [Formula: see text], for the classes A and B, respectively, and obeying three diffusive regimes, i.e., [Formula: see text], [Formula: see text], and [Formula: see text]. Into the same site i, the reaction occurs according to the dynamical rule based on Gillespie's algorithm. Finite-size scaling analysis has shown that our model exhibits continuous phase transition to an absorbing state with a set of critical exponents given by [Formula: see text], [Formula: see text], and [Formula: see text] familiar to every investigated regime. In summary, the continuous phase transition, characterized by this set of critical exponents, does not have the same exponents of the mean-field universality class in both regular lattices and complex networks.

Nonstationary but quasisteady states in self-organized criticality.

The notion of self-organized criticality (SOC) was conceived to interpret the spontaneous emergence of long-range correlations in nature. Since then many different models have been introduced to study SOC. All of them have a few common features: externally driven dynamical systems self-organize themselves to nonequilibrium stationary states exhibiting fluctuations of all length scales as the signatures of criticality. In contrast, we have studied here in the framework of the sandpile model a system that has mass inflow but no outflow. There is no boundary, and particles cannot escape from the system by any means. Therefore, there is no current balance, and consequently it is not expected that the system would arrive at a stationary state. In spite of that, it is observed that the bulk of the system self-organizes to a quasisteady state where the grain density is maintained at a nearly constant value. Power law distributed fluctuations of all lengths and time scales have been observed, which are the signatures of criticality. Our detailed computer simulation study gives the set of critical exponents whose values are very close to their counterparts in the original sandpile model. This study indicates that (i) a physical boundary and (ii) the stationary state, though sufficient, may not be the necessary criteria for achieving SOC.

Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions.

Recently, we argued [Chin. Phys. Lett. 39, 080502 (2022)0256-307X10.1088/0256-307X/39/8/080502] that the Ising model simultaneously exhibits two upper critical dimensions (d_{c}=4,d_{p}=6) in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions d from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for 4<d<6 and d≥6, and thus strongly support the argument that 6 is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.

Anomalous scaling at nonthermal fixed points of the sine-Gordon model

We extend the theory of non-thermal fixed points to the case of anomalously slow universal scaling dynamics according to the sine-Gordon model. This entails the derivation of a kinetic equation for the momentum occupancy of the scalar field from a non-perturbative two-particle irreducible effective action, which re-sums a series of closed loop chains akin to a large-$N$ expansion at next-to-leading order. The resulting kinetic equation is analyzed for possible scaling solutions in space and time that are characterized by a set of universal scaling exponents and encode self-similar transport to low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest that kinetic scattering here is dominated by rather non-local processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and position space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.

Three-dimensional critical behavior and anisotropic magnetic entropy change in the axion insulator candidate EuSn2P2

The magnetic properties of the axion insulator candidate ${\mathrm{EuSn}}_{2}{\mathrm{P}}_{2}$ are comprehensively investigated. Under an applied external field, the antiferromagnetic phase below 30 K is driven to a forced ferromagnetic state. The yielded Rhodes-Wolfarth ratio (RWR) is equal to 1.13 (1.12) with $H//ab$ $(H//c)$, indicating the forced ferromagnetism in ${\mathrm{EuSn}}_{2}{\mathrm{P}}_{2}$ is weak itinerant. A large magnetocrystalline anisotropy with ${K}_{u}=521$ $\mathrm{kJ}/{\mathrm{m}}^{3}$ at 2 K is evaluated. Analysis on the critical behavior generates a set of critical exponents $\ensuremath{\beta}=0.281(2)$ with ${T}_{C}=30.2(1)$ K, $\ensuremath{\gamma}=0.77(3)$ with ${T}_{C}=30.1(1)$ K, and $\ensuremath{\delta}=3.76(4)$ at ${T}_{C}=30$ K, exhibiting a three-dimensional critical behavior. A phase diagram of $H$ versus $T$ is established, where a tricritical point at temperature and field of (30 K, 16.7 kOe) is determined on the boundaries of the paramagnetic, antiferromagnetic, and forced ferromagnetic states. In addition, the magnetic entropy change exhibits anisotropic behavior. The maximal value of the magnetic entropy $\ensuremath{-}\mathrm{\ensuremath{\Delta}}{S}_{M}^{\text{max}}$ reaches 7.36 (6.41) J ${\mathrm{kg}}^{\ensuremath{-}1}\phantom{\rule{4pt}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ with a magnetic field change of 50 kOe in $H//ab$ $(H//c)$, which is comparable with that of well-known materials suitable for magnetic refrigeration with second-order ferromagnetic transitions. The plots of $\ensuremath{-}\mathrm{\ensuremath{\Delta}}{S}_{M}(T,H)$ of high field can be rescaled into a universal curve, verifying the high-field paramagnetic to forced ferromagnetic transition is of a second order. The well-established scaling behavior of the magnetization and the magnetic entropy change further supports ${\mathrm{EuSn}}_{2}{\mathrm{P}}_{2}$ as an axion insulator.

Post-Growth Dynamics and Growth Modeling of Organic Semiconductor Thin Films.

The ability to control the properties of organic thin films is crucial for obtaining highly performant thin-film devices. However, thin films may experience post-growth processes, even when the most sophisticated and controlled growth techniques such as organic molecular beam epitaxy (OMBE) are used. Such processes can modify the film structure and morphology and, thus, the film properties ultimately affecting device performances. For this reason, probing the occurrence of post-growth evolution is essential. Equally importantly, the processes responsible for this evolution should be addressed in view of finding a strategy to control and, possibly, leverage them for driving film properties. Here, nickel-tetraphenylporphyrin (NiTPP) thin films grown by OMBE on highly oriented pyrolytic graphite (HOPG) are selected as an exemplary system exhibiting a remarkable post-growth morphology evolution consistent with Ostwald-like ripening. To quantitatively describe the growth, the height-height correlation function (HHCF) analysis of the atomic force microscopy (AFM) images is carried out, clarifying the role of the post-growth evolution as an integral part of the whole growth process. The set of scaling exponents obtained confirms that the growth is mainly driven by diffusion combined with the presence of step-edge barriers, in agreement with the observed ripening phenomenon. Finally, the results together with the overall approach adopted demonstrate the reliability of the HHCF analysis in systems displaying post-growth evolution.

Emergence of Polarization in Coevolving Networks.

Polarization is a ubiquitous phenomenon in social systems. Empirical studies document substantial evidence for opinion polarization across social media, showing a typical bipolarized pattern devising individuals into two groups with opposite opinions. While coevolving network models have been proposed to understand polarization, existing works cannot generate a stable bipolarized structure. Moreover, a quantitative and comprehensive theoretical framework capturing generic mechanisms governing polarization remains unaddressed. In this Letter, we discover a universal scaling law for opinion distributions, characterized by a set of scaling exponents. These exponents classify social systems into bipolarized and depolarized phases. We find two generic mechanisms governing the polarization dynamics and propose a coevolving framework that counts for opinion dynamics and network evolution simultaneously. Under a few generic assumptions on social interactions, we find a stable bipolarized community structure emerges naturally from the coevolving dynamics. Our theory analytically predicts two-phase transitions across three different polarization phases in line with the empirical observations for the Facebook and blogosphere data sets. Our theory not only accounts for the empirically observed scaling laws but also allows us to predict scaling exponents quantitatively.

Axial constants and sectional regularity of homogeneous ideals

We introduce a notion of sectional regularity for a homogeneous ideal I I , which measures the regularity of its general sections with respect to linear spaces of various dimensions. It is related to axial constants defined as the intercepts on the coordinate axes of the set of exponents of monomials in the reverse lexicographic generic initial ideal of I I . We show the equivalence of these notions and several other homological and ideal-theoretic invariants. We also establish that these equivalent invariants grow linearly for the family of powers of a given ideal.

Kibble−Zurek scaling of the dynamical localization−skin effect phase transition in a non-Hermitian quasi-periodic system under the open boundary condition

In the present study, the driven dynamics in a non-Hermitian Aubry–André (AA) model under the open boundary condition (OBC) are studied. For this model, non-Hermiticity is introduced by the non-reciprocal hopping, and this model undergoes a localization–skin effect phase transition depending on the strength of the quasi-periodic potential. Although the properties of non-Hermitian systems are very sensitive to the imposed boundary conditions, we find that the scaling behavior can also be described by the same set of the exponents under the periodic boundary condition (PBC). When the initial state is prepared deep in the localized phase and the potential strength is slowly driven through the critical point, we find that the driven dynamics of the localization length ξ and the inverse participation ratio (IPR) could be described by the Kibble–Zurek scaling (KZS). Then, we numerically verify these predictions for different initial states. Finally, the dynamical emergence of the skin effect state is found, and the dynamics can also be described by the Kibble−Zurek scaling with the same set of critical exponents.

Quench dynamics and scaling laws in topological nodal loop semimetals

We employ quench dynamics as an effective tool to probe different universality classes of topological phase transitions. Specifically, we study a model encompassing both Dirac-like and nodal loop criticalities. Examining the Kibble-Zurek scaling of topological defect density, we discover that the scaling exponent is reduced in the presence of extended nodal loop gap closures. For a quench through a multicritical point, we also unveil a path-dependent crossover between two sets of critical exponents. Bloch state tomography finally reveals additional differences in the defect trajectories for sudden quenches. While the Dirac transition permits a static trajectory under specific initial conditions, we find that the underlying nodal loop leads to complex time-dependent trajectories in general. In the presence of a nodal loop, we find, generically, a mismatch between the momentum modes where topological defects are generated and where dynamical quantum phase transitions occur. We also find notable exceptions where this correspondence breaks down completely.

Phase transition in the majority rule model with the nonconformist agents

Independence and anticonformity are two types of social behaviors known in social psychology literature and the most studied parameters in the opinion dynamics model. These parameters are responsible for continuous (second-order) and discontinuous (first-order) phase transition phenomena. Here, we investigate the majority rule model in which the agents adopt independence and anticonformity behaviors. We define the model on several types of graphs: complete graph, two-dimensional (2D) square lattice, and one-dimensional (1D) chain. By defining p as a probability of independence (or anticonformity), we observe the model on the complete graph undergoes a continuous phase transition where the critical points are pc≈0.334 (pc≈0.667) for the model with independent (anticonformist) agents. On the 2D square lattice, the model also undergoes a continuous phase transition with critical points at pc≈0.0608 (pc≈0.4035) for the model with independent (anticonformist) agents. On the 1D chain, there is no phase transition either with independence or anticonformity. Furthermore, with the aid of finite-size scaling analysis, we obtain the same sets of critical exponents for both models involving independent and anticonformist agents on the complete graph. Therefore they are identical to the mean-field Ising model. However, in the case of the 2D square lattice, the models with independent and anticonformist agents have different sets of critical exponents and are not identical to the 2D Ising model. Our work implies that the existence of independence behavior in a society makes it more challenging to achieve consensus compared to the same society with anticonformists.

Effective-dimension theory of critical phenomena above upper critical dimensions

Phase transitions and critical phenomena are among the most intriguing phenomena in nature and the renormalization-group theory for them is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an upper critical dimension d c seriously disagree with reality. In addition to its fundamental significance, the problem is also of practical importance because both complex classical systems with long-range spatial or temporal interactions and quantum phase transitions with long-range interactions can substantially lower d c . The extant scenarios built on a dangerous irrelevant variable (DIV) to resolve the problem introduce two sets of critical exponents and even two sets of scaling laws whose origin is obscure. Here, we consider the DIV from a different perspective and clearly unveil the origin of the two sets of exponents and hence the intrinsic inconsistency in those scenarios. We then develop an effective-dimension theory in which critical fluctuations and system volume are fixed at an effective dimension by the DIV. This enables us to account for all the extant results consistently. A novel asymptotic finite-size scaling behavior for a correlation function together with a new anomalous dimension and its associated scaling law is also predicted. We also apply the theory to quantum phase transitions.

Theory of Critical Phenomena with Memory

Memory is a ubiquitous characteristic of complex systems, and critical phenomena are one of the most intriguing phenomena in nature. Here, we propose an Ising model with memory, develop a corresponding theory of critical phenomena with memory for complex systems, and discover a series of surprising novel results. We show that a naive theory of a usual Hamiltonian with a direct inclusion of a power-law decaying long-range temporal interaction violates radically a hyperscaling law for all spatial dimensions even at and below the upper critical dimension. This entails both indispensable consideration of the Hamiltonian for dynamics, rather than the usual practice of just focusing on the corresponding dynamic Lagrangian alone, and transformations that result in a correct theory in which space and time are inextricably interwoven, leading to an effective spatial dimension that repairs the hyperscaling law. The theory gives rise to a set of novel mean-field critical exponents, which are different from the usual Landau ones, as well as new universality classes. These exponents are verified by numerical simulations of the Ising model with memory in two and three spatial dimensions.