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Single crystal Cr1.27Te2 samples were synthesized by using the chemical vapor transport method. Single crystal x-ray diffraction studies show a trigonal crystal structure with a P 3 ̄ m 1 symmetry space group. We then systematically investigate magnetic properties and critical behaviors of single crystal Cr1.27Te2 around its paramagnetic-to-ferromagnetic phase transition. The Arrott plot indicates a second-order magnetic phase transition. We estimate critical exponents β = 0.2631 ± 0.002, γ = 1.2314 ± 0.007, and TC = 168.48 ± 0.031 K by using the Kouvel–Fisher method. We also estimate other critical exponents δ = 5.31 ± 0.004 by analyzing the critical isotherm at TC = 168.5 K. We further verify the accuracy of our estimated critical exponents by the scaling analysis. Further analysis suggests that Cr1.27Te2 can be best described as a quasi-2D Ising magnetic system. [ABSTRACT FROM AUTHOR]

Single crystal Cr1.27Te2 samples were synthesized by using the chemical vapor transport method. Single crystal x-ray diffraction studies show a trigonal crystal structure with a P 3 ̄ m 1 symmetry space group. We then systematically investigate magnetic properties and critical behaviors of single crystal Cr1.27Te2 around its paramagnetic-to-ferromagnetic phase transition. The Arrott plot indicates a second-order magnetic phase transition. We estimate critical exponents β = 0.2631 ± 0.002, γ = 1.2314 ± 0.007, and TC = 168.48 ± 0.031 K by using the Kouvel–Fisher method. We also estimate other critical exponents δ = 5.31 ± 0.004 by analyzing the critical isotherm at TC = 168.5 K. We further verify the accuracy of our estimated critical exponents by the scaling analysis. Further analysis suggests that Cr1.27Te2 can be best described as a quasi-2D Ising magnetic system. [ABSTRACT FROM AUTHOR]

Ferromagnetic single crystalline MnNiSi samples were first fabricated through a Sn-flux growth technique, followed by measurements of their structural characteristics and intrinsic magnetic properties. Additionally, the critical behavior for second-order ferromagnetic to paramagnetic phase transition was investigated through utilization of techniques such as the modified Arrott plot, the Kouvel–Fisher method, and the magnetocaloric effect scaling law method. Through different methods of analysis, reliable critical exponents were obtained. Renormalization of interactions around the Curie temperature indicates the reliability of the obtained exponents. The obtained critical exponents are close to those theoretically predicted for a three-dimensional isotropic short-range Heisenberg ferromagnet but shift toward the long-range mean-field estimates. This may arise from the coupling of short- and long-range interactions as well as the competition between localized Mn–Mn magnetic interactions and the hybridization between p- and d-type orbitals. [ABSTRACT FROM AUTHOR]

The all-temperature magnon (ATM) theory [Datta and A. Panda, J. Phys. Condens. Matter 21, 336003 (2009)] has been used to analyze the temperature dependence of magnetization and internal energy components of a mono-domain ferromagnetic solid. One impact of the ATM formulation is that calculated critical exponents are in better agreement with experiments than their counterparts from mean-field and critical phenomenon theories. These exponents can vary from one ferromagnet to another of similar symmetry and dimensionality but differing in spin and can be field-dependent. The ATM finding is that exponent β depends on spin and increases as T approaches TC, whereas the exponent γ is weakly dependent on spin and the applied field but relies on crystal symmetry. The main thrust of the present work has been to derive the thermally averaged spin-center force constants in terms of the baseline related (solid) and exchange-cum-field mediated (magnetic) components and to formulate phonon frequencies and their modifications by magnon–phonon coupling. The derived expressions are suitable for correct quantum chemical evaluation. A detailed calculation on different spin configurations at varying geometries is still hardly possible and beyond the scope of the present work that emphasizes the correctness of formulas and has the significance of explaining properties. The phonon frequency shift due to lattice expansion is always negative. It is also clarified that frequency modification by the magnon–phonon interaction is negative for certain phonon branches near TC, and the ratio of frequency modification and phonon frequency is approximately proportional to the ratio of curvatures of involved energy surfaces. [ABSTRACT FROM AUTHOR]

In this work, we have studied the effect of annealing temperature (T S) on the structural properties and the critical behavior of Nd 0.5 Sr 0.5 MnO 3 compound prepared by sol-gel method (T S = 600, 700, 800 and 1000 °C). Scanning electronic microscope images indicate that all the samples are nanosized. The refinement of X-ray diffraction data shows that the samples sintered at 600 °C and 700 °C have three structural phases and that the monoclinic phase is linked to a superstructure (superlattice). The samples sintered at 800 °C and 1000 °C crystallize in the orthorhombic structure with Pnma space group, indicating the occurrence of a structural transition with increasing particle size. The magnetic measurements reveal that all our samples exhibit a paramagnetic to ferromagnetic transition with the absence of the long-range charge-ordered antiferromagnetic state. Through the magnetic field dependent magnetization, the observed overlapping between the isotherms suggests the persistence of some antiferromagnetic ordering in the nanosized samples. The reduction of saturation magnetization at 5 K for the sample sintered at 1000 °C indicates the beginning of the establishment of long-range charge ordering. The study of critical exponents has shown that the mean-field model is the most suitable model to describe the magnetic interactions for all our samples, suggesting the presence of long-range ferromagnetic interactions. [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]

The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with p(x)-Laplace operator containing k distinct critical Sobolev-Hardy exponents combined with singularity points U ... on Ω, where Ω ⊂ ℝN is a bounded domain with 0 ∈ Ω and 1 < p- ≤ p(x) ≤ p+ < N. The real function M is bounded in [0, +∞) and the functions hi (i = 1,...,k) and f are functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space W01,p(x)(Ω) due to Yu, Fu and Li, and the Fountain Theorem. [ABSTRACT FROM AUTHOR]

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the existence of a weak solution for a general class of critical semilinear elliptic equations related to the polyharmonic operator. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the phenomena of order-disorder phase transition and the universality of the majority-rule model defined on three complex networks, namely the Barabási–Albert, Watts–Strogatz and Erdős–Rényi networks. Assume each agent holds two possible opinions randomly distributed across the networks' nodes. Agents adopt anticonformity and independence behaviors, represented by the probability p, where with a probability p, agents adopt anticonformity or independence behavior. Based on our numerical simulation results and finite-size scaling analysis, it is found that the model undergoes a continuous phase transition for all networks, with critical points for the independence model greater than those for the anticonformity model in all three networks. We obtain critical exponents identical to the opinion dynamics model defined on a complete graph, indicating that the model exhibits the same universality class as the mean-field Ising model. [ABSTRACT FROM AUTHOR]

This paper is concerned with the bifurcation properties of the standing wave solutions for the Schrödinger–Poisson system with doubly critical case The study of system (P ) is motivated by its important applications in many physical models, such as the quantum mechanical systems under external influences. Here, 3 ≤ N ≤ 6 , 0 < α < N , λ ∈ R , g is a nonnegative weight function, and 2 α ♯ and 2 α ∗ are the lower and upper Hardy–Littlewood–Sobolev critical exponents, respectively. Moreover, when N = 6 and 0 < α < 2 existence of the (weak) solutions of the system under consideration is also proved via the global bifurcation theorem due to Rabinowitz. [ABSTRACT FROM AUTHOR]

The general epidemic process (GEP), also known as susceptible-infected-recovered model, provides a minimal model of how an epidemic spreads within a population of susceptible individuals who acquire permanent immunization upon recovery. This model exhibits a second-order absorbing state phase transition, commonly studied assuming immobile healthy individuals. We investigate the impact of mobility on the scaling properties of disease spreading near the extinction threshold by introducing two generalizations of GEP, where the mobility of susceptible and recovered individuals is examined independently. In both cases, including mobility violates GEP's rapidity reversal symmetry and alters the number of absorbing states. The critical dynamics of the models are analyzed through a perturbative renormalization group (RG) approach and large-scale stochastic simulations using a Gillespie algorithm. The RG analysis predicts both models to belong to the same novel universality class describing the critical dynamics of epidemic spreading when the infected individuals interact with a diffusive species and gain immunization upon recovery. At the associated RG fixed point, the immobile species decouples from the dynamics of the infected species, dominated by the coupling with the diffusive species. Numerical simulations in two dimensions affirm our RG results by identifying the same set of critical exponents for both models. Violation of the rapidity reversal symmetry is confirmed by breaking the associated hyperscaling relation. Our study underscores the significance of mobility in shaping population spreading dynamics near the extinction threshold. [ABSTRACT FROM AUTHOR]

Fractional quantum Hall (FQH) phases emerge due to strong electronic interactions and are characterized by anyonic quasiparticles, each distinguished by unique topological parameters, fractional charge, and statistics. In contrast, the integer quantum Hall (IQH) effects can be understood from the band topology of non-interacting electrons. We report a surprising super-universality of the critical behavior across all FQH and IQH transitions. Contrary to the anticipated state-dependent critical exponents, our findings reveal the same critical scaling exponent κ = 0.41 ± 0.02 and localization length exponent γ = 2.4 ± 0.2 for fractional and integer quantum Hall transitions. From these, we extract the value of the dynamical exponent z ≈ 1. We have achieved this in ultra-high mobility trilayer graphene devices with a metallic screening layer close to the conduction channels. The observation of these global critical exponents across various quantum Hall phase transitions was masked in previous studies by significant sample-to-sample variation in the measured values of κ in conventional semiconductor heterostructures, where long-range correlated disorder dominates. We show that the robust scaling exponents are valid in the limit of short-range disorder correlations. The relationship between the integer and fractional quantum Hall states are of fundamental importance in condensed matter physics. Here, the authors report a universality of phase transitions in both regimes in ultraclean trilayer graphene samples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem 0.1 −▵u−12x·∇u=λ|u|2∗−2u+μ|u|p−2uinR+N,∂u∂n=λ|u|2∗−2uon∂R+N,$$\begin{equation} \hspace*{40pt}\left\{ \def\eqcellsep{&}\begin{array}{rcll} -\triangle u-\displaystyle\frac{1}{2}\left(x\cdot \nabla u\right)& =& \lambda |u{|}^{{2}^{*}-2}u+\mu |u{|}^{p-2}u & \text{in }{\mathbb{R}}_{+}^{N},\\ [12pt] \displaystyle\frac{\partial u}{\partial n}& =& \sqrt{\lambda}|u{|}^{{2}_{*}-2}u & \text{on }\partial {\mathbb{R}}_{+}^{N}, \end{array} \right. \end{equation}$$where R+N={(x′,xN):x′∈RN−1,xN>0}$ \mathbb {R}^{N}_{+}=\lbrace (x{^{\prime }}, x_{N}): x{^{\prime }}\in {\mathbb {R}}^{N-1}, x_{N}>0\rbrace$, N≥3$N\ge 3$, λ>0$\lambda >0$, μ∈R$\mu \in \mathbb {R}$, 20$\lambda >0$ and the different values of the parameters p$p$ and μ>0${\mu }>0$. Particularly, for λ>0$\lambda >0$, N≥4$N\ge 4$, 20$\mu >0$. Moreover, the existence of multiple solutions for (0.1) is also obtained by dual variational principle for all μ>0$\mu >0$ and suitable λ$\lambda$. [ABSTRACT FROM AUTHOR]

Abstract After obtaining an exact regular-AdS black hole resulting from the coupling of general relativity with nonlinear electrodynamics (NED), we explore the thermodynamics of the extended phase space, treating the cosmological constant ( $$\Lambda$$ Λ ) as the pressure (P) of the black holes and its conjugate as thermodynamic volume (V). Considering the NED parameter (g), we investigate the Hawking temperature, entropy, Gibb’s free energy and specific heat at the horizon radius. Due to the presence of NED charge, the black hole exhibits van der Waals-like phase transition instead of Hawking-Page phase transition, which could be observed through the $$G-T$$ G - T plots, which display a swallowtail pattern below the critical pressure, and it gives rise to second-order phase transitions when pressure attains its critical value. The first-order phase transition shares similarities with the liquid-gas phase transition. We determine the exact critical points and explore the influence of NED on $$P-V$$ P - V criticality, revealing that the isotherms undergo a liquid-gas-like phase transition for temperatures below its critical value $$T_C$$ T C , especially at lower $$T_C$$ T C . The identical critical exponent to that of the van der Waals fluid suggests that the NED does not alter the critical exponents, as observed in other arbitrary AdS black holes.

In this paper, we consider the existence and nonexistence of positive solution for the following critical Neumann problem (0.1) { − Δ u − 1 2 (x ⋅ ∇ u) = λ u + a | u | 2 ⁎ − 2 u in R + N , ∂ u ∂ n = μ | u | q − 2 u + | u | 2 ⁎ − 2 u on ∂ R + N , where R + N = { (x ′ , x N) : x ′ ∈ R N − 1 , x N > 0 } is the upper half-space, N ≥ 3 , λ , μ ∈ R are parameters, a ∈ { 0 , 1 } , n is the outward normal vector at the boundary ∂ R + N , 2 ≤ q < 2 ⁎ , 2 ⁎ = 2 N N − 2 is the usual critical exponent for the Sobolev embedding D 1 , 2 (R + N) ↪ L 2 ⁎ (R + N) and 2 ⁎ = 2 (N − 1) N − 2 is the critical exponent for the Sobolev trace embedding D 1 , 2 (R + N) ↪ L 2 ⁎ (∂ R + N). By applying the Mountain Pass Theorem without (PS) condition and the delicate estimates for Mountain Pass level, we obtain the existence of a positive solution under different assumptions on λ , μ and q. Meanwhile, some nonexistence results for problem (0.1) are also obtained by an improved Pohozaev identity and Hardy inequality according to the value of the parameters λ , μ and q. Particularly, for N ≥ 3 , λ ≥ 0 , μ > 0 and q ∈ (2 , 2 ⁎) , we obtain that problem (0.1) has a positive solution if and only if λ ∈ [ 0 , N 2) ; On the other hand, for N ≥ 3 and μ = 0 , we find a lower bound Λ ⁎ ∈ [ N 4 , N 2) depending on N such that problem (0.1) has a positive solution if λ ∈ (Λ ⁎ , N 2) and has no positive solution if λ ∈ (− ∞ , N 4) ∪ [ N 2 , + ∞). Moreover, we estimate that Λ ⁎ ∈ (N 4 , N − 2 2) if N ≥ 5 and Λ ⁎ = N 4 = 1 if N = 4 which gives the best lower bound of λ for the existence of a positive solution of problem (0.1) if N = 4. [ABSTRACT FROM AUTHOR]

The equilibrium behaviors of the anisotropic XY ferromagnet, with nonmagnetic impurity, have been investigated in three dimensions by Monte Carlo simulation using Metropolis algorithm. Two different types of anisotropy, namely, the bilinear exchange type and single-site anisotropy are considered here. The thermodynamic behaviors of the components of the magnetizations (M), susceptibility (χ) and the specific heat (C) have been studied systematically through extensive Monte Carlo simulations. The ferro–para phase transition has been observed to take place at a lower temperature for impure anisotropic XY ferromagnet. The pseudocritical temperature ( T c ∗ ) has been found to decrease as the system gets more and more impure (impurity concentration p increases). In the case of bilinear exchange type of anisotropy (λ), the pseudocritical temperature ( T c ∗ ) increases linearly with λ for any given concentration of nonmagnetic impurity (p). The slope of this linear function has been found to depend on the impurity concentration (p). The slope decreases linearly with the impurity concentration (p). In the case of the single-site anisotropy (D), the pseudocritical temperature ( T c ∗ ) has been found to decrease linearly with p for fixed D. The critical temperature (for a fixed set of parameter values) has been estimated from the temperature variation of fourth-order Binder cumulants ( U L ) for different system sizes (L). The critical magnetization (M (T c)) and the maximum value of the susceptibility ( χ p ) are calculated for different system sizes (L). The critical exponents for the assumed scaling laws, M (T c) ∼ L − β ν and χ p ∼ L γ ν , are estimated through the finite size analysis. We have estimated, β ν , equals 0. 4 8 ± 0. 0 5 and 0. 3 7 ± 0. 0 4 for bilinear exchange and single-site anisotropy, respectively. We have also estimated, γ ν equals 1. 7 8 ± 0. 0 5 and 1. 8 1 ± 0. 0 5 for bilinear exchange and single-site anisotropy, respectively. [ABSTRACT FROM AUTHOR]

We study the existence of solutions to the problem - Δ u + u p - M ∇ u q = 0 in Ω , (1) u = μ on ∂ Ω in a bounded domain Ω, where p > 1, 1 < q < 2, M > 0, μ is a nonnegative Radon measure in ∂Ω, and the associated problem with a boundary isolated singularity at a ∈ ∂Ω, - Δ u + u p - M ∇ u q = 0 in Ω , (2) u = 0 on ∂ Ω \ α. The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to (1) is obtained under a capacitary condition μ K ≤ c min cap 2 p , p ′ ∂ Ω , cap 2 - q q , q ′ ∂ Ω for\;all\;compacts\; K ⊂ ∂ Ω. Problem (2) depends on several critical exponents on p and q as well as the position of q with respect to 2 p p + 1 . [ABSTRACT FROM AUTHOR]

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The P S c sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized. [ABSTRACT FROM AUTHOR]

This paper is devoted to multiple solutions to a Kirchhoff–Schrödinger type problem of fractional p-Laplacian involving the Sobolev–Hardy critical exponent and a parameter $ \lambda >0 $ λ > 0. With some suitable assumptions on the potential $ V(x) $ V (x) and the nonlinearity $ f(x,u) $ f (x , u) , the Krasnoselskii's genus argument is exploited to show the existence of infinitely many solutions if λ is sufficiently large. Furthermore, we employ a fractional version of the concentration-compactness to prove that there are m-pairs solutions of the problem provided that λ is small enough and the nonlinear force $ f(x,\cdot) $ f (x , ⋅) is odd. [ABSTRACT FROM AUTHOR]

The investigation of site percolation on Sierpinski carpets is carried out through comprehensive numerical simulations. We utilize finite- size scaling theory, staying within the constraints of our computational resources, to determine critical exponents and percolation thresholds. Moreover, we employ an approach developed by Elliot et al. (Phys Rev C 6:3185, 1994; Phys Rev C 55:1319, 1997), which streamlines the process by eliminating the necessity of dealing with large lattices. This method facilitates the extraction of critical quantities that characterize the transition from a single generation within a given structure. By implementing this procedure, we enhance efficiency and accuracy in analyzing the percolation phenomenon on Sierpinski carpets. The obtained values of the percolation thresholds are plotted as a function of the fractal dimensions in order to determine the lower critical dimension of the site percolation problem which is calculated to be d c L = 1.52 . In addition, the behavior of the critical exponents as a function of the fractal dimension is also shown and discussed. [ABSTRACT FROM AUTHOR]

While Ising criticality in classical liquids has been firmly established both theoretically and experimentally, much less is known about criticality in liquids in which the growth of the correlation length is frustrated by finite-size effects. A theoretical approach for dealing with this issue is the random-field Ising model (RFIM). While experimental critical-exponent values have been reported for magnetic samples (here, we consider γ , ν and η), little experimental information is available for critical fluctuations in corresponding liquid systems. In this paper, we present a study on a binary liquid consisting of 3-methyl pyridine and heavy water in a very light-weight porous gel. We find that the experimental results are in agreement with the theoretical predictions from the RFIM. [ABSTRACT FROM AUTHOR]

Mn5Ge3 compound is a well-established rare-earth-free magnetocaloric material with a near-room temperature magnetocaloric effect. Chemical modification could increase its relative cooling power and to achieve that we have substituted copper and synthesized Mn4.8Cu0.2Ge3 polycrystalline compound through arc-melting process. Mn4.8Cu0.2Ge3 is an itinerant ferromagnet with a second-order magnetic transition in the vicinity of room temperature. In comparison with the pure Mn5Ge3 compound, the temperature span of the magnetocaloric effect has widened, which is a favorable requirement in practical devices. The magnetization critical exponents were estimated through the modified Arrott plot and Kouvel-Fischer methods. An unconventional critical behavior was noticed in this compound, in which the critical exponents β and γ fall close to the standard 3D-XY and 3D-Ising models, respectively. On the other hand, the pure Mn5Ge3 single crystal shows 3D-Ising critical behavior, and the deviation from the standard universality class in Mn4.8Cu0.2Ge3 could be attributed to the modification in Mn–Mn distance induced by the copper. On analyzing the magnetic properties of Mn5-xCuxGe3 (x = 0.1, 0.2, and 0.3) series, we speculate that x = 0.2 compound breaks the series monotonicity due to the unique influence of Cu substitution on Mn–Mn interactions. [ABSTRACT FROM AUTHOR]

The Biswas–Chatterjee–Sen (BChS) model of opinion dynamics has been studied on three-dimensional Solomon networks by means of extensive Monte Carlo simulations. Finite-size scaling relations for different lattice sizes have been used in order to obtain the relevant quantities of the system in the thermodynamic limit. From the simulation data it is clear that the BChS model undergoes a second-order phase transition. At the transition point, the critical exponents describing the behavior of the order parameter, the corresponding order parameter susceptibility, and the correlation length, have been evaluated. From the values obtained for these critical exponents one can confidently conclude that the BChS model in three dimensions is in a different universality class to the respective model defined on one- and two-dimensional Solomon networks, as well as in a different universality class as the usual Ising model on the same networks. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Schrödinger-Poisson system with a critical nonlinearity and a critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality$ \begin{equation*} \begin{cases} -\Delta u +u +\lambda\phi|u|^3u = |u|^4u +|u|^{q-2}u, \ \ &\ x\in\mathbb{R}^3,\\ -\Delta\phi = |u|^5, \ \ &\ x\in\mathbb{R}^3, \end{cases} \end{equation*} $where $ \lambda\in \mathbb{R} $ is a parameter and $ q\in(2,6) $. By employing variational methods, we present the existence and nonexistence results (depending on the parameters $ \lambda $ and $ q $) as follows:($ i $) if $ \lambda\ge (\frac{q+2}{8})^2 $ and $ q\in(2,6) $, there is no nontrivial solution of the above system;($ ii $) if $ \lambda\in (-\lambda^*,0] $ for some $ \lambda^*>0 $, we obtain a least energy radial sign-changing solution $ u_\lambda $ of the above system;($ iii $) we consider $ \lambda $ as a parameter and analyze the asymptotic behavior of $ u_\lambda $ as $ \lambda\to 0^- $.Moreover, a Schrödinger-Poisson system with a critical nonlocal term differs completely from a system with a subcritical nonlocal term. [ABSTRACT FROM AUTHOR]

We investigate the multiplicity results of positive solutions for a Kirchhoff-type problems with supercritical and critical nonlinear terms in a ball. By employing the Nehari method and Lusternik–Schnirelmann category theory to an auxiliary problems, we note that there is a relationship between the number of maxima in the coefficient function of the critical term and the number of positive solutions for the problems. [ABSTRACT FROM AUTHOR]

Recent cellular-level volumetric brain reconstructions have revealed high levels of anatomic complexity. Determining which structural aspects of the brain to focus on, especially when comparing with computational models and other organisms, remains a major challenge. Here we quantify aspects of this complexity and show evidence that brain anatomy satisfies universal scaling laws, establishing the notion of structural criticality in the cellular structure of the brain. Our framework builds upon understanding of critical systems to provide clear guidance in selecting informative structural properties of cellular brain anatomy. As an illustration, we obtain estimates for critical exponents in the human, mouse and fruit fly brains and show that they are consistent between organisms, to the extent that data limitations allow. Such universal quantities are robust to many of the microscopic details of the cellular structures of individual brains, providing a key step towards generative computational models of the cellular structure of the brain, and also clarifying in which sense one animal may be a suitable anatomic model for another. Cellular-level partial brain reconstructions have revealed the anatomic complexity of the brains of multiple organisms. Here, the authors quantify aspects of this complexity, demonstrating that across organisms the cellular structure of the brain shows universal scaling properties associated with being in the vicinity of criticality. [ABSTRACT FROM AUTHOR]

The study focused on the binary system 6OCB and 8OCB, which is well-suited for investigating stable re-entrant nematic-phase transitions. High-resolution measurements of heat capacity and birefringence are used to analyse critical behaviours near the nematic – smectic-A (N-SmA) and the smectic-A – re-entrant nematic (SmA- ${{\rm{N}}_{\rm{R}}}$ N R ) phase transitions. The critical behaviour of these transitions has been explored with the aid of a differential quotient extracted from the birefringence $\left({\rm \Delta {{n}}} \right)$ Δ n values. Simultaneously, critical exponents extracted from the specific heat capacity anomaly are also used to analyse the critical behaviour of the phase transition. The results indicate that the SmA to ${{\rm{N}}_{\rm{R}}}$ N R phase transition exhibits non-universal behaviour, with effective exponents falling between the tricritical (TCP) and 3D-XY values, similar to the N-SmA phase transition. The study reveals the existence of two tricritical points: one associated with the N-SmA transition and the other corresponding to the SmA- ${{\rm{N}}_{\rm{R}}}$ N R transition. This is supported by plotting the McMillan ratio against the critical exponents obtained from both birefringence and specific heat capacity anomaly. Again, with an increase in the SmA range, the SmA- ${{\rm{N}}_{\rm{R}}}$ N R transition tends towards the tricritical point (TCP), suggesting a relationship between the SmA range and the transitional behaviour. [ABSTRACT FROM AUTHOR]

The equilibrium and phase properties of the Ising model with three-spin interaction and an external field are studied within the framework of mean-field approximation. The thermodynamic properties of the model reveals two coexistence curves, signifying two distinct second-order phase transitions, dependent on the domain of the interaction parameter. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class. [ABSTRACT FROM AUTHOR]

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field - (a ε 2 + b ε [ u ] A / ε 2) Δ A / ε u + V (x) u = f (| u | 2) u , x ∈ R 2 , where ε > 0 is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both V : R 2 → R and A : R 2 → R 2 are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect. [ABSTRACT FROM AUTHOR]

The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents − Δ u − μ u x 2 = u 2 * s − 2 x s u + g (x , u) , x ∈ Ω ∖ 0 , u = 0 , x ∈ ∂ Ω have been investigated, where Ω is an open-bounded domain in R N N ≥ 3 , with a smooth boundary ∂ Ω , 0 ∈ Ω , 0 ≤ μ < μ ¯ : = N − 2 2 2 , 0 ≤ s < 2 , and 2 * s = 2 N − s / N − 2 is the Hardy–Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrödinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on g , by a detailed estimation of the extremum function and using mountain pass lemma with P S c conditions, we obtained that: (a) If μ ≤ μ ¯ − 1 , and λ < λ 1 μ , then the above problem has at least a positive solution in H 0 1 Ω ; (b) If μ ¯ − 1 < μ < μ ¯ , then when λ * μ < λ < λ 1 μ , the above problem has at least a positive solution in H 0 1 Ω ; (c) if μ ¯ − 1 < μ < μ ¯ and Ω = B (0 , R) , then the above problem has no positive solution for λ ≤ λ * μ. These results are extensions of E. Jannelli's research ( g (x , u) = λ u ). [ABSTRACT FROM AUTHOR]

We have systematically investigated the critical behavior around paramagnetic-ferromagnetic (PM-FM) phase transition temperature, TC, and attempted to determine the Curie temperatures and critical exponents β, γ and δ of perovskite manganites La0.67Ba0.33–xSrxMnO3 (x = 0.1, 0.2 and 0.25).The TC determined from M(T) is observed to increase from 336.7 to 368.4 K, with Sr doping at Ba. The critical exponents β, γ and δ are determined from the magnetization, M(T,H), using Modified Arrott Plots for theoretically predicted models and critical M(H) isotherms at T = TC. From the critical exponents we have inferred that La0.67Ba0.33–xSrxMnO3 behave as a three-dimensional (3D) isotropic nearest heighbor Heisenberg ferromagnet in the critical region. We have further verified that the behavior of 3D isotropic nearest neighbor Heisenberg is in agreement with the values of exponent 'n' determined based on the relation between field dependent change in magnetic entropy at the Curie temperature, T = TC. Sr doping at Ba site in La0.67Ba0.33–xSrxMnO3 does not lead to any crossover in the type of magnetic interactions present in the materials, indicating a stable behavior of 3D isotropic nearest neighbor Heisenberg ferromagnet in the critical region. The differences in the values of parameters may be attributed to the complex nature of interactions present in these materials. [ABSTRACT FROM AUTHOR]

We discuss the distribution of partition function zeros for the grand-canonical ensemble of the zeta-urn model, where tuning a single parameter can give a first or any higher order condensation transition. We compute the locus of zeros for finite-size systems and test scaling relations describing the accumulation of zeros near the critical point against theoretical predictions for both the first and higher order transition regimes.

We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities $ \begin{equation*} \left\{\begin{aligned} -\Delta_{\mathbb{G}}u-\mu_{1}\frac{\psi^{2}u}{\text{d}(z)^{2}} = \lambda_{1}\frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{\text{d}(z)^{\alpha}}+\beta p_{1}f(z)\frac{\psi^{\gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{\text{d}(z)^{\gamma}}\,\,\, \text{in } \mathbb{G},\\ -\Delta_{\mathbb{G}}v-\mu_{2}\frac{\psi^{2}v}{\text{d}(z)^{2}} = \lambda_{2}\frac{\psi^{\alpha}|v|^{2^*(\alpha)-2}v}{\text{d}(z)^{\alpha}}+\beta p_{2}f(z)\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{\text{d}(z)^{\gamma}}\,\,\, \text{in } \mathbb{G}, \end{aligned}\right. \end{equation*} $ where $ -\Delta_{\mathbb{G}} $ is the sub-Laplacian on Carnot group $ \mathbb{G} $, $ \mu_{1} $, $ \mu_{2}\in [0, \mu_{\mathbb{G}}) $, $ \alpha, \, \gamma\in (0, 2) $, $ \lambda_{1} $, $ \lambda_{2} $, $ \beta $, $ p_{1} $, $ p_{2} > 0 $ with $ 1 < p_{1}+p_{2} < 2 $, $ \text{d}(z) $ is the $ \Delta_{\mathbb{G}} $-gauge, $ \psi = |\nabla_{\mathbb{G}}\text{d}(z)| $, $ 2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2} $ is the critical Sobolev-Hardy exponents, and $ \mu_{\mathbb{G}} = (\frac{Q-2}{2})^{2} $ is the best Hardy constant on $ \mathbb{G} $. By combining a variant of the symmetric mountain pass theorem with the genus theory, we prove the existence of infinitely many weak solutions whose energy tends to zero when $ \beta $ or $ \lambda_{1} $, $ \lambda_{2} $ belong to a suitable range.

In this paper, the compensation characteristics and hysteresis loops of a ferrimagnetic mixed-spin diluted fullerene-like (XY)540 are studied based on Monte Carlo method. We show the ground-state phase diagrams in order to explore the stable configuration states. The temperature dependence of magnetization, magnetic susceptibility, internal energy, specific heat, and entropy are presented. The phase diagram including compensation temperature Tcomp and phase change temperature Tc is obtained. The multi-loop hysteresis phenomena under the influence of diverse physical parameters are observed. Finally, the critical exponents of the fullerene-like (XY)540 are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following elliptic system of Hamiltonian type on a bounded domain: (0.1) { − Δ u = K 1 (| y |) | v | p − 1 v , in B 1 (0) , − Δ v = K 2 (| y |) | u | q − 1 u , in B 1 (0) , u = v = 0 on ∂ B 1 (0) , where K 1 (r) and K 2 (r) are positive bounded functions, B 1 (0) is the unit ball in R N , (p , q) is a pair of positive numbers lying on the critical hyperbola: 1 p + 1 + 1 q + 1 = N − 2 N. We investigate the local uniqueness and the nondegeneracy of the bubbling solution for problem (0.1). Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of Greens function. Our results are quite different from single equation, which is mainly caused by the non-cooperative nonlinear terms and (p , q) lying on the critical hyperbola. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems of Hamiltonian type with interact critical exponents. [ABSTRACT FROM AUTHOR]

We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For fractal time processes, we not only discover new universality classes with a dimensional constant but also decompose the dangerous irrelevant variables to obtain corrections for critical dynamic behavior and static critical properties. This contrasts with the traditional theory of critical phenomena, which posits that static critical exponents are unrelated to the dynamical processes. Simulations of the Landau–Ginzburg model for fractal time processes and the Ising model with temporal long-range interactions both show good agreement with our set of critical exponents, verifying its universality. The discovery of this new universality class provides a method for examining whether a system is undergoing a fractal time process near the critical point. [ABSTRACT FROM AUTHOR]

We consider the following singular quasilinear Schrödinger equations involving critical exponent - Δ u - α 2 Δ (| u | α ) | u | α - 2 u = θ | u | k - 2 u + | u | 2 ∗ - 2 u + λ f (u) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where 0 < α < 1 . By using the variational methods, we first prove that for small values of λ and θ , the above problem has infinitely many distinct solutions with negative energy. Besides, we point out that odd assumption on f is required; the problem has at least one nontrivial solution. Finally, a new modified technique is used to consider the existence of infinitely many solutions for far more general equations. [ABSTRACT FROM AUTHOR]

Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum–optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality. [ABSTRACT FROM AUTHOR]

Many physical properties behave differently in disorder systems from the order one. The effects of atomic disorder on the magnetic properties have been explored on Co2TiSi and Co2TiAl full Heusler alloys. Co2TiSi crystallizes in an ordered L21 crystal symmetry, and Co2TiAl mainly possesses a B2 structure. The B2 disorder affects not only the magnetic ordering but also the saturation magnetization, causing the deviation from the Slater-Pauling prediction for Co2TiAl. The critical exponents of the magnetic phase transition have been analyzed by employing different techniques like, the Arrott plot, Kouvel–Fisher plot, critical isotherm analysis, scaling magnetic behavior, and magnetocaloric analysis as well. The calculated critical exponents are close to the mean field theory which implies long-range ferromagnetic ordering in Co2TiSi. However, the exchange interaction in Co2TiAl deviates from the long-range mean field theory. This may be due to the magnetic contribution of the B2-disordered phase. In addition to the magnetic ordering, the magnetocaloric analysis provides the applicability of Co2TiSi alloy as a low-cost, hysteresis-free magnetic refrigerant with a large working temperature near room temperature. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of nonlocal p(x)-Laplace equations with nonlinear boundary conditions. When the nonlinear boundary involves critical exponents, using the concentration compactness principle, mountain pass lemma, and fountain theorem, we can prove the existence and multiplicity of solutions. [ABSTRACT FROM AUTHOR]

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 . [ABSTRACT FROM AUTHOR]

Discontinuous shear thickening (DST) is associated with a sharp rise in a suspension's viscosity with increasing applied shear rate or stress. Key signatures of DST, highlighted in recent studies, are the very large fluctuations of the measured stress as the suspension thickens with increasing rate. A clear link between microstructural development and the dramatic increase in stress fluctuations has not been established yet. To identify the microstructural underpinnings of this behavior, we perform simulations of sheared dense suspensions. Through an analysis of the particle contact network, we identify a subset of constrained particles that contributes directly to the rapid rise in viscosity and large stress fluctuations. Indeed, both phenomena can be explained by the growth and percolation of constrained particle networks—in direct analogy to rigidity percolation. A finite size scaling analysis confirms this to be a percolation phenomenon and allows us to estimate the critical exponents. Our findings reveal the specific microstructural self-organization transition that underlies DST. [ABSTRACT FROM AUTHOR]

In this note, we revisit the scaling relations among "hatted critical exponents", which were first derived by Ralph Kenna, Des Johnston, and Wolfhard Janke, and we propose an alternative derivation for some of them. For the scaling relation involving the behavior of the correlation function, we will propose an alternative form since we believe that the expression is erroneous in the work of Ralph and his collaborators. [ABSTRACT FROM AUTHOR]

We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $ m = \frac{2d}{d+2s} $ in such a way that the associated free energy is conformal invariant and there is a family of stationary solutions $ U(x) = c\left(\frac{\lambda}{\lambda^2+|x-x_0|^2}\right)^{\frac{d+2s}{2}} $ for some constant $ c $ and $ \lambda>0, x_0 \in \mathbb{R}^d$. We analyse under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of dynamical solutions by virtue of stationary solutions. Precisely, solutions exist globally in time if the $ L^m $ norm of the initial data $ \|u_0\|_{L^m(\mathbb{R}^d)} $ is less than the $ L^m $ norm of stationary solutions $ \|U(x)\|_{L^m(\mathbb{R}^d)} $. Whereas there are blowing-up solutions for $ \|u_0\|_{L^m(\mathbb{R}^d)}>\|U(x)\|_{L^m(\mathbb{R}^d)} $. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups. [ABSTRACT FROM AUTHOR]