Explicit Form Research Articles

Magnetic skyrmions: From lumps to supercompactons

The magnetic skyrmion is described by one control parameter and one length scale. We study the two extreme limits of the control parameter, infinitely large and vanishing, and find that the magnetic skyrmion becomes a “restricted” magnetic skyrmion and an O(3) sigma model lump, respectively. Depending on the potential under consideration, the restricted limit manifests differently. In the case of the Zeeman term, the restricted magnetic skyrmion becomes a “supercompacton” that develops a discontinuity, whereas for the Zeeman term to the power 32 it becomes a normal compacton. In both the lump and the restricted limit the solution is given in exact explicit form. We observe that the case of the Zeeman term squared, which can also be understood as a special combination of the Zeeman term and the easy-plane potential (realizable in the laboratory), the analytically exact solution for all values of the coupling, including the Bogomol'nyi-Prasad-Sommerfield case, is also of the lump type. Finally, we notice that certain materials (e.g., Fe1−xCoxSi or Mn1−xFexGe) have a rather large control parameter ε of order 100, making the restricted limit a suitable rough approximation. Published by the American Physical Society 2024

Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering

Abstract This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u ″ ″ + q u = f u^{\prime\prime} ^{\prime\prime} +qu=f . The equation entails two distinct boundary conditions: pinned–pinned conditions on u u and u ′ u^{\prime} , and clamped–clamped conditions on u u and u ″ {u}^{^{\prime\prime} } . To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

Ground and Excited States from Ensemble Variational Principles

The extension of the Rayleigh-Ritz variational principle to ensemble states ρw≡∑kwk|Ψk⟩⟨Ψk| with fixed weights wk lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state ρw as well as the individual eigenstates |Ψk⟩ and eigenenergies Ek. To be more specific, we derive linear bounds d−ΔEw≤ΔQ≤d+ΔEw on the errors ΔQ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of d±≡d±(Q)(w,E) provide valuable insights into the optimal choice of the auxiliary weights wk in practical applications.

Quantum Fisher information of a $\Diamond$-type four level atom interacting with a single-mode quantized field in an optomechanical cavity

Abstract In science and technology, precision measurement of physical quantities is crucial, and the quantum Fisher information (QFI) plays a significant role in the study of quantum systems. In this work, we explore the dynamics of QFI for a hybrid optomechanical system, which consists of a $\Diamond$-type four level atom interacting with a single-mode quantized field through multi-photon process. We account for various sources of dissipation, including the decay rates of the atom, the cavity, and the mechanical modes. Using an effective Hamiltonian, we analytically derive the explicit form of the state vector of the entire system via the time-dependent Schr"{o}dinger equation. We then investigate the atomic QFI for the estimation precision of the decay rate of the mechanical oscillator. Furthermore, we examine how optomechanical and atom-field coupling strengths, dissipation parameters, and multi-photon transition influence the dynamics of atomic QFI. Our numerical results suggest that the estimation precision of the decay rate of the mechanical oscillator can be controlled by these parameters.

Skew-Symmetric Generalized Normal and Generalized t Distributions

In this paper, we introduce the skew-symmetric generalized normal and the skew-symmetric generalized t distributions, which are skewed extensions of symmetric special cases of generalized skew-normal and generalized skew-t distributions, respectively. We derive key distributional properties for these new distributions, including a recurrence relation and an explicit form for the cumulative distribution function (cdf) of the skew-symmetric generalized t distribution. Numerical examples including a simulation study and a real data analysis are presented to illustrate the practical applicability of these distributions.

Derivation of a general and explicit equation for the force between identical cylindrical permanent magnets with the help of Wolfram|Alpha application

Abstract Until recently, there has been no single, explicit equation that explains how force between identical permanent magnets varies with distance over any distance. Typically, observation data is fitted using two curve segments: at large and small distances. In this work, we derive a general equation to determine the force between identical cylindrical permanent magnets at any distance with the help of the WolframAlpha application to get the explicit form of an integral problem introduced by Vokoun et al. (J. Magn. Magn. Mater. 321 3758–63, 2009) that appears to have no analytical solution. The resulting equation accurately fits observational data and can be used to estimate the magnetization of permanent magnets.

Thermodynamically consistent phase-field model for pressure-induced multiphase phase transformation of metals under intensive dynamic loading

Understanding the pressured-induced multiphase phase transformation (MPT) of metals at high strain rate is of great technical and scientific interest. Despite of several decades of studies, the pressured-induced MPT of metals at high strain rate is far from well understood because the deformation at high strain rate is so complex that present in situ diagnostic techniques or atomistic simulations cannot provide the details of the MPT at high strain rate. In the current work, a multiphase phase-field (MPF) model for the dynamic phase transformation (PT) of metals is developed, in which the volume fraction of each phase is chosen as the order parameter. The driving force of the MPT consists of the Gibbs free energy, the gradient energy and the penalizing energy. In particular, a smooth and normalized interpolation function is introduced to establish the Gibbs free energy at the multiphase junction. With the aid of this interpolation function, the explicit forms of the kinetics equation and constitutive relations of the MPT at the multiphase junction are also derived. It is demonstrated that the model satisfies all seven criteria of a consistent MPF model. Compared to previous models, the thermodynamic stability of this model is improved significantly. When applied to address the MPT of bismuth subjected to ramp-wave loading (RWL), the model can quantify the connection between the microstructural evolution of multiple phases and the experimentally measured multiwave structures.Moreover, new insights concerning the MPT-related thermodynamic paths of bismuth at elevated temperatures and the metastable phase during dynamic MPT are gained.

Medium Amplitude Field Susceptometry (MAFS) for magnetic nanoparticles

The linear dynamic susceptibility is arguably the most important property to specify the magnetization dynamics of a given system. Therefore, this quantity has been studied in great detail and the corresponding measurements known as susceptometry are well-established. Notwithstanding its relevance, the linear susceptibility is inherently limited to describe the magnetization response to weak external fields only. Here, we suggest the framework of Medium Amplitude Field Susceptometry (MAFS) to study the non-linear response which complements the linear susceptibility and provides additional information on the magnetic properties of the system and is applicable to magnetic fields of medium amplitudes. In particular, we introduce the general third-order nonlinear susceptibility χˆ3 as a central quantity that completely specifies the lowest-order non-linear response to arbitrary time-dependent magnetic fields. We show that response functions in medium amplitude oscillatory magnetic fields and parallel superposition susceptometry are contained in χˆ3 as special cases. Also included in χˆ3 are interesting intermodulation effects when the system is probed by a superposition of oscillating magnetic fields with different frequencies. We work out the explicit form of χˆ3 for several model systems for the dynamics of magnetic nanoparticles (MNPs). We expect this unifying framework to be not only of theoretical interest, but also useful for a deeper characterization of MNP systems, giving additional information on their suitability for various applications.

Simulations of nerve signal propagation in axons

PurposeIn the literature, soliton solutions of the Heimburg–Jackson model have been proposed by Drab et al. (2022), but for the considered models, i.e. Eq.(1) and Eq.(2), the existence of solitons, the dispersion analysis and the pseudospectral method have been studied (Engelbrecht et al., 2006, 2018, 2020; Tamm et al., 2017, 2022; Peets et al., 2013). Therefore, the gap should be filled by this work.Design/methodology/approachWhen nonlinear terms, dissipative terms and forcing terms are ignored, the system (Eq.(2)) reduces to a single, sixth-order partial differential equation (Tamm et al., 2022). In this work, our aim is to propose analytical solutions in the explicit form via ansatz-based method. Therefore, the parameter effects in wave profile will be proposed clearly in figures.FindingsWhile progress has been made in signal propagation in nerves, thanks to many experimental studies and theoretical predictions over the last two centuries, the results obtained in this study may answer new questions that arise.Originality/valueIn the literature, the existence of solitons, dispersion analysis and pseudospectral method have been investigated for the Heimburg-Jackson model (Engelbrecht et al., 2006, 2018, 2020; Tamm et al., 2017, 2022; Peets et al., 2013), and this study fills the gap in soliton solutions. Additionally, when nonlinear terms, dissipative term and forcing terms are ignored, the system (Eq.(2)) reduced to a single equation that is sixth-order partial differential equation (Tamm et al., 2022). In this work, our aim is to propose analytical solutions in the explicit form via ansatz-based method. Therefore, the parameter effects in wave profile will be proposed clearly in figures.

Supersymmetry of the Robinson-Trautman solution

The Robinson-Trautman solution in the Einstein-Maxwell-Λ system admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. Restricting to the case where the Maxwell field is aligned, i.e., the spacetime is algebraically special, we provide an exhaustive classification of supersymmetric Robinson-Trautman spacetimes in the four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauged supergravity. The differential constraints that arise from the integrability conditions of the Killing spinor equation enable us to systematically reconstruct the metric. We derive the explicit form of the Killing spinor either by directly integrating the Killing spinor equation or by casting the solution into the canonical form of supersymmetric solutions given in [13]. In any case, the supersymmetric Robinson-Trautman solution generically exhibits a naked singularity.

All-loop Heavy-Heavy-Light-Light correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory

We study Heavy-Heavy-Light-Light (HHLL) correlators HHO2O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle \\mathcal{HH}{\\mathcal{O}}_2{\\mathcal{O}}_2\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory with SU(N) gauge group at generic N. The light operator O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} is the dimension two superconformal primary in the stress tensor multiplet and H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document} is a general half-BPS superconformal primary operator with dimension (or R-charge) ∆H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\\mathcal{H}} $$\\end{document}. We consider the large-charge ’t Hooft limit, where ∆H→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\\mathcal{H}}\ o \\infty $$\\end{document} with fixed ’t Hooft-like coupling λ≔∆HgYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\lambda := {\\Delta }_{\\mathcal{H}}{g}_{\ extrm{YM}}^2 $$\\end{document}. We show that the L-loop contribution to the HHLL correlators in the leading large-charge limit is universal for any choice of the heavy operator H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{H} $$\\end{document}, given as λL∑ℓ=0LΦℓΦL−ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\lambda}^L{\\sum}_{\\ell =0}^L{\\Phi}^{\\left(\\ell \\right)}{\\Phi}^{\\left(L-\\ell \\right)} $$\\end{document} with an SU(N) colour factor coefficient, where Φ(ℓ) is the ladder Feynman integral, which is known to all loops. The dependence on the explicit form of the heavy operator lies only in the colour factor coefficients. We determine such colour factors for several classes of heavy operators, and show that the large charge limit leads to minimal powers of N. For the special class of “canonical heavy operators”, one can even resum the all-loop ladder integrals and determine the correlators at finite λ. Furthermore, upon integrating over the spacetime dependence the resulting integrated HHLL correlators agree with the existing results derived from supersymmetric localisation. Finally, as an application of the all-loop analytic results, we derive exact expressions for the structure constants of two heavy operators and the Konishi operator, finding intriguing connections with the integrated HHLL correlators.

Likelihood-Based Inference of Log-Logistic Accelerated Hazards Regression Models for Cross Hazards Data

ABSTRACT The proportional hazards model and accelerated failure time model are two important classes for analyzing survival data. However, the two models cannot capture time-scaled effects such as crossing hazard or the gradual effect of treatment over time. For the purpose of this detection, accelerated hazards model has been studied. In this paper, we propose the use of log-logistic distribution for the baseline hazard function of the accelerated hazards model which allows for time-dependent group effect. In particular, the log-logistic distribution gives various forms (e.g. non-monotone) of hazard function and also an easy implementation for likelihood-based model fitting due to an explicit form of hazard and survival functions. The likelihood-based estimation procedures of the accelerated hazards models are derived. Our method is demonstrated with two practical data sets, via the procedures of systematic model checking.

Unpacking the complexity of online incivility: an analysis of characteristics and impact of uncivil behavior during the Hong Kong protests

PurposeThis study seeks to establish a new framework for categorizing incivility, differentiating between explicit and implicit forms, and to investigate their respective abilities to proliferate and mobilize conversations, along with behavioral outcomes in various social contexts.Design/methodology/approachEmploying computational techniques, this research analyzed 10,145 protest-related threads from the HK Golden Forum, a prominent online discussion board in Hong Kong.FindingsOur analysis revealed divergent effects of explicit and implicit incivility on their diffusion, influences on deliberative discussions, and user participation. Explicit incivility was found to impede deliberative conversations, while implicit incivility tended to provoke more responses. Explicit uncivil expressions encouraged the propagation of incivility but reduced the likelihood of individual involvement. In contrast, implicit incivility had a stronger dampening effect on further uncivil comments and achieved greater thread popularity. The results showed strong associations between uncivil expressions and the contextual norms surrounding social movements.Originality/valueTheoretically, this research introduced a classification of incivility and underscored the importance of differentiating between implicit and explicit incivility by examining their effects on deliberation and engagement. Although previous studies have extensively covered explicit incivility, this study goes further by analyzing implicit incivility and comparing both forms of uncivil discourse in a less-studied context. Methodologically, the study developed a Cantonese dictionary to differentiate between two types of incivility, providing a practical reference for more nuanced analyses. By revealing how varying movement norms moderate the interplay between deliberative and uncivil expressions, the study drew attention to the highly situational nature of incivility.

Spectral tau explicit form for approximating solutions to real-life IBVPs using Chebyshev derivatives

This paper established a functional design of the spectral Tau method (STM) upon the first derivatives of Chebyshev polynomials (FDCHPs). A new linearization relation has been developed. Hence, this relation and another investigated one for integration have been used via the Tau method to execute the Tau integration analytically. Moreover, explicit algebraic systems for solving the Lane–Emden and Riccati equations and the contamination of a system of three artificial lakes are introduced. The convergence and error analyses are explored in depth. Some enlightening boundary value problems (BVPs) for real-life and physical applications, as singularly perturbed equations, are provided to guarantee that this approach is authentic, reliable, and appropriate. Accurate results are obtained using only a few number of retained nodes. The different outcomes, patterns, and better results showed that the FDCHPs differ from Chebyshev polynomials of the second kind.

A new semi-analytical approach for nonlinear dynamic responses of functionally graded porous graphene platelet-reinforced circular plates and spherical shells

This paper presents the semi-analytical approach for nonlinear dynamic responses of porous graphene platelet-reinforced circular plates (CiPs) and spherical shells (SpSs) resting on the Pasternak foundation in thermal environment. The graphene platelets (GPLs) are distributed in the functionally graded (FG) distribution patterns and uniform distribution pattern through the CiP and SpS thickness direction. The governing equations of the GPL-reinforced structures subjected to time-dependent external pressure respected to the harmonic and linear functions of time are established based on the geometrically nonlinear higher-order shear deformation theory. A simple and effective semi-analytical solution for the considered problem is established using the Euler-Lagrange equations, and the damping viscosity of the foundation can be counted using the Rayleigh dispassion function. The Runge-Kutta method is employed to determine the dynamic responses of the GPL-reinforced structures, while the fundamental frequencies of free and linear vibration, and frequency-amplitude curves are obtained in explicit form. The numerical results obtained reveal that the GPL and porous distributions, geometrical and material properties can significantly affect the frequency-amplitude curves, and dynamic responses of harmonically loaded and linearly loaded structures, specially, the dynamic buckling phenomenon is clearly observable with the SpSs but not with the CiPs.

Spin-2 Green’s functions on Kerr in radiation gauge

Abstract We construct retarded and advanced Green’s functions for gravitational perturbations in Kerr in an ingoing radiation gauge. Our Green’s functions have a frequency domain piece that has previously been obtained by Ori (2003 Phys. Rev. D 67) based on the Chrzanowski-Cohen-Kegeles metric reconstruction method. As is well known, this piece by itself is not sufficient to obtain an actual Green’s function. We show how to complete it with a piece based on a method by Green et al (2020 Class. Quantum Grav. 37). The completion piece has a completely explicit form in the time-domain and is supported on pairs of points on the same outgoing principal null geodesic which are in the appropriate causal order. We expect our Green’s functions to be useful for gravitational self-force calculations and other perturbation problems on Kerr spacetime.

Light-ray wave functions and integrability

Using integrability, we construct (to leading order in perturbation theory) the explicit form of twist-three light-ray operators in planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM. This construction allows us to directly compute analytically continued CFT data at complex spin. We derive analytically the “magic” decoupling zeroes previously observed numerically. Using the Baxter equation, we also show that certain Regge trajectories merge together into a single unifying Riemann surface. Perhaps more surprisingly, we find that this unification of Regge trajectories is not unique. If we organize twist-three operators differently into what we call “cousin trajectories” we find infinitely more possible continuations. We speculate about which of these remarkable features of twist-three operators might generalize to other operators, other regimes and other theories.

Fast Convex Optimization via Time Scale and Averaging of the Steepest Descent

In a Hilbert setting, we develop a gradient-based dynamic approach for fast solving convex optimization problems. By applying time scaling, averaging, and perturbation techniques to the continuous steepest descent (SD), we obtain high-resolution ordinary differential equations of the Nesterov and Ravine methods. These dynamics involve asymptotically vanishing viscous damping and Hessian-driven damping (either in explicit or implicit form). Mathematical analysis does not require developing a Lyapunov analysis for inertial systems. We simply exploit classical convergence results for SD and its external perturbation version, then use tools of differential and integral calculus, including Jensen’s inequality. The method is flexible, and by way of illustration, we show how it applies starting from other important dynamics in optimization. We consider the case in which the initial dynamic is the regularized Newton method, then the case in which the starting dynamic is the differential inclusion associated with a convex lower semicontinuous potential, and finally we show that the technique can be naturally extended to the case of a monotone cocoercive operator. Our approach leads to parallel algorithmic results, which we study in the case of fast gradient and proximal algorithms. Our averaging technique shows new links between the Nesterov and Ravine methods. Funding: The research of R.I. Boţ and D.-K. Nguyen was supported by the Austrian Science Fund (FWF), projects W 1260 and P 34922-N.

Психолого-педагогическая помощь детям, находящимся в условиях деструктивной социальной среды

The article examines the problems of a destructive social environment, which includes not only explicit forms of violence, but also more subtle ones, such as neglect. These are no less destructive factors: emotional alienation, information saturation with negative content, lack of stable teaching and educational models. In this context, the justification of the need to protect children from a psychological and pedagogical point of view becomes especially relevant. The main problem is to find effective methods and approaches to ensuring psychological and pedagogical protection of children in a destructive social environment. Key issues that need to be addressed within the framework of this problem: 1. Which theoretical models most adequately describe the mechanisms of the destructive environment's impact on the child's psyche? 2. What diagnostic tools can effectively identify children in need of psychological and pedagogical protection? 3. What forms of psychological and pedagogical assistance are most effective for different age groups and types of problems? 4. How to organize a system of training specialists capable of providing qualified assistance to children in a destructive environment? The results of the study allowed us to formulate conclusions. A destructive social environment should be considered as a multifactorial phenomenon, including not only social, but also psychological, economic and cultural factors. A literature review has shown that most approaches to helping children in a destructive social environment focus on creating a supportive and structured environment, as well as on developing self-regulation skills in children. Our results indicate that a successful intervention should include an integrated approach combining emotional support with cognitive development and social skills, including the implementation of programs aimed at developing emotional intelligence and social skills, and programs to improve children's adaptive abilities have proved effective. The results of the study indicate the need for individualization of approaches depending on the specifics of the problems faced by the child. The data confirm that children with different levels of social disadvantage require different types of support.

Explicit inverse Shapiro isomorphism and its application

We find an explicit form of the inverse isomorphism from Shapiro’s lemma in terms of inhomogeneous cocycles and apply it to construct special nonsplit coverings of groups with a unique conjugacy class of involutions.