Embeddings between generalized weighted Lorentz spaces

The generalized Chaplygin gas is usually defined as a barotropic perfect fluid with an equation of state $p=-A \rho^{-\alpha}$, where $\rho$ and $p$ are the proper energy density and pressure, respectively, and $A$ and $\alpha$ are positive real parameters. It has been extensively studied in the literature as a quartessence prototype unifying dark matter and dark energy. Here, we consider an extended family of generalized Chaplygin gas models parameterized by three positive real parameters $A$, $\alpha$ and $\beta$, which, for two specific choices of $\beta$ [$\beta=1$ and $\beta=\left(1+\alpha\right)/(2\alpha)$], is described by two different Lagrangians previously identified in the literature with the generalized Chaplygin gas. We show that, for $\beta > 1/2$, the linear stability conditions and the maximum value of the sound speed $c_s$ are regulated solely by $\beta$, with $0 \le c_s \le 1$ if $\beta \ge 1$. We further demonstrate that in the non-relativistic regime the standard equation of state $p=-A \rho^{-\alpha}$ of the generalized Chaplygin gas is always recovered, while in the relativistic regime this is true only if $\beta=\left(1+\alpha\right)/(2\alpha)$. We present a regularization of the ($\alpha\rightarrow 0$, $A \rightarrow \infty$) limit of the generalized Chaplygin gas, showing that it leads to a logarithmic Chaplygin gas model with an equation of state of the form $p = {\mathcal A} \ln\left(\rho/\rho_{*}\right)$, where ${\mathcal A}$ is a real parameter and $\rho_*>0$ is an arbitrary energy density. We finally derive its Lagrangian formulation.

In this paper, we discuss the rational maps $$F_\lambda (z) = z^n + \lambda /z^n ,n \geqslant 2$$ with the positive real parameter λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Furthermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpinski curve is given.

In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration–compactness principle for variable exponent under suitable assumptions on the nonlinearities.

This paper proposes a rational approximation-based approach to find positive real parameters for the extended Debye model (EDM), aimed at condition assessment of insulation systems of power transformers. The EDM can model the slow and fast polarization phenomenon, including relaxation mechanisms with different relaxation times within a composite dielectric material. In the proposed approach, the complex permittivity of the transformer’s composite insulation is approximated via rational functions, as given by the vector fitting (VF) software tool, and the EDM parameters are identified from the obtained poles/residues. To guarantee positive real parameters, i.e., a physically realizable circuit, VF is internally modified to calculate the final residues of the rational approximation via a constrained linear least-squares problem without resorting to further post-processing algorithms, as in existing methods, hence without affecting fitting accuracy. The effectiveness of the parametrized EDM is demonstrated in two ways: (a) by reconstructing frequency domain spectroscopy (FDS) curves provided via measurements in new oil-immersed power transformers and (b) by the comparison of the calculated polarization current given by EDM versus real measurements in time domain. The achieved fitting accuracy in most of the cases is above 99 percent for the reconstructed FDS curves, while the polarization current waveform is reproduced with good agreement.

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

On the period five trichotomy of all positive solutions of xn+1 =( p+ xn−2 )/ xn

A uniform LMI formulation for tuning PID, multi-term fractional-order PID, and Tilt-Integral-Derivative (TID) for integer and fractional-order processes

A program is described which enables performing of genetic algorithms for the determination of two positive real parameters. These new types of procedures are tested on a software of determination of flame temperatures previously developed in a fully classic way. The genetic operators used are crossover and mutation. They perform operations on a binary coded form of the parameters. The goal of the present study consists in developing and optimizing a genetic determination of the parameters at a given temperature. We succeed in selecting the general architecture of the procedure and implementing it in our main software of calculation of flame temperature. We have chosen this pyrotechnic field of application because we knew the behaviour of the real parameters, so the debugging operations were easier.

Homogeneous operators on Hilbert spaces of holomorphic functions

Asymptotic expansion of polyanalytic Bergman kernels

The purpose of the present paper is to carry out a detailed study of a sequence of positive linear operators acting on continuous function spaces on an arbitrary real interval and constructed by means of (Borel) integrated means with respect to two families of probability Borel measures on the underlying interval and a positive real parameter. The study is mainly focused on their approximation properties in weighted spaces of continuous functions with respect to wide classes of weights. Pointwise estimates as well as weighted norm estimates are also established. In the final section a weighted asymptotic formula is obtained.

We consider linear normed spaces of measurable functions dominated by positive measurable function powered by real positive parameter. Also, we consider its dual and predual, and we propose a method for constructing a limit spaces of these functional spaces taken by power parameter. We prove that these limit spaces are (LF)-spaces and also prove that the limit spaces presume the relation of duality, i.e., the limit space of predual spaces is predual for the limit space of dominated functions, and the limit space of duals is dual for it. Also, the limit space of predual spaces is embedded into the limit space of dual spaces.

In this paper, we consider a sequence of operators as a wavelet type extension of univariate generalized Kantorovich operators depending on a positive real parameter given in [3]. We establish quantitative estimates for the rate of convergence of these operators in the continuous functions space and $L^{p}$-spaces in terms of modulus of continuity and $K$-functionals, respectively. Furthermore, some inequalities such as Bernstein-Markov type for continuous functions and variation preservation type property of the operators when the involved function is of bounded variation are provided.

In this paper, we investigate the subelliptic nonlocal Brezis–Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, [Formula: see text] [Formula: see text] where [Formula: see text] is the fractional [Formula: see text]-sub-Laplacian on a stratified Lie group [Formula: see text] with homogeneous dimension [Formula: see text] [Formula: see text] is an open bounded subset of [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] is subelliptic fractional Sobolev critical exponent, [Formula: see text] are real parameters and [Formula: see text] is a lower order perturbation of the critical power [Formula: see text]. Utilizing direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter [Formula: see text] is sufficiently small. Additionally, we examine the problem for [Formula: see text], representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for [Formula: see text] even for the Heisenberg group.

Let X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.