We classify surjective maps that preserve both order and spectrum on positive unbounded self-adjoint operators. As a consequence, we provide a comprehensive description of such maps on the set of all lower semi-bounded self-adjoint operators.
We present a method of finding non-Liouvillian first integrals of rational two-dimensional differential systems. The method is based on the existence of two independent invariants that satisfy a linear second-order ordinary differential equation with respect to one of the variables. We call systems with this property R-integrable. These invariants are not necessarily polynomial; they can be expressed via special functions. We demonstrate that if there exists a rational transformation that maps some Riccati system to a given rational system, then the latter is always R-integrable. We prove that the famous Van der Pol systems and their generalizations are not R-integrable. We find novel integrable polynomial Liénard differential systems without invariant algebraic curves. The related first integrals are expressible via the Airy functions. We prove that the Kudashev system arising in the theory of the Korteweg–de Vries equation is R-integrable.
In the first part, we study different geometric properties of some well-known families of polyharmonic mappings due to Li et al. (J. Math. Anal. Appl., 434, 1462–1473 [28]). In particular, the sufficient coefficient conditions for the starlikeness and convexity, the conditions on the q^th - integral mean, and the bounds on arclength and Bloch constant of a polyharmonic mapping are obtained. The Bohr phenomenon for harmonic and polyharmonic mappings is studied in the next part. We have refined two results of Kayumov et al. (Math. Nachr., 291(11-12), 1757–1768 [25]) and found their polyharmonic analogue. The article concludes with an improved Bohr phenomenon of K-quasiconformal sense-preserving harmonic mappings.
This paper investigates a novel class of generalized fractional Musielak-Sobolev spaces denoted by ℋ^s/2;Ψ _ξ ,y_ϕ . Key properties such as separability and reflexivity are analyzed in detail. Furthermore, continuous and compact embeddings are rigorously established, together with additional functional-analytic properties. As an application, the framework is used to study the existence and multiplicity of solutions to a class of nonlocal generalized problems, focusing in particular on the nonlocal Schrödinger problem with discontinuous nonlinearity.
We introduce the classes B_p,q (φ , ψ ) and ℬ_p,q (φ , ψ ) of pairs (u, v) of weights u and v on ℝ^n and prove two-weight norm estimates between two local generalized Morrey spaces for the multi-dimensional Hardy operator and its adjoint for weights in these classes. Such estimates were earlier known in the one-weight case and for radial weights. The proof is based on point-wise estimates for the Hardy operator and its adjoint via the norm of the domain space. A number of theorems on inclusion of pairs of radial weights into the classed B_p,q (φ , ψ ) and ℬ_p,q (φ , ψ ) is proved. Belonging of pairs of weights to the introduced classes proves to have a form of criterion for two-weight norm estimates in the case of classical Morrey spaces and power weights. We also apply the obtained results to two-weight study of multidimensional versions of Calderón and Stieltjes operators in Morrey spaces.
This paper is devoted to the study of finite time blow-up for initial-boundary value problem to a class of semilinear parabolic equation under the influence of a linear memory term u_t-Δ u+∫ _0^tg(t-s)Δ u(x,s)ds =|u|^p-2u, in Ω× (0, T), where parameter p>2 and Ω is a smooth bounded domain in ℝ^n, n≥ 1, T∈ (0, ∞ ] is the maximal existence time of solution. By virtue of the concavity method, variational method, an improved potential well method involving time variable t and some new differential inequality techniques, we obtain three finite time blow-up results for the problem under different initial energy levels and suitably assumptions on the relaxation function g, we also derive the upper bound estimation for the blow-up time. Regarding the hypotheses on initial data and function g, our findings improve several existing finite time blow-up conclusions.
We construct a finite-support geometric framework in which electromagnetic coupling arises as the affine projection of an admissible interface geometry. The underlying structure is defined by a local admissibility partition on sign-chains of an ordering field, yielding a spectrum with eigenvalues λ _adm and λ _res = 1 - λ _adm . For the electromagnetic channel, this spectrum is fixed by admissibility to λ _adm = 2/3 and λ _res = 1/3 . We show that the electromagnetic interface has intrinsic Hausdorff dimension D_H^EM = 1 + λ _res = 4/3 , and that the low-energy coupling is the Hausdorff density of this interface under inverse-power smoothing at the Thomson scale: α (μ _0) = 1/4π μ _0^-1/3 ℋ^4/3( ℒ_∞ ^EM) . The scale evolution of the coupling follows from the same admissibility spectrum: dα/dlnμ = α ^2/2π λ _adm∑ _f Q_f^2, with λ _adm = 2/3 for the electromagnetic channel. Both the geometric scaling and the coupling structure are determined by a single admissibility partition. Conventional electromagnetic expressions arise as affine readouts of this finite-support geometry.
We study the geometric properties of the mappings having a bounded Dirichlet integral and we continue the new research on mappings satisfying an inverse Poletsky modular inequality. We extend on very general metric measure spaces the theorems of Phragmen-Lindelöf, Iversen-Tsuji and Koebe from the classical complex analysis and our results are new even for analytic mappings.
This paper focuses on the effective dynamics of a class of stochastic parabolic equations with a fast oscillation driven by fractional Brownian motion and Poisson jumps under non-Lipschitz conditions. This model is designed to describe systems that exhibit long-range dependence and emergency impact. In this work, we demonstrate the tightness of the slow component and the ergodicity of the fast component. Furthermore, it is shown that the slow component converges to the solution of the corresponding effective equation. The averaging principle simplifies the computational complexity, enabling us to bypass the intricacies of the original system and instead focus on analyzing the behavior of the effective equation. In contrast to studies that rely on Lipschitz conditions, the results presented here extend beyond these restrictions by considering non-Lipschitz conditions, which are more flexible and widely applicable.
In this paper, we investigate the boundedness and compactness of the inclusion mapping i_d: F_log(p,p-2,s) →𝒯_log , p, s^∞(μ ) . As an application, we establish complete characterizations for the boundedness, compactness, and operator norm of the Volterra integral operators J_g and I_g acting on F_log(p,p-2,s) .
Quantum calculus facilitates new methods for computing and classifying q-special functions. This study establishes some novel q-integral inequalities of Newton and Ostrowski types for several classes of convex functions. These findings illustrate how q-calculus and q-integral inequalities work together, offering new applications in electronics, chemistry and optimization. We prove novel auxiliary results related to quantum differentiable functions fundamental to deriving these improved inequalities. We also provide graphical illustrations of established results for validation using quadratic, trigonometric, exponential, and logarithmic cases.
We develop a coordinate-anchored pointwise theory of fractional tangent functionals on bounded rectangles. By working on the range of the coordinatewise Riemann–Liouville integral, we define anchored fractional operators by exact inversion and prove a representation theorem: every coordinate fractional tangent functional is a scalar multiple of evaluation of the inverse operator at the base point, and the normalized one is unique. At interior points, the resulting fractional tangent space acts faithfully on the common anchored space. We then construct an exterior algebra over the polynomial algebra generated by the fractional coordinate primitives. Its fractional exterior differential defines a nilpotent algebraic chain complex, admits an explicit polynomial Poincaré homotopy, and is related to the ordinary polynomial de Rham complex by a positive diagonal rescaling. Since this fractional differential is not a graded derivation for the ordinary wedge product, the global object is a de Rham-type chain complex. We also prove a rigidity theorem for positive-cone-preserving linear coordinate changes preserving the fractional coframe and formulate a Lorentzian coframe-attached Maxwell-type system, deriving charge conservation and fractional wave equations within the polynomial coefficient class.
This paper investigates the theoretical foundations of Bessel fractional calculus. Distinctions are established between Bessel fractional operators and their classical counterparts. Semigroup theory is developed for the Bessel fractional integral operator, showing that it generates a C_0 -semigroup under specific conditions. Fundamental connections between differential and integral operators are analyzed, with comparisons drawn to classical theory. Furthermore, the structure of the solution space for homogeneous fractional differential equations is characterized. The influence of operator parameters–specifically the order and Bessel index ν on the resulting integral/differential operations is examined.
In this work, we consider an inverse problem for the discontinuous Dirac operator. It is shown that the unknown potential functions can be uniquely determined by a particular set of eigenvalues for an infinite set of different boundary conditions.
This paper is devoted to the study of a particular type of starlike mappings, namely, elastically starlike mappings on the unit ball of complex Banach spaces. We first introduce the concept of elastically starlikeness on the unit ball 𝔹 of a complex Banach space X, starting from the notion of elastically starlikeness in one complex variable, recently introduced by Aron [6]. In this more general setting, we present methods for constructing such mappings, using results for the class ℳ(𝔹) and for starlike mappings. Furthermore, we establish strict inclusions between the classes of α -elastically starlike mappings for different values of the real parameter α , and between these and some classical classes of biholomorphic mappings. A central part of the paper is devoted to obtaining sharp coefficient estimates and Fekete-Szegö type inequalities for the class of elastically starlike mappings. We first consider the case of one complex variable, followed by the general setting of complex Banach spaces. Moreover, using the results obtained in the general case, we derive estimates for particular elastically starlike mappings, and for elastically starlike mappings on the unit polydisc U^n in ℂ^n .
We establish the well-posedness and exponential stability of mild solutions for the Patlak-Keller-Segel-Navier-Stokes system on the real hyperbolic space ℍ^d(ℝ) (d⩾ 2) . We consider the system obtained by coupling a parabolic-elliptic Keller–Segel-type model with a Navier–Stokes-type equation. First, We employ the L^p-L^q estimates for both scalar and vectorial heat semigroups to derive useful interpolation inequalities, including L^p,r-L^q,r estimates and Yamazaki-type estimates on ℍ^d(ℝ) . Using these interpolation estimates, we establish the global well-posedness of mild solutions for the corresponding linear systems in the framework of Lorentz space L^p,r(ℍ^d) for d
Let 0 < q <= p <= r <= infinity and tau is an element of (0, infinity]. In this article, we introduce a local variant MBq,r p,tau of Besov-Bourgain-Morrey spaces M B-center dot (p,tau) (q,r) , whose special case tau = r was originally introduced by J. Bourgain and has proved to play an important role in the study related to the Strichartz estimate and some partial differential equations. These local spaces MBp,tau (q,r) include Bourgain-Lebesgue, local Morrey, and amalgam spaces as special cases. We find the sufficient and necessary conditions, respectively, for their nontriviality, for MB center dot p,tau.(q,r) to be properly contained in MB (p,tau) (q,r) , and also for both the boundedness and the Fefferman-Stein vector-valued maximal inequality about the local Hardy-Littlewood maximal operator on MB (p,tau)(q,r). Moreover, on MBp,tau (q,r) we study their diversity, their duality, and their interpolation in terms of Calder & oacute;n products. Using the Calder & oacute;n product and a new pointwise sparse domination, we obtain the boundedness of local fractional integrals from Calder & oacute;n products to MBp,tau (q,r) . Moreover, we also obtain the boundedness of local Calder & oacute;n-Zygmund operators on MB (p,tau) (q,r ).
We study the behavior near the boundary angular or conical point of weak solu-tions to the Robin problem for an elliptic quasi-linear second-order equation with theanisotropic variable (->)p(x)-Laplacian.
In this paper, we introduce and study a family of multidimensional mixed exponential Kantorovich operators on the hypercube [0,1]^N . The idea is to define a vector of operators where the integral mean replaces the sample values of the function just with respect to a single variable, letting the exponential sampling-type structure for the other N-1 variables. For such operators, we compute the first multidimensional exponential moments and then we prove both pointwise and uniform convergence: the latter result is obtained by means of Korovkin type theorems. Furthermore, we prove a quantitative estimate of the order of approximation in terms of the modulus of continuity of the approximated function.
