For the first time, a nonlinear Schrödinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally generalizes a number of simpler nonlinear partial differential equations encountered in various fields of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. Two- and one-dimensional reductions are described, which reduce the studied nonlinear Schrödinger equation to simpler equations of lower dimension or ordinary differential equations (or systems of ordinary differential equations). In addition to the general Schrödinger equation with two arbitrary functions, related nonlinear partial differential equations are also examined, in which the dispersion function is specified arbitrarily while the potential function is expressed in terms of it. For all considered classes of nonlinear PDEs, using the methods of generalized and functional separation of variables, as well as the semi-inverse approach and the principle of structural analogy of solutions, many new exact solutions have been found, which are expressed in terms of elementary or special functions, or in the form of quadratures. Both Cartesian and polar coordinate systems are employed to analyze the equations under consideration. Special attention is paid to finding solutions with radial symmetry. It is shown that the nonlinear Schrödinger equation, in which the functions defining the potential and dispersion are linearly related (one of these functions can be chosen arbitrarily), can be reduced to a two-dimensional nonlinear PDE that admits exact linearization. The exact solutions obtained in this work can be used as test problems intended for verifying the adequacy and assessing the accuracy of numerical and approximate analytical methods for solving complex nonlinear PDEs of mathematical physics.

We study \(T\)-periodic solutions of cooperative non-autonomous systems of the form \[u'(t)=f(t,u(t))+F(t), \qquad t\in(0,T),\] in the ordered Banach space \(C_{\mathrm{per}}([0,T];\mathbb{R}^{m})\). Using the explicit periodic resolvent kernel \(K_\lambda\) associated with \(u'+\lambda u=g\), we recast the problem as a fixed-point equation \(u=\mathcal{T}u\) and work in a fully specified Carathéodory framework. More precisely, under assumptions (A1)–(A4) on measurability, regularity, cooperativity and local growth, and a structural condition (H\(_\lambda\)) on the diagonal derivatives of \(f\), we define a monotone, completely continuous operator \(\mathcal{T}\) that leaves invariant the order interval generated by a weak \(T\)-periodic sub- and supersolution. A monotone iteration scheme then yields the existence of weak \(T\)-periodic solutions trapped between the barriers, and we prove the existence of extremal (minimal and maximal) periodic solutions in this interval (Theorem 2). Under an additional Lipschitz condition (A5), we obtain a contraction property for \(\mathcal{T}\), which implies uniqueness and order-stability of the periodic orbit (Proposition 1). As an application, we revisit a water–solute cell-volume model with \(T\)-periodic influx and efflux, derive explicit parameter and bounding conditions ensuring the existence of a strictly positive periodic regime (Theorem 3), and illustrate the qualitative behaviour by a numerical simulation.

It is shown that every 3-perfect number in its prime factorization has the exponent of the number 2 which is greater than 1.

The study of approximation theory and the asymptotic behavior of random variables are conventionally predicated on the assumption of classical convergence. Nevertheless, the attainment of classical convergence to a unique limit is frequently impeded in various physical and stochastic processes by measurement errors or inherent system roughness. To mitigate this issue, we introduce the concept of rough asymptotically deferred weighted statistical equivalence of order α in probability. This novel structure generalizes classical asymptotic equivalence through the incorporation of a roughness degree r. We further define the notion of minimal roughness degree and scrutinize the algebraic properties of this new relation such as convexity. Moreover, we establish a rough Korovkin type approximation theorem for sequences of positive linear operators and provide an estimate regarding the rate of convergence. The manuscript concludes by presenting a numerical simulation to visualize our findings which serves to demonstrate strictly stronger generalizations of existing theories.

We introduce \(r\)-Fock space \(\mathscr{F}_{r}\) which generalizes some previously known Hilbert spaces, and study the \(r\)-derivative operator \(\frac{\mbox{d}^r}{\mbox{d}z^r}\) and the multiplication operator by \(z^r\). A general uncertainty inequality of Heisenberg-type is obtained. We also consider the extremal functions for the \(r\)-difference operator \(D_r\) on the space and obtain approximate inversion formulas.

Active lock-in options are a class of complex derivatives characterized by pronounced path dependence and optimal decision making features, and they possess significant application value in the design of structured financial products and risk management. This paper investigates the pricing of active lock-in call options under a stochastic volatility framework. The lock-in decision is formulated as an optimal stopping problem and is further reformulated as a partial differential equation with obstacle constraints. By introducing a linear complementarity problem formulation, the structural properties of the option value function and the optimal lock-in boundary are systematically characterized. From a numerical perspective, an IMEX time discretization scheme is employed to transform the continuous problem into a sequence of time-layered discrete complementarity systems. These systems are efficiently solved using the projected successive over relaxation (PSOR) algorithm. Numerical experiments are conducted to analyze the structural features and economic interpretations of the value function and the associated free boundary surface.

In this paper, we extend the classical logistic law by incorporating autonomously evolving, time-dependent coefficients that allow both the intrinsic growth rate \(\gamma(t)\) and the carrying capacity \(K(t)\) to vary over time according to logistic modulated dynamics. In particular, the carrying capacity is modeled as a logistic process with intrinsic growth rate \(\alpha\) and saturation parameter \(\beta\), yielding an asymptotic level of \(\frac{\alpha}{\beta}\). The objective is to investigate how temporal variability in the governing coefficients influences both transient and asymptotic regimes of the population dynamics and to assess the extent to which the system behavior can be controlled through a reduced set of key parameters. Analytical results are derived in closed form, expressed in terms of hypergeometric functions, and compared with numerical integrations for validation purposes. It is shown that the model admits a long-term equilibrium determined by the ratio \(\frac{\alpha}{\beta}\), independently of the initial population size \(S_0\), while short- and medium-term dynamics are strongly shaped by the interplay between \(S_0\) and the non-autonomous logistic evolution of the carrying capacity \(K(t)\). These results illustrate how analytically tractable non-autonomous logistic models with internally generated coefficient trajectories can enhance the qualitative understanding of population dynamics and provide reliable benchmarks for numerical simulations, with potential applications in sustainable resource management, aquaculture, and ecological modeling.

Some new classes of inverse variational inequalities, which can be viewed as a novel important special case of general variational equalities, are investigated. Projection method, auxiliary principle and dynamical systems coupled with finite difference approach are used to suggest and analyzed a number of new and known numerical techniques for solving inverse variational inequalities. Convergence analysis of these methods is investigated under suitable conditions. One can obtain a number of new classes of inverse variational inequalities by interchanging the role of operators. Some important special cases are highlighted. Several open problems are suggested for future research.