Studies in Applied Mathematics最新文献

In [Journal of Mathematical Study, 57 (2024): 476–485], the authors studied a sequence of expected score functions arising from an adaptive algorithm motivated by the classical score-based secretary problem. More specifically, they proved that, for each $��n�\in �N�$, the corresponding expected score function admits a unique maximizer, and that the resulting sequence of maximizers is monotonically increasing and diverges to infinity as $��n�\to �\infty �$. In the present paper, we further investigate the asymptotic behavior of this maximizer sequence by establishing both upper and lower bounds for its growth rate.

In this work, we develop an integrable perturbation theory for the defocusing modified Korteweg–de Vries kink solution based on the squared eigenfunction expansion associated with the underlying Zakharov–Shabat scattering problem. We derive the completeness relation for the squared eigenfunctions appropriate to the kink background, establish the adjoint structure needed to handle perturbations of both the continuous and discrete spectral components, and obtain explicit evolution equations for the perturbed kink parameters at leading order. The study of the first-order correction shows that perturbations generically produce a radiative shelf in front of the kink. We also apply our results to certain physically relevant perturbations and show that the predictions are consistent with direct numerical simulations.

The propagation of transverse waves in an anisotropic elastic solid is investigated both in the presence and absence of viscous effects. After discussing in detail the appropriate mathematical models for transversely isotropic viscoelastic materials, we have studied the propagation of bulk shear waves. In particular, we have investigated the break of simple waves and the admissibility of shocks when dissipative effects are negligible, and have provided conditions on the existence of kinks waves (also known as viscous shocks) when dissipation has to be taken into account. Particular attention has been paid also to the propagation of small but finite amplitude waves within the fourth-order weakly nonlinear theory of viscoelasticity.

The classical heat conduction equation in one-space dimension does not have the property of a finite speed of information transfer. Maxwell found an extension of the equation which Cattaneo also obtained by generalizing the Fourier law of heat conduction. In this work, the time delay rate which Cattaneo defined is taken to be dependent on temperature, satisfying a thermodynamic constraint. The resulting generalized nonlinear Cattaneo system is analyzed in two directions. First, in supporting the existence of traveling waves and second by similarity methods, suitable in the study of an initial-value problem. The results show the emergence of wave fronts propagating from higher to lower temperature regions, as well as self-similar profiles describing diffusive-like behavior.

In this paper, we investigate the mean-field derivation of a multispecies nonlocal conservation laws from an $�N$ particle Liouville equation incorporating particle interactions. The convergence is established in a strong sense. First, a vanishing viscosity result is provided to demonstrate the convergence of the nonlocal viscous system to its associated nonlocal conservation laws, including both weak and strong convergence. Achieving strong $�L^1$ convergence in the vanishing viscosity limit requires a sequence of delicate uniform estimates and Helly's compactness argument. Second, the relative entropy method is employed to justify the strong $�L^1$ convergence to the mean field limit in the multispecies setting.

In this paper, we establish a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlevé (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain the noncommutative nonisospectral mixed relativistic Toda lattice and its Lax pair. Then, we perform a stationary reduction on this Lax pair to obtain a matrix dP-type equation. The validity of this reduction is demonstrated through a specific choice of weight function and the application of quasideterminant properties. In the scalar case, our matrix dP equation reduces to the known alternate dP II equation.

In this paper, we derive four new candidates for an intrinsic viscosity operator on an ellipsoid by using the heuristic of the thin shell limit along the scaling direction of the ellipsoid. We show that the general method of the thin shell limit through the asymptotic expansion depends on the averaging method used. We consider both the homogeneous Navier and Hodge boundary conditions. We also obtain a geometric representation of these two boundary conditions.