Related papers: On matrix Lax representations for (1+1)-dimensiona…
Machine-learned force fields (MLFFs), especially pre-trained foundation models, are transforming computational materials science by enabling ab initio-level accuracy at molecular dynamics scales. Yet their rapid rise raises a key question:…
We consider (3+1)-dimensional second-order evolutionary PDEs where the unknown $u$ enters only in the form of the 2nd-order partial derivatives. For such equations which possess a Lagrangian, we show that all of them have a symplectic…
In this letter, a definition of the higher dimensional Lax pair for a lower dimensional system which may be a chaotic system is given. A special concrete (2+1)-dimensional Lax pair for a general (1+1)-dimensional three order autonomous…
Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. For any (1+1)-dimensional scalar evolution equation $E$, we…
It is shown that time-harmonic hypersurface motions in various, conformally flat, N-dimensional manifolds admit a multilinear description, dL/dt={ L, M_1, ... , M_{N-2} }, automatically generating infinitely many conserved quantities, as…
In this note, we use give some algebraic applications of a previous result by the author which compares the deformations parameterized by the Maurer-Cartan elements of a differential graded Lie algebra, and a differential graded Lie…
Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…
In this manuscript, we propose matrix- and tensor-oriented methods for the numerical solution of the multidimensional evolutionary space-fractional complex Ginzburg--Landau equation. After a suitable spatial semidiscretization, the…
A $2n$-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux…
Large Language Models (LLMs) have achieved impressive results in Machine Translation (MT). However, careful evaluations by human reveal that the translations produced by LLMs still contain multiple errors. Importantly, feeding back such…
Eigenvalue perturbation theory is applied to justify using complex-valued linear scalar test equations to characterize the stability of implicit-explicit general linear methods (IMEX GLMs) solving autonomous linear ordinary differential…
In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central…
Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation…
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…
Deep metric learning (DML) has received much attention in deep learning due to its wide applications in computer vision. Previous studies have focused on designing complicated losses and hard example mining methods, which are mostly…
Consistency is a key requirement of high-quality translation. It is especially important to adhere to pre-approved terminology and adapt to corrected translations in domain-specific projects. Machine translation (MT) has achieved…
The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R \simeq \mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose…
Multifold observations are common for different data modalities, e.g., a 3D shape can be represented by multi-view images and an image can be described with different captions. Existing cross-modal contrastive representation learning…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…
Logistic equations play a pivotal role in the study of any non linear evolution process exhibiting growth and saturation. The interest for the phenomenology, they rule, goes well beyond physical processes and cover many aspects of ecology,…