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In this paper, a probabilistic representation of the tau functions of KP (Kadomtsev-Petviashvili) solitons in terms of stochastic areas will be presented.
The Eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely…
We derive an analytic connection between the screened self-consistent effective potential from density functional theory (DFT) and atomic effective pseudopotentials (AEPs). The motivation to derive AEPs is to address structures with…
The KP-I equation \[ (u_t-2uu_x+\tfrac{1}{2}(\beta-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0 \] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number $\beta>1/3$). This equation admits ---…
The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-$k$-CSPs and Tensor PCA. On the flip side, a SoS lower…
Inspired by the closed contour of momentum conservation in an interaction, we introduce an integrable one-dimensional theory that underlies some integrable models such as the Kadomtsev-Petviashvili (KP)-hierarchy and the amplituhedron. In…
Spectral parameter power series (SPPS) method is a recently introduced technique for solving linear differential equations and related spectral problems. In the present work we develop an approach based on the SPPS for analysis of…
The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the KP hierarchy: by a generalized Jacobi-Trudy formula, and a generalized Giambelli formula. We use the latter formula in the second part to…
We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order…
The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating…
We study the series expansion of the tau function of the BKP hierarchy applying the addition formulae of the BKP hierarchy. Any formal power series can be expanded in terms of Schur functions. It is known that, under the condition…
The transmission eigenvalue problem (TEP) plays a central role in inverse scattering theory. Despite substantial theoretical progress, the numerical solution of direct and inverse TEP in spherically symmetric domains with variable…
The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct…
Ab initio quantum chemistry calculations for systems with large active spaces are notoriously difficult and cannot be successfully tackled by standard methods. In this letter, we generalize a Green's function QM/QM embedding method called…
Okamoto has obtained a sequence of $\tau$-functions for the \PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric…
We show that (i) any constrained polynomial optimization problem (POP) has an equivalent formulation on a variety contained in an Euclidean sphere and (ii) the resulting semidefinite relaxations in the moment-SOS hierarchy have the constant…
The complex absorbing potential (CAP) formalism has been successfully employed in various wavefunction-based methods to study electronic resonance states. In contrast, Green's function-based methods are widely used to compute ionization…
By using pseudo-differential operators containing two derivations, we extend the Kadomtsev-Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we obtain its B\"{a}cklund transformations, bilinear equations…
A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional ``phase space'' variables $(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions…
We study the spectrum of the Markov matrix of the totally asymmetric exclusion process (TASEP) on a one-dimensional periodic lattice at ARBITRARY filling. Although the system does not possess obvious symmetries except translation…