相关论文: Generalized Master Function Approach to Quasi-Exac…

We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…

A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy…

We consider the double sinh-Gordon potential which is a quasi-exactly solvable problem and show that in this case one has two sets of Bender-Dunne orthogonal polynomials . We study in some detail the various properties of these polynomials…

We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases n-th harmonic…

The relationship between the quasi-exactly solvable problems and W-algebras is revealed. This relationship enabled one to formulate a new general method for building multi-dimensional and multi-channel exactly and quasi-exactly solvable…

Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance…

In this article we show that separation of variables for second-order superintegrable systems in two-dimensional Euclidean space generates both exactly solvable (ES) and quasi-exactly solvable (QES) problems in quantum mechanics. In this…

In this paper, the authors propose the utilization of Fibonacci Neural Networks (FNN) for solving arbitrary order differential equations. The FNN architecture comprises input, middle, and output layers, with various degrees of Fibonacci…

The quantum singular value transformation has revolutionised quantum algorithms. By applying a polynomial to an arbitrary matrix, it provides a unifying picture of quantum algorithms. However, polynomials are restricted to definite parity…

We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a…

In our preceding paper, we have proposed an algorithm for obtaining finite-norm solutions of higher-order linear ordinary differential equations of the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial with…

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations,…

Consider the model where we can access a parity function through random uniform labeled examples in the presence of random classification noise. In this paper, we show that approximating the number of relevant variables in the parity…

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ matrices as well as quantized factor algebras of $M_q(n)$ are analyzed. The latter are the quantized function…

The solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of…

Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…

Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…

In a recent preprint by Deutsch et al. [1995] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on $n$ qubits by 2-qubit unitary operations. We address that comment by proving strong lower…