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In this paper, we propose a computationally valid and theoretically justified methods, the likelihood ratio scan method (LRSM), for estimating multiple change-points in a piecewise stationary generalized conditional integer-valued…
We present a novel numerical method for solving the elliptic partial differential equation problem for the electrostatic potential with piecewise constant conductivity. We employ an integral equation approach for which we derive a system of…
The differential transformation method (DTM) enables the easy construction of a power-series solution to a nonlinear differential equation. The exponentiation operation has not been specifically addressed in the DTM literature, and…
Recently, a novel method based on coding partitions [1]-[4] has been used to derive power series expansions to previously intractable problems. In this method the coefficients at $k$ are determined by summing the contributions made by each…
We provide algorithms computing power series solutions of a large class of differential or $q$-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.
In this contribution we consider localized, robust and efficient a-posteriori error estimation of the localized reduced basis multi-scale (LRBMS) method for parametric elliptic problems with possibly heterogeneous diffusion coefficient. The…
The Process Reward Model (PRM) plays a crucial role in mathematical reasoning tasks, requiring high-quality supervised process data. However, we observe that reasoning steps generated by Large Language Models (LLMs) often fail to exhibit…
Spectral parameter power series (SPPS) representations for solutions of Sturm-Liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems. They are based on a computation of…
Hypercomplex signal processing (HSP) offers powerful tools for analyzing and processing multidimensional signals by explicitly exploiting inter-dimensional correlations through Clifford algebra. In recent years, hypercomplex formulations of…
The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform.…
This paper deals the implementation of \emph{homotopy perturbation transform method} (HPTM) for numerical computation of initial valued autonomous system of time-fractional partial differential equations (TFPDEs) with proportional delay,…
A representation in the form of spectral parameter power series (SPPS) is given for a general solution of a one dimension Dirac system containing arbitrary matrix coefficient at the spectral parameter, \[ B \frac{dY}{dx} + P(x)Y = \lambda…
This work introduces an ensemble parameter estimation framework that enables the Lumped Parameter Linear Superposition (LPLSP) method to generate reduced order thermal models from a single transient dataset. Unlike earlier implementations…
Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs)…
The random feature method (RFM) has demonstrated great potential in bridging traditional numerical methods and machine learning techniques for solving partial differential equations (PDEs). It retains the advantages of mesh-free approaches…
Digital System Research has pioneered the mathematics and design for a new class of computing machine using residue numbers. Unlike prior art, the new breakthrough provides methods and apparatus for general purpose computation using several…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
A form of the Laplace transform is reviewed as a paradigm for an entire class of fractional functional transforms. Various of its properties are discussed. Such transformations should be useful in application to differential/integral…
Meshless methods are often used in numerical simulations of systems of partial differential equations (PDEs), particularly those which involve complex geometries or free surfaces. Here we present a novel compact scheme based on the local…
We present a novel technique for constructing differentiable order-type operations, including soft ranking, soft top-k selection, and soft permutations. Our approach leverages an efficient closed-form formula for the inverse of the function…