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We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including…

The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It states that CNF formulas cannot be analyzed for satisfiability with a speedup over exhaustive search. This hypothesis and its…

We study in this paper the function approximation error of multivariate linear extrapolation. The sharp error bound of linear interpolation already exists in the literature. However, linear extrapolation is used far more often in…

Conformal prediction has emerged as a widely used framework for constructing valid prediction sets in classification and regression tasks. In this work, we extend the split conformal prediction framework to hierarchical classification,…

We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on…

The challenge of delivering efficient explanations is a critical barrier that prevents the adoption of model explanations in real-world applications. Existing approaches often depend on extensive model queries for sample-level explanations…

Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…

Motivated by problems where the response is needed at select localized regions in a large computational domain, we devise a novel finite element discretization that results in exponential convergence at pre-selected points. The two key…

We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the…

Bayesian methods are useful for statistical inference. However, real-world problems can be challenging using Bayesian methods when the data analyst has only limited prior knowledge. In this paper we consider a class of problems, called…

This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove…

In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we…

This summary of the doctoral thesis provides a comprehensive formulation of the Extended Discrete Fourier Transform (EDFT), derived directly from the Fourier integral and its orthogonality properties. The method is obtained by solving…

In this paper we study the cumulative conditional expectation function (CCEF) in the copula context. It is shown how to compute CCEF in terms of the cumulative copula function, this natural representation allows to deduce some useful…

We describe a systematic expansion for full QCD. The leading term in the expansion gives the valence approximation. The expansion reproduces full QCD if an infinite number of higher terms are included.

We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of the work are enumerated as follows: (i) We develop an approximate inversion technique for variable-exponent…

Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…

Despite recent progress, AI still struggles on advanced mathematics. We consider a difficult open problem: How to derive a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic? The CEEF problem does not have known…

Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show…

This paper presents a communication efficient distributed algorithm, $\mathcal{CIRFE}$ of the \emph{consensus}+\emph{innovations} type, to estimate a high-dimensional parameter in a multi-agent network, in which each agent is interested in…