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The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
For interior-point algorithms in linear programming, it is well-known that the selection of the centering parameter is crucial for proving polynomility in theory and for efficiency in practice. However, the selection of the centering…
The use of autonomous vehicles for target localization in modern applications has emphasized their superior efficiency, improved safety, and cost advantages over human-operated methods. For localization tasks, autonomous vehicles can be…
The kernel polynomial method based on Jacobi polynomials $P_n^{\alpha,\beta}(x)$ is proposed. The optimal-resolution positivity-preserving kernels and the corresponding damping factors are obtained. The results provide a generalization of…
When using images to locate objects, there is the problem of correcting for distortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their…
This paper considers the problem of minimizing the ordered weighted average (or ordered median) function of finitely many rational functions over compact semi-algebraic sets. Ordered weighted averages of rational functions are not, in…
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. New one-parameter generating functions for these polynomials are derived. Asymptotical formulas for…
Motivated by previous investigations on the radiative effects of the electric dipoles embedded in structured cavities, localization of electromagnetic waves in two dimensions is studied {\it ab initio} for a system consisting of many…
Special bases of orthogonal polynomials are defined, that are suited to expansions of density and potential perturbations under strict particle number conservation. Particle-hole expansions of the density response to an arbitrary…
Deep networks often exhibit a preference for "simple" solutions, and such a simplicity bias is widely believed to play a key role in generalization. Yet a broadly applicable, quantitative measure of simplicity remains elusive. We introduce…
The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These…
Kernel mean embedding is a useful tool to represent and compare probability measures. Despite its usefulness, kernel mean embedding considers infinite-dimensional features, which are challenging to handle in the context of differentially…
An efficient technique to solve precision problems consists in using exact computations. For geometric predicates, using systematically expensive exact computations can be avoided by the use of filters. The predicate is first evaluated…
In this paper we study the scale-space classification of signals via the maximal set of kernels. We use a geometric approach which arises naturally when we consider parameter variations in scale-space. We derive the Fourier transform…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
Optimization seeks extremal points in a function. When there are superextensively many optima, optimization algorithms are liable to get stuck. Under these conditions, generic algorithms tend to find marginal optima, which have many nearly…
Zernike polynomials are widely used in optics and ophthalmology due to their direct connection to classical optical aberrations. While orthogonal on the unit disk, their application to discrete data or non-circular domains--such as…
A set of orthogonal polynomials on the unit disk $B(0,1)$ known as Zernike polynomials are commonly used in the analysis and evaluation of optical systems. Here Zernike polynomials are used to construct wavelets for polynomial subspaces of…
In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…