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We study the polyregular string-to-string functions, which are certain functions of polynomial output size that can be described using automata and logic. We describe a system of combinators that generates exactly these functions. Unlike…
We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive…
We present a variational approach to a general Lienard-type equation in order to linearize it and, as an example, the Van der Pol oscillator is discussed. The new equation which is almost linear is factorized. The point symmetries of the…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
We introduce a rotation-invariant representation of planar shapes. In particular, this representation encodes shapes as vectors such that the Euclidean distance between them serves as a valid shape distance. For standardized, star-shaped…
A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line…
We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions.…
For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute…
We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a…
For a rational function of several variables with nonnegative imaginary part on the upper poly-half-plane, the matrix representations are obtained.
Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal…
In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic…
We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational…
The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
We study generalized regular bent functions using a representation by bent rectangles, that is, special matrices with restrictions on rows and columns. We describe affine transformations of bent rectangles, propose new biaffine and bilinear…
We present a general approach for collaborative filtering (CF) using spectral regularization to learn linear operators from "users" to the "objects" they rate. Recent low-rank type matrix completion approaches to CF are shown to be special…
We define and characterize the $f$-matrices associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. These generalize the face matrices of simplices and hypercubes. Their generating functions can be…
In this paper, we introduce a new concept, namely $\epsilon$-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the…