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We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the…
Naively trained AI models can be heavily biased. This can be particularly problematic when the biases involve legally or morally protected attributes such as ethnic background, age or gender. Existing solutions to this problem come at the…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame…
We revisit the notion of individual fairness proposed by Dwork et al. A central challenge in operationalizing their approach is the difficulty in eliciting a human specification of a similarity metric. In this paper, we propose an…
Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region $\Omega$ of certain special quadratic functions $f(\bf{E})$ where $\bf{E}(\bf{x})$ derives from a potential $\bf{U}(\bf{x})$.…
The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids,…
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In…
The approximation of a general $d$-variate function $f$ by the shifts $\phi(\cdot-\xi)$, $\xi\in\Xi\subset \Rd$, of a fixed function $\phi$ occurs in many applications such as data fitting, neural networks, and learning theory. When…
By the help of power series f we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f. Utilising these functions we prove some inequalities for the spectral radius of the bounded…
In this paper we define an internal binary operation between functions called in the text \emph{fractal convolution}, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We…
We study the following functional equation that has arisen in the context of mechanical systems invariant under the Poincare algebra: \sum\limits_{i=1}^{n+1}\dfrac{\partial}{\partial x_{i}}\prod\limits_{j\neq i}f(x_{i}-x_{j}) =0,\qquad n…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities.…
The paper is organized as a self-contained literate Prolog program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The complete Prolog code is available at…
The theory of complete generalized Jordan sets is employed to reduce the PDE with the irreversible linear operator $B$ of finite index to the regular problems. It is demonstrated how the question of the choice of boundary conditions is…
Machine learning algorithms are being used in high-stakes decisions, including those in criminal justice, healthcare, credit, and employment. The research community has responded with two largely independent research fields:…
In this paper, we cast fair machine learning as invariant machine learning. We first formulate a version of individual fairness that enforces invariance on certain sensitive sets. We then design a transport-based regularizer that enforces…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…