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We prove an asymptotic formula for the shifted convolution of the divisor functions $d_3(n)$ and $d(n)$, which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates…

Number Theory · Mathematics 2019-09-26 Berke Topacogullari

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the…

Mathematical Physics · Physics 2013-01-22 Giulio Cottone , Mario Di Paola

This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional…

Optimization and Control · Mathematics 2022-04-22 Ashkan Mohammadi

Because the stochastic calculus yields rarely random variables with laws defined by explicit closed formulas, probabilistic numerical computations are done most often by simulation. The simulation by the shift, whose field of application is…

Probability · Mathematics 2007-05-23 Nicolas Bouleau

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial…

Numerical Analysis · Mathematics 2021-11-12 Carolina Vittoria Beccari , Giulio Casciola , Lucia Romani

The singular values of convolutional mappings encode interesting spectral properties, which can be used, e.g., to improve generalization and robustness of convolutional neural networks as well as to facilitate model compression. However,…

Machine Learning · Computer Science 2025-06-09 Antonia van Betteray , Matthias Rottmann , Karsten Kahl

In a seminal paper, Valiant (2006) introduced a computational model for evolution to address the question of complexity that can arise through Darwinian mechanisms. Valiant views evolution as a restricted form of computational learning,…

Machine Learning · Computer Science 2014-04-07 Elaine Angelino , Varun Kanade

Fa\`a di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Fa\`a di Bruno's formula in higher dimensions, providing a…

Combinatorics · Mathematics 2022-10-14 Samuel G. G. Johnston , Joscha Prochno

In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. Some…

Rings and Algebras · Mathematics 2017-12-22 Laurent Veysseire

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…

Computational Complexity · Computer Science 2007-05-23 Luca Trevisan

Boolean functions are mathematical objects used in diverse applications. Different applications also have different requirements, making the research on Boolean functions very active. In the last 30 years, evolutionary algorithms have been…

Neural and Evolutionary Computing · Computer Science 2024-02-16 Claude Carlet , Marko Ðurasevic , Domagoj Jakobovic , Stjepan Picek , Luca Mariot

We discuss and prove a number of results for calculating characteristic cycles, or graded, enriched characteristic cycles. We concentrate particularly on results related to hypersurfaces.

Algebraic Geometry · Mathematics 2016-11-16 David B. Massey

Inspired by two basic mechanisms in animal visual systems, we introduce a feature transform technique that imposes invariance properties in the training of deep neural networks. The resulting algorithm requires less parameter tuning, trains…

Computer Vision and Pattern Recognition · Computer Science 2021-12-10 Chengxi Ye , Xiong Zhou , Tristan McKinney , Yanfeng Liu , Qinggang Zhou , Fedor Zhdanov

We discuss the functional lazy techniques in generation and handling of arbitrarily long sequences of derivatives of numerical expressions in one ``variable''; the domain to which the paper belongs is usually nicknamed ``Automatic…

Programming Languages · Computer Science 2025-01-24 Jerzy Karczmarczuk

Probabilistic circuits (PCs) represent a probability distribution as a computational graph. Enforcing structural properties on these graphs guarantees that several inference scenarios become tractable. Among these properties, structured…

Machine Learning · Computer Science 2020-09-03 Meihua Dang , Antonio Vergari , Guy Van den Broeck

The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…

Optimization and Control · Mathematics 2015-07-21 Boris S. Mordukhovich , M. Ebrahim Sarabi

A unification of characteristic mode decomposition for all method-of-moment formulations of field integral equations describing free-space scattering is derived. The work is based on an algebraic link between impedance and transition…

Classical Physics · Physics 2023-01-04 Mats Gustafsson , Lukas Jelinek , Kurt Schab , Miloslav Capek

Functions are a fundamental object in mathematics, with countless applications to different fields, and are usually classified based on certain properties, given their domains and images. An important property of a real-valued function is…

Quantum Physics · Physics 2024-09-06 Nhat A. Nghiem , Tzu-Chieh Wei

Catamorphisms are functions that are recursively defined on list and trees and, in general, on Algebraic Data Types (ADTs), and are often used to compute suitable abstractions of programs that manipulate ADTs. Examples of catamorphisms…

Logic in Computer Science · Computer Science 2025-02-19 Emanuele De Angelis , Fabio Fioravanti , Alberto Pettorossi , Maurizio Proietti

This paper characterizes the inherent power of evolutionary algorithms. This power depends on the computational properties of the genetic encoding. With some encodings, two parents recombined with a simple crossover operator can sample from…

Neural and Evolutionary Computing · Computer Science 2022-08-04 Elliot Meyerson , Xin Qiu , Risto Miikkulainen
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