Related papers: Complete Reduction for Derivatives in a Primitive …
The arithmetic derivative is a function from the natural numbers to itself that sends all prime numbers to $1$ and satisfies the Leibniz rule. The arithmetic partial derivative with respect to a prime $p$ is the $p$-th component of the…
The Number Field Sieve and its numerous variants is the best algorithm to compute discrete logarithms in medium and large characteristic finite fields. When the extension degree n is composite and the characteristic p is of medium size, the…
Gradients of neural networks can be computed efficiently for any architecture, but some applications require differential operators with higher time complexity. We describe a family of restricted neural network architectures that allow…
Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
Containers conveniently represent a wide class of inductive data types. Their derivatives compute representations of types of one-hole contexts, useful for implementing tree-traversal algorithms. In the category of containers and cartesian…
Consider a normal function $f$ on the ordinals (i. e. a function $f$ that is strictly increasing and continuous at limit stages). By enumerating the fixed points of $f$ we obtain a faster normal function $f'$, called the derivative of $f$.…
The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems…
This paper introduces the $f$-EI$(\phi)$ algorithm, a novel iterative algorithm which operates on measures and performs $f$-divergence minimisation in a Bayesian framework. We prove that for a rich family of values of $(f,\phi)$ this…
The Artificial Intelligence field seldom address the development of a fundamental building piece: a framework, methodology or algorithm to automatically build hierarchies of abstractions. This is a key requirement in order to build…
In this short note a differential version of the classical Weil descent is established in all characteristics. This yields a ready-to-deploy tool of differential restriction of scalars for differential varieties over finite differential…
Higher derivative corrections are ubiquitous in effective field theories, which seemingly introduces new degrees of freedom at successive order. This is actually an artefact of the implicit local derivative expansion defining effective…
This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate…
We provide a recipe to construct towers of fields producing high order elements in $\mathrm{GF}(q,2^n)$, for odd $q$, and in $\mathrm{GF}(2,2 \cdot 3^n)$, for $n \ge 1$. These towers are obtained recursively by $x_{n}^2 + x_{n} = v(x_{n -…
A meadow is a zero totalised field (0^{-1}=0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational…
This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD)…
We consider $\min\{f(x):g(x) \le 0, ~x\in X\},$ where $X$ is a compact convex subset of $\RR^m$, and $f$ and $g$ are continuous convex functions defined on an open neighbourhood of $X$. We work in the setting of derivative-free…
We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} \phi = P(\phi,\nabla\phi) + f(\phi,\nabla\phi)\xi \] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $\xi$ is an…
Superiorization reduces, not necessarily minimizes, the value of a target function while seeking constraints-compatibility. This is done by taking a solely feasibility-seeking algorithm, analyzing its perturbations resilience, and…
In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional…
We generalize the concept of a number derivative, and examine one particular instance of a deformed number derivative for finite field elements. We find that the derivative is linear when the deformation is a Frobenius map and go on to…