Related papers: A General Implicit/Inverse Function Theorem

We give an explicit implicit function theorem for formal power series that is valid for all fields, which implies in particular Lagrange inversion formula and and Flajolet-Soria coefficient extraction formula known for fields of…

We prove some statements of left- and right-continuous variants of generalized inverses of non-decreasing real functions.

Hadamard's global inverse theorem provides conditions for a function to be globally invertible on Rn. In this note we show that the conditions are robust enough for the conclusion to hold even if we relax the conditions by removing the…

An implicit operation of a class of similar algebras $\mathsf{K}$ is a collection of first order definable partial functions on the members of $\mathsf{K}$ that is globally preserved by homomorphisms. For instance, "taking inverses" can be…

Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…

We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.

We introduce an asymmetric operator of generalised translation, define the generalised modulus of smoothness by its means, and obtain the direct and inverse theorems in approximation theory for it.

Using a new technique involving integration it is possible to find the exact roots of simple functions. In this case, simple functions are defined as smooth functions having an inverse, and that inverse having an antiderivative. This…

We give a natural notion of (non-exact) integral functor in the context of k-linear and graded categories. In this broader sense, we prove that every k-linear and graded functor is integral.

This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory,…

Khabibullin's conjecture deals with two linear integral inequalities for some non-negative continuous function $q(t)$. The integral in the first of these two inequalities converts $q(t)$ into another function of one variable $g(t)$. This…

Sufficient conditions are given for a hard implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when…

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ…

Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical…

We present a theorem about irreducibility of a polynomial that is the resultant of two others polynomials. The proof of this fact is based on the field theory. We also consider the converse theorem and some examples.

After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types. Besides axioms and inference rules with the usual meaning, a generalized inference…

We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.

This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently [2] and developed in [4], [5] and [6]. Their peculiarity is…

This article presents an elementary proof of the Implicit Function Theorem for differentiable maps F(x,y), defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the…

We show that the main results of the expected utility and dual utility theories can be derived in a unified way from two fundamental mathematical ideas: the separation principle of convex analysis, and integral representations of continuous…