# Questions tagged [implicit-function-theorem]

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### Function composition inside implicit function equation

For a function $f: R^n \times R^n \times R^n \rightarrow R^n$ analytic in a neighborhood of $(0,0,0)$ such that $f(0,0,0)=0$, I would like to show the existence of a function $\varphi$ analytic on a ...
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### Analytic analogue of implicit functions for differential operators

Let $p\colon \mathbb{R}^2 \to \mathbb{R}$ be a polynomial with a non-vanishing gradient at $p^{-1}(0)$. Then, the implicit function theorem says that $S = \{(x,y) \in \mathbb{R}^2 \mid p(x,y) = 0\}$ ...
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### Second order differentiability of solution operator to nonlinear boundary value problem

I am currently reading the paper [1]. The paper deals with the BVP: \begin{align*} \partial_t u - \Delta u + u(u-a) & = -qu \text{ in } ]0,T\mathclose[ \times \Omega \\[6pt] \partial_{n}u & = ...
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### Implicit function theorem for non $C^1$ mappings

I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
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### Implicit function theorem without uniqueness?

Imagine you are given $f(x,y) := y^2-\sin(x)^2$ and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$. One idea that comes to mind is the ...
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### Recovering the openness of a map from the openness of its scalar projections

Good morning. I have been thinking about the following question for a while without much success, therefore I'm starting to doubt its validity, although I don't have a clear counterexample in mind. ...
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### Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar. It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
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### Maximizing the ratio of multilinear polynomials

Consider two multilinear Polynomials $A(x_1,x_2,x_3,\dotsc,x_n)$ and $B(x_1,x_2,x_3,\dotsc,x_n)$ of $n > 2$ variables $x_i \in \mathbb{R}$ and their ratio \begin{equation*} F(x_1,x_2,x_3,\dotsc,...
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### Inverse and implicit function theorems with domain

I have seen an author use the Implicit Function Theorem for a map whose second partial derivative has a bounded inverse, but is unbounded. The map itself is not defined on an open set, but only on a ...
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### Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$

In financial mathematics, the inverse series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see this post for ...
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### Taylor Series expansion for an implicitely defined family of functions

Can we find a Taylor Series expansion for $y(x)$ implicitly defined by: $$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$ In financial mathematics, the two-additive-factors Model G2++ is commonly used for ...
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### Is there a full-rank map with connected graph and simply connected image that is not injective?

I want to find a continuously differentiable function $F:X\to Y$, where $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$ are open ($n\le m$) with ${\rm rk}\, \frac{\partial F}{\partial x}(x) = n$...
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### Factoring constant rank maps into a submersion and an immersion

Let $X$ and $Z$ be smooth manifolds and $\phi: X \to Z$ a smooth map so that the differential $D \phi$ is everywhere of rank $d$. Is there necessarily a $d$-fold $Y$ so that $\phi$ factors as a ...
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### Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild". For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
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### Does the implicit function theorem hold for discontinuously differentiable functions?

(This was posted on math.SE over 5 days ago and has not been answered, although a comment mentioned a similar question on this site.) Wikipedia's statement of the implicit function theorem requires ...
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### Concavity of the solution of a parametric implicit function

Suppose $F(x,y;k)=f(x,y)+kg(x,y)=0$ uniquely defines the solution $y(x;k)$ for $x\in \mathbb{D}$, a compact domain, and $0\leq k \leq 1$ is a parameter. We know that for $k=0,1$, $y(x;0)$ and $y(x,1)$ ...
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Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it? to clearly express the problem assume that $$z(\mathbf{a,B,c})=\mathop {\inf} ... • 275 4 votes 0 answers 860 views ### A strong form of implicit function theorem (what happens when the derivative is degenerate?) (this can be considered as some ad) Consider the system of equations F(x,y)=0. (Here x, y are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/C^\infty ... • 2,222 1 vote 1 answer 336 views ### Implicit function theorem for boundary value problems I have a nonlinear, two point boundary value problem of the form F(x, y(x), y'(x); \Omega ) = y'' along with some boundary conditions of the form y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega. ... • 229 9 votes 2 answers 1k views ### Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem? From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ... • 2,019 2 votes 2 answers 499 views ### What is the closure of space of polynomials in a dense subspace along with a marked point equal to? EDIT Let \mathbb{C}^{m*} be the space of non zero polynomials of degree at most d in two variables. So an element of this space is essentially$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots f_{0d}...
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Let $V_1, V_2 \rightarrow M$ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section ...