Questions tagged [implicit-function-theorem]

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Full-rank matrix

I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem). $$\left[\begin{array}{cccccccccc} 0 & 1 & 1 & 1 & 0 & 0 & ...
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Bounded derivative of diffeomorphism from implicit function theorem

Let us define $f$ and a set of functions $\{g_n\}$ as follows: $$f:[0,T]\mapsto\mathbb{R}\textrm{ s.t. } f(t)>0,\forall t \in[0,T]$$ $$\{g_n:\mathbb{R}\mapsto\mathbb{R}^+\}\textrm{ a set of ...
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10 votes
2 answers
341 views

Implicit function theorem with continuous dependence on parameter

Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$. Let $f:X\times P\to Y$ be a continuous map such that for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
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1 vote
1 answer
175 views

Does the Implicit Function Theorem in Banach spaces holds if the differential is only one-to-one (not onto!)?

Is the Implicit Function Theorem in the following form correct: Let $V_1,V_2,W$ be Banach spaces, and $Ω⊂V_1×V_2$ an open subset containing $(x_0,y_0)$. Let consider a continuously differentiable map $...
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Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$

I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....
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2 votes
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68 views

Implicit function theorem for stochastic differential equation

For each $\theta\in \mathbb{R}$, we consider a stochastic differential equation (SDE): $$ d X_t =b(t,X_t,\theta)dt+\sigma dW_t,\; t\in [0,T];\quad X_0=x_0\in \mathbb{R}, $$ where $\sigma\ge 0$ and ...
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Arranging the $k$ solutions of $r(z)=te^{i\theta}$ into $k$ continuous functions of $(t,\theta)$

I have originally opened this question on MSE, but I migrated here, since I realized this environment is more suitable. Let $r$ be a rational function, that is, quotient of two coprime polynomials $p,...
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Is there a version of Hamilton's infinite dimensional family of implicit function theorems which gives us a submersion map?

Hamilton, in his notes on "Inverse Function Theorem of Nash and Moser" states a theorem(1.1.3 on Page 172), where a given nonlinear map between tame Frechet spaces is locally surjective, if ...
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3 votes
1 answer
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Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ...
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3 votes
1 answer
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Tangent space of smooth Hilbert submanifolds

Let $X, Y$ be Hilbert spaces and $F:X \rightarrow Y$ smooth. Assume that $M := F^{-1}(0) \subset X$ is a smooth submanifold. Is it true that for any $x\in M$, the tangent space $T_xM$ is a Hilbert ...
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3 votes
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Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
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2 votes
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141 views

Continuity of a constrained parameterized convex optimization problem

Consider the parameterized optimization problem: \begin{align} \boldsymbol{s}(p)= &\arg \min_{ \boldsymbol{x}} \quad g( \boldsymbol{x})\\ \text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...
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1 vote
1 answer
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Is this operator invertible?

Let $T(t)$ be a strongly continuous semi-group on a Banach space $X$, and let $A(\cdot)\in C(0,\tau; \mathcal{L}(X))$ for some $\tau>0$. The operator $G:C(0,\tau;X)\to C(0,\tau;X)$ maps every $h\in ...
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1 answer
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Continuity of a parameterized convex optimization problem

I have a parameterised optimization problem: \begin{align} \boldsymbol{S}(p)= &\arg \min_{ \boldsymbol{x}} g( \boldsymbol{x})\\ \text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \...
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12 votes
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Understanding a certain algebraic set arising in Deep Learning

I'm not a professional geometer. Thanks in advance for your patience. So, let $n$, $k$, $p_0,\ldots,p_{k}$ be positive integers. Let $X$ (resp. $Y$) be an $p_0$-by-$n$ (resp. an $p_{k}$-by-$n$) real ...
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An application of Implicit Function Theorem for Curve Shortening Flow on plane

I'm trying understand the article "Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem" by Ben Andrews and Paul Bryan and they stated that ...
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1 answer
286 views

Continuity of implicitly defined function

Consider a function $g(x)$ defined implicitly via $$\int_x^{x + g(x)} f(\xi) \,d \xi - u(x) = 0. $$ I know that for every $x$ a unique $g(x)$ exists. Furthermore $f$ is locally integrable and $u$ is ...
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3 votes
1 answer
117 views

Continuity in the roots of an algebraic variety with respect to the coordinates

If I'm slightly misusing definitions forgive me I'm not an algebraist. I have $N$ polynomials $f_n(x)$, $n=1,\ldots,N$ where $x\in \mathbb{R}^N$ and the set $\{x:f_n(x)=0\text{ for all }n\}$ is ...
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2 votes
0 answers
194 views

Implicit Function Theorem, parametrized - how can we get uniform domains? (from math.se)

(This question is a duplicate from here) Consider a family of continously differentiable functions $F_r\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (where $r\in[0,1]$). For every parameter $r$, we ...
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Implicit Function Theorem

Let $f$ be a $C^2$ function defined on a neighborhood of $0$ in $\mathbb R^n$ such that $f(0)=0, df(0)\not=0$. By the Implicit Function Theorem, it is easy to get (after a rotation) that near 0 $$ f(x)...
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2 votes
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Roots of a partially holomorphic function

Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\...
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2 votes
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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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2 votes
3 answers
339 views

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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  • 171
1 vote
1 answer
540 views

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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  • 171
3 votes
2 answers
126 views

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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2 votes
0 answers
79 views

Implicit function theorem and selfadjoint operators

Is there a version of the implicit function theorem in some space of functions from $[0,1]$ to a Hilbert space $H$ that contains as a special case the unique solvability of the initial-value problem $\...
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3 votes
0 answers
107 views

A measurable implicit function with differentiated arguments

I have encountered the need for an unusual implicit function theorem, about which I know very little. I would appreciate it if someone could help me with a few pointers. The setup is as follows. Let $...
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13 votes
2 answers
784 views

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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3 votes
0 answers
304 views

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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3 votes
1 answer
2k views

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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3 votes
1 answer
532 views

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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1 vote
1 answer
568 views

a closed form lower bound solution for linear programming

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} ...
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3 votes
0 answers
661 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/$C^\infty$ ...
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1 vote
1 answer
268 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 $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. ...
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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 ...
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2 votes
2 answers
469 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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  • 3,215
1 vote
1 answer
307 views

A version of implicit function theorem when sections are not everywhere smooth?

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 ...
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