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Questions tagged [lipschitz-functions]

For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.

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Uniformly Continuous and locally Lipschitz but not Globally Lipschitz Function on a "Connected but not compact" set

I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and ...
máthēma's user avatar
-1 votes
1 answer
17 views

Composition of asymmetric contraction mappings [closed]

Let $(M,d)$ and $(N,q)$ be metric spaces. The operator $T:M\longrightarrow N$ is contractive in the sense that $q(T(m_1),T(m_2)) \leq c d(m_1, m_2)$ for some $c\in [0,1)$. Similarly, the operator $J:N\...
phil's user avatar
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0 answers
181 views

Sign permanence of locally Lipschitz functions calculated on a sequence

Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I ...
Fra's user avatar
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1 vote
0 answers
23 views

How to conduct error analysis for gradient descent with a function not differentiable everywhere

I am trying to bound the error of a method which uses gradient descent on a function which has the term $\lVert Ax - b \rVert$. The analytical solution of the derivative of $\lVert Ax - b \rVert$ is $\...
beanbanx's user avatar
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1 answer
55 views

Counterexample of function that is hard to find

Is there a continuous function $f:\left [0,\dfrac{1}{2}\right ]\to \mathbb{R}$ for which there is no constant $c>0$ such that: $$|f(x)-f(y)|\leq\dfrac{c}{-\ln(|x-y|)},\ \forall\ x,y\in\left [0,\...
Bogdan's user avatar
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2 votes
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37 views

Lipschitz function on the n-dimensional sphere

I am working through the proof of the concentration of measure phenomenon as in Albiac's Topics in Banach space theory (theorem 13.2.2 in the 2nd edition). We are working with a Lipschitz function $f: ...
abbaaaa's user avatar
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1 answer
39 views

Global existence for quadratic ODE

It is known systems of ODEs with a locally Lipschitz vector field can only have local existence results, as the solutions may blow up in finite time. I wonder if anything can be said on selected ...
Guran Semiotovic's user avatar
1 vote
1 answer
29 views

Lipschitz Continuity of inf mappings

For simplicity, let $\mathcal{B} = \mathcal{C}[[-\tau, 0], \mathbb{R}]$ for some $\tau \geq 0$. I am working with time delay differential equations of the form $$ \dot{x}(t) = F(x_t) = \inf_{\theta \...
Olayo's user avatar
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0 answers
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Prove that a certain function is 2-lipschitzienne

I'm trying to prove the following proposition. Prove that g:E×E→ℝ+, (x,y)→d(x,y) is a 2-lipschitz function. The distance on ℝ+ is the usual distance The distance on E×E is defined as d'((x,y),(x',y'))=...
Idoia's user avatar
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How to bound scalarized gradient difference norm in terms of smoothness in convex optimization?

We know if a convex function is $\mu$-smooth, the following inequality is true: $\| \nabla g (u) - \nabla g(v) \| \leq \mu \|u-v\|$ I want to derive an bound for the following slightly different term ...
randomprime's user avatar
1 vote
0 answers
31 views

Chain rule with Sobolev functions

Let $f: \mathbb{R} \to \mathbb{R} $ be Lipschitz and let $u\in W^{1,p}(\Omega)$, with $1\le p <\infty$ and $\Omega\subset \mathbb{R}^N$ open. Assuming that $f(0)=0$ I have to show that $f\circ u\in ...
Shiva's user avatar
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31 views

Smoothness of a cylinder

I´m trying to check the smoothness property of a cylinder $D:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2\leq 1, z\in [0, H]\}$, but I´m having a problem understanding the definition of Lipschitz continuous ...
oli H.'s user avatar
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-1 votes
1 answer
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Determining the Rectangular Region where the Function Satisfies the Lipschitz Condition [closed]

This is my first time asking a question on Stack Exchange, and I'm currently facing a problem related to an ordinary differential equation. I am trying to determine the rectangular region where the ...
storm Tom's user avatar
2 votes
1 answer
92 views

Relation between $L$-Lipschitz continuity and sequential weak continuity.

Def1: Let $A:\mathcal{H} \to \mathcal{H}$ is $L$-Lipschitz continuous operator, where $\mathcal{H}$ is real Hilbert space, if for all $x,y \in \mathcal{H}$, $\|Ax-Ay\| \leq L \|x-y\|$. Here $L>0$ ...
Watanjeet's user avatar
3 votes
0 answers
128 views

Largest RKHS norm of 1-Lipschitz functions on bounded domain and range

The objective is to find the largest RKHS norm of 1-Lipschitz functions on bounded domain and range: $$\sup_{f \in \mathcal{F}} \langle f, f \rangle_\mathcal{H}$$ The domain is the p-dimensional ...
Rob Romijnders's user avatar
4 votes
1 answer
44 views

How to find examples of $L^p$ converging random variables where a specified non-Lipschitz continuous function does not converge in $L^p$?

Background: I am preparing for a probability theory exam, and am struggling with a particular type of problem. The questions involve showing that if $X_n \xrightarrow{L^p} X$, then it does not ...
FD_bfa's user avatar
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1 vote
1 answer
60 views

$C^1$ convex function on a compact set is Lipschitz continuous

Let $K \subseteq \mathbb{R}^n$ be compact and convex and let $f: K \rightarrow \mathbb{R}$ be a $C^1$ convex function. I wish to use the $C^1$ characterisation of convexity, i.e., $$f(y) \geq f(x) + \...
V. Elizabeth's user avatar
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2 answers
46 views

Showing a certain type of function on $\mathbb R ^d$ is Lipschitz?

I want to prove the following: Assume that $f:\mathbb R ^d \to \mathbb R$ is continuous, convex and $|f(x)|\leq a+b|x|$. Then $f$ is Lipschitz. I thought it would follow immediately from the ...
J.R.'s user avatar
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1 vote
0 answers
21 views

Is there a way to efficiently estimate the maximum singular value of a symmetric, non-constant matrix (given some constraints)?

Given a symmetric n x n matrix - whose entries are polynomial functions (let's say of degree 1 but ideally higher degrees as well) of n variables $x_0 ... x_n$ - is there some established method to ...
ufghd34's user avatar
  • 81
3 votes
1 answer
28 views

How does the largest eigenvalue of symmetric matrix associated with a quadratic form change with dimension n.

This is likely not the smartest question but I just wanted to ask if someone could explain why the Lipschitz constant of a quadratic $x^TAx$ where A has entries randomly generated from the Uniform ...
ufghd34's user avatar
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1 vote
1 answer
82 views

The Medians of Lipschitz Functions on $(X,d,\mu)$ (Existence and Uniqueness)

Let $\varphi:(X,d,\mu)\to \Bbb R$ be a Lipschitz function, where $\mu$ is a probability measure on the metric space $(X,d)$. The median $m_\varphi$ of $\varphi$ is defined as the real number such that ...
stoic-santiago's user avatar
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0 answers
30 views

Please help with literature for Lyapunov stability for non-linear observer

I'm working with an approximation-based observer design for reaction-diffusion PDE, where I apply the Petrov-Galerkin approximation to a non-linear PDE and get the following ODE: $$\dot{\beta} = A\...
Áron Fehér's user avatar
0 votes
1 answer
34 views

Proving a bound on $|\nabla w(0)|$ for a solution to $\Delta w = f(w)$

Let $f \in C_c^{\infty}(\mathbb R)$ with $0 \leq f \leq 1$ on $\mathbb R$. I am trying to prove the following: Suppose $w \in C^\infty(B_3(0))$, $w \geq 0$, and solves $\Delta w = f(w)$ on $B_3(0)$. ...
Luke's user avatar
  • 765
1 vote
0 answers
20 views

Region around previous trajectory which contains optimal solution [closed]

I am currently working on Model Predictive Control. I somehow want to bound the region in which the optimal solution of the optimization problem will lie in. I though that this could maybe be done ...
L208's user avatar
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2 votes
0 answers
17 views

Lipschitz Vector field which acts positively on the gradient

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a Lipschitz function. Let $U_\delta=\{|f|<\delta\}$ be open. I want to show that given any $\epsilon>0$, there is a $\delta>0$ such that there is a ...
stratified's user avatar
1 vote
0 answers
26 views

Product $f=gh$ is Lipschitz, what about $g$?

Given two functions $g,h$, such that their product $f(x)=g(x)h(x)$ is Lipschitz of constant $L_f$. Suppose we know that $h$ is Lipschitz too, what can be said about $g$? From the following (very) ...
user8354084's user avatar
0 votes
0 answers
20 views

Prove that for $f$ a Lipschitz function on $\mathbb{R}^d$, $\overline{\mu}(A) := \int|f^{-1}(y) \cap A|dy$ extends to a Radon measure

Let $f: \mathbb{R}^d \to \mathbb{R}^d$ be an $l$-Lipschitz function, and let $\mathcal{L}$ be the $d$-dimensional Lebesgue measure on $\mathbb{R}^d$. We may verify that the function $y \mapsto |f^{-1}(...
Squirrel-Power's user avatar
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32 views

On Lipschitz Matrix-valued Functions

In the context of matrix-valued functions, i.e. $f:\mathbb{R}^n\rightarrow \mathbb{R}^{m\times n}$, we say that $f$ is Lipschitz if $\exists L>0$, s.t. $$ \|f(x)-f(y)\|_F \leq L\|x-y\|_2 $$ with $\|...
user8354084's user avatar
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0 answers
30 views

Compute the norm of a bilinear operator on the space of Lipschitz functions vanishing at $0$

Let $M=[-1,1]\subset ℝ$. A function $f: M → ℝ$ is a Lipschitz function if there exists a finite non-negative constant $C$ such that $|f(s)-f(t)| ⩽ C\|s-t\|$ for all $s, t ∈ M$. Denote the set of ...
hbghlyj's user avatar
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0 answers
19 views

locally Lipschitz of a epidemiological system

let $$ \left\{\begin{array}{l} d X(t)=F(X(t)), \\ X(0)=X_0 \in \mathbb{R}_{+}^7 . \end{array}\right. $$ where $$ X(t)=\left(X_i(t)\right)_{1 \leqslant i \leqslant 7}=\left(M(t), S(t), H_1(t), H_2(t), ...
mouad lmazini's user avatar
1 vote
1 answer
70 views

Showing $L^2$ - convergence of $\phi_n(X_n)$ to $\phi(X)$ when $\phi_n \to \phi$ and $X_n\to X$

Let $(X_n)$ be a sequence of square integrable random variables converging to $X$ in $L^2$ and $(\phi_n)$ a sequence of smooth functions converging to $\phi$ uniformly on compact sets where $\phi$ is $...
Snildt's user avatar
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1 vote
1 answer
34 views

$f$ is continuous and $|f(x) - f(y)| \leq L|x - y|$ , $\int_{0}^{T} f(x)dx = 0$, then prove $\sup_{x\in[0,T]}|f(x)|\leq \frac{1}{2}LT$.

It's easy to see $\sup_{x\in[0,T]}|f(x)|\leq LT$. But I'm having trouble with the coefficients $ \frac{1}{2}$. Suppose $f_{\max}(x)=f(a)\geq 0, f_{\min}(x)=f(b)\leq 0$, then $$|f(a)-f(b)|=|f(a)|+|f(b)|...
Ychen's user avatar
  • 594
1 vote
0 answers
59 views

Lipschitz weaker than continuity?

In my differential equations class, our Professor gave the following exercise: (1) Suppose that the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies the Lipschitz Condition on the whole ...
rudinable's user avatar
1 vote
0 answers
17 views

About the sigma algebra generated by the Hausdorff measure on $\mathbb R^n$

Let $\mathcal{H}^k$ be the $k-$dimensional Hausdorff measure on $\mathbb R^n$, with $k \in \{1, \ldots n\}$. By Carathéodory's theorem we know that there exists a sigma algebra $\mu(\mathcal{H}^k)$ of ...
Nick_W's user avatar
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0 votes
1 answer
58 views

Give an example of an origin-passing function that is continuous not Lipschitz and its integral on a interval $[0,b]$, $b>0$ is infinite

Does there exist a function $f(x)$ such that it is continuous but not Lipschitz on $[0,b],b>0$ (not Lipschitz near $0$) and $f(0)=0, f(x) >0, \forall x \in (0,b]$. the integral $\int_{0}^{b}\...
Ren De jin's user avatar
2 votes
1 answer
48 views

An integral inequality with lipschitz function

Suppose that $m=\min_{x\in[a,b]}f(x),K=\int_a^b\frac{dx}{f(x)}$, $f(x)>0$ and it satisfies $\forall x,y\in[a,b],|f(x)-f(y)|\le L|x-y|$. Prove that $\int_a^bf(x)dx\le \frac{e^{2LK}-1}{2L}m^2$. I've ...
TaD's user avatar
  • 133
1 vote
0 answers
41 views

Lipschitz approximation of the identity in metric spaces

Let $(X,d)$ be a complete and separable metric space and let $Y \subset X$ be a sigma compact subset. Does there exist a sequence of functions $(T_\epsilon)_{\epsilon>0}$ such that $T_\epsilon: Y \...
Bremen000's user avatar
  • 1,456
4 votes
1 answer
102 views

$1-$Lipschitz function of two variables as limit of separable functions

Let $X,Y$ be compact metric spaces. Suppose $m: X \times Y \to [0,1]$ is $1-$Lipschitz, where $X \times Y$ is given the taxicab metric $d((x,y),(x',y'))=d_X(x,x')+d_Y(y,y')$. Can one write $m$ as a ...
Oddly Asymmetric's user avatar
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0 answers
10 views

Example 1.2 Nonlinear Control Khalil

$f( x) =\begin{bmatrix} x_{2}\\ -sat( x_{1} +x_{2}) \end{bmatrix}$ is not continuously differentiable on $R^2$. Using the fact that the saturdation function sat(.) satisfies $|sat(\eta)-sat{\xi}|$, we ...
SS1's user avatar
  • 79
0 votes
0 answers
22 views

How do we combine sensitivity in differential privacy with Lipschitz continuous?

The definition of sensitivity in differential privacy is: $s_{GS}(f) = max_{D1,D2}\|f(D_1)-f(D2)\|$ for two neighboring datasets $D_1, D_2$ (take global $l_1$ sensitivity as an example here), many ...
Shan Sha's user avatar
1 vote
0 answers
38 views

Upper bound for supremum of Lipschitz function

Given a Lipschitz function $L$, I want to show the following: For $\gamma >0$, let $\chi(\gamma)$ be an equidistant partition on $[0,1]$ with grid length $n^{-\gamma}$ where $n \ge 1$. Then \begin{...
WeakLearner's user avatar
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0 votes
0 answers
18 views

prove existence, uniqueness and non-negativity of the system of ODE

I have an epidemic model with 9 compartments, where all epidemic parameters are non-negative. \begin{equation} \label{model} \begin{split} &\frac{dS}{dt} = \beta N -\delta S-\phi \frac {SI}{...
HRAUNMAKASH's user avatar
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0 answers
40 views

Smooth function times continuous is Lipschitz?

Is a smooth compactly supported function multiplied by a continuous but not necessarily differentiable function already lipschitz continuous? right now Im just interested in the answer. If you know ...
Perelman's user avatar
  • 269
3 votes
1 answer
61 views

If $f$ is Lipschitz and $g \in C^\infty_c(\mathbb{R})$, is $g \circ f$ Lipschitz? [closed]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lipschitz continuous and $g \in C^\infty_c(\mathbb{R})$ (i.e. smooth with compact support). Is the composition $g \circ f$ Lipschitz? I have tried proving ...
CBBAM's user avatar
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0 votes
0 answers
20 views

Fixed point theorem with OSL condition

Use Banach's CMP (Fixed Point Principle) to show existence of a unique fixed point of the $d$-dimensional integral equation $$ x(t) = T[x](t) = x(0) + \int_{0}^{t} f(x(s))\,ds, \quad t \in [0,T] $$ ...
Lelouch's user avatar
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0 answers
22 views

Relation between local Lipschitz continuity constant and Lipschitz smoothness constant of a function.

Using local Lipschitz continuity of $f(\cdot)$: \begin{align} f(\mathbf{a}) &\leq f(\mathbf{b})+ L_0 \lVert{\mathbf{a}-\mathbf{b}}\rVert \end{align} In the FedProx Paper (https://arxiv.org/pdf/...
Nazreen Shah's user avatar
0 votes
1 answer
81 views

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a Lipschitz function, and let $N=f^{-1}(0)$. For a.e. $x \in N$ $Df(x)$ exists and is equal to zero

I am studying measure theory, and especially Lebesgue differentiation theorem. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a Lipschitz function, and let $N=f^{-1}(0)$. Show that for almost every point $...
love and light's user avatar
0 votes
0 answers
30 views

Convergence of $C^2$ functions with fixed point iteration

In Ridgway Scott's Numerical Analysis, the following two problems appear: In the first exercise, there was a clear mention of $x_0$ being sufficiently close to $\alpha$. So I knew that I should be ...
modz's user avatar
  • 101
2 votes
1 answer
38 views

Lipschitzity of the minimal coordinate

I am interested in the smallest coordinate of a vector. Given a positive real vector $x=(x_1,...,x_n)$, I need your help to determine if the function $f(x)= min_{1\leq i \leq n}x_i$ is Lipschitz. If ...
RiezFrechetKolmogorov's user avatar
0 votes
1 answer
28 views

Showing a function $\mathbb R \to \mathbb R^2$ is bilipschitz

Problem: Is the function $f\colon \mathbb R \to \mathbb R^2$, defined by $f(x)=(x, \cos x)$, $M$-bilipschitz for some $M\geq 1$? (Using the standard metric) So far: By the MVT we have $\lvert \cos x -\...
categoricallystupid's user avatar

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