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Questions tagged [nonlinear-analysis]

For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.

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2answers
42 views

Nonlinear transformation of region from $\mathbb R^2\to\mathbb R^2$

If I have a given continuous nonlinear map $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$, and a region $D \subset \mathbb{R}^2$, is it necessarily true that $T(\partial D)=\partial T(D)$? That is, do ...
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0answers
36 views

Integration of first order second degree partial differential equation [on hold]

How to evaluate it's integral? $$ \int \left( \frac{dp}{dx} \right)^2 \, dx $$
1
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1answer
48 views

Squared-derivative PDEs

Is there a general theory for equations of the type $ f_y^2 = A(x,y) f_x$? where one first derivative is expressed as a multiple of the other one. Concretely, I'm interested in the equation $$ ( x+...
1
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0answers
27 views

How to do a sensitivity analysis on a non-linear equation?

In the company, it is very difficult to actually do quotations for our customers properly because we do not have perfect information regarding the factors that affect the cost and profit. So I created ...
2
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0answers
97 views

Nonexistance result of elliptic equation

I want to prove that there is no $L^2(\mathbb{R}^N)$-solution of the equation $−\Delta \phi =\lambda \phi$ for every $\lambda \in \mathbb{R}$. I know that the Pohozaev identity asserts for $N\geq 3$, $...
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0answers
17 views

About the Frechet derivative of a functional

How can I compute the Frechet derivative of the functional $$ I(u)=\frac{1}{2}\int_{\Omega} \vert \nabla u\vert ^2\ dx \ + \ \int_{\Omega}\left[1-|u|^2\right]^2\ dx$$ for $u$ in a functional ...
2
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1answer
17 views

How to show uniqueness of the non-degenerate solution of $g(x)=0$?

Let $g(x): \mathbb{R}^n \rightarrow \mathbb{R}^n$. We call a root $x_*$ of $g(x)=0$ non-degenerate if $Jg(x_*)$ is invertible, where $Jg(x_*)$ is the Jacobian at $x_*$. How can we show if $x_*$ is ...
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0answers
23 views

Trajectory of points at infinity

If two points initially starts at zero and travel to infinity,what is the nature of the space if the trajectory where to: 1) converge 2) diverge & what will happen to the trajectory of two points ...
2
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0answers
22 views

A continuity result for functions on a Sobolev space

Let $$W^{1,p}_T = \{u \in W^{1,p}([0,T];\mathbb{R}^N) \mid u(0) = u(T)\},$$ where $W^{1,p}([0,T],\mathbb{R}^n)$ is the usual Sobolev space of functions from $[0,T]$ to $\mathbb{R}^N$. Let $$F:[0,T] \...
2
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0answers
37 views

Is there such branch as Nonlinear Matrix Algebra?

Is there such branch as nonlinear matrix algebra, that researches nonlinear functions with matrix (tensor) valued arguments and outputs where nonlinearities are applied componentwise. Such functions ...
2
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1answer
35 views

How can a function including a sin operation be linearly transformable for any offset?

In the paper "Attention is all you need" the authors have chosen a function to encode the position of a word in a sequence (section 3.5). The following encoding is chosen: $ PE(pos, 2dim) = sin(pos / ...
1
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1answer
15 views

Inner direct sum and nonlinear operators

Suppose we have a not necessarily linear operator $A:X\to X'$, where X is a real Hilbert space that can be written as the inner direct sum $X=V \oplus V^{\perp}$. Consider $u\in X$, so $u$ can be ...
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0answers
45 views

Optimization Methods in Banach Spaces

does anyone know if there's a theory for the following problem: Optimize the task $\begin{align*} T_\phi(\tilde{u})&=\inf\limits_u T_\phi(u)\\ Au&=b\\ u&\in L^p(\Omega),\,\Omega\subset \...
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0answers
13 views

Solving non-linear Leonard Jones potential using approximations

Can you help me solving this problem, I want to find the values of $K$ and $r_{\max}$ in which $f(K)=0$, to simplify the calculations let $a=b=1$ $$f(K) = \int_{0.73}^{r_{\max}}\sqrt{K-\Big(\dfrac{a}{...
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1answer
25 views

Solution of a system of linear equations when matrix is a function of a vector variable

My problem is actually a nonlinear equation: $A(x)x=0$, where x is a vector and matrix A is a function of vector variable x. Are there any works related to this kind of problem??
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0answers
24 views

The application of Gronwall Inequality in $y(t)<\lambda(t)+\rho(t)\int\mu(\tau)y(\tau)d\tau$

Assume $y(t)$ is nonnegative scalar function and: $y(t)\leq k_1 \exp(-\alpha(t-t_0))+\int_{t_0}^t \exp(-\alpha(t-\tau))[k_2y(\tau)+k_3]d\tau$ where $k_1,k_2,k_3$ are all nonnegative and $\alpha>...
3
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1answer
73 views

Solve differential equation $xyy'=x^4+y^4$

How to find general solution to this differential equation (if it exists): $$ xyy'=x^4+y^4 ?$$ I do not know how to even approach it since I never dealt with nonlinear equations. Only thing that I ...
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0answers
23 views

How to find explicit solution for recursive sequences with nonlinear terms?

Are there any general known methodologies or algorithms to derive an explicit equation for a nonlinear sequence? As an example, I am struggling to find an explicit expression for the sequence below: $...
0
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1answer
38 views

Riccati and Linear 1st Order ODE Parallel

I have noticed a certain similarity between Riccati 1st Order ODEs and linear 1st Order ODEs. Specifically, the general solution for each is given by any particular solution plus some function of the ...
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0answers
12 views

Lookup tables inversion

I'm having troubles solving the following problem. I have 2 2D lookup tables of the form $\lambda_1 = f_1(x, y) \\ \lambda_2 = f_2(x, y)$ where $x\in \{x_1, x_2,...,x_N\}$ and $y\in \{y_1, y_2,...,...
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1answer
35 views

Continuity for a nonlinear functional

I'd like to check the continuity for the nonlinear functional $T: (C^{0}([0,1],\Vert \cdot \Vert_{\infty}) \rightarrow (C^{0}([0,1],\Vert \cdot \Vert_{\infty}) $ , with $T(f)(x)=\arctan(f(x))$. I ...
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1answer
23 views

Determining components of Halley's method

I'm working on a homework problem about Halley's method and I'm not quite sure where to start and how I prove a cubic convergence. Consider the iterative method $$x_{n+1} = x_{n} − \frac{h_n}{Ah_n + ...
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1answer
50 views

What algorithm to use to find the non-linear mapping function between 2D shapes generated from biosignals attractors?

I have two biosignals recording the same phenomenon with different methods. Target signal is a reference signal and I would like to find some non-linear mapping from the original signal to the ...
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0answers
18 views

Uniqueness and existence of this system, verifying my answer

I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve. 1)Getting the Jacobian, I obtain $$ J= \begin{bmatrix} 0 & 1\\ -1-2xy & 1-...
-1
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1answer
42 views

Nonlinear equation analysis withe epsilon value [closed]

Consider the nonlinear equation $$\frac{d^2x}{dt^2}+\epsilon\sin(x)=0,~~\epsilon \ll 1\\ x(0)=0,~~\dot{x}(0)=1$$ and find... A. The value of $x_0$ as $\epsilon$ goes to $0$ B. The first order term $...
1
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1answer
41 views

Solve an equation involving the error function

Let $0<a<1$ be given. The equation: $$a = 1 - \frac{2\sqrt{x/\pi}}{\mathrm e^x \mathrm{erf}(\sqrt x)}$$ has a unique root $x$, because the right-hand side is increasing in $x$, and goes to $0,...
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0answers
27 views

boundary problem: use main theorem of monotone operators

I am trying to investigate for which $\alpha \in \mathbb R$ the boundary problem $$-u''(x)+\alpha sin(u(x))u'(x)=f(x)$$ $$u(a) = u(b) = 0$$ is weakly solvable using the main theorem of monotone ...
2
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0answers
29 views

How to prove that a linear differential operator generates a semigroup?

I am working on a nonlinear PDE and I should now prove that the linear operators $L:=\partial_x^3+\partial_x^2$, $S(u):=u L$ and $T(u):=L+S(u)$ are generators of a $C_0$ semigroup such that $\|e^{-s A}...
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0answers
23 views

Fourier Transform of B-splines linear combination, any application to spectral analysis of non stationary series?

In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the ...
1
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1answer
49 views

Transform a non-linear differential equation into a linear equation

Following the work of Yaoji Lu - 1967 (here's a link to the full paper) I got stuck at the step when the author transform a non-linear differential equation into a linear equation (eq. 3.9 pag. 19). ...
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0answers
46 views

How to prove the best polynomial approximation operator is continuous?

For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this ...
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0answers
9 views

Positive and negative eigenvalues via Brouwer degree

The problem is: if $B=B(0,1)\subset\mathbb{R}^n$ is the open unit ball and $f$ is a continuous function defined on the closed unit ball $\overline{B}$ with $0\notin f(\overline{B})$, then there exists ...
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0answers
28 views

Property of continuous functions defined on a sphere of R^n where n is odd

If $B=B(0,1)$ is the unit open ball in $\mathbb{R}^n$ with $n$ odd and $f:\partial B \rightarrow \partial B$ is continuous, then exists $x\in \partial B $ such that $f(x)=x$ or $f(x)=-x$. Any idea of ...
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0answers
34 views

Finding a conjugation given a first integral

In the ODE given by: $x'=X(x)$ , where $X$ is my vectorial field $\in C^1$ in an open subset of $\mathbb R^n$ , If $X$ have $f$ a first integral and $df(p)\neq0$ then there exists a neighborhood $V_{...
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0answers
835 views

How to determine linear terms from the nonlinear dataset?

Let us take the parametric curve r($t$) = [$t^2$;$t$], $t$ = [0,1]. Using this equation, I generate 1000 points. Now my goal is to determine the value of $t$ for each point on the curve without using ...
1
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4answers
99 views

Polynomial growth implies locally Lipschitz?

Let $f:\mathbb{R}^m\to\mathbb{R}^m$ satisfy $\|f(x)\|\le c\|x\|^n$ for some (re-edit:) $n\in\mathbb{N}$ with some constant $c>0$. Is $f$ locally Lipschitz? I see that it is around $x=0$: $\|f(x)-f(...
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0answers
9 views

Preservation of Minima with a Non-Linear Monotonic Mapping

I am trying to make a transformation on the set of parameters within the Ising model namely, $a_i$ and $b_{i,j}$. The Hamiltonian is: $H = \sum_{i} a_ix_i + \sum_i \sum_j b_{i,j}x_ix_j $ They need to ...
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0answers
29 views

Non-linear Basis Functions for PDE

An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not an expert on this subject so I don't know if it's any good. From ...
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0answers
34 views

Hessian Metric and Bregman divergence

I read from a paper that Bregman divergence is an approximation to the Hessian metric when the two points are nearby. What is the definition of Hessian metric? How can we derive this approximation?
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1answer
51 views

Regularity of coefficients in Galerkin method

Let $V\subset H\subset V^*$ be an evolution triple and suppose that $u\in W^{1,2}(0,T;V,H)$, where $$W^{1,2}(0,T;V,H)=\{f\in L^2(0,T,V)\,|\,f'\in L^2(0,T,V^*)\}.$$ Now, let $\{w_1, w_2,...\}$ be a ...
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0answers
25 views

When a nonlinear equation is regular?

I was solving a nonlinear equation in matlab using the function fsolve. Matlab says "the problem appears regular as measured by the gradient". In the definition of problem appears regular it is ...
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0answers
53 views

Solve the nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form $$f(x)=f(0)+D(x)+N(x).$$ Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...
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1answer
22 views

Source of non-linear Laplace equation

Consider the following non-linear generalization of the Laplace equation $$\Delta \phi - \frac{\sum_i (\partial_i \phi)^2}{2 \phi} = 0$$ I am looking for spherically symmetric solutions, so I assume ...
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0answers
52 views

mapping degree of $f: X \rightarrow Y$ with $\dim X \neq \dim Y$

I am writing about the mapping degree (also called Brouwer degree or topological degree). When calculating the degree for a function $f$, one has to use the determinant of the Jacobian of $f$. This ...
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0answers
15 views

Basic example of evolution triple

Let $G$ be a bounded region of $\mathbb{R}^n$ with $n\ge1$. We set $$V=\dot{W}^{m}_{p}(G),\qquad H=L^2(G),$$ with $2\le p<\infty$ and $m\ge 1$. Then "$V\subset H\subset V^*$" is an evolution triple....
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0answers
53 views

On solution of a nonlinear differential inequality

I have the following differential inequality: $$f'(x)\geq cf(x)^a,\quad \forall x\in[0,1]$$ where $0<a<1,\, f(x)\geq 0$. I'm taking the following approach to solve the problem: $$f'(x)\geq cf(x)...
2
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0answers
47 views

To show that a 2D nonlinear ODE undergoes a pitchfork bifurcation

Show that a pitchfork bifurcation occurs when $\mu = 0$ for the system $$\dot{x} = \mu x + xy + 3y^2$$ $$\dot{y} = -2y + x^2 + 2xy^2$$. Attempt at a solution If $\dot{x} = 0$ then $$x = \frac{-3y}{\...
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1answer
35 views

To show that $\dot{x} = -y - x +y^2 - x^2,\; \dot{y} = xy$ has no periodic orbits.

I've tried using index theory, but there's a non-isolated fixed point at $(0, 0)$ (the remaining fixed points at $(0, 1)$ and $(-1, 0)$ are saddles). The terms in the equations have even indices and ...
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0answers
131 views

Proof check: Show that $x_{n + 1} = f(x_n) = 15 - x_n^2$ is a (Smale) horseshoe

A solution is given here: However, I believe it to be incorrect because $f(K_1) = f(K_2) \neq (-\sqrt{15}, \sqrt{15})$. Attempt at a solution: Referring to the diagram from the question... By the ...
1
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4answers
115 views

Example of nonexpansive mapping.

I am trying to construct some examples of the nonexpansive mapping $T$ from $R^2$ to $R^2$ such that $T$ should have fixed points more than one. But I could not construct. Can somebody help me? Please....