The stability tag has no usage guidance.

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### Constructing Lyapunov Function [on hold]

I have the following system
\begin{align}\frac{dx}{dt}&=x(y^2 +1) +y \\ \frac{dy}{dt} &= x^2y +x\end{align}
I want to try and prove that it has no limit cycles by showing that the system ...

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157 views

### Existense of semi-stable vector bundles on smooth curves in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...

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82 views

### Direct limit of coherent sheaves and semi-stability

Let $R$ be a discrete valuation ring, $\{B_i\}_{i\in I}$ be an inductive system of $R$-algebras of finite type and $B$ the direct limit of the inductive system. Let $X$ be a regular, projective ...

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### Choosing a group action to do GIT of hypersurfaces

When studying GIT stability of hypersurfaces $d$ of $\mathbb P^n$ we look at the Hilbert Scheme $H=\mathbb P^N$ parametrizing homogeneous polynomials $f_d(x_0,\ldots,x_n)$ of degree $d$. There is ...

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60 views

### How to derive explicit bound for the solution of following equation?

Let's have equation
$$
y''(t) + \left(\frac{3}{16t^{2}} + \frac{a}{t} -\frac{b}{t^{\frac{5}{4}}}cos(2t)\right)y(t) = 0, \quad t \in (1, \infty), \quad a, b > 0
$$
How to derive explicit upper bound ...

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31 views

### Mathieu equation instability

Let's have Mathieu equation:
$$
\tag 1 \frac{d^2y(x)}{dx^2} +(A -2q\times cos(2x))y(x) = 0, \quad q\geqslant 1, \quad A > 2q
$$
This case is different from the famous case $q << 1, A> ...

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27 views

### A Statement from Brauer and Nohel's book on stability of time-depending linear systems

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a 2x2 couterexample by Vinograd to the system $y'=A(t)y$ where $A(t)$= \begin{matrix} -1 -9 \cos^2 6t + ...

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78 views

### Nonlinear ODE system: stability

I've got this 4x4 system that should model the wine fermentation process. All the $\mu, K_N, k_d$ etc are positive constants. Of course I have no idea of how to solve it. But at least I would like to ...

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70 views

### Show that 0 is Lyapunov stable by using the given Hamiltonian $H(z)$ as a Lyapunov-function

Good day,
This is my first question, I hope all information is given. If not, feel free to ask. Currently I am reading the paper "Stability of relative equilibria in the problem of N+1 vortices" by ...

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61 views

### Examples of systems with stable equilibria at the boundary of the phase space

Hopfield networks are gradient dynamical systems, used (among other things) to solve combinatorial optimization problems, because stable equilibria are at vertices of the hypercube $[-1,1]^n$. They ...

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175 views

### Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science.
I am interested in proving or obtaining a counterexample to the following conjecture.
Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...

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39 views

### How to analyse the stability of hyperbolic balance laws with diffusion?

Assume we have the following system of balance laws:
$$ U_t+\partial_x F(U,x)=S(U,x)+\partial_x(D(U,x)U_x). $$
Is there any method to analyse the stability of its solution (assume that the solution ...

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276 views

### Difference between Gieseker semistable and slope semistable

Let $X$ be a projective reduced (not necessarily irreducible) curve over an algebraically closed field and $\mathcal{F}$ be a pure coherent sheaf on $X$. Is it true that $\mathcal{F}$ is Gieseker ...

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80 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

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59 views

### Is a general extension of general stable sheaves on $\mathbb P^2$ stable?

Theorem 2 in this paper by Bhosle gives a nice condition on slopes for when a general extension of general stable bundles on curves is stable. Does anyone know whether there is an analogous result for ...

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21 views

### On the stability analysis of a discrete difference system with multiplicative noise

If we assume that
\begin{equation*}
\rho \{\phi \otimes \phi+\psi \otimes \psi\}<1
\end{equation*}
where $\rho$ denotes the spectral radius, then can we verify the following inequality holds
...

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82 views

### How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in ...

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384 views

### Semistability in GIT

If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...

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65 views

### GKS stability of a finite difference scheme

In this paper, I can not reproduce the results obtained equation 62.
I have tried to reproduce it using Wolfram alpha but the results are different.
However, using equation (40) instead of the one ...

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219 views

### Stability of minimal surfaces

Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...

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76 views

### Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:
One factor would be the condition number of the approximation operator. The other factor ...

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199 views

### What are the applications of Grillakis Shatah and Strauss paper?

I am studying the following paper.
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197.
...

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98 views

### Lyapunov stability, nonlinear system

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article.
...

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123 views

### Lyapunov stability of linear system

Consider a linear ODE system
$$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$
Proposition. Suppose that $$\sup_{t\ge ...

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164 views

### Prove that origin is globally exponentially stable with Lyapunov Indirect Method

I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially ...

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451 views

### Derived categories of singular varieties

Given my limited knowledge on derived categories, all the results on derived categories of complex of bounded sheaves are build upon smooth varieties, and people literally avoid singular case (as in ...

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313 views

### Non-linear state-space model system stability using Lyapunov?

I have a non-linear system modelled in state-space as follow:
$$
\mathbf {\dot x} = \mathbf A(x) \mathbf x
$$
I need to find out if this system is stable, so I was thinking in using the Lyapunov ...

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102 views

### Stability of a system of ODEs

It is well known that for a system of ODEs, $\dot{\boldsymbol{y}} = \boldsymbol{Ay}$, the global stable equlibrium point is given by the eigenvector that correponds to the largest eigenvalue of ...

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366 views

### Existence of constant scalar curvature Kahler metrics on projective manifolds

It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler ...

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128 views

### Mumford's vector bundle stability equivalent the notion orbit stability for a G-space?

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class ...

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### Ampleness on the P^1 bundle over Siegel threefold

I am looking at the Shimura variety for $\mathrm{GSp}_4(\mathbb Q)$, with hyperspecial level structure at $p$. Let $X$ denote the special fiber over $\mathbb F_p$. For simplicity, let us pretend ...

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173 views

### How many different states of Nash equilibrium?

So there is this quite well known Prisoner's dilemma where two parties can both defect and cooperate (and get points based on their decisions). In my presently used example I take it that cooperating ...

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131 views

### Stability analysis of ODE disturbed by a random variable [closed]

I have a question concerning the stability analysis for a kind of differential equation taking the form
$$\dot x=Ax+Bw,$$
where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ ...

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187 views

### dp-minimality and stability

What are some of the common popular stable theories that are known to be dp-minimal (or not dp-minimal)?
Some dp-minimal examples I am aware of are strongly minimal theories, superstable theories of ...

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227 views

### In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation
$0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...

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98 views

### Compute roots of sum_i c_i/(a_i + b_i x)^p

How to compute the (real) roots of
$$\sum_{i=1}^n \frac{c_i}{(a_i + b_i \cdot x)^p}$$
for given reals $a_i, b_i, c_i$, and positive integers $n, p$? The cases $p=1, ..., 5$ and $n=6, ..., 20$ would ...

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289 views

### well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...

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370 views

### Higgs bundle and stable bundle

Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X.
I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus.
In particuler, this bundle ...

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210 views

### Nonlinear stability

What is the difference between linear stability and nonlinear stability of numerical schemes for the solution of time dependent PDEs?

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248 views

### Direct sum of two stable bundles of same slope

How to prove that the direct sum of two stable vector bundles of the same slope over a smooth curve is a semistable bundle?

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### What's “bad” about unstable sheaves?

To construct a (coarse or fine) moduli space that is separated, one usually throw away some class of the object in question. For moduli of sheaves people talk about (semi-)stability. A coherent sheaf ...

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### Moduli spaces of vector bundles and stability conditions

Let $C$ be an algebraic curve. One of the easiest examples of stabilty functions is
$$Z:Coh(C)/ \{ 0 \} \rightarrow \overline{\mathbb{H}};\ \ \ \ Z(E):=-deg(E)+i\cdot rk(E).$$
This induces the ...