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171 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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25 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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53 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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17 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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0answers
60 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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2answers
348 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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0answers
56 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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2answers
208 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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1answer
59 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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1answer
175 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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84 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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107 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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3answers
244 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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0answers
154 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 ...
5
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1answer
396 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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1answer
94 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 ...
2
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2answers
123 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 $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ ...
2
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1answer
328 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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124 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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0answers
138 views

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 ...
4
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1answer
172 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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1answer
220 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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1answer
172 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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1answer
833 views

Schur-Cohn Stability Test.

Where can I find a proof for the Schur-Cohn stability test?
3
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0answers
261 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 ...
2
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1answer
91 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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2answers
731 views

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 ...
3
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1answer
332 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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1answer
204 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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1answer
240 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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2answers
480 views

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 ...