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The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post): $ $ \mathcal{A}p=0, \quad p\in C^2(\mathbb{R}\times[0,1]) \tag1$ $ where $ $ \mathcal{A}=-x\partial_y[y(1-y)\cdot]+\frac12 x^2\partial^2_y[y^2(1-y)^2\cdot]+x\partial_x+\frac12\partial^2_x+q(x)\partial_x. \tag2 $ $ Here, $ q(x)=\partial_x\log p_1(x)$ , where $ p_1(x)\propto e^{-x^2}$ . The role of $ p_1(x)$Read more

I want to solve a little bit complicated system of ODEs, where I have to vary initial conditions. My attempt to solution is like this (similar to what I found here Using ndsolve within manipulate): ClearAll[w, x, y, t, k, x0, x1, y0, y1]; k = 2.5; x0 = 5.5828*10^(-5); x1 = 5.5828; y0 =Read more

Let $ (V,0)\subset (\mathbb{C}^n,0)$ , $ n\geq 3$ a (germ of) hypersurface given by $ V =\{f=0\}$ , $ f$ a germ of holomorphic function. A (germ of) holomorphic vector field $ X$ on $ (\mathbb{C}^n,0)$ is “tangent to $ V$ ” if $ X(f) = {\rm d}f(X)$ belongs to the ideal generated by $Read more

May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references? In particular, let $ S$ be a scheme that is torus invariant with finitely many zero and one dimensional orbits. What can you say about the singularity at a fixed point if the number of torusRead more

Let $ f(x,y) \in \mathbb{C}[x,y]$ be a polynomial whose vanishing set $ C:= \{f=0\} \subset \mathbb{C}^2$ is a plane curve singular at the origin (i.e. $ f(0) = f_x(0)=f_y(0)=0$ ). One can attach the following invariants: Tjurina number: $ \tau := dim_{\mathbb{C}} (\mathbb{C}[x,y]/(f,f_x,f_y))$ Milnor number: $ \mu: = dim_{\mathbb{C}} (\mathbb{C}[x,y] /(f_x,f_y))$ Apriori it is notRead more