We study propagation property for one-dimensional nonlinear diffusion equations with convection-absorption, including the prototype model ∂t(u m)− ∂x(|∂xu|p−1∂xu)− μ|∂xu|q−1∂xu+ λu = 0, where m, p, q, k > 0, and n-dimensional simplified variant ∂t(u )−Δp+1u = 0, where Δp+1u = div (|∇u|p−1∇u). Among the conclusions, we make complete classification of the parameters in the first equation to disti...

We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an “effective fluctuation tensor” that we denote by Q. We pro...

We reformulate the integrality property of the Poincaré inner product in the middle dimension, for an arbitrary Poincaré Q-algebra, in classical terms (discriminant and local invariants). When the algebra is 1-connected, we show that this property is the only obstruction to realizing it by a closed manifold, up to dimension 11. We reinterpret a result of Eisenbud and Levine on finite map germs,...

For projective varieties with a certain class of ‘mild’ isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property (i.e. h st ≤ h p+1,q+1 st for p+ q ≤ d− 2, where d is the dimension of the variety). This result fits nicely with a 6-dimensional counterexample of Mustaţă and Pa...

We show that orbital ω-stability is upwards absolute for א0-presented abstract elementary classes satisfying amalgamation and the joint embedding property (each for countable models). We also show that amalgamation does not imply upwards absoluteness of orbital ω-stability by itself. Suppose that k = (K, k) is an abstract elementary class (or AEC; see [1, 8] for a definition), and let (M,a,N) a...

Abstract We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in works by D. Guan and 1st author. Any such manifold $Q$ dimension $2n-2$ is obtained as a finite degree $n^2$ cover some $W_F$, which we call base $Q$. show that algebraic reduction its projective space $n-1$. Besides, give partial classification submanifolds $Q$, describe degen...

We investigate which elliptic PDEs have the property that every viscosity supersolution belongs to a Sobolev space $$W^{1,q}_{loc}(\Omega )$$ , $$\Omega \subseteq \mathbb {R}^n$$ . The asymptotic cone of operator’s sublevel set seems be essential. It turns out much can said if we know how is related dominative p-Laplacian, with $$p = \frac{n-1}{n}q + 1$$ show that, in certain sense, this minima...

We prove the following main theorem of the theory of (r, q)-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions: (1) each internal face is an r-gon, where r ≥ 3 ; (2) the degree of each internal vertex is q , where q ≥ 3 , and the degree of each boundary vertex is at most q and at least 2 . Then it also possesses the following third property: (3) the vertices, ...

We construct an analogue of the Lyndon-HochschildSerre spectral sequence in the context of polynomially bounded cohomology. For G an extension of Q by H, this spectral sequences converges to the polynomially bounded cohomology of G, HP ∗(G). If the extension is a polynomial extension in the sense of Noskov with H and Q isocohomological and Q of type HF∞, the spectral sequence has E 2 -term HP (...

Let Vn(q) be the n-dimensional vector space over the finite field with q elements, and let T1, T2, . . . , Tr+1 be r+1 subspaces of Vn(q) such that Vn(q) = ⊕r+1 i=1 Ti. Assuming that dim(Ti) = 2 for 1 ≤ i ≤ r, let F be the collection of all subspaces S’s of Vn(q) such that S = ⊕r+1 i=1 (S ∩ Ti) and S contains at least one Ti for some 1 ≤ i ≤ r. In this paper we will prove that F is log concave ...