862 research outputs found

    A note on the divergence-free Jacobian Conjecture in R^2

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    We give a shorter proof to a recent result by Neuberger, in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results presented in Neuberger

    Stationary layered solutions for a system of Allen-Cahn type equations

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    We consider a class of semilinear elliptic system of the form −Δu(x,y)+∇W(u(x,y))=0,(x,y)∈R2-\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in\R^{2} where W:R2→RW:\R^{2}\to\R is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system −q¨(x)+∇W(q(x))=0, x∈R-\ddot q(x)+\nabla W(q(x))=0,\ x\in\R, which connect the two minima of WW as x→±∞x\to\pm\infty has a discrete structure, then the given system has infinitely many layered solutions

    Average resistance of toroidal graphs

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    The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on dd-dimensional toroidal grids and we exploit the connection between resistance and Laplacian eigenvalues. Our analysis provides tight estimates of the average resistance, which are key to study its asymptotic behavior when the number of nodes grows to infinity. In dimension two, the average resistance diverges: in this case, we are able to capture its rate of growth when the sides of the grid grow at different rates. In higher dimensions, the average resistance is bounded uniformly in the number of nodes: in this case, we conjecture that its value is of order 1/d1/d for large dd. We prove this fact for hypercubes and when the side lengths go to infinity.Comment: 24 pages, 6 figures, to appear in SIAM Journal on Control and Optimization (SICON

    A (short) survey on Dominated Splitting

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    We present here the concept of Dominated Splitting and give an account of some important results on its dynamics.Comment: 19 page

    Combustion dynamics in steady compressible flows

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    We study the evolution of a reactive field advected by a one-dimensional compressible velocity field and subject to an ignition-type nonlinearity. In the limit of small molecular diffusivity the problem can be described by a spatially discretized system, and this allows for an efficient numerical simulation. If the initial field profile is supported in a region of size l < lc one has quenching, i.e., flame extinction, where lc is a characteristic length-scale depending on the system parameters (reacting time, molecular diffusivity and velocity field). We derive an expression for lc in terms of these parameters and relate our results to those obtained by other authors for different flow settings.Comment: 6 pages, 5 figure

    Nivat's conjecture holds for sums of two periodic configurations

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    Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps Z2→A\mathbb Z^2 \rightarrow \mathcal A where A\mathcal A is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let Pc(m,n)P_c(m,n) denote the number of distinct m×nm \times n block patterns occurring in a configuration cc. Configurations satisfying Pc(m,n)≤mnP_c(m,n) \leq mn for some m,n∈Nm,n \in \mathbb N are said to have low rectangular complexity. Nivat conjectured that such configurations are necessarily periodic. Recently, Kari and the author showed that low complexity configurations can be decomposed into a sum of periodic configurations. In this paper we show that if there are at most two components, Nivat's conjecture holds. As a corollary we obtain an alternative proof of a result of Cyr and Kra: If there exist m,n∈Nm,n \in \mathbb N such that Pc(m,n)≤mn/2P_c(m,n) \leq mn/2, then cc is periodic. The technique used in this paper combines the algebraic approach of Kari and the author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with proofs. 12 pages + references + appendi
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