Artificial neural network-based low-dimensional model for spatio-temporally varying cellular flames

In general obtaining a mathematical model from experimental data of a system with spatio-temporal variation is a challenging task. In this article Karhunen-Loéve (KL) decomposition and artificial neural networks (ANN) are used to extract a simple and accurate dynamic model from video data from experiments of two-dimensional flames of a radial extinction mode regime. The KL decomposition is used to identify coherent structures or eigenfunctions of the system. Projections onto these eigenfunctions reduce the data to a small number of time series. The ANN is then used to process these time series. As a result a low-dimensional, nonlinear dynamic model is obtained.     

On semi-R-boundedness and its applications

R-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier's property (α) are required to guaranty the R-boundedness of a relevant set of operators. In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the H^∞-calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions.     

On the Hamilton–Waterloo Problem with Odd Orders

Given nonnegative integers , the Hamilton–Waterloo problem asks for a factorization of the complete graph into α ‐factors and β ‐factors. Without loss of generality, we may assume that . Clearly, v odd, , , and are necessary conditions. To date results have only been found for specific values of m and n. In this paper, we show that for any integers , these necessary conditions are sufficient when v is a multiple of and , except possibly when or 3. For the case where we show sufficiency when with some possible exceptions. We also show that when are odd integers, the lexicographic product of with the empty graph of order n has a factorization into α ‐factors and β ‐factors for every , , with some possible exceptions.     

Sparsity-certifying Graph Decompositions

We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. Ileana Streinu; Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author.     

Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

We show that the stability conditions on the Kuznetsov component of a Gushel–Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of ${\text{GL}}_2^{+}(\mathbb{R})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel–Mukai fourfolds.     

The expansion of a non-ideal gas around a sharp corner for 2-D compressible Euler system

In this paper, we consider the problem of expansion of a non-ideal gas turning around corresponding large or small sharp corner for 2-D compressible Euler system. We focus on extending the results of this problem for polytropic gas to that for a more realistic gas, that is, van der Waals gas. In this case, rarefaction waves, shock wave, fan–jump composite waves, or fan–jump–fan composite waves may be appeared. The flow expands to vacuum state when the inclination angle $\theta$ is small enough, that is, a zone of cavitation will come into being. Otherwise, the flow will arrive at a zone of constant state. The corresponding problem can be transformed into interaction of a planar rarefaction wave with a planar centered wave and interaction of a simple wave with rigid wall, which are actually characteristic boundary value problems for 2-D self-similar Euler system. The estimates of solution are obtained by making use of characteristic analysis, corresponding characteristic decompositions and invariant region of solution. Furthermore, by extending the local solution, and combining with those estimates and hyperbolicity, the existence of global classical solution up to the interface of non-ideal gas with vacuum is obtained.