Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Optimization of heterogeneous Catalyst-assisted fatty acid methyl esters biodiesel production from Soybean oil with different Machine learning methods

There is a growing attention to the bio and renewable energies due to fast depletion of fossil fuels as well as the global warming problem. Here, we developed a modeling and simulation method by means of artificial intelligence (AI) for prediction of the bioenergy production from vegetable bean oil. AI methods are well known for prediction of complex and nonlinear process. Three distinct Adaptive Boosted models including Huber regression, LASSO, and Support Vector Regression (SVR) as well as artificial neural network (ANN) were applied in this study to predict actual yield of Fatty acid methyl esters (FAME) production. All boosted utilizing the Adaptive boosting algorithm. The important influencing parameters on the biodiesel production such as the catalyst loading (CAO/Ag, wt%) and methanol to oil (Soybean oil) molar ratio were selected as the input variables of models while the yield of FAME production was selected as output. Model hyper-parameters were tuned to maintain generality while improving prediction accuracy. The models were evaluated using three distinct metrics Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and R². Error rates of 8.16780E-01, 4.43895E-01, 2.06692E + 00, and 3.92713 E-01 were obtained with the MAE metric for boosted Huber, SVR, LASSO and ANN models. On the other hand, the RMSE error of these models were about 1.092E-02, 1.015E-02, 2.669E-02, and 1.01174E-02, respectively. Finally, the R-square score were calculated for boosted Huber, boosted SVR, and boosted LASSO as 0.976, 0.990, 0.872, and 0.99702, respectively. Therefore, it can be concluded that although the boosted SVR and ANN models were better models for prediction of process efficiency in terms of error, but all algorithms had high accuracy. The optimum yield of 83.77% and 81.60% for biodiesel production were observed at optimum operating values from boosted SVR and ANN models, respectively.     

Universality of deep convolutional neural networks

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. Deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the approximation or generalization ability of deep learning methods generated by the network architectures such as deep convolutional neural networks. Here we show that a deep convolutional neural network (CNN) is universal, meaning that it can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. This answers an open question in learning theory. Our quantitative estimate, given tightly in terms of the number of free parameters to be computed, verifies the efficiency of deep CNNs in dealing with large dimensional data. Our study also demonstrates the role of convolutions in deep CNNs.     

Approximation of quantum assemblages

In this paper, we derive a set of projectors on a large Hilbert space which can universally work for approximating quantum assemblages with binary inputs and outputs. The dimension of the Hilbert space depends on the accuracy of the approximation.     

Separation between second price auctions with personalized reserves and the revenue optimal auction

When buyer valuations are drawn IID from a known regular distribution, a second price auction with a symmetric reserve price is the revenue-optimal single-item auction. When this distribution is irregular, we provide the first separation result showing that a second price auction with reserves earns at most 0.778 times the revenue of Myerson’s optimal auction, even when the reserves can be asymmetric. Since the lower bound is 0.745 for i.i.d. buyers, our result is nearly tight.