We prove the complete monotonicity on $${(0, \infty)^{n}}$$ for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that $${P^{-\beta}}$$ is completely monotone on a convex cone$C$: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of $${P^{-\beta}}$$ for some $${\beta > 0}$$ can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for$P$, and is also related to the Rayleigh property for matroids.

3-D CAD models are an important digital resource in the manufacturing industry. 3-D CAD model retrieval has become a key technology in product lifecycle management enabling the reuse of existing design data. In this paper, we propose a new method to retrieve 3-D CAD models based on 2-D pen-based sketch inputs. Sketching is a common and convenient method for communicating design intent during early stages of product design, e.g., conceptual design. However, converting sketched information into precise 3-D engineering models is cumbersome, and much of this effort can be avoided by reuse of existing data. To achieve this purpose, we present a user-adaptive sketch-based retrieval method in this paper. The contributions of this work are twofold. First, we propose a statistical measure for CAD model retrieval: the measure is based on sketch similarity and accounts for users’ drawing habits. Second, for 3-D CAD models in the database, we propose a sketch generation pipeline that represents each 3-D CAD model by a small yet sufficient set of sketches that are perceptually similar to human drawings. User studies and experiments that demonstrate the effectiveness of the proposed method in the design process are presented.

The objective of this paper is to design novel multi-layer neural network architectures for multiscale simulations of flows taking into account the observed data and physical modeling concepts. Our approaches use deep learning concepts combined with local multiscale model reduction methodologies to predict flow dynamics. Using reduced-order model concepts is important for constructing robust deep learning architectures since the reduced-order models provide fewer degrees of freedom. Flow dynamics can be thought of as multi-layer networks. More precisely, the solution (eg, pressures and saturations) at the time instant n+ 1 depends on the solution at the time instant n+ 1 and input parameters, such as permeability fields, forcing terms, and initial conditions. One can regard the solution as a multi-layer network, where each layer, in general, is a nonlinear forward map and the number of layers relates to the internal time steps. We will rely on rigorous model reduction concepts to define unknowns and connections for each layer. In each layer, our reduced-order models will provide a forward map, which will be modified (" trained") using available data. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Because of the lack of available data, the training will be supplemented with computational data as needed and the interpolation between data-rich and data-deficient models. We will also use deep learning algorithms to train the elements of the reduced model discrete system. We will present main ingredients of our approach and numerical results. Numerical

We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.

In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\R^2\times H^1(\R^3,\R^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\R^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\R^3\\ u,v>0&\hbox{in}\;\R^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\R^3}u^2=a^2\quad\hbox{and}\;\int_{\R^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the case of fixed frequencies $\lambda_1,\lambda_2$. When the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$.