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The spectral p -norm of r -matrices generalizes the spectral 2-norm of 2-matrices. In 1911 Schur gave an upper bound on the spectral 2-norm of 2-matrices, which was extended in 1934 by Hardy, Littlewood, and Pólya to r -matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for 2-matrices. The main result of this paper extends the latter result to r -matrices, thereby improving the result of Hardy, Littlewood, and Pólya. The proof is based on combinatorial concepts like r-partite r-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral p -norm in general. Thus, another application shows that the spectral p -norm and the p -spectral radius of a symmetric nonnegative r -matrix are equal whenever p ≥ r . This result contributes to a classical area of analysis, initiated by Mazur and Orlicz back in 1930. Additionally, a number of bounds are given on the p -spectral radius and the spectral p -norm of r -matrices and r -graphs. [ABSTRACT FROM AUTHOR]

In this note we characterize, within the framework of the theory of finite set, those categories of graphs that are algebraic universal in the sense that every concrete category fully embeds in them. The proof of the characterization is based on the sparse–dense dichotomy and its model theoretic equivalent. [ABSTRACT FROM AUTHOR]

In the Lorentzian product G n × R 1 , we give a comparison theorem between the f -volume of an entire f -maximal graph and the f -volume of the hyperbolic H r + under the condition that the gradient of the function defining the graph is bounded away from 1. This condition comes from an example of non-planar entire f -maximal graph in G n × R 1 and is equivalent to the hyperbolic angle function of the graph being bounded. As a consequence, we obtain a Calabi–Bernstein type theorem for f -maximal graphs in G n × R 1 . [ABSTRACT FROM AUTHOR]

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red–blue edge coloring of an optimal 1-plane graph G partitions the edge set of G into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: ( i ) Every optimal 1-plane graph has a red–blue edge coloring such that the blue subgraph is maximal planar while the red subgraph has vertex degree at most four; this bound on the vertex degree is worst-case optimal. ( ii ) A red–blue edge coloring may not always induce a red forest of bounded vertex degree. Applications of these results to graph augmentation and graph drawing are also discussed. [ABSTRACT FROM AUTHOR]

We present a novel approach for optimizing acoustic parameters using sensitivity analysis for computer-aided design and analysis of architectural models. Our approach builds on recent low-dispersion wave-based acoustic solvers that can accurately compute the pressure field in large models. We present an efficient technique to compute the gradient of the pressure field using automatic differentiation and combine that with a quasi-Newtonian optimization method to automatically compute the optimal material properties. We highlight the performance on many complex CAD models to optimize the strength and clarity acoustic parameters, and thereby improve the acoustic characteristics of large models. To the best of our knowledge, this is the first practical and accurate approach for acoustic material optimization of large indoor CAD models. [ABSTRACT FROM AUTHOR]