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Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

In this paper, the authors present selected information on accelerated reliability tests of special off-road vehicles and their parts. The vehicles were tested in standard operation and on a polygon using accelerated durability tests of the bearings of the vehicles. The paper presents the theoretical basis of accelerated testing of the reliability properties of the parts and the possibilities of calculating the acceleration coefficient. The authors assumed the linearity of the acceleration function, which is proved in practice in most cases. The authors also present the possibility of using a special polygon for accelerated tests. [ABSTRACT FROM AUTHOR]

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last of these, for which the least is known, and in particular on understanding which polytopes are complex psd-minimal. We prove the existence of an obstruction to complex psd-minimality which is efficiently computable via lattice membership problems. Using this tool, we complete the classification of complex psd-minimal polygons (geometrically as well as combinatorially). In dimension three we exhibit several new examples of complex psd-minimal polytopes and apply our obstruction to rule out many others. [ABSTRACT FROM AUTHOR]

We investigate the conformal mapping of a rectangle onto a polygon. Using theta elliptic functions, we represent such a mapping via an integral of the Schwarz–Christoffel type; this representation is convenient for finding the conformal modulus of a quadrilateral or a generalized quadrilateral. We also write the Schwarz–Christoffel formula for a mapping from the triangle with angles , , . We present several examples of mapping from a rectangle. Besides, we give an implicit formula for the conformal modulus of a quadrilateral. [ABSTRACT FROM AUTHOR]

EBSDPolygonizer has catalyzed advancements in microstructural magnetic modelling by enabling the intricate conversion of EBSD data into detailed microstructures with precise grain orientation and boundary data. This paper elucidates the software’s pivotal role in enhancing the prediction and correlation of magnetic anisotropy and permeability with material properties. Demonstrated through substantial research, including that from the University of Warwick, it showcases how EBSDPolygonizer significantly refines the analysis of magnetic anisotropy and its relation to mechanical properties in steels, aligning closely with traditional r-value methods. The tool’s integrative approach, which considers both crystallographic textures and stress effects, heralds a new era in multiphysics simulations that bridge micromechanical and magnetic models. EBSDPolygonizer stands as a potent enabler for future-focused, detailed microstructural modelling, marking a leap forward for material science research and application.