Concert recording 2013-04-27a

[Track 01]. Chromatic fantasia and fugue, BWV 903 / J.S. Bach -- [Track 02]. Sonata in C major, Hob. XVI/50. Allegro ; [Track 03]. Adagio ; [Track 04]. Allegro molto / F.J. Haydn -- [Track 05]. Variations serieuses, op. 54 / Felix Mendelssohn -- [Track 06]. Piano sonata no. 1, op. 22. Allegro marcato ; [Track 07]. Ruvido ed ostinato / Alberto Ginastera

Noetherian π\pi-bases and Telg\'arsky's Conjecture

We investigate Noetherian families. By using a special Noetherian $\pi$-base, we give a result which states that existence of NONEMPTY's 2-tactic in the Banach-Mazur game on a space $X$, BM(X), if NONEMPTY has a winning strategy in $BM(X)$ and $X$ has the special Noetherian $\pi$-base. This result includes one of the Galvin's theorems which is important in this topic. From this result, we prove that for any topological space $X$ if $\pi w(X)\leq \omega_1$ and NONEMPTY has a winning strategy in $BM(X)$, then NONEMPTY has a 2-tactic in $BM(X)$. As a result of this fact, under $\mathsf{CH}$, we show that for any separable $T_3$ space $X$ if NONEMPTY has a winning strategy in $BM(X)$, then NONEMPTY has a 2-tactic in $BM(X)$. So, we prove that Telg{\'a}rsky's conjecture cannot be proven true in the realm of separable $T_3$ spaces, and more generally, in the class of spaces $X$ with $\pi w(X)\leq \omega_1$. We pose some questions about this topic

Physics-informed machine learning in asymptotic homogenization of elliptic equations

We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation's versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements.</p

Reply to Nadler: Spinoza’s Free Person and Wise Person Reconsidered

This article addresses Steven Nadler’s response to my objections to his reading of Spinoza’s free person (homo liber). Nadler argues that there are no clear and significant differences between the free person and the wise person (vir sapiens) in their character or in the role they play in Spinoza’s moral philosophy; in fact, they are one and the same. I begin by critically examining three inferences which Nadler’s reading in part relies on. I then address the differences between the contexts in which Spinoza explicitly invokes the free person and the wise person. I argue that even though there may not be significant differences between the free person and the wise person in terms of their character and comportment, there is still reason to think that the free person plays a particular role in Spinoza’s moral philosophy—one which does not hinge on the attainment of the cognitive and affective excellence represented by the wise person at the end of the Ethics

Physics-informed machine learning in the determination of effective thermomechanical properties

We determine the effective (macroscopic) thermoelastic properties of two-phase composites computationally. To this end, we use a physics-informed neural network (PINN)-mediated first-order two-scale periodic asymptotic homogenization framework. A diffuse interface formulation is used to remedy the lack of differentiability of property tensors at phase interfaces. Considering the reliance on the standard integral solution for the property tensors on only the gradient of the corresponding solutions, the emerging unit cell problems are solved up to a constant. In view of this and the exact imposition of the periodic boundary conditions, it is merely the corresponding differential equation that contributes to minimizing the loss. This way, the requirement of scaling individual loss contributions of different kinds is abolished. The developed framework is applied to a planar thermoelastic composite with a hexagonal unit cell with a circular inclusion by which we show that PINNs work successfully in the solution of the corresponding thermomechanical cell problems and, hence, the determination of corresponding effective properties

Elastostatics of star-polygon tile-based architectured planar lattices

We showed a panoptic view of architectured planar lattices based on star-polygon tilings. Four star-polygon-based lattice sub-families were investigated numerically and experimentally. Finite element-based homogenization allowed computation of Poisson's ratio, elastic modulus, shear modulus, and planar bulk modulus. A comprehensive understanding of the range of properties and micromechanical deformation mechanisms was developed. By adjusting the star angle from $0^\circ$ to the uniqueness limit ($120^\circ$ to $150^\circ$), our results showed an over 250-fold range in elastic modulus, over a 10-fold range in density, and a range of $-0.919$ to $+0.988$ for Poisson's ratio. Additively manufactured lattices showed good agreement in properties. The additive manufacturing procedure for each lattice is available on www.fullcontrol.xyz/#/models/1d3528. Three of the four sub-families exhibited in-plane elastic isotropy. One showed high stiffness with auxeticity at low density with a primarily axial deformation mode as opposed to bending deformation for the other three lattices. The range of achievable properties, demonstrated with property maps, proves the extension of the conventional material-property space. Lattice metamaterials with Triangle-Triangle, Kagome, Hexagonal, Square, Truncated Archimedean, Triangular, and Truncated Hexagonal topologies have been studied in the literature individually. We show that all these structures belong to the presented overarching lattices

Physics-informed machine learning in asymptotic homogenization of elliptic equations

We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation’s versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements