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Brittle solids are often toughened by adding a second-phase material. This practice often results in composites with material heterogeneities on the meso scale: large compared to the scale of the process zone but small compared to that of the application. The specific configuration (both geometrical and mechanical) of this mesoscale heterogeneity is generally recognized as important in determining crack propagation and, subsequently, the (effective) toughness of the composite. Here, we systematically investigate how dynamic crack propagation is affected by mesoscale heterogeneities taking the form of an array of inclusions. Using a variational phase-field approach, we compute the apparent crack speed and fracture energy dissipation rate to compare crack propagation under Mode-I loading across different configurations of these inclusions. If fixing the volume fraction of inclusions, matching the inclusion size to the K-dominance zone size gives rise to the best toughening outcome. Conversely, if varying the volume fraction of inclusions, a lower volume fraction configuration can lead to a better toughening outcome if and only if the inclusion size approaches from above the size of the K-dominance zone. Since the size of the K-dominance zone can be estimated \textit{a priori} given an understanding of the application scenario and material availability, we can, in principle, exploit this estimation to design a material's mesoscale heterogeneity that optimally balances the tradeoff between strength and toughness. This paves the way for realizing functional (meta-)materials against crack propagation in extreme environments.

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441{467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution.

A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lam\'{e} coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.

We present here a new splitting method to solve Lyapunov equations in a Kronecker product form. Although this resulting matrix is of order $n^2$, each iteration demands two operations with the matrix $A$: a multiplication of the form $(A-\sigma I) \tilde{B}$ and a inversion of the form $(A-\sigma I)^{-1}\tilde{B}$. We see that for some choice of a parameter the iteration matrix is such that all their eigenvalues are in absolute value less than 1. Moreover we present a theorem that enables us to get a good starting vector for the method.

Many mechanisms behind the evolution of cooperation, such as reciprocity, indirect reciprocity, and altruistic punishment, require group knowledge of individual actions. But what keeps people cooperating when no one is looking? Conformist norm internalization, the tendency to abide by the behavior of the majority of the group, even when it is individually harmful, could be the answer. In this paper, we analyze a world where (1) there is group selection and punishment by indirect reciprocity but (2) many actions (half) go unobserved, and therefore unpunished. Can norm internalization fill this "observation gap" and lead to high levels of cooperation, even when agents may in principle cooperate only when likely to be caught and punished? Specifically, we seek to understand whether adding norm internalization to the strategy space in a public goods game can lead to higher levels of cooperation when both norm internalization and cooperation start out rare. We found the answer to be positive, but, interestingly, not because norm internalizers end up making up a substantial fraction of the population, nor because they cooperate much more than other agent types. Instead, norm internalizers, by polarizing, catalyzing, and stabilizing cooperation, can increase levels of cooperation of other agent types, while only making up a minority of the population themselves.

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result for generalizations of continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and show that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves the way towards uncertainty quantification for signature kernel regression.

We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.

A sequential pattern with negation, or negative sequential pattern, takes the form of a sequential pattern for which the negation symbol may be used in front of some of the pattern's itemsets. Intuitively, such a pattern occurs in a sequence if negated itemsets are absent in the sequence. Recent work has shown that different semantics can be attributed to these pattern forms, and that state-of-the-art algorithms do not extract the same sets of patterns. This raises the important question of the interpretability of sequential pattern with negation. In this study, our focus is on exploring how potential users perceive negation in sequential patterns. Our aim is to determine whether specific semantics are more "intuitive" than others and whether these align with the semantics employed by one or more state-of-the-art algorithms. To achieve this, we designed a questionnaire to reveal the semantics' intuition of each user. This article presents both the design of the questionnaire and an in-depth analysis of the 124 responses obtained. The outcomes indicate that two of the semantics are predominantly intuitive; however, neither of them aligns with the semantics of the primary state-of-the-art algorithms. As a result, we provide recommendations to account for this disparity in the conclusions drawn.

Solving linear systems is of great importance in numerous fields. In particular, circulant systems are especially valuable for efficiently finding numerical solutions to physics-related differential equations. Current quantum algorithms like HHL or variational methods are either resource-intensive or may fail to find a solution. We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems whose non-zero terms are within distance $K$ of the main diagonal. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$, significantly improving over previous convergence guarantees, which require quantum states exponential to $K$. We propose a hybrid quantum-classical algorithm using the Hadamard test and the quantum Fourier transform as subroutines and show its PromiseBQP-hardness. Additionally, we introduce a quantum-inspired algorithm with similar performance given sample and query access. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.

A method of numerically solving the Maxwell equations is considered for modeling harmonic electromagnetic fields. The vector finite element method makes it possible to obtain a physically consistent discretization of the differential equations. However, solving large systems of linear algebraic equations with indefinite ill-conditioned matrices is a challenge. The high order of the matrices limits the capabilities of the Gaussian method to solve such systems, since this requires large RAM and much calculation. To reduce these requirements, an iterative preconditioned algorithm based on integral Laguerre transform in time is used. This approach allows using multigrid algorithms and, as a result, needs less RAM compared to the direct methods of solving systems of linear algebraic equations.