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Computer-Aided Design (CAD) models are defined by their construction history: a parametric recipe that encodes design intent. However, existing large-scale 3D datasets predominantly consist of boundary representations (B-Reps) or meshes,…
We present a novel efficient implementation of the flexible boundary condition (FBC) method, initially proposed by Sinclair et al., for large single-periodic problems. Efficiency is primarily achieved by constructing a hierarchical matrix…
An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin condition. We introduce the Reconstruction Off-site Data (ROD)…
Boundary Representation (B-Rep) is the widely adopted standard in Computer-Aided Design (CAD) and manufacturing. However, generative modeling of B-Reps remains a formidable challenge due to their inherent heterogeneity as geometric cell…
This paper presents the first systematic study on the fundamental problem of seeking optimal cell average decomposition (OCAD), which arises from constructing efficient high-order bound-preserving (BP) numerical methods within Zhang--Shu…
In 2021, Casares, Colcombet and Fijalkow introduced the Alternating Cycle Decomposition (ACD), a structure used to define optimal transformations of Muller into parity automata and to obtain theoretical results about the possibility of…
We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application,…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous…
The correct treatment of boundary conditions is a key step in the development of the SPH method. The SPH community has to face several challenges in this regard - in particular, a primordial aspect for any boundary formulation is to ensure…
This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with "dirty geometries". FCM, being a fictitious domain approach based on higher order finite…
The finite cell method (FCM) belongs to the class of immersed boundary methods, and combines the fictitious domain approach with high-order approximation, adaptive integration and weak imposition of unfitted Dirichlet boundary conditions.…
Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a…
We introduce a finite-dimensional algebra that controls the possible boundary conditions of a conformal field theory. For theories that are obtained by modding out a Z_2 symmetry (corresponding to a so-called D_odd-type, or half-integer…
To any fixed, finite relational structure, $\mathbb{D}$, there is an associated decision problem, CSP$(\mathbb{D})$, which is a restricted version of the constraint satisfaction problem. In [8], the so called "algebraic approach" to the…
We review the condensation completion of a modular tensor category $\mathcal{C}$, which yields a fusion 2-category $\Sigma\mathcal{C}$ of separable algebras, bimodules over algebras and bimodule maps in $\mathcal{C}$. Physically,…
The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving…
We provide sharp cylindrical parametrizations of cylindrical cell decompositions by maps with bounded $C^{r}$ norm in the sharply o-minimal setting, thus generalizing and strengthening the Yomdin-Gromov Algebraic Lemma. We introduce forts,…
Canonical Polyadic (or CANDECOMP/PARAFAC, CP) decompositions (CPD) are widely applied to analyze high order tensors. Existing CPD methods use alternating least square (ALS) iterations and hence need to unfold tensors to each of the $N$…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…