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Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of…

We establish global stability for a chemotaxis-growth model with logarithmic sensitivity under dynamic Dirichlet boundary conditions on a 1D domain. We analyze both parabolic-parabolic and parabolic-hyperbolic systems. The key challenge is…

We investigate the Hitchin hyperk\"ahler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as…

The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as $\Gamma$-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness $h$ and…

We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables,…

We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that…

Protein inverse folding is a fundamental problem in bioinformatics, aiming to recover the amino acid sequences from a given protein backbone structure. Despite the success of existing methods, they struggle to fully capture the intricate…

The non-Euclidean geometry of hyperbolic spaces has recently garnered considerable attention in the realm of representation learning. Current endeavors in hyperbolic representation largely presuppose that the underlying hierarchies can be…

We investigate the global well-posedness and large-time dynamics of the pressureless Euler--Monge--Amp\`ere (EMA) system with velocity damping in multidimensions, subject to radially symmetric initial data. We first establish the phenomenon…

We give a lower bound for the widths of the collars of certain short partial pants decomposition of the surface. Then we apply this to obtain upper bounds of the renormalized volume of certain Schottky manifolds in terms of the hyperbolic…

We considered the question of applying the multiscale entanglement renormalization ansatz (MERA) to describe chiral topological phases. We defined a functional for each layer in the MERA, which captures the correlation length. With some…

We study one dimensional models of diatomic molecules where both the electrons and nuclei are treated as quantum particles, going beyond the usual Born-Oppenheimer approximation. The continuous system is approximated by a grid which…

We propose a model reduction technique for parametrized partial differential equations arising from scalar hyperbolic conservation laws. The key idea of the technique is to construct basis functions that are local in parameter and time…

We study expansions near the boundary of solutions to the Dirichlet problem for the constant mean curvature equation in the hyperbolic space. With a characterization of remainders of the expansion by multiple integrals, we establish optimal…

Recent advancements in multi-modal large language models have propelled the development of joint probabilistic models capable of both image understanding and generation. However, we have identified that recent methods suffer from loss of…

While deep learning has achieved remarkable success in medical imaging, the "black-box" nature of backpropagation-based models remains a significant barrier to clinical adoption. To bridge this gap, we propose Aristotelian Rapid Object…

In this paper we derive several (and in many cases sharp) estimates for the $\mathrm{L}^2$-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and…

A micromorphic computational homogenization framework has recently been developed to deal with materials showing long-range correlated interactions, i.e. displaying patterning modes. Typical examples of such materials are elastomeric…

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ be the set of all fixed points induced by the action of…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter at least three and standard module $V$. We introduce two direct sum decompositions of $V$. We call these the displacement decomposition for $\Gamma$ and the split…