Related papers: Effective counting on translation surfaces

Recently, parametric mappings have emerged as highly effective surface representations, yielding low reconstruction error. In particular, the latest works represent the target shape as an atlas of multiple mappings, which can closely encode…

We prove that every finite subgroup of $GL_{2}(\mathbb{R})$ can be realized as the Veech group of some translation surface.

We prove a general counting result for arcs of the same type in compact surfaces. Wealso count infinite arcs in cusped surfaces and arcs in orbifolds. These theorems are derived from aresult that guarantees the convergence of certain…

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…

Positional encodings are a common component of neural scene reconstruction methods, and provide a way to bias the learning of neural fields towards coarser or finer representations. Current neural surface reconstruction methods use a…

Motivated by the effective bounds of ordinary differential equations, we prove an effective version of uniform bounding for partial differential fields with commuting derivations. More precisely, we provide an upper bound for the size of…

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become…

We give cohomological criteria for logarithmic good reduction of elliptic surfaces up to modification. Along the way, we prove several more general results about such surfaces in positive characteristic, as well as about log smooth…

Transformer model has been widely used on machine translation tasks and obtained state-of-the-art results. In this paper, we report an interesting phenomenon in its encoder-decoder multi-head attention: different attention heads of the…

Given $M_\varphi$, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy $\varphi$ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the…

For any algebraic variety $V$ defined over a number field $k$, and ample height function $H$ on $V$, one can define the counting function $N_V(B) = #{P\in V(k) \mid H(P)\leq B}$. In this paper, we calculate the counting function for Kummer…

Transformers serve as the foundational architecture for large language and video generation models, such as GPT, BERT, SORA and their successors. Empirical studies have demonstrated that real-world data and learning tasks exhibit…

Convolutional Neural Networks (CNNs) are extremely efficient, since they exploit the inherent translation-invariance of natural images. However, translation is just one of a myriad of useful spatial transformations. Can the same efficiency…

In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We also obtain an upper bound to the maximum height that a compact embedded translator in…

We present a method for establishing invariant manifolds for saddle--center fixed points. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs, and does not require rigorous…

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two…

We prove effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.

We present a three-pronged approach to improving Statistical Machine Translation (SMT), building on recent success in the application of neural networks to SMT. First, we propose new features based on neural networks to model various…

Language models can perceive visual properties of text despite receiving only sequences of tokens-we mechanistically investigate how Claude 3.5 Haiku accomplishes one such task: linebreaking in fixed-width text. We find that character…

Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a…