相关论文: Some characterizations of affinely full-dimensiona…
Let $P$ be a finite set of points in $\mathbb{R}^d$ or $\mathbb{C}^d$. We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by $P$ is at least the number of $(d-2)$-flats spanned by $P$. In answering…
We introduce a new class of non-linear models for functional data based on neural networks. Deep learning has been very successful in non-linear modeling, but there has been little work done in the functional data setting. We propose two…
In this paper, using group actions, we introduce a new method for constructing partial geometric designs (sometimes referred to as $1\frac{1}{2}$-designs). Using this new method, we construct several infinite families of partial geometric…
In a physical design problem, the designer chooses values of some physical parameters, within limits, to optimize the resulting field. We focus on the specific case in which each physical design parameter is the ratio of two field…
We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of…
Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the…
We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and…
We construct examples of noetherian three-dimensional local geometrically normal domains of prime characteristic which are $F$-injective but not $F$-full. Along the way, we find examples of two-dimensional local geometrically normal domains…
Airfoil shape design is a classical problem in engineering and manufacturing. In this work, we combine principled physics-based considerations for the shape design problem with modern computational techniques using a data-driven approach.…
We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and…
Deep learning methods are notoriously data-hungry, which requires a large number of labeled samples. Unfortunately, the large amount of interactive sample labeling efforts has dramatically hindered the application of deep learning methods,…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
Among the major difficulties that one may encounter when estimating parameters in a nonlinear regression model are the nonuniqueness of the estimator, its instability with respect to small perturbations of the observations and the presence…
We argue that the theory and practice of diffusion-based generative models are currently unnecessarily convoluted and seek to remedy the situation by presenting a design space that clearly separates the concrete design choices. This lets us…
We consider the multigraded Hilbert scheme corresponding to the Hilbert function of a finite number of points in general position in a smooth projective complex toric variety. We develop several criteria for a point of that parameter space…
In this article we study optimization problems ruled by $\alpha$-fractional diffusion operators with volume constraints. By means of penalization techniques we prove existence of solutions. We also show that every solution is locally of…
Transformers pretrained via next token prediction learn to factor their world into parts, representing these factors in orthogonal subspaces of the residual stream. We formalize two representational hypotheses: (1) a representation in the…
We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic.…
A leading-twist factorization formula is derived for the longitudinal structure function in the x -->1 limit of deeply inelastic scattering. This is achieved by defining a new jet function which is gauge independent and probes the…
This work concerns the design of perfectly conducting objects that are invisible to an incident transverse magnetic plane wave. The object in question is a finite planar waveguide with a finite periodic array of barriers. By optimizing this…