Related papers: Lexicographic Generation of Projective Spaces

Conditional neural text generation models generate high-quality outputs, but often concentrate around a mode when what we really want is a diverse set of options. We present a search algorithm to construct lattices encoding a massive number…

Recent empirical evidence shows that LLM representations encode human-interpretable concepts. Nevertheless, the mechanisms by which these representations emerge remain largely unexplored. To shed further light on this, we introduce a novel…

We give a geometric characterisation of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line…

Despite great advances in computation, materials design is still science fiction. The construction of structure-property relations on the quantum scale will turn computational empiricism into true design.

There is given the program realization system "lexicon" by the using lexicographical procedure.

Direct prediction of material properties from microstructures through statistical models has shown to be a potential approach to accelerating computational material design with large design spaces. However, statistical modeling of highly…

We present an approximation to topological spaces by {\it noncommutative} lattices. This approximation has a deep physical flavour based on the impossibility to fully localize particles in any position measurement. The original space being…

Volumetric design is the first and critical step for professional building design, where architects not only depict the rough 3D geometry of the building but also specify the programs to form a 2D layout on each floor. Though 2D layout…

One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper…

For a vector space V of homogeneous forms of the same degree in a polynomial ring, we investigate what can be said about the generic initial ideal of the ideal generated by V, from the form of the generic initial space gin(V) for the revlex…

It can be difficult to tell whether a trained generative model has learned to generate novel examples or has simply memorized a specific set of outputs. In published work, it is common to attempt to address this visually, for example by…

Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…

By virtue of linguistic compositionality, few syntactic rules and a finite lexicon can generate an unbounded number of sentences. That is, language, though seemingly high-dimensional, can be explained using relatively few degrees of…

The paper contains two natural constructions of extreme hyperspace selections generated by special ordinal decompositions of the underlying space. These constructions are very efficient not only in simplifying arguments but also in…

Writers generally rely on plans or sketches to write long stories, but most current language models generate word by word from left to right. We explore coarse-to-fine models for creating narrative texts of several hundred words, and…

This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several classes of such objects using such…

We automatically verify the crucial steps in the original proof of correctness of an algorithm which, given a geometric graph satisfying certain additional properties removes edges in a systematic way for producing a connected graph in…

One of the best things about geometry is that it's cool! Geometry enables us to create incredible designs and astounding patterns. This article shows how to use a simple technique (iteration) to create designs that are both cool and…

We study pseudorandomness and pseudorandom generators from the perspective of logical definability. Building on results from ordinary derandomization and finite model theory, we show that it is possible to deterministically construct, in…

We study (strong) first countability of locally solid convergence structures on Archimedean vector lattices. Among other results, we characterise those vector lattices for which relatively unform-, order-, and $\sigma$-order convergence,…