Related papers: On Euler's inequality and automated reasoning with…
Implicit neural representations have emerged as a powerful tool in learning 3D geometry, offering unparalleled advantages over conventional representations like mesh-based methods. A common type of INR implicitly encodes a shape's boundary…
In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting.…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be…
Modeling scene geometry using implicit neural representation has revealed its advantages in accuracy, flexibility, and low memory usage. Previous approaches have demonstrated impressive results using color or depth images but still have…
Capacity control, the bias/variance dilemma, and learning unknown functions from data, are all concerned with identifying effective and consistent fits of unknown geometric loci to random data points. A geometric locus is a curve or surface…
Across machine learning (ML) sub-disciplines, researchers make explicit mathematical assumptions in order to facilitate proof-writing. We note that, specifically in the area of fairness-accuracy trade-off optimization scholarship, similar…
[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
It has been noticed since around 2007 that certain enumeration problems can be solved when an analytic or algebraic curve is identified. This curve is the key to the problem. In these lectures, a few such examples are presented. One is a…
We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the…
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the…
We study the use of the Euler characteristic for multiparameter topological data analysis. Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, including in the context of…
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…
The present survey is devoted to results on Trudinger-Moser inequalities in two dimension. We give a brief overview of the history of these celebrated inequalities and, starting from the geometric problem that motivated Moser's original…
We present a mechanical proof of the Cauchy-Schwarz inequality in ACL2(r) and a formalisation of the necessary mathematics to undertake such a proof. This includes the formalisation of $\mathbb{R}^n$ as an inner product space. We also…
The local Euler obstructions and the Euler characteristics of linear sections with all hyperplanes on a stratified projective variety are key geometric invariants in the study of singularity theory. Despite their importance, in general it…
We study evolutes and involutes of space curves. Although much of the material presented is not new and can be found in classic treatises, we believe that a modern and unified treatment, complemented with several novel observations, may be…
Domain of mathematical logic in computers is dominated by automated theorem provers (ATP) and interactive theorem provers (ITP). Both of these are hard to access by AI from the human-imitation approach: ATPs often use human-unfriendly…
The Euler characteristic (EC) is a powerful topological descriptor that can be used to quantify the shape of data objects that are represented as fields/manifolds. Fast methods for computing the EC are required to enable processing of…