The study counts 23 maximal 1-systems on a torus with 2 punctures.
problem Counting maximal 1-systems on a torus with punctures.
method Defined and analyzed 1-systems, generalized results to surfaces with boundary.
result There are exactly 23 maximal 1-systems on a torus with 2 punctures.
Study on loops on non-orientable surfaces, determining cardinality and order.
problem Determining the cardinality and order of maximal complete 1-systems of loops on non-orientable surfaces.
method Proved the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is 2∣χ∣(∣χ∣+1), and used this to determine the cardinality of maximal complete 1-systems of loops. result Exact cardinality of maximal complete 1-systems of loops on punctured projective planes is determined.
Study shows how learning and analytical models affect reneging and jockeying in a dual M/M/1 system.
problem How do learning and analytical models affect reneging and jockeying in a dual M/M/1 system?
method Analytical and online trained actor-critic models were used to study reneging and jockeying in a dual M/M/1 system.
result Both analytical and online trained actor-critic models yield the same asymptotic limits for reneging and jockeying, but differ in practical sizes.
There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n variables, the so called U/K-system. Let G_{m,n} denote the Grassmannian of n-d…
Unique maximal curve systems found for up to 5 punctures.
problem Finding unique maximal curve systems in punctured projective planes.
method Analyzing mapping class group action on maximal 1-systems of loops. result Maximal 1-system is unique for up to 5 punctures. Considering a lead-time-and price-sensitive demand, we investigate whether a client rejection policy, modeled as M/M/1/K system, can be more profitable than an all-client acceptance policy, modeled as M/M/1 system. We provide analytical insights for the cases with and without holding and penalty costs by comparing M/M/…
33 curves on a 3-genus surface, all intersecting at most once.
problem Finding a saturated system of curves on a surface of genus 3.
method Constructing 33 essential curves pairwise non-homotopic and intersecting at most once.
result The constructed system is saturated, not properly contained in any other system.
New solutions to SU(n+1) Toda system found on compact Riemann surfaces with cone singularities.
problem Solving SU(n+1) Toda system with cone singularities on compact Riemann surfaces.
method Character n-ensembles and toric curves on compact Riemann surfaces.
result Established a correspondence between character n-ensembles and toric solutions to SU(n+1) system with cone singularities.
Study curves in non-orientable surfaces with specific intersection properties.
problem Enumerate and understand curves in non-orientable surfaces with intersection constraints.
method Generalized construction of Malestein-Rivin-Theran to non-orientable surfaces.
result Lower bound for maximum number of curves in generic non-orientable surface.
Quadratic growth of intersecting curves on surfaces resolved.
problem Understanding the largest size of intersecting simple closed curves on surfaces.
method Introduced almost nibs, flowers, and stem systems to analyze curve intersections.
result The size of intersecting curves grows quadratically with the surface's Euler characteristic.
Przytycki has shown that the size Nk(S) of a maximal collection of simple closed curves that pairwise intersect at most k times on a topological surface S grows at most as a polynomial in ∣χ(S)∣ of degree k2+k+1. In this paper, we narrow Przytycki's bounds by showing that $$ \mathcal{N}_{k}(S)…
Let Gr(d,n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X⊂Gr(d,n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this cons…
New approach to disentangle utility from impulse in recommendation systems.
problem Difficulty in inferring user utility from engagement signals.
method Generative model based on self-exciting Hawkes process to infer utility from return probability.
result It is possible to disentangle System-1 and System-2 decision processes to optimize content based on user utility.