Clarifies definitions of global hyperbolicity in various spaces.
problem Clarifying terminology in recent literature on global hyperbolicity.
method Comparing definitions in Lorentzian length spaces, optimal transport, and topological preordered spaces.
result The causal relation is a closed order and preserves compactness in all cases.
The paper finds maximal metrics on Euclidean spaces.
problem Finding maximal elements in moduli spaces of Riemannian metrics.
method Defining a preorder on moduli space by isometry groups and identifying maximal elements.
result Constructs many examples of maximal metrics on Euclidean spaces.
The paper defines a preorder on links and explores its implications for symmetric unions.
problem Understanding the relationships between links and their symmetric unions.
method Defining a preorder relation and proving properties of links and their orbifold groups.
result If a link L is a Montesinos link with r≥3 rational tangles, then L′ is either a Montesinos link with at most r+1 rational tangles or a certain connected sum. Upper bounds for Legendrian links in tight contact 3-manifolds.
problem Bounding the complexity of Legendrian links in tight contact 3-manifolds.
method Constructing exact Lagrangian cobordisms and defining minimal Lagrangian genus.
result Established upper bounds for Legendrian links with a common rotation number.
Study vector-valued robust control under uncertainty.
problem Dynamic stochastic control with multi-objective criteria under model uncertainty.
method Robust minimax approach, set-valued framework, dynamic programming principle.
result Derived weak and strong versions of dynamic programming principle for vector-valued control problems.
Given two families of continuous functions u and v on a topological space X, we define a preorder R=R(u,v) on X by the condition that any member of u is an R-increasing and any member of v is an R-decreasing function. It turns out that if the topological space X is quasi-compact and sequentially com…
This study approximates neural network features for modeling relations and attention mechanisms.
problem Approximating neural network features for modeling relations and attention mechanisms.
method Analyzes inner products of multi-layer perceptrons for universal approximation of symmetric and asymmetric relation functions.
result Universal approximation of relation functions and attention mechanisms using inner products of neural networks.
In this paper we shall prove that the plane of financial events, introduced and applied to financial problems by the author himself (see [2], [3] and [4]) can be considered as a fibration in two different ways. The first one, the natural one, reveals itself to be isomorphic to the tangent- bundle of the real line, when…