The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.
problem Finding minimal hypersurfaces in wedge-shaped manifolds with boundary.
method Developed a min-max theory for locally wedge-shaped manifolds with boundary.
result Proved existence of smooth free boundary minimal hypersurfaces in wedge-shaped manifolds.
Proof of Gromov's theorem on convex polytopes with acute angles.
problem Gromov's conjecture on extremal scalar curvature of convex polytopes.
method Smoothing construction using Dirac operator techniques.
result Detailed proof of Gromov's theorem.
Let C(L) be the right-angled Coxeter group defined by an abstract triangulation L of S2. We show that C(L) is isomorphic to a hyperbolic right-angled reflection group if and only if L can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an …
Flat minimal hypersurfaces found in wedge-shaped domains.
problem Finding minimal surfaces in wedge-shaped domains.
method Proving stability and flatness of C1,1-to-edge minimal hypersurfaces. result Stable minimal hypersurfaces are flat in wedge-shaped domains.
Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
problem Estimating curvature of stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
method Compactness theorem and Schoen-Simon-Yau estimates.
result Curvature estimate for free boundary minimal hypersurfaces in wedge-shaped manifolds.
Let Σ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in Rn+1. Suppose that Σ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if ∂Σ is embedded for n=2, or if ∂Σ is convex…
Maps between acute triangles with minimal stretch found and studied.
problem Finding the minimal stretch between acute triangles.
method Formula for the smallest Lipschitz constant and analysis of the metric space.
result Metric space of pairs of acute triangles with fixed area is Finsler and geodesics determined.
Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.
problem Finding optimal shapes to minimize the average length of billiard trajectories.
method Used techniques from Teichmüller theory.
result Optimal shapes minimize average lengths of billiard trajectories in specific polygons.
Minimal surfaces with dihedral symmetry are studied as angles converge to zero.
problem Understanding minimal surfaces with dihedral symmetry as angles approach zero.
method Analyzing the limit of minimal surfaces in wedges with varying angles and using the implicit function theorem.
result New minimal surfaces are discovered and existence proofs are simplified.
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and bound…
We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in R3 of constant mean curvature which meet planes Π1 and Π2 in constant contact angles γ1 and γ2 and bound, together with those planes, a…
Study on Teichmüller space of acute triangles using explicit calculations.
problem Analogue of Thurston's metric on acute triangles.
method Direct calculation of distance function and Finsler structure.
result Explicit expressions and properties of the metric on acute triangles.
In a 2013 paper, Gromov proves that if smooth Riemannian metrics gi converge to a smooth Riemannian metric g uniformly, and gi have scalar curvature uniformly bounded below, then g shares the same scalar curvature lower bound. In some places in the paper, the proofs are only sketched. In this paper we explain…
AI system predicts acute critical illness from EHRs with explainability.
problem Lack of clinical interpretability in AI predictions for acute critical illness.
method Developed an explainable AI early warning score (xAI-EWS) system.
result System provides clinicians with insights into EHR data explaining predictions.
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
Paper proves inequality for capillary hypersurfaces in a wedge.
problem Proving a best version of Heintze-Karcher inequality for capillary hypersurfaces.
method Utilized Heintze-Karcher method and modified parallel hypersurfaces.
result Classified capillary constant mean curvature hypersurfaces hitting the edge in a wedge.
New groups derived from square configurations have right-angled and HNN structures.
problem Understanding the fundamental groups of square configurations and their homotopy properties.
method Analyzing configuration spaces and their fundamental groups, proving group presentations and homotopy equivalences.
result The fundamental groups of certain square configurations have minimal presentations with commutator relators and are HNN extensions of specific meta-square groups.
We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat ker…
Study on scalar curvature in wedge spaces with existence and obstruction results.
problem Existence and obstructions of scalar curvature in wedge spaces.
method Utilized established tools for wedge spaces including Yamabe, elliptic, and index theories.
result Provided existence and obstruction results for scalar curvature under suitable positivity assumptions.
The paper explores Kähler and anti-Kähler structures on quasi-statistical manifolds.
problem Investigating Kähler and anti-Kähler structures on quasi-statistical manifolds.
method Analyzing conditions for integrability of almost complex structures and defining Kähler and anti-Kähler manifolds.
result Conditions for (Nˊ,h,abla,L) to be an anti-Kähler manifold are identified. Improved rigidity of Delaunay triangulated plane.
problem Rigidity of Delaunay triangulated plane under discrete conformality.
method Modifying Wu's proof to weaken the uniformly acute condition to the uniformly Delaunay condition.
result Improved rigidity result for Delaunay triangulated plane.
Paper equates torsions on wedge singularities.
problem Analyzing spaces with wedge singularities.
method Equating analytic and intersection Reidemeister torsions.
result Equation of torsions in specific wedge singularities.
Consider a financial market in which an agent trades with utility-induced restrictions on wealth. By introducing a general convex-analytic framework which includes the class of umbrella wedges in certain Riesz spaces and faces of convex sets (consisting of probability measures), together with a duality theory for polar…
Bayesian optimization improves classifier selection for acute infection and mortality.
problem Improving accuracy of acute infection and mortality prediction.
method Comparison of hyperparameter optimization methods (grid search, random sampling, Bayesian optimization).
result Bayesian optimization outperforms grid search or random sampling for in-hospital mortality classifiers.
In this paper, the development of a probabilistic network for the diagnosis of acute cardiopulmonary diseases is presented. This paper is a draft version of the article published after peer review in 2018 (https://doi.org/10.1002/bimj.201600206). A panel of expert physicians collaborated to specify the qualitative part…
Study developed phenotypes for ICU patients' brain dysfunction states.
problem Underdiagnosis of acute brain dysfunction in ICU patients.
method Created algorithms to quantify and cluster brain dysfunction states.
result Developed three phenotypes of ICU patients' brain dysfunction states.
Geometrodynamics derived from Riemannian manifolds using geospin matrix.
problem Formulating dynamics on Riemannian manifolds using Cartan structural equations.
method Introducing four real dynamical variables and applying them to Cartan structural equations.
result Rewritten Cartan structural equations in a real geometrodynamical form.
New insights into Khovanov homology complexity and topological structure.
problem Complexity of computing Khovanov homology for closed braids.
method Analysis of independence simplicial complexes and polynomial time algorithms.
result Independence simplicial complexes associated to 4-braid diagrams are homotopy equivalent to wedges of spheres.
We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbound…
Background: Fluctuating hearing loss is characteristic of Meniere's Disease (MD) during acute episodes. However, no reliable audiometric hallmarks are available for counselling the hearing recovery possibility. Aims/Objectives: To find parameters for predicting MD hearing outcomes. Material and Methods: We applied mach…
A fake wedge is a diagram of spaces K <- A -> C whose double mapping cylinder is contractible. The terminology stems from the special case A = K v C with maps given by the projections. In this paper, we study the homotopy type of the moduli space D(K,C) of fake wedges on K and C. We formulate two conjectures concerning…
New steady Kähler-Ricci solitons found on orbifolds.
problem Constructing steady Kähler-Ricci solitons on orbifolds.
method Using asymptotically cylindrical structures and orbifold actions.
result Found new examples of gradient steady Kähler-Ricci solitons converging to a Ricci-flat cylinder.
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
Examines how irreducibility and rigidity affect digital images.
problem Understanding interactions between irreducibility and rigidity in digital images.
method Analyzes Cartesian products, wedges, and cold and freezing sets.
result Interactions between irreducibility and rigidity in digital images.
The paper classifies Poincaré complexes as topological manifolds.
problem Classifying Poincaré complexes as topological manifolds.
method Using spherical fibrations and CW-complexes, the paper proves stability and homotopy equivalence.
result A sufficient condition for Poincaré complexes to be homotopy types of topological manifolds.
Study boundary actions of CAT(0) spaces and their C∗-algebras.
problem Investigate boundary actions of CAT(0) spaces and their associated C∗-algebras. method Topological dynamics and C∗-algebras, focusing on actions of specific groups and their properties. result Established (strongly) pure infiniteness results for reduced crossed product C∗-algebras of boundary actions. Wedge Sampling improves tensor completion with nearly-linear sample complexity.
problem Efficiently completing low-rank tensors from a subset of entries.
method Non-adaptive wedge sampling to promote structured connections in tensor completion.
result Polynomial-time algorithms achieve weak and exact recovery with nearly linear sample complexity.
It has been shown that the Alvarez-Gaumeˊ-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…
We discuss the Ricci-flat `model metrics' on C2 with cone singularities along the conic {zw=1} constructed by Donaldson using the Gibbons-Hawking ansatz over wedges in R3. In particular we describe their asymptotic behavior at infinity and compute their energies.
Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.
problem Understanding capillary surfaces with anisotropic forces.
method Generalized Minkowski norm on the unit sphere, new Minkowski formulae, Heintze-Karcher inequality.
result Proved an Alexandrov-type theorem in the anisotropic setting.
Algorithm finds characteristic maps over complex shapes.
problem Finding non-singular characteristic maps over wedged simplicial complexes.
method Constructive puzzle algorithm based on linear algebra.
result Algorithm performs well compared to other methods.
Large vessel occlusion (LVO) plays an important role in the diagnosis of acute ischemic stroke. Identifying LVO of patients in the early stage on admission would significantly lower the probabilities of suffering from severe effects due to stroke or even save their lives. In this paper, we utilized both structural and …
Adaptive-SGD method optimizes machine learning training with dynamic batch and step sizes.
problem Optimizing machine learning training with adaptive batch and step sizes.
method Adaptive-SGD method that dynamically adjusts batch size and step size based on local curvature and probability of descent directions.
result Adaptive-SGD achieves global linear convergence on self-concordant functions and compares favorably to fine-tuned methods.
We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this approach with explicit examples. In particular, we describe a short program to determin…
We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions. These correlation functions a…
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus g with one boundary component to ∧3H, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…