Ancient solutions and translators identified for Lagrangian flow.
problem Characterizing ancient solutions and translators of Lagrangian mean curvature flow.
method Analyzing almost calibrated, exact, ancient solutions with specific geometric properties.
result All ancient solutions with entropy less than 3 are special Lagrangian, planes, or translators in \(\mathbb{C}^2\).
Study shows how neck pinches occur in Lagrangian flows and their continuation.
problem Understanding and continuation of Lagrangian mean curvature flows with singularities.
method Analyzes zero Maslov, rational Lagrangian flows in compact Calabi-Yau surfaces.
result Tangent flow is unique and can be continued past singularities.
Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smoot…
PGNs dynamically infer and use graph structures to improve model generalization.
problem Static graph structures inferred by machine learning practitioners are often suboptimal for tasks.
method PGNs augment graphs with dynamically inferred pointers for improved model generalization.
result PGNs outperform unrestricted GNNs and Deep Sets on dynamic graph connectivity tasks.
Study shows not all ribbon knots can be symmetric unions.
problem Whether every ribbon knot can be a symmetric union.
method Exhibited a specific ribbon Montesinos knot that cannot be a symmetric union.
result Found a ribbon knot that is not a symmetric union.
Study of symmetric unions of knots with new inequality and epimorphism results.
problem Understanding the genera of symmetric unions of knots.
method Introduced symmetric unions inspired by earlier work, showed an identity between twisted Alexander polynomials and genera, and established an epimorphism between knot groups.
result Obtained an inequality concerning the genera of symmetric unions and provided a positive answer to an old problem.
We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions appears only in a single hyperbolic 3-manifold or Dehn fillings of a single hyp…
Tanaka shows amphichiral symmetric unions of the unknot are trivial.
problem Understanding amphichiral symmetric unions and their Jones polynomials.
method Analyzing the Jones polynomial of amphichiral symmetric unions of the unknot and generalizing to other knots.
result Amphichiral symmetric unions of any knot with one twist region are trivial.
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of un…
We prove that all 2-bridge ribbon knots are symmetric unions.
Let two Heegaard splittings V1∪W1 and V2∪W2 of a 3-manifold M be given. We consider the union stabilization M=V∪W which is a common stabilization of V1∪W1 and V2∪W2 having the property that V=V1∪V2. We show that any two Heegaard splittings of a 3-manifold have a uni…
Study classifies static potentials on 3-manifolds, proving one-dimensionality under specific conditions.
problem Classifying the dimension of static potentials on 3-manifolds.
method Analysis of relative zero sets of static potentials, using Miao and Tam's technique.
result Proves one-dimensionality of static potentials under specific conditions.
Examines insurance market development and similarity post-2004 EU enlargement.
problem Comparing insurance markets of EU old and new members post-enlargement.
method Analyzes data from 2004 to present to compare insurance markets.
result Identifies similarities and differences in insurance markets post-2004 enlargement.
New static vacuum metrics confirmed for near Euclidean boundary data.
problem Establishing sufficient conditions for near Euclidean boundary data in static vacuum metrics.
method Using new arguments from studying the conjecture for arbitrary static vacuum metrics.
result Any hypersurface in a dense subfamily is static regular.
Study AFPP of unions of convex digital disks in 2D.
problem Conditions for AFPP of union of convex disks in digital plane.
method Use results from [6] to analyze AFPP.
result Conditions for AFPP of union of convex disks.
Extended symmetric unions extend properties of Alexander polynomials.
problem Properties of Alexander polynomials of symmetric unions.
method Constructing pairs of knots with specific epimorphisms.
result Alexander polynomials of constructed knots exhibit extended properties.
A short proof for a theorem about composite knots.
problem Proving a theorem about composite knots with symmetric union presentations.
method Presenting a concise proof of Tanaka's theorem.
result Composite knots with symmetric union presentations have non-trivial connected summands.
Extends static vacuum metrics with specific boundary conditions.
problem Proving the existence of static vacuum metrics with prescribed boundary data.
method Introducing static regular types (I) and (II), showing local well-posedness, and confirming Bartnik's conjecture.
result Confirms Bartnik's static vacuum extension conjecture for a broad range of boundary conditions.
Two-bridge ribbon knots have symmetric union presentations.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
We classify static manifolds which admit more than one static decomposition whenever a condition on the curvature is fullfilled. For this, we take a standard static vector field and analyze its associated one parameter family of projections onto the base. We show that the base itself is a static manifold and the warpin…
In this paper, we investigate a multivariate multi-response (MVMR) linear regression problem, which contains multiple linear regression models with differently distributed design matrices, and different regression and output vectors. The goal is to recover the support union of all regression vectors using l1/l2-reg…
Study of 3D vacuum static spaces with specific curvature properties.
problem Classifying 3D vacuum static spaces with certain curvature conditions.
method Used generalized maximum principle to classify 3D spaces.
result Gave a complete classification of 3D complete vacuum static spaces.
New rigidity theorem on static manifolds with boundary.
problem Static metrics on manifolds with boundary.
method Obata-type rigidity theorem, sufficient geometric conditions.
result Scalar curvature map can be locally surjective at static metrics on manifolds with boundary.
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
The paper classifies vacuum static spaces with harmonic curvature.
problem Classifying vacuum static spaces with harmonic curvature.
method Extending the 4-dimensional work by Kim-Shin, the paper classifies n-dimensional spaces (n≥5). result New counterexamples to the Fischer-Marsden conjecture on compact vacuum static spaces.
The paper investigates geometrical aspects of static spacetime with almost gradient Ricci solitons.
problem Geometrical properties of static spacetime with almost gradient Ricci solitons.
method Analyzing conditions and properties of static spacetime with almost gradient Ricci solitons.
result Conditions and properties of static spacetime with almost gradient Ricci solitons are determined.
The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.
problem Proving geometric inequalities for static convex domains in static rotationally symmetric spaces.
method Locally constrained curvature flow in a static rotationally symmetric space Nn+1, proving graphical solutions and static convexity preservation. result Proves weighted geometric inequalities for static convex domains close to a slice of Nn+1. Proves equality in Minkowski inequality for static, flat manifolds.
problem Proving equality in Minkowski inequality for static, flat manifolds.
method Analyzes quasi-spherical metrics and static manifolds.
result Equality in Minkowski inequality achieved only by Schwarzschild space slices.
An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence…
We consider Killing vector fields on standard static space-times and obtain equations for a vector field on a standard static space-time to be Killing. We also provide a characterization of Killing vector fields on standard static space-times with compact Riemannian parts.
Proves existence of static vacuum metrics with specific boundary data.
problem Existence of static vacuum metrics with prescribed boundary data.
method Proves existence and local uniqueness of static vacuum metrics close to the Euclidean metric.
result Existence of static vacuum metrics with prescribed Bartnik boundary data.
Analyzing static solutions in Finsler gravity, extending known results.
problem Extending the analyticity of static vacuum solutions to Finsler spacetimes.
method Examining Finsler spacetimes with properties similar to static Lorentzian spacetimes.
result Finsler spacetimes with vanishing Ricci scalar are analytic.
Extended symmetric union with multiple tangle regions and Alexander polynomial properties.
problem Characterizing knots with multiple tangle regions.
method Generalizing the symmetric union construction to include multiple tangle regions and analyzing the Alexander polynomial.
result The Alexander polynomial of the constructed knot is the product of the Alexander polynomials of the tangles and the square of the partial knot's Alexander polynomial.
Paper derives Riccati equation for static spaces and proves its applications.
problem Deriving Riccati equation for static spaces.
method Proving splitting theorem and connectivity of conformal boundary.
result Establishes compactness of universal covering for static triples.
We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework where a set of non-dominated models is given. The concept of semi-static completeness is introduced: it corresponds to having exact re…
In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum…
Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.
Study on stellar models' topology and mass using minimal surfaces.
problem Investigating the topology and mass of static stellar models.
method Analyzing stable free boundary minimal surfaces in static perfect fluid spaces.
result Proved non-existence of stable free boundary minimal surfaces and derived upper bounds for Hawking mass.
Classifies vacuum static spaces with harmonic curvature.
problem Classifying vacuum static spaces with harmonic curvature.
method Thorough classification through geometric analysis.
result Spaces are locally isometric to four types.
Paper proves a rigidity result for static perfect fluids.
problem Proving a rigidity result for static perfect fluids.
method Robinson's divergence formula and boundary conditions.
result Rigidity result for static perfect fluids.
Existence proved for static vacuum extensions near Schwarzschild spheres.
problem Proving existence of static vacuum extensions near Schwarzschild spheres.
method Existence and local uniqueness of static vacuum extensions for Bartnik data on a sphere near a Schwarzschild sphere.
result Existence of static vacuum extensions near Schwarzschild spheres.
The paper refutes the manifold hypothesis for image data and proposes the union of manifolds hypothesis.
problem The manifold hypothesis fails to capture the structure of image data.
method Empirical verification of the union of manifolds hypothesis on image datasets.
result Image data lies on a disconnected set with varying intrinsic dimensions.
Paper proves rigidity of static manifolds and applies to metric extensions.
problem Detecting rotational symmetry in static systems.
method Conformal techniques and Minkowski-type inequalities.
result Proves global uniqueness of static metric extensions.
Alternative proof for static black hole uniqueness with nonpositive mass.
problem Proving static uniqueness for Kottler spacetimes with nonpositive mass.
method Alternative, more elementary proof of static uniqueness theorem.
result Alternative proof of static uniqueness for Kottler spacetimes with nonpositive mass.
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. The study proves unique static manifolds with positive scalar curvature and boundary.
problem Characterizing static three-manifolds with boundary and positive scalar curvature.
method Analyzing Ricci curvature bounds and quotient spaces.
result The only orientable quotient of the Nariai static manifold with boundary Nar−1,1(S2) is the only such manifold with connected boundary under certain conditions. This paper presents some partial answers to the following question. QUESTION. If a normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, must X be contractible? The main results of the paper are: THEOREM 1. If a normal space X is the union…
We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas u…