The study examines topological surfaces using grid structures in Euclidean and hyperbolic spaces.
problem Understanding topological surfaces in geometric spaces.
method Using gridded surfaces in cubic honeycombs of Euclidean and hyperbolic spaces.
result Topological surfaces can be represented as gridded structures in these spaces.
We explore visual representations of tilings corresponding to Schläfli symbols. In three dimensions, we call these tilings "honeycombs". Schläfli symbols encode, in a very efficient way, regular tilings of spherical, euclidean and hyperbolic spaces in all dimensions. In three dimensions, there are only a finite number …
The paper analyzes the emergence of almost-honeycomb structures in low-energy planar clusters.
problem Understanding the formation of shapes resembling honeycombs in low-energy configurations.
method Detailed quantitative estimates and a revision of the global isoperimetric principle for honeycomb clusters.
result The majority of chambers in low-energy planar clusters are generalized hexagons, closely resembling regular hexagons.
Sharp decay estimates for critical Dirac equations on \(\mathbb{R}^n\).
problem Analyzing decay behavior of critical Dirac equations.
method Proving sharp pointwise decay estimates.
result Explicit asymptotic behavior of excited states in honeycomb structures.
Project aims to diagnose and track IPF disease using deep learning.
problem Diagnosing and tracking IPF disease in lung images.
method Developed a deep learning model for identifying honeycombing and ground glass patterns in HRCT lung images.
result Deep learning model achieved high accuracy in identifying specific lung regions.
3D self-affine tiles with specific digit sets have boundary homeomorphic to a 2-sphere.
problem Characterizing 3D self-affine tiles with collinear digit sets whose boundary is a sphere.
method Using lattice tiling combinatorics and topological properties of spheres, the paper characterizes such tiles.
result The boundary of these tiles is homeomorphic to a 2-sphere under certain conditions.
We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the 2-dimensional case, the lattice is regular and it incorporates dyna…
It is known that every ribbon category with unimodality allows symmetrized 6j-symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category E. We define the mirror conjugate symmetry of 6j-symbols instead and sho…
{\em Riemannian cubics} are curves in a manifold M that satisfy a variational condition appropriate for interpolation problems. When M is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are know…
Computes the monodromy of cubic surfaces branching over smooth cubic curves.
problem Understanding the monodromy of cubic surfaces.
method Computational approach using the relationship between inflection points and lines on cubic surfaces.
result The monodromy map is surjective onto the centralizer of the image of a generator of the deck group.
Study of sub-Riemannian cubics in SU(2) for long-term dynamics.
problem Optimizing curves in a sub-Riemannian manifold with constraints.
method Analysis of sub-Riemannian Lie quadratics in SU(2).
result Characterization of sub-Riemannian Lie quadratics in SU(2).
Study shows minimum δ-invariant for smooth cubic surfaces.
problem Determining the minimum δ-invariant for smooth cubic surfaces. method Analyzing properties of smooth cubic surfaces to find δ-invariants. result Smooth cubic surfaces have a δ-invariant of at least 6/5. The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformati…
The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.
problem Understanding conformal properties of cubic metrics with isotropic scalar curvature.
method Analyzing the conformal flatness and isotropic scalar curvature of cubic metrics.
result Cubic metrics with weakly isotropic scalar curvature must be Minkowski metrics.
Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.
problem Understanding K-stability and Kähler-Einstein metrics for cubic fourfolds.
method Local volume estimates and Ambro-Kawamata's non-vanishing theorem.
result All smooth cubic fourfolds admit Kähler-Einstein metrics.
The study shows that certain cubical presentations lead to aspherical spaces.
problem Understanding the asphericity of cubical presentations in 2D.
method Analyzing the second homotopy group of coned-off spaces associated with cubical presentations.
result The coned-off space is aspherical under specific conditions.
New cubic forms linked to η-invariants and mod 2 indices.
problem Anomaly cancellation and modularity in 12-manifolds.
method Combination of Witten classes and affine E8 character. result Relates cubic forms to η-invariants and mod 2 indices.
It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.
Solves infinite family of cubic polynomial problems.
problem Infinite family of twisted polynomial problems.
method Using Dehn twists and 9-adic expansions.
result Result of twisting depends on 9-adic expansion.
We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.
The flow and Yamada polynomials of cubic graphs are studied, extending known identities and conjectures.
problem Characterizing the structure of flow and Yamada polynomials for cubic graphs.
method Establishing quadratic identities and proving conjectures about flow and Yamada polynomials.
result The golden identity for the flow polynomial characterizes planarity of cubic graphs for a certain family.
Computes cohomology of smooth cubic surfaces using simplicial resolution.
problem Computing the cohomology of smooth cubic surfaces.
method Vassiliev's simplicial resolution method.
result Cohomology isomorphic to that of P2 modulo embedding. Study de Rham theory for cubical manifolds and quandles.
problem Rational homotopy type of classifying spaces of smooth quandles.
method Show de Rham theory for cubical manifolds and study secondary characteristic classes.
result Secondary characteristic classes produce cocycles of quandles.
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with norm…
Complete classification of centroaffine hypersurfaces with parallel cubic form.
problem Characterizing centroaffine hypersurfaces with specific geometric properties.
method Analyzing hypersurfaces with respect to the Levi-Civita connection of the centroaffine metric.
result A complete classification of locally strongly convex centroaffine hypersurfaces with parallel cubic form.
A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6].
We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation ∣∇f∣2=9∣x∣4 must be rotationally equivalent to either xn3−3xn(x12+...+xn−12), or to one of four exceptional Cartan cubic polynomials…
Researchers classify and visualize 5-cube cubical surfaces.
problem Classifying and visualizing surfaces in a 5-dimensional cube.
method Exhaustive search, classification by genus and demigenus, 3D visualization, reinforcement learning for optimization.
result 2690 connected closed cubical surfaces in the 5-cube, visualized and optimized for 3D printing.
Study on choosing points on cubic curves, answering some questions about their flexibility.
problem Determining if algebraic structures can continuously choose points on cubic plane curves.
method Analyzing the flex points and sextatic points of cubic plane curves.
result Affirmative answer for n=9 and 18, negative for infinitely many n. Proves K-stability of cubic threefolds and calculates their Kähler-Einstein metrics.
problem K-stability of cubic threefolds and explicit calculation of Kähler-Einstein metrics.
method Detailed study of three-dimensional canonical and terminal singularities, estimate of Kawamata log terminal volumes.
result All smooth cubic threefolds admit Kähler-Einstein metrics, and a precise list of singular KE ones is provided.
No nontrivial automorphisms for cubic surfaces moduli space.
problem Understanding automorphisms of cubic surfaces moduli space.
method Analyzing the fundamental group of the moduli space.
result No nontrivial biholomorphic automorphisms for cubic surfaces moduli space.
Study of algebraic dynamics on Markov cubics in tropical geometry.
problem Understanding the dynamics of Markov cubics over non-archimedean fields.
method Tropicalization and (∞,∞,∞)-triangle reflection group on hyperbolic plane. result Existence of Fatou domain and finitude of orbits with rational points over prime power denominators.
Origami solves real cubic equations, revealing a specific curve.
problem Solving real cubic equations using origami.
method Investigating a specific real cubic curve F(x,y)=0 and analyzing its properties. result The shape of Beloch's curve is determined by the Hessian at its singular point.
Study of choosing distinct points on cubic curves, proving impossibility.
problem Choosing distinct points on cubic plane curves continuously.
method Topological fiber bundle analysis, proving non-existence of continuous sections.
result Proves impossibility for non-multiple of 9 points.
Constructs independent bases for cubic curve families using Hessian structures.
problem Finding independent bases for cubic curve families.
method Uses a Hessian structure to define a cost function for constructing bases.
result Constructs valuatively independent bases for H0(X,Lk). We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …
The paper studies 8D manifolds with a specific tensor field called a cubic discriminant.
problem Characterizing and understanding 8D Riemannian manifolds with reduced structure groups.
method Introducing an almost quaternion-Hermitian structure and a cubic discriminant tensor field.
result Only two non-flat, integrable examples of these structures are found: quaternion-Kähler symmetric spaces.
The study connects cubic differentials to convex RP^2-structures and their ends.
problem Understanding the relationship between cubic differentials and convex RP^2-structures.
method Affine sphere construction and analysis of poles of cubic differentials.
result Poles of cubic differentials correspond to ends of convex RP^2-structures.
Operational guide to discrete exterior calculus on cubic cells.
problem Applying calculus on discrete manifolds.
method Defining discrete exterior calculus on cubic cells for discrete manifolds.
result Gauss and Stokes theorems are recovered on the discrete torus.
Lower bound for complexity of finding flex points on cubic curves.
problem Finding flex points on cubic plane curves.
method Bounding the Schwarz genus of a cover associated to the problem.
result Lower bound for topological complexity close to optimal.
Up to isomorphism there are six fixed-point free crystallographic groups in Euclidean Space generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of E3 by such a group. The cubic tessellation of E3 induces tessellations on each such manifold. These tessellations of the …
The article introduces a combinatorial differential algebra for cubic planar graphs.
problem Understanding the structure of cubic planar graphs.
method Defining a combinatorial differential graded algebra based on binary sequences and counting.
result The algebra's graded augmentation variety's rational points match (q+1)-colorings of the dual graph.
Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.