Graphs with maximum degree Δ have at most O(1) equiangular lines for λ < 3/sqrt(2).
problem Finding the maximum number of equiangular lines in graphs with a given maximum degree.
method Using eigenfunctions and nodal domains to estimate the multiplicity of eigenvalues.
result The maximum multiplicity of λ as the second largest eigenvalue is O(1) for graphs with maximum degree Δ and cyclomatic number.
Six quaternionic lines with optimal angles found in 2D quaternion space.
problem Finding optimal configurations of quaternionic lines in 2D space.
method Simple presentation of lines as orbit of a reflection group, finding other optimal designs.
result Optimal spherical designs of 10, 15, and 20 lines in quaternion space.
An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to π/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperboli…
We investigate three-dimensional surfaces where the normal vector forms a constant angle with the radius vector. These surfaces naturally extend equiangular (logarithmic) spirals in the plane.
We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration spa…
We study pairs of curves with Poncelet's porism properties and compute their vertex curves.
problem Understanding pairs of curves with Poncelet's porism properties.
method Developed formulas to compute vertex curves for given envelope curves and vice versa, for all sufficiently regular pairs of Poncelet curves.
result Formulas produce all possible sufficiently regular pairs of Poncelet curves, including sets of curves analogous to pencils of conic sections.
New hyperbolic polyhedra with π/3 angles and volumes calculated.
problem Finding new hyperbolic polyhedra with specific dihedral angles.
method Constructed a new sequence of hyperbolic polyhedra with π/3 angles and determined their volumes. result Volumes of some constructed polyhedra determined.
Language models allocate information storage, not collapsing into uniform representations.
problem Incomplete neural collapse in language model representations.
method Analyzing variance and information sharing across 14 models, proving an information floor.
result Within-class variance is allocated information storage, not collapsed into uniform representations.
A twisted curve in Euclidean 3-space E^3 can be considered as a curve whose position vector can be written as linear combination of its Frenet vectors. In the present study we study the twisted curves of constant ratio in E^3 and characterize such curves in terms of their curvature functions. Further, we obtain some re…
New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.
problem Investigate geometric properties of kth Order Preserving Sets and ovals. method Introduce and analyze kth Order Preserving Sets and Midpoint Sets; study geometric properties and isoperimetric inequalities. result Established an isoperimetric-type inequality relating perimeter and area of ovals and their associated sets.
We adapt the method of Simon [JDG '93] to prove a C1,α-regularity theorem for minimal varifolds which resemble a cone C02 over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish C1,α-regularity near the cone $\bf{C}_0^2 \ti…
Dropout is a popular regularization technique in deep learning. Yet, the reason for its success is still not fully understood. This paper provides a new interpretation of Dropout from a frame theory perspective. By drawing a connection to recent developments in analog channel coding, we suggest that for a certain famil…
Let C be a smooth, convex curve on either the sphere S2, the hyperbolic plane H2 or the Euclidean plane E2, with the following property: there exists α, and parameterizations x(t),y(t) of C such that for each t, the angle between the chord connecting x(t) to y(t)…
Sparse coding in learned dictionaries has been established as a successful approach for signal denoising, source separation and solving inverse problems in general. A dictionary learning method adapts an initial dictionary to a particular signal class by iteratively computing an approximate factorization of a training …
We analyze deep neural networks using convex duality to reveal hidden layer structures.
problem Understanding the structure of deep neural networks.
method Introducing a convex analytic framework to characterize hidden layer weights.
result Optimal hidden layer weights align with previous layers via duality.
Performance of distributed optimization and learning systems is bottlenecked by "straggler" nodes and slow communication links, which significantly delay computation. We propose a distributed optimization framework where the dataset is "encoded" to have an over-complete representation with built-in redundancy, and the …
Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.
problem Understanding and optimizing deep learning training phases.
method Direct measurements on three deepnet architectures across seven datasets.
result Deep nets exhibit 'Neural Collapse' during training's final phase, simplifying decision-making.
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
Study restricts line arrangements with odd points using topological arguments.
problem Restrictions on line arrangements with singular points of odd multiplicity.
method Topological arguments on locally-flat spheres in 4-manifolds.
result No line arrangement with 13 lines and only triple points exists.
A special group of transformations of the real line cannot act effectively on it.
problem Understanding the limitations of transformations on the real line.
method Analyzing the group of orientation-preserving quasi-isometries of the real line.
result The group of quasi-isometries of the real line cannot act effectively on the line.
A new loss function HUG decouples and generalizes neural collapse.
problem Neural collapse limits in deep learning models.
method Hyperspherical uniformity gap (HUG) as a unified framework.
result HUG decouples and generalizes neural collapse, improving model flexibility and robustness.
Study Blaschke's asymptotic lines on surfaces in 3D space.
problem Characterize Blaschke's asymptotic lines on surfaces in 3D.
method Analyze binary differential equations near cusp and umbilic points.
result Describe Blaschke's asymptotic lines near Euclidean parabolic set.
This paper studies how to compress neural networks while maintaining accuracy.
problem Compressing a two-layer neural network with fewer nodes without losing accuracy.
method Using tools from high-dimensional probability, the authors minimize the L_2 loss between the target and compressed networks.
result The error rate of the approximation is shown as a function of input dimension and network size in the mean-field limit.
This work justifies neural collapse under MSE loss and analyzes the optimization landscape.
problem Understanding neural collapse in deep neural networks under MSE loss.
method Global landscape analysis of vanilla nonconvex MSE loss.
result The only global minimizers are neural collapse solutions.
Co-PLNet combines point and line predictions to improve wireframe parsing accuracy and efficiency.
problem Separate line and point predictions lead to inconsistent wireframes.
method Co-PLNet uses a Point-Line Prompt Encoder to convert early point detections into spatial prompts, which guide line refinement.
result Co-PLNet achieves better accuracy and robustness in wireframe parsing compared to existing methods.
Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.
problem Classifying metric lines in Engel-type groups.
method Sequence method to study metric lines in jet space.
result Classified metric lines of Engel-type groups $\Eng(n)$.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
For a pair of points in a smooth closed convex planar curve γ, its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines (EML) is formed by the union of three affine invariants sets: Affine Envelope Sy…
A line arrangement of 3n lines in CP2 satisfies Hirzebruch property if each line intersect others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in CP2 is…
The paper examines asymptotic lines of plane fields in 3D space.
problem Qualitative properties of asymptotic lines in plane fields.
method Analysis of null directions and Gaussian curvature.
result Asymptotic lines coincide with classical ones in completely integrable fields.
The paper explores reflection principles for lightlike line segments on maximal surfaces.
problem Reflection property does not hold for lightlike line segments on maximal surfaces.
method Analyzes reflection properties for lightlike line segments connecting shrinking singularities.
result Shows a kind of reflection principle for lightlike line segments on maximal surfaces.
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial,…
The paper finds two types of metric lines in curve spaces.
problem Classifying metric lines in jet spaces of curves.
method Established the existence of two families of metric lines in the 2-jet space of plane curves.
result Found precise criteria for identifying metric lines in sub-Riemannian geodesics.
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …
Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, let Pn be the boundary of a regular polygon in the plane…
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …
Geometric quantization extended to big line bundles.
problem Quantization of line bundles with large curvature.
method Proving asymptotic isometry and submultiplicative norms equivalence, showing Mabuchi geodesic rays.
result Bounded submultiplicative filtrations on big line bundles lead to Mabuchi geodesic rays.
There is a natural duality between line congruences in R3 and surfaces in R4 that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…
New BDEs reveal singular surfaces from line congruences.
problem Understanding binary differential equations associated with line congruences.
method Applied pointwise to quadratic differential forms, studying quotients of quadratic forms and associated polar lines.
result Introduced a new singular surface in Euclidean 3-space.
We give quantitative and qualitative results on the family of surfaces in CP3 containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterwards we prove that …
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
problem Limited theoretical understanding of line graph transformation's impact on GNN models.
method Examined CFI and strongly regular graphs, showing line graph transformation helps WL tests distinguish these graphs.
result Line graph transformation aids WL tests in distinguishing challenging graph properties.
Study conic-line arrangements of degree 7, finding their topology and components.
problem Understanding the topology and components of conic-line arrangements of degree 7.
method Examined the embedded topology using SageMath and Sirocco package.
result Determined the number of connected components and fundamental groups.