Solved Dudeney's 100-year-old puzzle about triangle to square dissection.
problem How to dissect an equilateral triangle into the fewest pieces to form a square.
method Reduced the problem to analyzing graph structures representing piece correspondences.
result Proved that four is the minimum number of pieces for an equilateral triangle to square dissection.
New method finds lattice polygons that can be dissected into triangles with integer areas.
problem Finding lattice polygons that can be dissected into triangles with integer areas.
method A new version of Sperner's Lemma.
result Simple and complete description of lattice polygons that can be dissected into triangles with integer areas.
The study proves conjecture for specific Artin groups.
problem Proving conjecture about Artin groups' properties.
method Analyzing Artin groups associated to triangle-free graphs and cones over square-free bipartite graphs.
result Proves conjecture for specific Artin groups.
We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degener…
An involutive diffeomorphism σ of a connected smooth manifold M is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the constru…
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
In this note, given a regular Courant algebroid, we compute its group of automorphisms relative to a dissection. We also propose an infinitesimal version and recover examples of the literature.
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…
Origamis with specific groups have Veech groups that surject onto SL(2, Z/nZ).
problem Characterizing Veech groups of origamis as totally non-congruence groups.
method Using results on SL(2, Z/nZ) and properties of origamis' deck transformation groups.
result Origamis with certain triangle group quotients have Veech groups that surject onto SL(2, Z/nZ).
Study predicts risk of true-lumen narrowing after ATAAD surgery using CT data.
problem Early post-surgery risk assessment for aortic dissection patients.
method Retrospective study with CT data, derived cross-sectional shapes, form factor (FF) for morphology assessment, linear discriminant analysis (LDA) for risk classification, LOPO-CV for prediction.
result Machine-learning model accurately predicts risk for all high-risk patients and low-risk patients, potentially reducing hospital visits.
Modeling aortic wall inhomogeneities to predict dissection risks.
problem Predicting localized stress accumulations in the aortic wall due to inhomogeneities.
method Stochastic constitutive model with random field realizations, coupled with a convolutional neural network surrogate.
result The neural network accurately predicts stress distributions and assesses uncertainty in aortic wall stress.
The self intersection of an immersion i : S^2 \to R^3 dissects S^2 into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular, for every n we construct an immersion i : S^2 \to R^3 with 2n triple points, for …
The paper generalizes linking number properties for complete graphs.
problem Understanding linking numbers in spatial complete graphs.
method Analyzing the sum of square linking numbers and triangle-triangle links.
result Explicit formula for the sum of square linking numbers in large complete graphs.
New algorithms improve agnostic learning for triangles and polygons, reducing time complexity.
problem Efficient agnostic learning for geometric concept classes.
method Data structures and algorithms from computational geometry, probabilistic combinatorics.
result Optimal time complexity improvements for agnostic learning of triangles and polygons.
This work makes vision networks more interpretable by identifying key neurons.
problem Complex deep networks are hard to interpret, hindering safety-critical applications.
method Uses stochastic local competition and multimodal models to dissect and interpret neurons.
result The method generates understandable descriptions for a few active neurons, revealing network decision-making.
Researchers dissect Neural ODEs to understand their dynamics.
problem Understanding the inner workings of Neural ODEs.
method Developing continuous-depth formulation to clarify design choices.
result Clarified the influence of design choices on Neural ODE dynamics.
Polygon p-widths are found via billiard trajectories.
problem Finding p-widths of polygons. method Proved via billiard trajectories and computed specific cases.
result Polygon p-widths are achieved by billiard trajectories. Paper introduces quadrilateral labyrinth fractals and their properties.
problem None explicitly stated; focus on fractal construction and properties.
method Construction of quadrilateral labyrinth fractals and study of their topological properties.
result Properties of quadrilateral labyrinth fractals are studied.
Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over…
Chain-ladder reserving is sensitive to outliers, leading to unreliable estimates.
problem Sensitivity of loss reserving techniques to outliers.
method Derivation of impact functions for reserves and mean squared errors of prediction under Mack's Model.
result Impact of outliers varies widely in a loss triangle and depends on other cells.
Napoleonic triangles don't exist in hyperbolic geometry.
problem The existence of Napoleonic triangles in hyperbolic geometry.
method Analyzing the construction of equilateral triangles on hyperbolic triangles.
result Hyperbolic triangles do not form Napoleonic triangles, except equilateral ones.
New bounds on inscribed triangles in arbitrary planar domains.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
Paper calculates eigenvalues of a specific triangle on a sphere.
problem Computing eigenvalues of a specific triangle on a sphere.
method Computed first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2) on the sphere.
result Computed the first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2).
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
New method shows any triangle group generating pair is related to special coverings.
problem Understanding generating pairs of triangle groups.
method Special almost orbifold coverings.
result Any generating pair of a triangle group is represented by a special covering.
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, th…
Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
Proves a new skein exact triangle for real monopole Floer homology.
problem None explicitly stated; focuses on proving a new mathematical structure.
method Introduces a new exact triangle for real monopole Floer homology.
result Proves an unoriented skein exact triangle for real monopole Floer homology.
The vast majority of market impact studies assess each product individually, and the interactions between the different order flows are disregarded. This strong approximation may lead to an underestimation of trading costs and possible contagion effects. Transactions in fact mediate a significant part of the correlatio…
New surgery exact triangles in Heegaard Floer homology for rational slopes.
problem Constructing new surgery exact triangles in Heegaard Floer homology.
method Combining combinatorial triangle and quadrilateral counting in genus 1 Heegaard diagrams.
result Solving the combinatorial problem for rational slopes, including tricky cases.
We answer the question "Does the Y-triangle move preserve intrinsic knottedness?" in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K_7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.
problem Determining Courant-sharp eigenvalues on a Möbius strip.
method Analyzing the eigenvalues and nodal patterns of the Möbius strip.
result Only the first and second eigenvalues are Courant-sharp on the Möbius strip.
We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.
We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unorde…
New method proves mateability of triangle groups with Blaschke products.
problem Proving mateability of triangle groups with Blaschke products.
method Associating two piecewise analytic circle maps to the triangle group, mating these with Blaschke products, and constructing a common lift.
result Proves mateability of all cusped triangle groups with suitable Blaschke products.
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…
In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this paper investigates the existence of the incenter and the isogonal conjugate of a…
A formula for Rademacher symbols in triangle groups is provided.
problem No specific problem stated; focuses on a mathematical formula.
method Presentation of an explicit formula for Rademacher symbols.
result Generalizes Ghys' proof of modular knot linking numbers.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
Criterion for stopping conjugacy class enumeration in triangle groups.
problem Enumerating all conjugacy classes in cocompact triangle groups.
method Encoding by P. Dehornoy and T. Pinsky; stopping criterion based on geometric length.
result Stopping criterion for the generation of conjugacy classes in cocompact triangle groups.
New skein exact triangles for link Floer homology.
problem Understanding link Floer homology through skein relations.
method Construction of skein triples for rational tangles.
result Established a framework for potential further skein exact triangles.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
The study examines Lipschitz normally embedded Hölder triangles in 4D space.
problem Comparing ambient and outer Lipschitz geometry of Hölder triangles.
method Analyzes Lipschitz normally embedded Hölder triangles in \(\mathbb{R}^4\).
result Infinitely many equivalence classes of microknots.
Study finds only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
problem Determining Courant-sharp eigenvalues for compact flat surfaces.
method Analyzing flat Klein bottle and cylinders, proving only first and second eigenvalues are Courant-sharp.
result Only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.