Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.
problem Homological stability in the space direction of square configurations.
method Analyzing the ordered configuration space of squares in a rectangle.
result Most rectangles can be almost entirely filled with squares and still have stable homology.
Optimal weight windows are symmetric rectangles centered at peak.
problem Finding the best weight windows for weighted least squares.
method Investigated symmetric and tapered rectangle window weights, showing the best rectangle window is optimal.
result The best rectangle window is optimal for all tapered rectangle window definitions.
The paper proves geometric properties of square tables and saddle surfaces.
problem The mathematical table problem from a geometric-topological perspective.
method Geometric-topological proofs on cylinder, saddle surfaces, and level sets of Fenn graphs.
result Zero-existence theorem on a cylinder, proving Fenn's square-table theorem under different boundary conditions.
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was ca…
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
Formula for interleaving distance of rectangle persistence modules.
problem Calculating distances between rectangle persistence modules.
method Formulas based on rectangle geometry, extended to decomposable modules.
result Closed formulas for interleaving and bottleneck distances.
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Paper finds a counterexample showing rectangle condition doesn't detect strong irreducibility.
problem Detecting strong irreducibility from the rectangle condition.
method Constructing a double branched cover of a knot in S^3.
result Found a genus 2 Heegaard splitting that is strongly irreducible but fails rectangle condition.
In this paper, we define the rectangle condition on the bridge sphere for a n-bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of 3-manifolds. We show that the satisfaction of the rectangle condition for a n-bridge decomposition can …
Curves inscribe rectangles with positive area.
problem Finding angles for inscribing rectangles within Jordan curves.
method Proving existence of a subset of angles with measure at least A/R^2.
result Angles inscribing rectangles have measure at least A/R^2.
The study describes a cell structure for multisets in a rectangle.
problem Understanding the space of multisets in a rectangle.
method Developed a piecewise Euclidean bi-simplicial cell structure.
result Connected to spaces of complex polynomials and permutahedra.
Rectangles can fit on smooth curves, proving a theorem about Klein bottles.
problem Fitting rectangles on smooth curves.
method Shevchishin's theorem about Klein bottle embeddings.
result Similar rectangles can be placed on smooth Jordan curves.
Paper classifies pillow box isometric deformations preserving crease patterns.
problem Classifying pillow box isometric deformations preserving crease patterns.
method Continuous isometric deformations from pillow boxes to double rectangles, preserving crease patterns.
result Such deformations necessarily change pillow box topology.
We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension.
A new ensemble model uses simple hyper-rectangles to improve gradient boosting machine performance.
problem Improving gradient boosting machine performance and avoiding overfitting.
method Proposes a new ensemble model with axis-parallel hyper-rectangles as base models, integrates into GBM, and uses SHAP for interpretation.
result GBM with HRBMs can be an effective and interpretable model for regression and classification problems.
Efficiently recovers piecewise linear functions from noisy samples.
problem Recovering a piecewise linear function from noisy samples with unknown segmentation.
method Iterative merging approach for multidimensional segmented regression.
result First sample and computationally efficient algorithm in any fixed dimension.
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.
Equal diagonal energies proven on Liouville surfaces.
problem Diagonal energies on Liouville surfaces.
method Analyzing parameter curves and rectangles on Liouville surfaces.
result Diagonal energies are equal in n-dimensional Liouville manifolds.
The study generalizes origamis to flat surfaces, exploring their combinatorial and geometric properties.
problem Understanding the geometric and combinatorial properties of flat surfaces.
method Developing a system of linear equations to represent flat surfaces and studying their Veech groups.
result Veech groups of certain flat surfaces are included under a specific covering relation.
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.
While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvili…
In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to…
We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a 2-fold covering of S3 branched along a link. The condition implies that any thin meridional level surface in the link complement is incom…
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.
problem Classifying pseudo-Anosov homeomorphisms up to topological conjugacy.
method Algorithmic approach using geometric Markov partitions.
result Geometric type is a complete invariant of conjugation.
Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.
problem Understanding free path lengths on flat surfaces with circular obstacles.
method Proved the existence of a limiting distribution using radius of obstacles as a parameter.
result Relates the limiting distribution to heights of zippered rectangle decompositions.
We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical…
Hawksmoor's ceiling and Pantheon dome cofferings are explained using conformal mappings and Mercator's projection.
problem Understanding the design of curved ceiling cofferings using mathematical methods.
method Differential geometry and Mercator's projection of curved surfaces onto planes.
result The cofferings are conformal images of square tilings, ensuring right-angle intersections and square coffers.
Fast BATLLNN speeds up verification of TLL NNs by 400x.
problem Verifying output constraints for TLL NNs.
method Uses TLL architecture and decoupled box constraints to improve verification performance.
result 400x faster than state-of-the-art verifiers.
Given i.i.d samples from some unknown continuous density on hyper-rectangle [0,1]d, we attempt to learn a piecewise constant function that approximates this underlying density non-parametrically. Our density estimate is defined on a binary split of [0,1]d and built up sequentially according to discrepancy crite…
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.
The paper studies geometric structures of polynomial spaces.
problem Understanding the geometric and combinatorial structures of polynomial spaces.
method Introducing and analyzing finite piecewise Euclidean cell complexes.
result The branched rectangle and annulus complexes are homeomorphic to specific polynomial spaces.
We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.
problem Classifying essential annuli in genus two handlebody-knots.
method Introducing τ- and ρ-tangles and good rectangles, classifying these structures.
result Categorization of atoroidal 3-decomposable genus two handlebody-knots based on essential annuli.
Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.
We prove that for every smooth Jordan curve γ, if X is the set of all r∈[0,1] so that there is an inscribed rectangle in γ of aspect ratio tan(r⋅π/4), then the Lebesgue measure of X is at least 1/3. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedde…
ARGEN method improves variable selection and regularization in high-dimensional sparse models.
problem Constrained variable selection and regularization in high-dimensional sparse linear models.
method ARGEN penalty method, variable selection and regularization.
result ARGEN method has variable selection and estimation consistency under certain conditions.
Let R be a compact, connected, orientable surface of genus g with n boundary components with g≥2, n≥0. Let N(R) be the nonseparating curve graph, C(R) be the curve graph and HT(R) be the Hatcher-Thurston graph of R. We prove that if $λ: \mathcal{N}(R) \rightarro…
Proposes adaptive ridge regression for functional linear models with piecewise shapes.
problem Functional linear regression with unknown coefficient function.
method Adaptive piecewise function template with L2 penalization. result Improves predictive power and interpretability compared to standard methods.
Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle 6π. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …
Efficiently price VIX options using multilevel Monte Carlo in rough Bergomi model.
problem Pricing VIX options in a rough Bergomi model with high computational complexity.
method Combining rectangle discretization, Cholesky sampling, and multilevel Monte Carlo.
result Reduced computational complexity to O(ε−2log2(ε)) and asymptotically optimal O(ε−2). We consider the problem of learning a sparse rule model, a prediction model in the form of a sparse linear combination of rules, where a rule is an indicator function defined over a hyper-rectangle in the input space. Since the number of all possible such rules is extremely large, it has been computationally intractabl…
Study critical points of Laplace eigenfunctions in polygons.
problem Characterize critical points of Laplace eigenfunctions in polygonal domains.
method Analyze components of the critical set with codimension 1.
result For simply connected polygons, if a second Neumann eigenfunction has infinitely many critical points, the polygon must be a rectangle.
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special ca…
New integral defined for Hölder continuous functions, characterizing distributional volume forms.
problem Defining and characterizing a new integral for Hölder continuous functions.
method Constructing a distribution from Hölder continuous functions and using integral properties.
result Characterizes the Hölder regularity of the constructed distribution.
We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of t…