Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
Study calculates slope gaps on polygon surfaces, finding non-unimodal distributions.
problem Understanding the distribution of slope gaps on polygon surfaces.
method Explicit computation of slope gap distributions for 2n-gons, providing bounds on non-differentiability points.
result Slope gap distributions are not always unimodal, answering a question by Athreya.
We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.
problem Understanding the slope gap distribution of Veech surfaces.
method Explicit parameterization of a Poincaré section to the horocycle flow, finiteness result for the first return map.
result The limiting gap distribution of slopes of saddle connections on Veech surfaces is piecewise real-analytic with finitely many points of non-analyticity and has quadratic tail decay.
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
The paper calculates gap distributions for translation surfaces, focusing on the double heptagon.
problem Calculating gap distributions for translation surfaces.
method Describes a procedure to find winning holonomy vectors and applies it to the double heptagon.
result Explicitly computed gap distribution for the regular double heptagon translation surface.
We give an explicit formula for the limiting gap distribution of slopes of saddle connections on the golden L, or any translation surface in its SL(2, R)-orbit, in particular the double pentagon. This is the first explicit computation of the distribution of gaps for a flat surface that is not a torus cover.
We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré se…
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
We construct a Poincaré section for the horocycle flow on the modular surface SL(2,R)/SL(2,Z), and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…
Study slopes of direct images in complex manifolds, proving a Mehta-Ramanathan type theorem.
problem Distribution of Harder-Narasimhan slopes in direct image sheaves.
method Analyzing asymptotic distributions of slopes under base changes of families of complex projective manifolds.
result Asymptotic distribution of slopes can be recovered from base changes over generic curves.
The paper improves precision matrix estimation by SLOPE, especially in high-dimensional settings.
problem Estimating precision matrices with structured edge patterns.
method Graphical SLOPE, focusing on sparsity and cluster recovery.
result The method converges to the optimal solution and accurately identifies cluster structures.
Neural networks' slope controls their generalization.
problem Controlling overfitting in neural networks.
method Control the slope of neural networks to avoid memorization.
result The slope of a well-trained neural network is generally independent of network width and architecture.
Modeling risk and performance with Levy-stable distributions.
problem Understanding risk and performance in financial markets with non-Gaussian distributions.
method Developed a finite-horizon model using Levy-stable scaling, identified parameters from data, derived formulas for various financial ratios.
result Horizon-correct formulas for risk measures are derived and validated across different horizons.
The CAP slope is Bayes' theorem in cumulative coordinates, unlocking the weight of evidence, Somers' D, and Gini coefficient.
problem Calibration and weight of evidence
method Identifying the CAP slope as Bayes' theorem
result Revealing the CAP slope as Bayes' theorem
The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.
Study slopes in 3-manifolds, proving conjectures about knots.
problem Understanding slopes in 3-manifolds and their implications for knots.
method Upper bounds on distances between slopes, applications to knots and surgeries.
result Bounds on boundary and degeneracy slopes for knots in 3-manifolds.
Using the Hatcher-Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such knots determine a boundary slope as predicte…
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies th…
Study provides concrete examples of knot slopes.
problem Finding explicit characterizing slopes for knots.
method Concrete examples for the (-2,3,7)-pretzel knot.
result Explicit characterizing slopes for the knot 12n242. 2-level SLOPE improves high-dimensional inference with fewer hyperparameters.
problem High-dimensional linear regression with complex penalty sequences.
method 2-level SLOPE with three hyperparameters, reducing penalty space.
result Sharp trade-off between TPP and FDP, computationally efficient.
Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the m…
A slope p/q is a characterising slope for a knot K in S3 if the oriented homeomorphism type of p/q-surgery on K determines K uniquely. We show that when K is a hyperbolic knot its set of characterising slopes contains all but finitely many slopes p/q with q≥3. We prove stronger results for hyper…
New research shows that many slopes are characterizing for satellite knots.
problem Characterizing slopes for satellite knots.
method Detailed examination of the JSJ decomposition of a surgery along a knot, combined with other authors' constraints on surgery slopes.
result Many non-integral slopes are characterizing for composite knots.
The Slope Conjecture relates the degree of the colored Jones polynomial to the boundary slopes of a knot. We verify the Slope Conjecture and the Strong Slope Conjecture for Montesinos knots M(r1,s−u11,t1) with r,u,t odd, s even and u≤−1, r<−1<1<s,t.
New slopes identified for torus knots, improving previous results.
problem Identifying characterizing slopes for torus knots.
method Applying recent work on L-space surgeries, improving previous bounds.
result Every nontrivial L-space slope of T5,2 is characterizing for T5,2. A non-trivial slope r on a knot K in S3 is called a characterizing slope if whenever the result of r-surgery on a knot K′ is orientation preservingly homeomorphic to the result of r-surgery on K, then K′ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with $|p| +…
In many physical, social or economical phenomena we observe changes of a studied quantity only in discrete, irregularly distributed points in time. The stochastic process used by physicists to describe this kind of variables is the Continuous Time Random Walk (CTRW). Despite the popularity of this type of stochastic pr…
New knots found with specific slope properties.
problem Finding knots with non-integral boundary slopes.
method Kabaya's method for boundary slopes and layered solid torus construction.
result Existence of knots with boundary slopes of any positive denominator.
New rules reduce SLOPE model fitting time by screening out irrelevant variables.
problem Expensive tuning of regularization parameter in penalized regression models.
method Strong screening rules for group-based SLOPE models.
result Significant acceleration of fitting process for Group SLOPE and sparse-group SLOPE.
Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict bound…
The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions.
Let r_m and r_M be the least and greatest finite boundary slopes of a hyperbolic knot K in S^3. We show that any cyclic surgery slopes of K must lie in the interval (r_m - 1/2, r_M + 1/2).
Study Mazur doubles of knots and their relation to the Slope Conjecture.
problem Proving the Strong Slope Conjecture for Mazur doubles of knots.
method Analyzing Mazur doubles under certain hypotheses and using the Slope Conjecture.
result Mazur doubles of knots satisfy the Strong Slope Conjecture under certain conditions.
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for L-space knots. Proves existence of Lefschetz fibrations with arbitrary slopes.
problem Finding Lefschetz fibrations with specific slopes.
method Proves existence for rational numbers in (2,8) with large genus.
result Arbitrary slopes (r in (2,8)) are possible for genus-g Lefschetz fibrations.
The paper examines slopes and their norms in exceptional Dehn fillings.
problem Understanding slopes and norms in exceptional Dehn fillings.
method Investigates the relationship between Euclidean length and Culler-Shalen norm on horotori.
result Establishes two inequalities between Euclidean length and Culler-Shalen norm.
The paper extends logistic regression for unbounded majority classes and derives asymptotic properties.
problem Infinitely imbalanced logistic regression inference.
method Derive a second order expansion for slope parameter under unbounded majority class.
result The second order term converges to a normal distribution with a variance depending only on the minority class's mean.
This paper completes proofs for left orderable slopes of double twist knots.
problem Proving the left orderability of slopes for double twist knots.
method Continuous families of hyperbolic SL2(R) representations of knot groups.
result Any slope in (-4n, 4m) is a left orderable slope of C(2m+1, -2n).
A slope qp is called a characterizing slope for a given knot K0 in S3 if whenever the qp-surgery on a knot K in S3 is homeomorphic to the qp-surgery on K0 via an orientation preserving homeomorphism, then K=K0. In this paper we try to find characterizing slopes for torus knots $…
Efficient packages solve SLOPE problem in multiple languages.
problem Efficiently solving SLOPE problem for various data types.
method Hybrid coordinate descent algorithm for GLMs with multiple loss functions.
result Packages outperform existing SLOPE implementations in speed.
New invariant slope for links helps complete signature formula.
problem Signature formula for splice of links.
method Developed slope invariant and computed it using various methods.
result Established close relation to Conway polynomials and Kojima-Yamasaki η-function.
Formula found for energy slope in complex geometry.
problem Calculating the asymptotic slope of a K-energy.
method Established a formula for the asymptotic slope.
result Found a formula for the asymptotic slope of α-K-energy.
No characterizing slopes for multi-component links.
problem No characterizing slopes for multi-component links.
method Demonstrated through infinitely many examples of non-homeomorphic complements.
result Infinitely many links with non-homeomorphic complements.
SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted l1 penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this pa…
The paper studies slopes for knot fillings with left-orderable fundamental groups.
problem Understanding slopes for knot fillings with left-orderable fundamental groups.
method Using the Riley polynomial and root analysis, the paper computes and conjectures on slopes.
result The paper computes the range of rational slope r for left-orderable fillings of two-bridge knots. We describe an algorithm for computing boundary slopes of 2-bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.
A new fast algorithm solves SLOPE optimization problem.
problem Inefficient algorithms for SLOPE optimization in high dimensions.
method Combines proximal gradient descent and proximal coordinate descent steps.
result Our method outperforms competing algorithms in benchmarks.