Solves Fu-Yau equation for negative slope parameters in arbitrary dimensions.
problem Solving the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
method Solves the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
result First non-trivial solutions of the Fu-Yau equation in any dimension strictly greater than 2.
Study shows existence of Strominger system solutions is not stable under complex structure deformations.
problem Stability of Strominger system solutions under complex structure deformations.
method Analyzes stability of solutions to the Strominger system in dimensions six, considering both positive and negative slope parameters.
result Existence of solutions to the Strominger system is neither open nor closed under holomorphic deformations of the complex structure.
We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.
We construct Lefschetz fibrations over S2 which do not satisfy the slope inequality. This gives a negative answer to a question of Hain.
The paper constructs metrics with negative curvature on complex manifolds.
problem Constructing complete Kähler metrics with negative bisectional curvature on hyperbolic complex manifolds.
method Introducing a mechanism for constructing complete Kähler metrics with negative bisectional curvature.
result Realized Chern slopes c12/c2 for surfaces with negative holomorphic sectional curvature. 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| +…
Study Fu-Yau equation on astheno-Kähler manifolds, proving existence and uniqueness.
problem Existence and uniqueness of solutions to Fu-Yau equation on astheno-Kähler manifolds.
method Analysis of Fu-Yau equation on compact Hermitian manifolds, proving existence and uniqueness for astheno-Kähler manifolds.
result Proved existence and uniqueness of solutions for Fu-Yau equation on astheno-Kähler manifolds.
We show that, for any given 3-manifold M, there are at most finitely many hyperbolic knots K in the 3-sphere and fractions p/q (with q > 22), such that M is obtained by p/q surgery along K. This is a corollary of the following result. If M is obtained by Dehn filling the cusps of a hyperbolic 3-manifold X, where each f…
The paper proves a slope equality for smooth plane curve fibrations and applies it to Durfee's conjecture.
problem Proving Durfee's conjecture for isolated hypersurface singularities.
method Using slope equality for fibered surfaces with smooth plane curve fibers.
result The strong Durfee-type inequality for isolated hypersurface singularities is proven, implying Durfee's conjecture.
Proving a conjecture for a specific type of knots.
problem Relating knot polynomial degrees to surface slopes.
method Using state-formula and Hatcher-Oertel algorithm.
result Matches parameters of knot polynomials with surface parameters.
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. 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.
The study bounds slopes for Dehn fillings of two-bridge knots with hyperbolic representations.
problem Bounding slopes for Dehn fillings of two-bridge knots with hyperbolic representations.
method Combining the Riley polynomial with Khoi's surgery-slope formula, and analyzing meridian and longitude translation parameters.
result The set of surgery slopes admitting hyperbolic PSL(2,R) representations is bounded. We give a formula of the Donaldson-Futaki invariants for certain type of semi test configurations, which essentially generalizes Ross-Thomas' slope theory. The positivity (resp. non-negativity) of those "a priori special" Donaldson-Futaki invariants implies K-stability (resp. K-semistability). We show its applicability…
In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evalu…
Bayesian SLOPE combines SLOPE with Bayesian methods for linear regression.
problem Estimating regression coefficients with variable selection and prediction properties.
method Introduces Bayesian SLOPE by treating the SLOPE estimate as the posterior mode and providing a Gibbs sampler for full Bayesian inference.
result Offers credible sets and standard error estimates for parameters, facilitating Bayesian prediction.
The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a …
New examples of positive twisted torus knots with same surface slope found.
problem Finding new examples of positive twisted torus knots with same surface slope.
method Extending Dean's conditions and using new four-parameter family.
result Four-parameter family of positive twisted torus knots with same surface slope.
The study examines methods to correct measurement error in nutritional epidemiology studies.
problem Measurement error in nutritional studies leads to biased and underconfident estimates.
method The article reviews various bias-correction models for exposure variables in nutritional epidemiology.
result Bias-correction methods are essential for accurate inference in nutritional studies.
Unified Kähler-Ricci and Anomaly flows with new formulation.
problem Unified Kähler-Ricci and Anomaly flows.
method New formulation of Anomaly flow with vanishing slope parameter, unifying with Kähler-Ricci flow.
result Unified Kähler-Ricci and Anomaly flows with new formulation.
SLOPE outperforms LASSO in low noise scenarios but is suboptimal in large noise scenarios.
problem Sparse linear regression with high-dimensional data.
method Characterized SLOPE's estimation error under specific conditions and compared it with LASSO and bridge regression.
result SLOPE is optimal for low noise scenarios but suboptimal in large noise scenarios.
Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, F, in S3; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yield…
We give examples of smooth surfaces with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kahler classes that do not admit any constant scalar curvature Kahler metrics. We also compare this to the work of Song-Weinkove on the J-flow.
Investment strategies derived from commodity futures curves exploit dynamics in price movements.
problem Modeling and predicting the term structure of commodity futures prices.
method Employed the Nelson-Siegel framework to model term structure, and developed investment strategies based on changes in slope and curvature parameters.
result Significant profits generated from systematic strategies based on the change in slope, unrelated to risk factors and robust to transaction costs.
This paper proposes Dropping Networks for improved transfer learning in natural language understanding tasks.
problem Transfer learning between natural language understanding tasks often suffers from negative transfer.
method Combines Dropout and Bagging (Dropping) for improved transferability in neural networks.
result Improves transfer learning performance and comparable results to state-of-the-art using a fraction of target task data.
Anomaly flow solves Fu-Yau equation on toric fibrations.
problem Existence and uniqueness of solutions for Fu-Yau equation.
method Anomaly flow approach, convergence on toric fibrations.
result First case where Anomaly flow exists for all time.
This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed …
Nested model averaging improves high-dimensional linear regression performance.
problem High-dimensional linear regression with predictor ordering impact.
method Combining model averaging with regularized estimators on the solution path.
result Nested model averaging with lasso and SLOPE outperforms competing methods.
New property helps show many knot fillings are not left-orderable.
problem Non-left-orderability of knot groups after Dehn filling.
method Introducing conjugate-slope property to study knot groups.
result The conjugate-slope property implies non-left-orderability of many knot fillings.
The anomaly flow on a complex 3-fold is studied with integral Shi-type estimates and long-time existence conditions.
problem Long-time existence of the anomaly flow on a compact complex 3-fold.
method Integral Shi-type estimates adapted from integration-by-parts arguments, with a smallness condition on the slope parameter.
result Long-time existence of the anomaly flow on a compact complex 3-fold under a smallness condition on the slope parameter.
We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-…
Let M be a compact, connected, orientable, irreducible 3-manifold and T an incompressible torus boundary component of M such that the pair (M,T) is not cabled. In the paper "Toroidal and Klein bottle boundary slopes" [arXiv:math/0601034] by the author it was established that for any K-incompressible tori F,F' in (M,T) …
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
problem Understanding Legendrian isotopy in cable links of uniformly thick knots.
method Introduced new technique of Legendrian surgeries to classify Legendrian knots in negative cables of twist knots.
result Found new phenomena of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.
We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we ve…
Let M be a compact, connected, orientable, irreducible 3-manifold and T' an incompressible torus boundary component of M such that the pair (M,T') is not cabled. By a result of C. Gordon, if S and T are incompressible punctured tori in M with boundary on T' and boundary slopes at distance d, then d is at most 8, and th…
Consider genus g curves that admit degree d covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family Y that naturally maps into the moduli space of stable genus g curves Mˉg. We study the geometry of Y, and pr…
The Slope Conjecture is verified for a specific family of Montesinos knots.
problem Relating the degree of the colored Jones polynomial to boundary slopes of knots.
method Verification for a specific family of Montesinos knots with given conditions.
result The Slope Conjecture and Strong Slope Conjecture are confirmed for the specified knots.
Adam optimization algorithm can have non-zero average regret under certain conditions.
problem Non-zero average regret in Adam optimization algorithm.
method Used a three-periodic sequence of linear functions on [-1,1] with slopes c, -1, -1, and analyzed Adam variants.
result Adam optimization algorithm can have non-zero average regret under certain conditions.
Proves conjectures for pretzel knots using polynomial degrees.
problem Proving conjectures about pretzel knots.
method Using Hatcher-Oertel algorithm and colored Jones polynomial.
result Maximal degrees of colored Jones polynomial determine boundary slopes.
New method optimizes neural network learning by adjusting random parameters to target function features.
problem Difficulty in setting optimal random parameters for neural network learning.
method Adjusts sigmoid slopes and positions to target function features in a randomized learning method.
result Significantly better approximation of complex target functions compared to standard methods.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
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.
New method corrects Laplace/BIC errors in singular models, revealing effective dimension.
problem Laplace/BIC errors in singular models due to incorrect effective dimension assumption.
method RLCT (real log canonical threshold) to correct effective dimension in linear models.
result Correct evidence slope and effective dimension estimation in linear settings.
The growth rate of real GDP per capita in the biggest OECD countries is represented as a sum of two components - a steadily decreasing trend and fluctuations related to the change in some specific age population. The long term trend in the growth rate is modelled by an inverse function of real GDP per capita with a con…
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.
The study identifies characterising slopes for hyperbolic knots and proves stronger results for L-space knots.
problem Characterizing slopes of hyperbolic knots and their properties.
method Combining Lackenby's proof and Heegaard Floer homology genus bounds.
result All but finitely many slopes are characterising for hyperbolic knots and L-space knots.
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.
Locally adaptive activation functions boost deep and physics-informed neural networks.
problem Improving the performance and training speed of deep and physics-informed neural networks.
method Layer-wise and neuron-wise locally adaptive activation functions with a slope recovery term.
result The proposed methods accelerate convergence and reduce training cost.