The paper studies stable mappings of plane curves using distance-squared functions.
problem Stability of mappings of plane curves.
method Investigation of compositions of plane curves and generic distance-squared mappings.
result Stable mappings of plane curves are explored.
We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…
Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.
problem Exploring conditions for maximality of Hilbert square of real surfaces.
method Analyzing Hilbert square of maximal real surfaces and examining specific examples.
result Hilbert square can be maximal even for surfaces with disconnected real locus.
Directly simulates squared Bessel processes efficiently.
problem Simulating squared Bessel processes accurately and efficiently.
method Two-dimensional Chebyshev expansion for non-central chi-square distribution inverse.
result Accurate and efficient simulation for various degrees of freedom.
The paper studies counting problems on square-tiled surfaces.
problem Understanding the frequency of properties in square-tiled surfaces.
method Examining properties of the square torus and their implications in translation surfaces.
result Implications between properties and their frequency in translation surfaces.
CD converges linearly for MCP/SCAD penalized least squares.
problem Recovering sparse signals from data.
method Coordinate descent for MCP/SCAD penalized least squares.
result CD converges linearly to solutions of MCP/SCAD penalized least squares.
Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
Study on Nijenhuis tensor forms and vanishing properties.
problem Understanding the properties of Nijenhuis tensor.
method Provided strong and weak forms of the square of Nijenhuis tensor, and identified vanishing results.
result Identified new vanishing results for the square of Nijenhuis tensor.
Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group Γ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of Γ is studied in some detail.
Square can fit inside curves close to smooth ones.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappin…
The study finds arithmetic groups often in square-tiled surface monodromies.
problem Understanding arithmetic properties of square-tiled surfaces.
method Analyzing variations of Hodge structures and Kontsevich-Zorich monodromies.
result Arithmetic groups are frequent in low genus square-tiled surfaces.
The heights of Alexandroff square transformation groups are computed and proven.
problem Computing possible heights of Alexandroff square transformation groups.
method Analyzing the heights of transformation groups for Alexandroff square, unit square with lexicographic order, and unit square with Euclidean topology.
result Proven heights for transformation groups of Alexandroff square, unit square with lexicographic order, and unit square with Euclidean topology.
Cross validation residuals extended to GLS models.
problem Validating models with correlated data.
method Leave-M-out cross validation for GLS models, demonstrating relationship with Cook's distance.
result No need to refit model for reduced datasets.
New expressions for Nijenhuis tensor squares found.
problem Understanding Nijenhuis tensor squares.
method Expressed dual forms in terms of exterior derivative derivatives.
result New vanishing results for Nijenhuis tensor squares.
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…
A new method simulates square-root processes efficiently.
problem Simulating square-root processes accurately and efficiently.
method Simulate the integrated square-root process instead of the square-root process itself.
result High precision with low number of time steps, and exact limiting Inverse Gaussian distributions.
Paper preserves properties of mappings through generic mappings.
problem Preserving properties of mappings in higher dimensions.
method Composing generic generalized distance-squared mappings.
result Non-singular or injective properties preserved.
New method corrects least-squares temporal difference for better lambda-return estimation.
problem Improving lambda-return estimation in reinforcement learning.
method Uncorrected least-squares temporal difference with a correction method.
result Enhanced accuracy in temporal difference learning.
The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.
problem Calculating the Smith-Thom deficiency of Hilbert squares and conditions for maximality.
method Using Mayer-Vietoris mapping and rank calculations.
result Established necessary and sufficient conditions for maximality of Hilbert squares in projective complete intersections.
Square percolation determines threshold for group divergence in random graphs.
problem Threshold for quadratic divergence in random right-angled Coxeter groups.
method Square-graph analysis of random graphs to determine connectivity and divergence.
result Threshold probability for quadratic divergence is \( p_c(n) = \sqrt{\sqrt{6}-2}/\sqrt{n} \).
Formulae count square-tiled surfaces in genus two.
problem Counting square-tiled surfaces in genus two.
method Parametrized classification into four diagrams, provided formulae for enumeration.
result Formulae for enumeration of square-tiled surfaces in four diagrams, completing the count for genus two.
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…
Paper characterizes square metrics with vanishing Douglas curvature.
problem Characterizing square metrics with vanishing Douglas curvature.
method Using β-deformations to achieve analytical examples.
result Characterization of singular square metrics with vanishing Douglas curvature.
Square-root impact law confirmed for option trades.
problem Is market impact similar for stocks and options?
method Analyzed proprietary data of option trades.
result Square-root law holds for option markets.
New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.
problem Modeling STSs with specific horizontal restrictions.
method Modified model with conjugacy classes of permutations to restrict horizontal gluings.
result Asymptotic analysis of components, genus distribution, and saddle connections.
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics in dimension n≥3, namely, an n(≥3)-dimensional square metric is locall…
Illustrates interleaved learning with Kalman Filter for linear least squares.
problem Improving machine learning algorithms through interleaved learning.
method Simple statistical and optimization framework based on Kalman Filter.
result Demonstrates the effectiveness of interleaved learning.
Sum-of-squares proofs help solve complex estimation problems.
problem Recovering hidden parameters from high-dimensional distributions.
method Sum-of-squares proofs for polynomial systems.
result Sum-of-squares proofs can be used to solve polynomial systems efficiently.
Computes Steenrod squares on Khovanov homology for knots up to 11 crossings.
problem Computing Steenrod squares on Khovanov homology.
method Spatial refinements of even and odd Khovanov homology, computation of Sq^2.
result Steenrod squares Sq^2 on Khovanov homology spaces are determined for knots up to 11 crossings.
We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.
New proof of four squares theorem using projective geometry.
problem Proving every natural number is a sum of four squares.
method Projective differential geometry and modular forms.
result A new geometric approach to Lagrange's theorem.
We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance meas…
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
Researchers find a Steenrod square for link Floer homology.
problem Computing the second Steenrod square for link Floer homology.
method Explicitly framing moduli spaces and constructing a framed 1-flow category.
result An algorithm for computing the second Steenrod square for all grid homology versions.
Square loss performs comparably or better than cross-entropy in neural architectures for various tasks.
problem The superiority of cross-entropy loss over square loss in classification tasks is debated.
method Comparison of several neural architectures on NLP, ASR, and computer vision datasets using both loss functions.
result Square loss often produces better results in the majority of tasks, especially in NLP and ASR.
Guarantees uniform convergence for square-root Lipschitz losses.
problem Uniform convergence guarantees for square-root Lipschitz losses.
method Using Rademacher complexity and square root of scalar loss function Lipschitz constant.
result Generalizes previous results and handles non-smooth loss functions.
We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that …
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.
Investigates hypersurfaces in Finsler spaces with generalized square metrics.
problem Classifying and understanding hypersurfaces in Finsler spaces with generalized square metrics.
method Examined the generalized square metric F(x,y) = (α(x,y) + β(x,y))^(n+1)/(α^n(x,y)) and its application to Finslerian hypersurfaces.
result Established the classification and existence of first, second, and third kind of hyperplanes in the Finsler manifold.
A new neural network using chi-square test for binary classification.
problem Improving binary classification accuracy.
method Backpropagation neural network with chi-square test redefined cost and error functions.
result Significantly improved classification accuracy compared to related approaches.
In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.
This is a survey article on distance-squared mappings and related topics.