A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of…
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Square-tiled torimath.GT
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.
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.
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 of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of is studied in some detail.
On the square peg problemmath.GT
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.
Distance-squared mappingsmath.GT
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…
Arithmeticity of the Kontsevich--Zorich monodromies of certain families of square-tiled surfacesmath.DS
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.
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.
Square metrics is an important class of Finsler metrics. Recently, we introduced a special class of non-regular Finsler metrics called singular square metrics. The main purpose of this paper is to provide a necessary and sufficient condition for singular square metrics to be of constant Ricci or flag curvature when dim…
Cross validation residuals are well known for the ordinary least squares model. Here leave-M-out cross validation is extended to generalised least squares. The relationship between cross validation residuals and Cook's distance is demonstrated, in terms of an approximation to the difference in the generalised residual …
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 and the threshold for quadratic divergence in random right-angled Coxeter groupsmath.PR
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 \( p_c(n) = \sqrt{\sqrt{6}-2}/\sqrt{n} \).
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…
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.
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.
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 , namely, an -dimensional square metric is locall…
In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products and tensor squares . Using these results we prove that tensor squares of some groups with one relation a…
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.
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.
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.
Iterative Hessian sketch: Fast and accurate solution approximation for constrained least-squaresmath.OC
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…
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property…
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.
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.
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 …
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.
Evaluation of Neural Architectures Trained with Square Loss vs Cross-Entropy in Classification Taskscs.LG
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.
Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. Af…
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.
This is a survey article on distance-squared mappings and related topics.
In the following text we compute possible heights of (Alexandroff square), (unit square with lexicographic order topology) and (unit square with induced topology of Euclidean plane). We prove , $P_h(\m…
On Einstein square metricsmath.DG
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.
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.
This book introduces linear models and their theories rigorously.
problem Understanding linear models and their theories.
method Explains linear models from three perspectives, introduces maximum likelihood estimation, and proves least squares is the best unbiased linear model.
result Least squares is the best unbiased linear model in terms of mean squared error.
Proposes a method to compute the second Steenrod square for odd Khovanov homology.
problem Computing the second Steenrod square for odd Khovanov homology.
method Proposes a new method to compute the second Steenrod square, showing it to be a link invariant.
result Shows the proposed method gives a refinement of the Rasmussen s-invariant with coefficients.
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.
We study the asymptotic behaviour of doubly periodic instantons with square-integrable curvature. Then, we establish the equivalence given by the Nahm transform between the doubly periodic instantons with square integrable curvature and the wild harmonic bundles on the dual torus.
We introduce the implicitly constrained least squares (ICLS) classifier, a novel semi-supervised version of the least squares classifier. This classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-s…