We consider a pseudo-Riemannian metric that changes signature along a smooth curve on a surface, called the discriminant curve. The discriminant curve separates the surface locally into a Riemannian and a Lorentzian domain. We study the local behaviour and properties of geodesics at a point on the discriminant where th…
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48 results for “discriminant surface”
Study infinite Euclidean distance discriminants of algebraic varieties.
problem Understanding the structure of data points with infinitely many critical points in Euclidean distance correspondence.
method Developed computer code to compute discriminants and proved properties of fibers.
result Infinite Euclidean distance discriminants contain all data points with infinitely many critical points for the nearest-point problem.
Permutation of Weierstrass points on Veech surfaces in classified by discriminant.
problem Classifying permutations of Weierstrass points on Veech surfaces in based on discriminant.
method Analyzing the permutation group induced by the affine group on Weierstrass points, considering both affine and Dehn multitwists.
result The permutation group is , , or depending on the discriminant value.
Study on the topology of twistor discriminant loci in 4-sphere.
problem Topology of twistor discriminant loci of algebraic surfaces.
method Classical algebraic geometry techniques applied to twistor fibration.
result Real dimension of twistor discriminant locus is always 2 except for two exceptional cases.
The study classifies points on ruled surfaces in 4-space based on geometric properties.
problem Characterizing points on smooth ruled surfaces in 4-space.
method Contact with transverse planes, binary differential equations, and projective transformations.
result Parabolic points on ruled surfaces in 4-space can be classified as butterfly hyperbolic, parabolic, or elliptic based on the discriminant of a binary differential equation.
Lower-dimensional video discriminators improve GAN performance.
problem High curvature in unconstrained video discriminator loss surfaces.
method Proposed Lower-Dimensional Video Discriminators (LDVD GANs).
result LDVD GANs double Temporal-GAN performance and achieve state-of-the-art performance.
This paper explores dual relations between line congruences and surfaces in 3D and 4D.
problem Understanding the duality between line congruences in and surfaces in .
method Analyzes the correspondence between principal lines and asymptotic lines, ridge curves and flat ridge curves, and subparabolic curves.
result Discusses the behavior of subparabolic curves at the discriminant curve of the line congruence and the parabolic curve of the dual surface.
We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and…
Connected Prym eigenforms found in a specific mathematical space.
problem Understanding the structure of Prym eigenforms in a particular mathematical space.
method Proving the connectedness of Prym eigenform loci and classifying square-tiled surfaces.
result The projection of Prym eigenform loci is a single Teichmüller curve.
We show that for every nonelementary representation of a surface group into there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.
Paper enhances haptic signals distinguishability with boosted technique.
problem Lack of large datasets in haptics domain limits feature extraction.
method General framework for haptic signal analysis, using spectral features and boosted embedding.
result Framework needs less training data and outperforms state-of-the-art.
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
We describe the structure of the asymptotic lines near an inflection point of a Lagrangean surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a comp…
We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recen…
We describe topologically the discriminant locus of a smooth cubic surface in the complex projective space that contains 5 fibres of the projection .
Algorithm finds periodic points on Veech surfaces.
problem Finding periodic points on non-square-tiled Veech surfaces.
method Developed an algorithm to compute periodic points.
result Proved that in low discriminant, non-square-tiled Veech surfaces have no periodic points, except for fixed points of the Prym involution.
Study bifurcations of curves on surfaces in Minkowski 3-space.
problem Understanding the behavior of curves on surfaces in Minkowski 3-space.
method Analyzing the degeneracy of induced pseudo metric, discriminant of principal curvatures, parabolic curve, and mean curvature vanishing points.
result Bifurcations of robust features on surfaces in Minkowski 3-space.
Deep clustering models are shown to be equivalent to K-means under certain conditions.
problem The challenge of unsupervised deep learning and clustering.
method The study proves the equivalence of recent discriminative models and K-means under specific conditions and parameters.
result Maximizing the L2 regularized mutual information is equivalent to a soft and regularized K-means loss.
Evolving Affine Evolutoidsmath.DG
The paper explains how curves evolve from one envelope to another using singularity theory.
problem Understanding the evolution of curves from one envelope to another.
method Singularity theory applied to plane curves without inflexions.
result Existence of the first slope at which singularities occur in the discriminant surface.
Paper develops a classification method using matrix-variate t-distributions.
problem Classifying matrix-valued observations with dependence structure.
method Develops an Expectation-Maximization algorithm for discriminant analysis.
result Method shows promise on various datasets.
Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfacesmath.AG
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
New inequality for K3 surfaces connects systoles and volumes.
problem Finding a constant relating systoles and volumes on K3 surfaces.
method Introducing stability conditions and proving inequalities on K3 surfaces.
result A systolic inequality for K3 surfaces with a constant C.
The paper studies skew ruled surfaces with specific normalization properties.
problem Characterizing skew ruled surfaces with relative normalizations.
method Analyzes skew ruled surfaces in with a specific form of support function.
result Investigates properties of Tchebychev and support vector fields for these surfaces.
For a generic embedding of a smooth closed surface into , the subset of which is the affine -equidistant of appears as the discriminant set of a stable mapping , hence their stable singularities are and . In this paper…
This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces …
Study extends symmetries of sphere points to surface mapping classes.
problem Understanding symmetries of points on spheres and their connections.
method Establishing isomorphisms between moduli spaces and mapping class groups.
result Finitely many integral braid group orbits in rank 4 Stokes matrices.
Study shows infinite kernels in topological monodromy for curve families.
problem Understanding kernels of topological monodromy representations.
method Extending Kuno's arguments and using Carlson-Toledo techniques.
result Kernels are infinite for certain linear systems on surfaces.
The paper calculates spectral determinants for two complex surfaces.
problem Calculating spectral determinants for complex surfaces.
method Closed explicit formulas, multiplicative relations, Belyi maps, and constant-curvature spheres.
result Spectral determinants of the Bolza surface and Klein quartic are calculated.
QC-SPHARM detects Alzheimer's Disease early using hippocampal surface geometry.
problem Early detection of Alzheimer's Disease (AD) using hippocampal surface geometry.
method Spherical harmonics registration, conformality and curvature distortions quantification, t-test feature selection, SVM classification.
result 85.2% testing accuracy on ADNI data, 81.2% on aMCI progression data.
Let V be a finite dimensional complex vector space and V^* its dual and let X in P(V) be a smooth projective variety of dimension n and degree d at least two. For a generic n-tuple of hyperplanes H_1,...,H_n in P(V^*)^n, the intersection of X with H_1,...,H_n consists of d distinct points. We define the "discriminant o…
Study examines curvature of Higgs bundle moduli space for large fields.
problem Analyzing curvature of Higgs bundle moduli space for large Higgs fields.
method Determined asymptotic behavior of sectional curvatures using hyperkähler metric.
result Leading order sectional curvatures are Dirac type contributions with explicit expressions.
Study of algebraic curves and surfaces in flag manifold using twistor geometry.
problem Understanding algebraic curves and surfaces in the flag manifold and their properties.
method Analysis of algebraic curves and surfaces in the flag manifold using twistor projection and anti-holomorphic involution.
result Bounds on the number of twistor fibres contained in algebraic surfaces of the flag manifold.
Paper tackles discrimination in predictions using causal modeling.
problem Mathematical guarantee for non-discrimination in predictions.
method Causal modeling to define discrimination, bounding prediction discrimination probability.
result Discrimination in predictions can still exist even if training data is free of discrimination.
Given a surface F, we are interested in Z/2 valued invariants of immersions of F into R^3, which are constant on each connected component of the complement of the quadruple point discriminant in Imm(F,R^3). Such invariants will be called ``q-invariants.'' Given a regular homotopy class A in Imm(F,R^3), we denote by V_n…
The study quantifies and compares aleatoric and epistemic discrimination in ML models.
problem Sources of discrimination in ML models and their impact on performance.
method Quantifying aleatoric and epistemic discrimination using statistical experiments and model accuracy.
result State-of-the-art fairness interventions are effective at removing epistemic discrimination but not aleatoric discrimination in datasets with missing values.
Slice-polynomial functions help compute twistor discriminant loci of cubic scrolls.
problem Computing the twistor discriminant locus of cubic scrolls in .
method Introduced slice-polynomial functions and their companions/extensions, used twistor theory.
result Generically constant cardinality of pre-images for slice-polynomial functions.
Geotagged data can be used to describe regions in the world and discover local themes. However, not all data produced within a region is necessarily specifically descriptive of that area. To surface the content that is characteristic for a region, we present the geographical hierarchy model (GHM), a probabilistic model…
The paper studies vector bundles over surfaces, focusing on singularity formation.
problem Understanding singularity formation in rank two holomorphic vector bundles over surfaces.
method Defining fertile families bearing bubbles and using elementary modifications to prove their existence.
result Existence of fertile families bearing bubbles for certain types of vector bundles.
Discriminator guidance improves autoregressive diffusion models for generating molecular graphs.
problem Improving the accuracy of autoregressive diffusion models for generating molecular graphs.
method Deriving ways to use a discriminator with a pretrained generative model in the discrete case, including optimal and sub-optimal scenarios.
result Using a discriminator can correct pretrained models and improve exact sampling from the data distribution.
This paper proposes a curriculum learning method for GANs using multiple discriminators.
problem Training GANs with sufficient convergence conditions and avoiding mode collapse.
method A framework for training the generator against an ensemble of discriminator networks, formalized in the full-information adversarial bandit framework.
result Our approach improves samples quality and diversity over existing baselines by effectively learning a curriculum.
Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane to M at a point p. Those vertices trace out a set, called the vertex set of M through p. We take p to be an isolated umbilic point on M and descr…
Solves GAN mode collapse by assigning minibatches to multiple discriminators.
problem Mode collapse in GANs where models generate similar samples.
method Multiple discriminators, microbatching, and changing tasks.
result Promotes sample diversity in generated sets.
Kernel discriminant analysis uses nonlinear embeddings to improve classification.
problem Limited effectiveness of linear discriminant analysis in capturing nonlinear features.
method Study of nonlinear embeddings in kernel discriminant analysis using polynomial and Gaussian kernels, solving generalized eigenvalue problems.
result Polynomial and Gaussian discriminants capture class differences through population moments and randomized projections.
Discriminative clustering uses mutual information to cluster data.
problem Clustering data into cohesive groups.
method Discriminative clustering using mutual information.
result Mutual information has been a cornerstone of discriminative clustering.
Generative adversarial networks enforce no-arbitrage in volatility surface computation.
problem Efficiently compute volatility surfaces without arbitrage violations.
method Generative adversarial network (GAN) with no-arbitrage constraints.
result Proposed GAN model outperforms ANN approaches in accuracy and computational time.
Pairwise discriminators stabilize adversarial training by ensuring the generator's alignment is preserved.
problem Stability issues in adversarial training when using sub-optimal discriminators.
method Introducing a family of objectives using pairwise discriminators, ensuring the generator's alignment is preserved.
result Only the generator needs to converge, and the alignment is preserved with any discriminator.
PBN combines generative and discriminative capabilities in a neural network.
problem Combining generative and discriminative capabilities in neural networks.
method Convolutional PBN, sharing FF-NN embodiment, combining generative and discriminative qualities.
result PBN shows excellent qualities from either generative or discriminative viewpoint.
Study on GANs under and overfitting using discriminator unseen data.
problem Understanding and mitigating under and overfitting in GANs.
method Using discriminators unseen by the generator to measure generalization.
result Model capacity of the discriminator affects generator quality; large discriminator capacities do not prevent overfitting.