Paper introduces Chern minimal surfaces in Hermitian surfaces and establishes identities related to their points and bundles.
problem Understanding the properties of Chern minimal surfaces in Hermitian surfaces.
method Using the Chern connection, the paper introduces Chern minimal surfaces and establishes identities related to their points and bundles.
result Established two identities relating the orders of complex and anticomplex points, the cap product of pull-back of first Chern class, and the Euler characteristics of tangent and normal bundles.
Estimator calculates surface curvature from point cloud samples.
problem Accurately estimating curvature from limited point cloud data.
method Algorithm using probability distribution and nearby points control.
result Controlled number of points ensures accurate curvature estimation.
New tools for constructing fixed point sets in digital topology.
problem Constructing fixed point sets in digital topology.
method Defining excludable points and articulation points, and showing their exclusion from freezing sets.
result Excludable points and articulation points can be excluded from all freezing sets.
While Multiple Instance (MI) data are point patterns -- sets or multi-sets of unordered points -- appropriate statistical point pattern models have not been used in MI learning. This article proposes a framework for model-based MI learning using point process theory. Likelihood functions for point pattern data derived …
Study on connection points on double regular polygons, providing coordinates and proving non-connection points.
problem Identifying connection points on double regular polygons.
method Examined coordinates in trace field, provided constructive proof for prime n. result For n=7, conjectured all remaining points are connection points; for n≥7 prime, provided explicit separatrix. New families of translation surfaces with multiple oblivious points discovered.
problem Identifying points on translation surfaces without nearby closed geodesics.
method Constructing new families of translation surfaces and proving existence in higher genera.
result Translation surfaces in every genus ≥3 have at least one oblivious point.
This paper reverses a construction by merging boundary critical points into an interior one.
problem Pushing interior critical points to the boundary and splitting them into two boundary points.
method Specific assumptions allow merging two boundary critical points into one interior critical point.
result Merging two boundary critical points into a single interior critical point.
A new model for point processes without intensity function trade-offs.
problem Inefficiency and trade-offs in existing point process models.
method Point Set Diffusion, a diffusion-based latent variable model.
result Achieves state-of-the-art performance in point process generation.
PINNACLE optimizes point selection for PINNs, improving accuracy.
problem Challenges in selecting points for training Physics-Informed Neural Networks (PINNs).
method Introduces PINNACLE, an algorithm that jointly optimizes collocation and experimental points selection, adjusting point proportions dynamically.
result PINNACLE outperforms existing methods in forward, inverse, and transfer learning problems.
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.
Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorith…
Investigates point spectra of vector fields and their properties.
problem Understanding the point spectra of vector fields.
method Define and study point spectra, prove properties under isometries, and analyze compactly supported fields.
result Point spectra are well-behaved under isometries and trivial for compactly supported fields.
We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we pr…
Hard to approximate critical points for simple nonconvex functions.
problem Approximating critical points of nonconvex functions.
method Proving hardness results for polynomial-time approximation of critical points.
result Proving that approximating critical points is intractable for simple nonconvex functions.
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.
Estimating boundaries from point clouds with improved accuracy and rigorous error estimates.
problem Identifying the boundary of a domain from point cloud samples.
method Developed new estimators for normal vectors, distances, and boundary tests; provided error estimates.
result Efficient and accurate estimators for boundary properties on point clouds.
The paper classifies circle actions on 6D manifolds with isolated fixed points.
problem Classifying circle actions on 6D manifolds with isolated fixed points.
method Performing equivariant connected sums at fixed points with specific manifolds.
result A sequence of operations can reduce the fixed point data to the empty collection.
PoPPy is a Point Process toolbox based on PyTorch, which achieves flexible designing and efficient learning of point process models. It can be used for interpretable sequential data modeling and analysis, e.g., Granger causality analysis of multi-variate point processes, point process-based simulation and prediction of…
The paper models user-advertiser interactions using point processes.
problem Causal inference problems in user-advertiser interaction.
method Temporal marked point processes and neural point processes.
result Neural point processes as practical solutions.
A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point…
The paper develops methods to create private synthetic spatial point patterns.
problem Generating private synthetic spatial point patterns.
method Developed differentially private Poisson and Cox point synthesizers.
result The synthesizers effectively maintain privacy and utility of synthetic data.
Self-focal points on ellipsoids of dimension 3 or higher are rare.
problem Existence of self-focal points on Riemannian manifolds of dimension 3 or higher.
method Analyzing geodesics and umbilic points on ellipsoids of various dimensions.
result Ellipsoids of dimension 3 or higher with at least 4 distinct axes have no self-focal points.
We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.
The paper introduces a method to probabilistically select inducing points in sparse Gaussian processes.
problem The challenge is selecting the optimal number of inducing points in sparse Gaussian processes.
method A point process prior is applied to the inducing points, and the posterior is approximated using stochastic variational inference.
result The model learns which and how many inducing points to use, leading to fewer inducing points being preferred as they become less informative.
Groups with special properties always have fixed points.
problem Groups acting on finite CW-complexes without fixed points.
method Exhibited specific groups with strong fixed-point properties.
result Groups with finite generation and torsion-freeness have global fixed points.
The paper proves Γ-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional. result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
The study confirms a conjecture about critical points of smooth functions.
problem Understanding isolated critical points of smooth functions.
method Investigated cone-like, reasonable, and Rothe H hypothesis critical points.
result The conjecture holds true for certain critical points.
New method upsamples sparse, non-uniform point clouds more accurately.
problem Suboptimal results from existing point cloud upsampling methods.
method Imposes manifold distribution constraints using Gaussian functions.
result Generates higher-quality, more uniformly distributed dense point clouds.
Characterizes Lebesgue points using nearest neighbor methods.
problem Consistency of classification algorithms based on nearest neighbors.
method Characterization of Lebesgue points via 1-Nearest Neighbor regression.
result Proves convergence of 1-Nearest Neighbor classification algorithms in metric spaces.
We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.
We propose to explain the predictions of a deep neural network, by pointing to the set of what we call representer points in the training set, for a given test point prediction. Specifically, we show that we can decompose the pre-activation prediction of a neural network into a linear combination of activations of trai…
In recent decades, the use of 3D point clouds has been widespread in computer industry. The development of techniques in analyzing point clouds is increasingly important. In particular, mapping of point clouds has been a challenging problem. In this paper, we develop a discrete analogue of the Teichmüller extremal mapp…
Estimates geodesics on surfaces without conjugate points.
problem Counting geodesics on surfaces without conjugate points.
method Margulis-type asymptotic estimates.
result Asymptotic estimates for geodesics on surfaces.
The main result is a direct proof of the implication (LVKFk,3)⇒(LT3k−1,3) below. Consider the following statements: (LVKF1,3) From any 11 points in R3 one can choose 3 pairwise disjoint triples whose convex hulls have a common point. (LVKFk,3) From any 6k+5 points in $ \m…
Study shows periodic points of Prym eigenforms in specific genera.
problem Understanding periodic points of Prym eigenforms in translation surfaces.
method Geometric proof using Prym involution and affine automorphism group.
result Fixed points of Prym involution are periodic points of Prym eigenforms.
Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only …
The wide adoption of DNNs has given birth to unrelenting computing requirements, forcing datacenter operators to adopt domain-specific accelerators to train them. These accelerators typically employ densely packed full precision floating-point arithmetic to maximize performance per area. Ongoing research efforts seek t…
Study finds central points of double heptagon surface are not connection points.
problem Identifying connection points on double heptagon translation surfaces.
method Used a gcd algorithm to determine hyperbolic directions and found non-connection points.
result Central points of heptagons are not connection points on double heptagon translation surfaces.
Quantized neural networks can represent all fixed-point functions under certain conditions.
problem Expressive power of quantized neural networks under fixed-point arithmetic.
method Analyzing necessary and sufficient conditions for quantized networks to represent all fixed-point functions.
result Various popular activation functions satisfy the sufficient condition for representing all fixed-point functions.
Study circle actions on unitary manifolds with discrete fixed points.
problem Understanding circle actions on compact unitary manifolds with discrete fixed points.
method Prove relationships between weights at fixed points and derive results regarding the first equivariant Chern class and Hirzebruch χy-genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S1-manifolds. Study on choosing points on cubic curves, answering some questions about their flexibility.
problem Determining if algebraic structures can continuously choose points on cubic plane curves.
method Analyzing the flex points and sextatic points of cubic plane curves.
result Affirmative answer for n=9 and 18, negative for infinitely many n. Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.
problem Characterize circle actions on oriented manifolds with exactly 3 fixed points.
method Analyzes manifold dimensions, isotropy submanifolds, and uses quaternionic projective space as a reference.
result For a manifold with three fixed points, its dimension must be a multiple of 4, and specific weights are unique.
Unified framework detects changes in complex system models.
problem Accurate identification of dynamic changes in simulation models.
method Combines machine learning and process-driven simulation modeling.
result Significantly improves change point detection accuracy.
Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular …
Study behavior of curvatures near singular points of frontals.
problem Understanding frontals near singular points.
method Investigate principal curvatures and vectors near singular points of frontals.
result Extend Ribaucour transformations to frontals with singular points.
Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.
problem Analyzing the growth of derivative maxima for C2 interval diffeomorphisms with parabolic fixed points. method Examining C2 diffeomorphisms with only parabolic fixed points, focusing on tangency and repelling behavior. result Maximal growth of derivative maxima is exactly quadratic for diffeomorphisms with a non-quadratic tangency to identity at a repelling fixed point.
Gradient-based optimization methods are the most popular choice for finding local optima for classical minimization and saddle point problems. Here, we highlight a systemic issue of gradient dynamics that arise for saddle point problems, namely the presence of undesired stable stationary points that are no local optima…
New proof for 6D symplectic manifold with 4 fixed points.
problem Classifying the integral cohomology ring and total Chern class for 6D symplectic manifolds with 4 fixed points.
method New different argument using moment map values and weights of fixed points.
result Determined the sets of weights and global invariants for the manifold.