arXiv research

For information retrieval and binary classification, we show that precision at the top (or precision at k) and recall at the top (or recall at k) are maximised by thresholding the posterior probability of the positive class. This finding is a consequence of a result on constrained minimisation of the cost-sensitive exp…

A new definition of umbilic points at infinity for polynomial surfaces.

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on whic…

We look at complete minimal surfaces of finite total curvature in $\mathbb{R}^4$. Similarly to the case of complex curves in $\mathbb{C}^2$ we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives a formula for the total normal curvature of the surface. The knowledge of the l…

The paper proves mapping class groups of closed surfaces are simply connected at infinity.

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high…

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs inter…

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

The metric jets, introduced in the first chapter, generalize the jets (at order one) of Charles Ehresmann. In short, for a "good" map $f$ (said to be "tangentiable" at $a$), we define its metric jet tangent at $a$ (composed of all the maps which are locally lipschitzian at $a$ and tangent to $f$ at $a$) called the "tan…

This research improves value-at-risk estimation during financial crises using non-extensive statistical methods.

In this paper, we are interested in short homologically and homotopically independent loops based at the same point on Riemannian surfaces and metric graphs. First, we show that for every closed Riemannian surface of genus $g \geq 2$ and area normalized to $g$, there are at least $\ceil{\log(2g)+1}$ homotopically indep…

In this paper, we define a reduced distance function based at a point at the singular time $T<\infty$ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than …

This paper explains robust overfitting in wide DNNs using adversarial training and NTK theory.

We introduce a new tool for predicting the evolution of an option for the cases where at some specific time, there is a high-degree of uncertainty for identifying its price. We work over the special case where we can predict the evolution of the system by joining a single price for the Option, defined at some specific …

Mathematical models reveal key factors for engaging Gen Z at work.

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …

Classifies arcs on a 4-punctured sphere that intersect at most once.

Study on scalar curvature decay on non-compact manifolds linked at infinity.

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…

Study asymptotic behavior of Weingarten surfaces at infinity.

Positive injectivity radius for manifolds with Lie structure at infinity.

Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set of arcs pairwise intersecting at most once, which start at p and end in Q, is |X|(|X| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise int…

Enhances robustness of AT frameworks to multiple perturbations without increasing training complexity.

Study examines uncertainty in adversarially trained models and proposes improved AT methods.

We study inverse scattering for $Δ_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(λ_1,$ $λ_2)$ in a suitable subset of $\mc$, de…

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with c…

Researchers extend CCVaR to multivariate data using Archimedean copulas.

Neural network predicts turbulence near-wall regions efficiently.

This paper develops a Carleman type estimate for immersed surface in Euclidean space at infinity. With this estimate, we obtain an unique continuation property for harmonic functions on immersed surfaces vanishing at infinity, which leads to rigidity results in geometry.

For a polynomial $f$ in two complex variables, we prove that the multi-link at infinity of the 0-fiber $f^{-1}(0)$ is a fibred multi-link if and only if all the values different from 0 are regular at infinity.

New method for semi-supervised learning in federated learning with and without labels at clients.

Analytic sets with unique infinite tangent cone are algebraic.

Adversarial training (AT) is one of the most effective defenses against adversarial attacks for deep learning models. In this work, we advocate incorporating the hypersphere embedding (HE) mechanism into the AT procedure by regularizing the features onto compact manifolds, which constitutes a lightweight yet effective …

We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a cl…

The study classifies graphs with specific curvature and maximum degree.

We present proofs of classical results in Poisson geometry using techniques from Dirac geometry. This article is based on mini-courses at the Poisson summer school in Geneva, June 2016, and at the workshop "Quantum Groups and Gravity" at the University of Waterloo, April 2016.

Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand,…

Improved multilevel scheme for value-at-risk computation.

A finitely presented group is semistable at infinity if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable at infinity. This article provides a brief introduction to the notion of semistability at infinity in geometric …

Ecker's and Huisken's quantities agree for ancient mean curvature flows.

The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.

In order to efficiently transmit and store speech signals, speech codecs create a minimally redundant representation of the input signal which is then decoded at the receiver with the best possible perceptual quality. In this work we demonstrate that a neural network architecture based on VQ-VAE with a WaveNet decoder …

Study of tangent cones at infinity for algebraic sets.

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study t…