arXiv research

New proof for symmetric spaces with rectangular lattices.

The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.

Square-root natural-gradient improves variational inference convergence.

Polynomials' roots count tied to surface umbilics.

The author suggests using non-Euclidean geometry for psychometric models.

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold whic…

In this note, we extend the notion of a Monge hypersurface from its roots in semi-Euclidean space to more general spaces. For the degenerate case, the geometry of these structures is studied using the Bejancu-Duggal method of screen distributions.

The study proves a geometric result related to Harish-Chandra's theorem.

In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometr…

In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible directions, meaning that all geodesics lines passing through the set K in these directions remain the same straight lines on exit. For example in the…

CROC identifies the earliest-changing stream as the root cause in multi-stream data.

New findings show Bregman proximal algorithms can get stuck near non-stationary points.

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal uppe…

This research simplifies Riemannian LBFGS for SPD manifolds.

Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example …

The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to early formulations of the problem - with an object to be classified described as …

New definition of patient-specific root causes of disease using counterfactuals.

A n n-body system is a labelled collection of n point masses in Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are n-configuration, configuration space, internal s…

New geometric object for polynomials simplifies complex data.

Unified treatment of elastic metrics for curves in any dimension.

Let $C$ be a curve in a closed orientable surface $F$ of genus $g \geq 2$ that separates $F$ into subsurfaces $\widetilde {F_i}$ of genera $g_i$, for $i = 1,2$. We study the set of roots in $\Mod(F)$ of the Dehn twist $t_C$ about $C$. All roots arise from pairs of $C_{n_i}$-actions on the $\widetilde{F_i}$, where $n=\l…

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on the unit circle in $\mathbb{C}$. Further we enumerate all non-trivial colorings of a torus knot diagram by the quandle u…

New method estimates root-directed tree from extreme data.

Proposes a new model to maximize out-of-sample Sharpe ratios by forecasting tangency portfolios.

The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.

The paper proposes a new probability distribution for rooted trees.

Guarantees uniform convergence for square-root Lipschitz losses.

The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the c…

Geometric approach connects Burau representation to sphere metrics, identifying kernels.

Study finds price impact follows a 'double' square-root law, suggesting mechanical origin.

The paper proposes a method to identify root causes of anomalies in time series data.

Study infers tree topology from customer data using contrastive learning.

Constructs Dirac generating operators for split Courant algebroids.

Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the Square-Root-Velocity-Function (SRVF) allows to define and efficiently compute a d…

A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the exi…

Many independent studies on stocks and futures contracts have established that market impact is proportional to the square-root of the executed volume. Is market impact quantitatively similar for option markets as well? In order to answer this question, we have analyzed the impact of a large proprietary data set of opt…

The notion of market impact is subtle and sometimes misinterpreted. Here we argue that impact should not be misconstrued as volatility. In particular, the so-called ``square-root impact law'', which states that impact grows as the square-root of traded volume, has nothing to do with price diffusion, i.e. that typical p…

Performance monitoring, anomaly detection, and root-cause analysis in complex cyber-physical systems (CPSs) are often highly intractable due to widely diverse operational modes, disparate data types, and complex fault propagation mechanisms. This paper presents a new data-driven framework for root-cause analysis, based…

The paper proposes a method to infer differentiation trees from RNA velocity data.

A new probability distribution on full rooted trees helps in model selection.

A translation surface of Euclidean space $\r^3$ is the sum of two regular curves $α$ and $β$, called the generating curves. In this paper we classify the minimal translation surfaces of $\r^3$ and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is pro…

Two-root Riemannian manifolds have no odd-dimensional examples.

In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them redu…

The study confirms that market volatility can be explained by correlated metaorders impacting prices in a square-root fashion.

Algorithm generates realistic metaorders from public trade data.

Novel approach integrates Multivariate Square-root Lasso into Synthetic Control for high-dimensional data.