A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a…
We show that, for each alpha in the interval (-1,1), the only Riemannian metrics on the space of positive definite matrices for which the alpha and -alpha-connections are mutually dual are matrix multiples fo the Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone, then this set is reduced to s…
Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorph…
GRAMPA algorithm recovers latent vertex correspondence in correlated graphs with high probability.
problem Recovering latent vertex correspondence between unlabeled, edge-correlated weighted graphs.
method Spectral graph matching algorithm, GRAMPA, with exact recovery guarantees for Erdős-Rényi graphs.
result GRAMPA exactly recovers latent vertex correspondence with high probability for Erdős-Rényi graphs with edge correlation coefficient 1−σ2 and average degree at least polylog(n) when σ≲1/polylog(n).
On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the α-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that i…
Extends particle classification to curved space-times using groupoids.
problem Classifying elementary particles in curved space-time.
method Developed a new definition of elementary particles as irreducible projective representations of kinematical groupoids, extending Wigner's program.
result Classification of elementary particles valid for a wide range of space-times, including new massless particles in magnetic-like backgrounds.
Study of correlated Wigner matrices with BBP transitions.
problem Understanding spectral transitions in correlated Wigner matrices.
method Analyzes a Wigner-type matrix with row/column correlations, decomposes into bulk and outliers, and uses integral operators to model transitions.
result Correlated Wigner matrices exhibit multiple BBP transitions at critical points.
We present an original and novel method based on random matrix approach that enables to distinguish the respective role of temporal autocorrelations inside given time series and cross correlations between various time series. The proposed algorithm is based on properties of Wigner eigenspectrum of random matrices inste…
The classical isoperimetric inequality in the Euclidean plane R2 states that for a simple closed curve M of the length LM, enclosing a region of the area AM, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…
We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this pu…
Simplified Ricci curvature for spherical fluid dynamics models.
problem Studying stability in incompressible fluid dynamics on a sphere.
method Definition and calculation of Ricci curvature for two-dimensional hydrodynamics using finite-dimensional Zeitlin models.
result Strong numerical evidence suggests convergence of finite-dimensional approximations to infinite-dimensional limit, indicating average instability for high-frequency modes.
PCA can detect a low-rank signal but is suboptimal for non-Gaussian matrices and synchronization problems.
problem Understanding the optimal and suboptimal performance of PCA in spiked random matrix models.
method Analysis of spiked random matrix models, including Gaussian and non-Gaussian Wigner ensembles, and synchronization problems.
result PCA achieves optimal detection for Gaussian Wigner ensembles under benign priors but is suboptimal for non-Gaussian matrices and synchronization problems.
The manipulation of LIBOR by a group of banks became one of the major blows to the remaining confidence in financial industry. Yet, despite an enormous amount of popular literature on the subject, rigorous time-series studies are few. In my paper, I discuss the following hypothesis. Namely, if we should assume for a st…
In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Meas…
For simple Lie groups, the only homogeneous manifolds G/K, where K is maximal compact subgroup,for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric spaces. An explicit calculation of the multiplicative factor on the c…
We consider properties of the measurement intensity ρ of a random variable for which the probability density function represented by the corresponding Wigner function attains negative values on a part of the domain. We consider a simple economic interpretation of this problem. This model is used to present the applic…