arXiv research

Study of discrete period matrices on embedded graphs, relating to Riemann surfaces.

Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.

The paper calculates Morse indices and nullities for triply periodic minimal surfaces.

The major study by Bordo and Helbing (2003) analyses the business cycle in Western economies 1881-2001. They examine four distinct periods in economic history, and conclude that there is a secular trend towards greater synchronisation for much of the 20th century. Their analysis, in common with the standard economic li…

New tensor-based method for estimating stock correlation matrices.

We perform a comparative analysis of the Chinese stock market around the occurrence of the 2008 crisis based on the random matrix analysis of high-frequency stock returns of 1228 stocks listed on the Shanghai and Shenzhen stock exchanges. Both raw correlation matrix and partial correlation matrix with respect to the ma…

We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…

Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.

We examine volatility of an Indian stock market in terms of aspects like participation, synchronization of stocks and quantification of volatility using the random matrix approach. Volatility pattern of the market is found using the BSE index for the three-year period 2000-2002. Random matrix analysis is carried out us…

A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of ${\mathbb C} P^1$ branched over 3 points. The algorithm yields a previously unknown symplectic basis of the hyperelliptic curve defined by the affine eq…

Using the correlation matrix formalism we study the temporal aspects of the Warsaw Stock Market evolution as represented by the WIG20 index. The high frequency (1 min) WIG20 recordings over the time period between January 2001 and October 2005 are used. The entries of the correlation matrix considered here connect diff…

We investigate the dynamics of correlations present between pairs of industry indices of US stocks traded in US markets by studying correlation based networks and spectral properties of the correlation matrix. The study is performed by using 49 industry index time series computed by K. French and E. Fama during the tim…

We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples, recovering known results to test the numerics and giving the yet unknown period m…

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…

A new machine learning model uses matrix exponentials for universal approximation.

If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlin…

Study examines how COVID-19 affects bond yields using network filtering methods.

Simulation of financial markets with 300 assets shows volatility clustering and unstable periods.

Complexity of periodic graphs linked to Mahler measure.

Study special Kähler geometry of Hitchin system using spectral curves and topological recursion.

We study the dynamic evolution of cross-correlations in the Chinese stock market mainly based on the random matrix theory (RMT). The correlation matrices constructed from the return series of 367 A-share stocks traded on the Shanghai Stock Exchange from January 4, 1999 to December 30, 2011 are calculated over a moving …

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the ent…

Paper tackles unpredictable feature evolution in learning.

Study solutions of elliptic sinh-Gordon equation, focusing on spectral genus two.

Method forecasts market states using sparse precision matrix and penalized Mahalanobis distance.

No Shimura-Teichmüller curves found in genus 5.

New algorithm reduces cold-start costs in multi-armed bandits for many products.

In this paper we analyzed dependencies in commodity markets investigating correlations of future contracts for commodities over the period 1998.09.01 - 2007.12.14. We constructed a minimal spanning tree based on the correlation matrix. The tree provides evidence for sector clusterization of investigated contracts. We a…

In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results.

The use of improved covariance matrix estimators as an alternative to the sample estimator is considered an important approach for enhancing portfolio optimization. Here we empirically compare the performance of 9 improved covariance estimation procedures by using daily returns of 90 highly capitalized US stocks for th…

The data in many disciplines such as social networks, web analysis, etc. is link-based, and the link structure can be exploited for many different data mining tasks. In this paper, we consider the problem of temporal link prediction: Given link data for times 1 through T, can we predict the links at time T+1? If our da…

Study on smooth moduli space of Riemann surfaces with uniformization theorem.

We prove the Livšic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a Hölder continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there …

New types of knot mosaics help count and analyze knots efficiently.

Develops a new non-abelian framework for Riemann surfaces and differential equations.

We analyze cross-correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994--95 (ii) 30-min returns of 881 US stock…

FL's early training phase significantly impacts final test accuracy.

This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.

This paper analyzes how periodic and soft target updates stabilize linear Q-learning.

Optimizes market-neutral portfolios using fractal models.

Holomorphic functions from knot complements link to quantum modular forms.

Quantum neural networks approximate periodic functions more efficiently.

We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a fe…

Study analyzes online student behavior patterns using log data.

This paper pretends to analyze the importance which the natural advantages and local resources are in the manufacturing industry location, in relation with the "spillovers" effects and industrial policies. To this, we estimate the Rybczynski equation matrix for the various manufacturing industries in Portugal, at regio…

Financial markets have been extensively studied as highly complex evolving systems. In this paper, we quantify financial price fluctuations through a coupled dynamical system composed of phase oscillators. We find a Financial Coherence and Incoherence (FCI) coexistence collective behavior emerges as the system evolves …

Study of loop braid groups for 3D manifolds, linking algebra and dynamics.

This paper classifies a specific weave type by their crossing number.