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.
Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n,1), n>0. Lehmer's question for Perron polynomials is equivalent to one about ge…
In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.
This paper addresses a long standing open problem due to Lehmer in which the triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap between 1 and the next smallest algebraic integer with respect to Mahler measure. The question has been studied in a wide range of contexts including number theory…
From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these d…
SmartDCA improves investment returns by adjusting purchases based on prices.
problem Inefficiency of Dollar-Cost Averaging (DCA) in volatile markets.
method Developed SmartDCA, a more efficient investment strategy that adjusts purchases based on price levels, and analyzed its performance using mathematical inequalities and mean definitions.
result SmartDCA outperforms DCA in long-term investments, with higher performance gains for higher power values of ρ.
Lueck expressed the Gromov norm of a knot complement in terms of an infinite series that can be computed from a presentation of the fundamental group of the knot complement. In this note we show that Lueck's formula, applied to torus knots, yields surprising power series expansions for the logarithm function. This gene…
This paper presents a construction of fibered links (K,Σ) out of chord diagrams $\sL$. Let Γ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of (K,Σ) equals the bilinear form of the simply-laced Coxeter system (W,S) associated to Γ; and the monodromy of $(K,Σ)…
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic…
The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph automorphisms, with finite quotient) the Mahler measure of its Lap…
The paper introduces models to learn generalized transformation equivariant representations.
problem Capturing intrinsic visual structures equivariant to various transformations.
method Deterministic and probabilistic AutoEncoding Transformations (AET and AVT) models trained to learn visual representations from generic groups of transformations.
result Generalized TERs (GTERs) that are equivariant to transformations in a more general fashion.
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a …
The conformal geometry of spacelike surfaces in 4-dimensional Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the know…
The paper examines how polarized curves behave near singular points.
problem Analyzing the behavior of polarized curves near singular points.
method Investigates the limiting behavior of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere.
result For a pole of first order, all transforms converge to the original curve. For a pole of second order, a generic Darboux transform converges, but a Calapso transform has a limit point or circle.
We study the dynamics of the discrete bicycle (Darboux, Backlund) transformation of polygons in n-dimensional Euclidean space. This transformation is a discretization of the continuous bicycle transformation, recently studied by Foote, Levi, and Tabachnikov. We prove that the respective monodromy is a Moebius transform…
The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform …
We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…
We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…