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.
We review the extent to which the universal enveloping algebra of a Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such Rinehart bialgebras.
This study connects ReLU neural networks to toric geometry to analyze function realization.
problem Determining which continuous piecewise linear functions can be realized by ReLU neural networks.
method Established a connection between toric geometry and ReLU neural networks, defining key structures like the ReLU fan, toric variety, and Cartier divisor.
result Proved a criterion for functions realizable by unbiased shallow ReLU networks using intersection numbers.
An elementary arbitrage principle and the existence of trends in financial time series, which is based on a theorem published in 1995 by P. Cartier and Y. Perrin, lead to a new understanding of option pricing and dynamic hedging. Intricate problems related to violent behaviors of the underlying, like the existence of j…
Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety…
The Cartier-Perrin theorem, which was published in 1995 and is expressed in the language of nonstandard analysis, permits, for the first time perhaps, a clear-cut mathematical definition of the volatility of a financial asset. It yields as a byproduct a new understanding of the means of returns, of the beta coefficient…
Recent advances in the understanding of time series permit to clarify seasonalities and cycles, which might be rather obscure in today's literature. A theorem due to P. Cartier and Y. Perrin, which was published only recently, in 1995, and several time scales yield, perhaps for the first time, a clear-cut definition of…
A new standpoint on financial time series, without the use of any mathematical model and of probabilistic tools, yields not only a rigorous approach of trends and volatility, but also efficient calculations which were already successfully applied in automatic control and in signal processing. It is based on a theorem d…
We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis (Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice…
This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particul…
We prove that if a Q-Fano variety V specially degenerates to a Kähler-Einstein Q-Fano variety V, then for any ample Cartier divisor H=−r−1KV with r∈Q>0, the normalized volume vol(v)=ACn(v)⋅vol(v) is globally minimized at the cano…
We give the definition of a duality that is applicable to arbitrary k-forms. The operator that defines the duality depends on a fixed form Ω. Our definition extends in a very natural way the Hodge duality of n-forms in 2n dimensional spaces and the generalized duality of two-forms. We discuss the properties of …
This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.
problem Investigates T-duality between hyperkähler structures and branes on algebraic integrable systems.
method Uses techniques of generalized geometry and Fourier-Mukai transform to show T-duality between semi-flat hyperkähler structures and generalized branes.
result Shows T-duality between semi-flat hyperkähler structures and generalized branes on algebraic integrable systems.
We study generalized complex structures and T-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal T-duality". As an application we deal with the problem of finding symplectic stru…
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality …
We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…
We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endo…
In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a δ-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.