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.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.
We compare various notions of weak subsolutions to degenerate complex Monge-Amp{è}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.
We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.
Study solves complex equation on specific types of manifolds.
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…
We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a -equivariant Fano compactification of a complex connected reductive group in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …
Solves complex Monge-Ampère equations on Kähler manifolds.
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…
We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…
N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if has sectional curvature between two constants and , then there exists such that $M…
The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
Study complex Monge-Ampère flows on Kähler manifolds using Perron method.
In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Ampre equation …
We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…
New method combines Monte Carlo and tensor networks for solving complex equations.
We compute the algebraic equation of the universal family over the Kenyon-Smillie -Teichmüller curve and give a nice geometric description of the torsion map. Moreover, we re-prove independently that the found algebraic equation describes a Teichmüller curve by computing the Picard-Fuchs equation associated to…
It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in , where is the dimensionality of the data, and show that a class of random matrices with indep…
Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…
The goal of this article is to describe the concepts of system dynamics and its applications to the simulation modeling of financial institutions daily activity. The hybrid method of the re-engineering of banking business processes based upon combination of system dynamics, queuing theory and tools of ordinary differen…
We investigate how to obtain various flows of Kähler metrics on a fixed manifold as variations of Kähler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi…
Study of polygon degeneration to segments in complex space.
The presentation of supergravity theories of our previous paper "Super-Poincare' algebras, space-times and supergravities (I)" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia and Scherk's theory of 11D-supergravity are equivalent…
The study finds conditions for a third rank Killing tensor field on a 2D Riemannian torus.
Reduces a complex hypersurface to a simplified equation with primary invariants.
A neural atlas simplifies 3D geometry simulation by avoiding meshing.
Let be a closed Riemannian manifold of positive scalar curvature and any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second Yamabe constant of as goes to . We obtain that $\lim_{t \to +\infty}Y^2(M\times N,[…
Study on colored Jones polynomial of figure-eight knot for complex parameters.
Mathematical models of a cellular action potential in cardiac modelling have become increasingly complex, particularly in gating kinetics which control the opening and closing of individual ion channel currents. As cardiac models advance towards use in personalised medicine to inform clinical decision-making, it is cri…
The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gau…
The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
The paper studies conditions for graphs connecting level sets of harmonic polynomials.
We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7,1) and L(7,2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e.algebraic relation corresponding to a ``2 tetrahedra to 3 t…
We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, w…
In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,)…
In this paper we describe how to include funding and margining costs into a risk-neutral pricing framework for counterparty credit risk. We consider realistic settings and we include in our models the common market practices suggested by the ISDA documentation without assuming restrictive constraints on margining proce…
We study and generalize in various ways the model of rational expectation (RE) bubbles introduced by Blanchard and Watson in the economic literature. First, bubbles are argued to be the equivalent of Goldstone modes of the fundamental rational pricing equation, associated with the symmetry-breaking introduced by non-va…
New PFPPs based on rank-dependent utility for better performance control.
This paper addresses detection of a reverse engineering (RE) attack targeting a deep neural network (DNN) image classifier; by querying, RE's aim is to discover the classifier's decision rule. RE can enable test-time evasion attacks, which require knowledge of the classifier. Recently, we proposed a quite effective app…
Mathematical modeling with Ordinary Differential Equations (ODEs) has proven to be extremely successful in a variety of fields, including biology. However, these models are completely deterministic given a certain set of initial conditions. We convert mathematical ODE models of three benchmark biological systems to Dyn…
Beta-SOD detects and corrects noisy object re-identification using cosine similarity and Beta mixtures.
A deep learning method solves nonlinear filtering problems efficiently.
We compare the isoperimetric profiles of $S^2 \times \re^3$ and of $S^3 \times \re^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2 \times \re^3$ and $S^3 \times \re^2$. Explicitly we show that $Y(S^3 \times \re^2, [g_0^3 +dx^2…
New PINN formulation respects causality for complex systems.
We prove that the three-sphere recognition problem lies in the complexity class NP. Our work relies on Thompson's original proof that the problem is decidable [Math. Res. Let., 1994], Casson's version of her algorithm, and recent results of Agol, Hass, and Thurston [ArXiv, 2002].
We mathematically analyze a simple market model where trading at each point in time involves only two agents with the sum of their money being conserved and with neither parties resulting with negative money after the interaction process. The exchange involves random re-distribution among the two players of a fixed fra…
Re-initializing neural networks improves generalization but not as much as other techniques.