arXiv research

Intertwining curvature bounds for graphs and quantum Markov semigroups verified.

Study quantum diffusion on spectral triples and spinor bundles.

We introduce imprecise Markov semigroups to handle uncertainty in Markov processes.

This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specif…

We extend the Feynman-Kac formula for Schrödinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be bounded from below or locally square integrable. This path integral formula is then u…

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving…

Quantum theory reinterprets financial pricing by focusing on observable price transitions.

In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM)…

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere …

RILA learns HQMMs robustly against adversarial corruption.

Quantum algorithms for financial derivatives and credit risk.

Hidden Quantum Markov Models (HQMMs) can be thought of as quantum probabilistic graphical models that can model sequential data. We extend previous work on HQMMs with three contributions: (1) we show how classical hidden Markov models (HMMs) can be simulated on a quantum circuit, (2) we reformulate HQMMs by relaxing th…

Flat semigroups can represent normal weighted homogeneous surface singularities.

New infinite family of hyperbolic L-space knots with specific semigroups.

Study dynamic risk measures with distributional uncertainty using optimal transport.

The paper characterizes stochastic completeness on Riemannian manifolds using nonlocal conditions.

Complex analysis techniques link Gaussian RBF kernels to quantum mechanics.

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

Proves representability of complex semigroup systems.

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference …

The problem behind this paper is the proper measurement of the degree of quality/acceptability/distance to arbitrage of trades. We are narrowing the class of coherent acceptability indices introduced by Cherny and Madan (2007) by imposing an additional mathematical property. For this, we introduce the notion of a conca…

A new diffusion method approximates Schrödinger bridge with improved convergence.

Paper improves variational inference on Boolean hypercube using quantum methods.

The paper studies a semigroup generated by finite intervals and characterizes its properties.

The paper uncovers the mathematical structure enabling value decomposition in multi-agent systems.

The paper studies dynamical properties in semigroups modulo ideals.

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely gene…

The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected compo…

Extending classical probabilistic reasoning using the quantum mechanical view of probability has been of recent interest, particularly in the development of hidden quantum Markov models (HQMMs) to model stochastic processes. However, there has been little progress in characterizing the expressiveness of such models and…

A reverse Riesz estimate and spectral gap imply a Poincaré inequality.

We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick…

Graphs approximate semigroups for diffusion on Riemannian manifolds.

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the…

A quantum reinforcement learning algorithm reduces sample complexity.

DM uses semigroup property to tune diffusion time for better data analysis.

Constructs free semigroups with critical exponents close to but less than ambient groups.

Paper proposes deep learning for operators in semigroups, improving dynamical system modeling.

The aim of this paper is to show that the dynamics of $L^p$ heat semigroups ($p>2$) on a symmetric space of non-compact type is very different from the dynamics of the $L^p$ heat semigroups if $p\leq 2$. To see this, it is shown that certain shifts of the $L^p$ heat semigroups have a chaotic behavior if $p>2$ and that …

We look at the semigroup generated by a system of heat equations. Applications to testing normality and option pricing are addressed.

Paper analyzes a new Hopf-Lax semigroup in metric spaces.

Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.

New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.

QBC uses quantum computers to speed up Bayesian computation.

Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rect…

Study estimates gaps in semigroup products, proving embedding properties.

Tensor-network techniques have enjoyed outstanding success in physics, and have recently attracted attention in machine learning, both as a tool for the formulation of new learning algorithms and for enhancing the mathematical understanding of existing methods. Inspired by these developments, and the natural correspond…