The paper generalizes the Black-Scholes model and inequality.
problem Developing a new model for call price surfaces.
method Introducing a noncommutative semigroup structure and using convex order compatibility.
result Each one-parameter semigroup corresponds to a unique log-concave probability density.
The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.
problem Analyzing perturbations of order ≤ 1 in noncommutative geometry.
method Develops a Feynman-Kac formula for differential operators of order ≤ 1 on complex metric vector bundles over Riemannian manifolds.
result Explicit Feynman-Kac type formula for holomorphic semigroups generated by Q. A reverse Riesz estimate and spectral gap imply a Poincaré inequality.
problem Establishing a Poincaré inequality using a reverse Riesz estimate and spectral gap.
method Combining a reverse Riesz estimate and spectral gap condition to prove a Poincaré inequality.
result A Poincaré inequality is derived from a reverse Riesz estimate and spectral gap condition.
An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree. This tree allows to define a family of spectral triples giving the Cantor set the structure of a noncommutative Riemanni…
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 …
Flat semigroups can represent normal weighted homogeneous surface singularities.
problem Representability of flat semigroups in normal weighted homogeneous surface singularities.
method Study of numerical semigroups associated with surface singularities and prove representability conditions.
result A numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
Quantum theory reinterprets financial pricing by focusing on observable price transitions.
problem Traditional financial models rely on latent variables; this paper proposes a new observable approach.
method Shift operators, spectral calculus, and Lindblad semigroups are used to define observable frequency operators and convolution generators.
result The framework leads to a nonlocal pricing equation that converges to classical Black-Scholes-Merton under small mesh limits.
Theory of covariant Schrödinger semigroups on Riemannian manifolds developed.
problem Developing theory for Schrödinger semigroups on Riemannian manifolds.
method Sobolev spaces, heat kernels, differential operators, Wiener measure, Dynkin and Kato potentials.
result Properties and continuity of covariant Schrödinger semigroups established.
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
Proves representability of complex semigroup systems.
problem Representability of systems of proportionally modular numerical semigroups.
method Canonical equivariant resolution of weighted homogeneous surface singularities.
result Every system of proportionally modular numerical semigroups is representable.
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…
The paper studies a semigroup generated by finite intervals and characterizes its properties.
problem Characterizing the semigroup generated by finite intervals.
method Analyzing the semigroup BωFn, showing Green relations coincide, isomorphic to partial convex order isomorphisms, and studying shift-continuous topologies. result The semigroup BωFn is isomorphic to the semigroup of partial convex order isomorphisms and admits only Rees congruences. The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
The study connects complex surface singularities to numerical semigroups and their properties.
problem Understanding the geometry and properties of strongly flat semigroups and their generalizations.
method Analyzing complex surface singularities and their associated semigroups, proving properties of Frobenius numbers.
result Strongly flat semigroups associated with negative definite Seifert homology spheres are numerical semigroups.
Heat semigroups used to solve geometric inequalities on manifolds.
problem Finding geometric inequalities on Riemannian and sub-Riemannian manifolds.
method Heat semigroups techniques applied to Riemannian and sub-Riemannian geometry.
result Applications of heat semigroups in geometric inequalities.
Intertwining curvature bounds for graphs and quantum Markov semigroups verified.
problem Intertwining curvature bounds for graphs and quantum Markov semigroups.
method Introducing and verifying curvature bounds in various examples.
result Improved entropic curvature bounds for depolarizing semigroups and qubits.
Develops noncommutative Cowen-Douglas theory for noncommuting operators.
problem Exploring noncommutative analogues of classical Cowen-Douglas theory.
method Defining noncommutative Cowen-Douglas class using matricial joint eigenvalues and showing equivalence classes are determined by associated noncommutative vector bundles.
result Unitary equivalence class of a tuple in the noncommutative Cowen-Douglas class is determined by the equivalence class of its associated noncommutative vector bundle.
The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.
problem Understanding the algebraic structure of numerical semigroups through topological representations.
method Associaing iterated torus knots to free numerical semigroups and analyzing their knot complements and Alexander polynomials.
result Alexander polynomials of knots associated with free numerical semigroups coincide with the semigroup's 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.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
Derive derivatives of Feynman-Kac semigroups on Riemannian manifolds.
problem Analyze the derivatives of Feynman-Kac semigroups on Riemannian manifolds.
method Use local martingales and geometric assumptions to derive Bismut-type formulae and local estimates.
result Prove Bismut-type formulae for first and second derivatives of Feynman-Kac semigroups.
Study of operators on loop spaces using Fermionic calculus and stochastic methods.
problem Analyzing operators on loop spaces arising from self-adjoint and closed operators.
method Fermionic calculus and stochastic methods to derive regularity and stochastic representations.
result Derivation of a stochastic refinement of the Duistermaat-Heckman localization formula.
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…
The paper studies Lévy processes on compact manifolds, proving properties of their semigroups.
problem Analyzing Lévy processes on compact Riemannian manifolds.
method Proving properties of Feller semigroups and generators on Lp spaces. result The generator has a discrete spectrum of eigenvalues and the semigroup is trace-class when the process has a non-trivial Brownian part.
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.
problem Approximating semigroups for diffusion on Riemannian manifolds.
method Discretized approximation using random walks on proximity graphs.
result Quantitative error estimates for convergence of discrete semigroups to continuous semigroups.
New bounds on manifold Betti numbers derived from semigroup norms.
problem Estimating the first Betti number of compact Riemannian manifolds.
method Birman-Schwinger principle and Schatten norm estimates for semigroup differences, without ultracontractivity assumptions.
result Explicit bounds on Betti numbers depend on Ricci tensor norms.
Study noncommutative coverings of irrational quantum tori.
problem Characterize noncommutative coverings of irrational quantum tori.
method Developed a framework for noncommutative coverings and studied irrational quantum tori.
result Characterized all connected noncommutative coverings of irrational quantum tori.
Constructs noncommutative spaces for D-branes on complex algebraic spaces.
problem Mathematical model for D-branes on noncommutative spaces.
method Toric geometry, Azumaya schemes, invertible sheaves.
result Embeds algebraic Calabi-Yau spaces into soft noncommutative schemes.
DM uses semigroup property to tune diffusion time for better data analysis.
problem Difficulty in tuning diffusion time for optimal data analysis.
method Proposes a semigroup criterion to select diffusion time.
result Effective and robust method for picking diffusion time.
Constructs free semigroups with critical exponents close to but less than ambient groups.
problem Creating free semigroups with critical exponents close to but less than ambient groups.
method Constructing finitely generated free subsemigroups with specific properties.
result Free semigroups with critical exponents arbitrarily close to but strictly less than ambient groups.
Curvature defined in noncommutative geometry for curved spaces.
problem Defining curvature in noncommutative geometry for curved spaces.
method Using spectral geometry and heat kernel asymptotic expansions.
result New ways to define local curvature invariants for noncommutative Riemannian spaces.
D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.
problem Exploring noncommutative mirror symmetry through D-branes on noncommutative Calabi-Yau spaces.
method Constructing noncommutative ringed spaces from local resolutions, realizing D-branes as morphisms, and defining kinetic energy.
result Dynamical D-branes on noncommutative spaces can be described by a Polyakov-like action, suggesting a bridge between string theory and noncommutative geometry.
Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metri…
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
problem Computing the Connes-Chamseddine cycle for 6D manifolds.
method Using noncommutative integral on 6D manifolds, they compute the cycle.
result The Connes-Chamseddine cycle on 6D manifolds is computed.
Paper proposes deep learning for operators in semigroups, improving dynamical system modeling.
problem Modeling unknown autonomous dynamical systems using time series data at varying time lags.
method Novel deep learning approach embedding semigroup property into data-driven learning process.
result Framework reduces data dependency, improves accuracy, robustness, and stability for long-time prediction.
A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this paper, we exte…
The aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if p≤2. To see this, it is shown that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2 and that …
Paper analyzes a new Hopf-Lax semigroup in metric spaces.
problem Analyzing a new Hopf-Lax semigroup in metric spaces.
method Using continuous sections of quotient maps and variational problems.
result The 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation.
We look at the semigroup generated by a system of heat equations. Applications to testing normality and option pricing are addressed.
New invariants for 3-manifolds from foliations and noncommutative geometry.
problem Creating new invariants for 3-manifolds.
method Using taut codim-1 foliations and noncommutative geometry techniques.
result Definition of new invariants for 3-manifolds.
Defines and proves generalized noncommutative residue theorems for specific dimensions.
problem Defining and proving residue theorems for noncommutative geometry.
method Defined generalized noncommutative residue of Dirac operator; proved Kastler-Kalau-Walze type theorems.
result Validated Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds.
Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
The paper explores noncommutative geometry of frame bundles using C*-algebras.
problem Understanding the noncommutative geometry of frame bundles.
method Using C*-algebras and unitary tensor functors, the paper constructs a free C*-dynamical system.
result Each C*-algebraic noncommutative principal SO(n)-bundle is uniquely determined by its associated noncommutative vector bundle.
The paper proves a Connes trace theorem for curved noncommutative tori.
problem Recovering scalar curvature in curved noncommutative tori.
method Proving a version of Connes' trace theorem for noncommutative tori of any dimension.
result Establishes a curved version of Connes' integration formula for scalar curvature.
New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.