We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis consists of two analytical techniques, the replica trick (or replica analytic con…
Study Cowen-Douglas operators from analytic function spaces.
problem Analytic continuation and spectrum of Cowen-Douglas operators.
method Investigate Banach spaces of analytic functions and their operators.
result Analytic continuations of functions relate to the spectrum of Cowen-Douglas operators.
This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.
Zeta functions for non-unitary twists are shown to have analytic continuation.
problem Analytic continuation of zeta functions for non-unitary twists.
method Analytic continuation for compact locally-symmetric spaces with non-unitary twists.
result Zeta functions admit analytic continuation as meromorphic functions.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
Improved neural network predicts spectral functions more accurately than traditional methods.
problem Reconstructing real-time spectral functions from imaginary-time Green's functions is ill-posed and challenging.
method Feature Learning Network (FL-net) for enhanced prediction accuracy.
result FL-net achieves at least 20% improvement over traditional methods like MEM.
New method stabilizes quantum Monte Carlo data analysis.
problem Noise in imaginary-time data impacts real-frequency spectra.
method Sparse modeling and regularization technique.
result Stable analytical continuation achieved with minimal bases.
Analytic plane curves determine unique conformal coordinates.
problem Determining a conformal coordinate system for analytic plane curves.
method Holomorphic continuation of the Frenet curvature form.
result Holomorphic continuation of the curvature form uniquely determines a conformal coordinate net.
Analytic convex bodies' Poincaré series extended holomorphically.
problem Analytic continuation of Poincaré series for convex bodies.
method Analytic continuation of Laplace transforms, holomorphic functions, and resolvent of multiplication operators.
result Poincaré series continues holomorphically to a conical neighborhood of the right half-plane, removing countable cuts and points.
Resurgent analysis reveals full partition function for 3-manifold invariants.
problem Analyzing resurgence in 3-manifold invariants for SL(2,C). method Resurgent analysis applied to infinite families of Seifert manifolds and torus knot complements.
result The contribution from abelian flat connections contains information of all non-abelian flat connections, indicating a full partition function.
Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We prove the existence of a meromorphic continuation of the resolvent $(Δ-\ev)^{-1}$ across the continuous spectrum to a Riemann surface multiply covering the plane. The methods are purely analytic and are adapted fr…
The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in …
The paper studies graphs minimizing Dirichlet energy with analytic boundaries, confirming a conjecture about singularities.
problem Understanding the singularities of area-minimizing currents with real analytic boundaries.
method Analyzing multi-valued graphs with real analytic interfaces that minimize Dirichlet energy.
result Dirichlet energy-minimizers with analytic boundary singularities are discrete in 2 dimensions, confirming a conjecture by B. White.
Proves real analyticity on surfaces based on restrictions.
problem Determining analyticity on non-continuous functions on manifolds.
method Analyzes restrictions to submanifolds homeomorphic to 2-sphere.
result Analyticity on manifold follows from analytic restrictions.
New method calculates eta invariant without analytic continuation.
problem Spectral asymmetry of non-semibounded systems.
method Direct pseudodifferential technique for curl operator.
result Eta invariant can be traced as spectral projection difference.
We derive the Chern-Gauss-Bonnet Theorem for manifolds with smooth non-degenerate boundary in the pseudo-Riemannian context from the corresponding result in the Riemannian setting by examining the Euler-Lagrange equations associated to the Pfaffian of a complex "metric" on the tangent space and then applying analytic c…
Proves solution uniqueness for biomembrane shape prediction.
problem Proving solution uniqueness for the genus one Canham variational problem.
method Combining numeric analytic continuation and singularity analysis to prove non-negativity of a sequence.
result Proves positivity of the sequence, leading to solution uniqueness.
The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite o…
Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the…
The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral form…
In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. In our previous work we described the resolvent, and specifically the asymptotic behavior of the Green'…
Analytic solutions found for a financial market model with traders.
problem Analyzing stability and price dynamics in a financial market model.
method Developed a continuous-time financial market model with two types of traders and proved stability conditions.
result Analytic formulae derived for price dynamics and trader profitability.
Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.
Analytic completeness criterion applied to constant mean curvature surfaces.
problem Determining the analytic completeness of constant mean curvature surfaces.
method Defining arc-properness and applying it to surfaces in de Sitter 3-space.
result A criterion for the analytic completeness of G-catenoids and their extensions.
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.
Researchers compute knot invariants in Seifert manifolds using Wilson loops.
problem Calculating BPS invariants for knots in Seifert manifolds.
method Analytically continued Chern-Simons level and representation from Wilson loop operator.
result Homological blocks for knots in Seifert manifolds and Seifert integer homology spheres.
Let φ(t),t∈[0,T] be a smooth solution to the Laplacian flow for closed G_2 structures on a compact 7-manifold M. We show that for each fixed positive time t∈(0,T], (M,φ(t),g(t)) is real analytic, where g(t) is the metric induced by φ(t). Consequently, any Laplacian soliton is real a…
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
Continuity of delta invariant leads to uniform Kähler-Einstein metrics.
problem Continuity of delta invariant in Kähler and twisted Kähler-Einstein metrics.
method Analytic delta invariant and uniform Yau-Tian-Donaldson theorem.
result Uniform Yau-Tian-Donaldson theorem for twisted Kähler-Einstein metrics.
Geometric analysis on real analytic manifolds using seminorms.
problem Characterizing operations on real analytic manifolds and vector bundles.
method Using seminorms and geometric decompositions of jet bundles.
result New characterizations of real analytic mappings and operations.
Analytic formulae for point vortex dynamics on surfaces with symmetry.
problem Understanding point vortex behavior on surfaces with continuous symmetry.
method Derived analytic formulae for Green and Robin functions on surfaces with hydrodynamic Killing vector fields.
result Unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry.
Real analyticity proved for modified Laplacian coflow solutions.
problem Analyzing the modified Laplacian coflow on compact 7-manifolds.
method Improved Chen's Shi-type estimate to show real analyticity.
result The modified Laplacian coflow solution is real analytic.
Continuity method proves existence of Mabuchi solitons on Fano manifolds.
problem Existence of Mabuchi solitons on extremal Fano manifolds.
method Continuity method applied to Mabuchi's generalized Kähler-Einstein metrics.
result Analytic proof of Mabuchi soliton existence without minimal model program.
Introduces a Boltzmann machine with Riemann-Theta functions for continuous and discrete states.
problem Modeling continuous and discrete states in neural networks.
method Develops a Boltzmann machine with continuous visible and discrete hidden states, solving probability density and conditional expectation analytically.
result Derives a novel parametric density function involving Riemann-Theta functions and uses it as an activation function in a feedforward neural network.
Homotopy method removes isolated zeroes in continuous maps.
problem Removing isolated zeroes in continuous maps with local obstructions.
method Constructing a continuous homotopy to a nonvanishing map.
result Existence of a continuous homotopy to a nonvanishing map for certain dimensions.
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
We consider the class of affine LIBOR models with multiple curves, which is an analytically tractable class of discrete tenor models that easily accommodates positive or negative interest rates and positive spreads. By introducing an interpolating function, we extend the affine LIBOR models to a continuous tenor and de…
We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recentl…
Real analytic functions can be extended on manifolds with normal crossings.
problem Extending continuous functions to Cω functions on manifolds with normal crossings. method Employing Cartan Theorems A and B from real analytic geometry.
result Continuous functions on the union of submanifolds with normal crossings can be extended to Cω functions on the entire manifold. New EM algorithm improves deep generative network training.
problem Training deep generative networks with complex posterior and likelihood distributions.
method Derive analytical posterior and marginal distributions using CPA property, derive analytical EM algorithm.
result EM training yields higher likelihood than Variational Autoencoders (VAEs).
Study short maturity Asian options under CEV model, presenting an analytical approximation.
problem Analyzing short maturity behavior of Asian options in CEV model.
method Presented an analytical approximation for Asian options prices under CEV model.
result Good numerical agreement with Monte Carlo simulations and benchmark test cases.
Let (M,g) be a simple Riemannian manifold. Under the assumption that the metric g is real-analytic, it is shown that if the geodesic ray transform of a function f∈L2(M) vanishes on an appropriate open set of geodesics, then f=0 on the set of points lying on these geodesics. The approach is based on a micr…
Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.
problem Proving the existence of bounding cochains for unobstructed Lagrangians.
method Introducing non-archimedean analytic structure and using family Floer techniques.
result All Lagrangians in a connected family are unobstructed if one is.
The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalize…
Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.
problem Computing the Wodzicki residue for pseudo-differential operators on compact Lie groups.
method Analytic continuation of traces and matrix-valued symbols.
result Main theorem complementary to [2], removing ellipticity hypothesis.
The Brylinski beta function is extended for coaxial layers on submanifolds.
problem Extending the Brylinski beta function to coaxial layers on submanifolds.
method Analytic continuation and computation of residues for the function.
result The Brylinski beta function has an analytic continuation with simple poles.
Analytic submanifolds of cocycles reveal discrete cohomology spaces.
problem Analyzing cocycles and their coboundaries on Lie groups.
method Defining and studying cocycles as maps with specific properties, showing they form submanifolds and decomposing them into bundles.
result Cohomology spaces are discrete, with cocycles forming analytic submanifolds and their orbits open.
Machine learning for solving ill-conditioned Fredholm integrals.
problem Analytical continuation of quantum many-body physics spectra.
method Projected regression from a large database of solutions.
result Method performs as well or better than Maximum Entropy method.