Study Frobenius determinants and their connections to Bessel functions.
problem Understanding Frobenius determinants and their geometric and differential equations.
method Investigation of Frobenius determinants and their relation to twistor theory and Bessel functions.
result Extensions and new insights into Bessel functions.
The tetrahedral index connects to a q-Bessel function, revealing new mathematical techniques.
problem Exploring connections between the tetrahedral index and Hahn-Exton q-Bessel function.
method Establishing a correspondence between the tetrahedral index and the q-Bessel function.
result New techniques and conjectures in q-hypergeometric theory.
Estimates eigenvalues using Bessel functions on manifolds.
problem Estimating eigenvalues of differential operators on manifolds.
method Using mean value lemma and curvature assumptions, derive differential inequalities involving Bessel functions.
result Establishes new estimates for eigenvalues involving positive roots of Bessel functions.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
Directly simulates squared Bessel processes efficiently.
problem Simulating squared Bessel processes accurately and efficiently.
method Two-dimensional Chebyshev expansion for non-central chi-square distribution inverse.
result Accurate and efficient simulation for various degrees of freedom.
Paper develops a new method for pricing complex financial options.
problem Pricing European-style double barrier knock-out options for homogeneous diffusions.
method Neumann series of Bessel functions representation for one-dimensional time-homogeneous diffusions.
result Efficient and simple numerical method for pricing and hedging.
Constructs surfaces with constant mean curvature from Bessel equation.
problem Creating surfaces with constant mean curvature.
method From the Bessel equation, constructs immersions of the twice-punctured Riemann sphere into R^3.
result Family of constant mean curvature surfaces constructed.
New simulation method simplifies Heston model with Poisson conditioning for better accuracy and efficiency.
problem Computational expense in exact simulation schemes for Heston model.
method Proposes a new exact simulation scheme without modified Bessel function evaluations, leveraging conditional integrated variance simplification.
result Good performance in terms of accuracy, efficiency, and reliability compared to existing methods.
The study reveals scaling limits of spectral projectors on Riemannian manifolds.
problem Understanding spectral function scaling limits on Riemannian manifolds.
method New off-diagonal estimates and Weyl law application.
result Scaling limit of spectral projector is a normalized Bessel function.
Following Donaldson's oppenness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use t…
Paper analyzes multidimensional PIDEs for financial modeling, proving existence and uniqueness in Bessel spaces.
problem Analyzing solutions of non-local nonlinear PIDEs in multidimensional spaces.
method Employing abstract semilinear parabolic equations theory in Bessel potential spaces.
result Existence and uniqueness of solutions for a wide class of Lévy measures in multidimensional spaces.
Global harmonic maps into SU(1,1) constructed from Smyth potentials using DPW method.
problem Globality of harmonic maps constructed from Smyth potentials in SU(1,1).
method Construct harmonic maps into SU(1,1) using the DPW method, solving a Riemann-Hilbert problem to achieve global Iwasawa factorization.
result Globality of the constructed harmonic maps proved using Bessel functions and asymptotic expansions.
We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale …
Study Hardy identities and inequalities on Cartan-Hadamard manifolds.
problem Existence and nonexistence of extremal functions in Hardy inequalities.
method Using the notion of a Bessel pair, we derive Hardy identities and inequalities.
result Established several Hardy type inequalities with improvements and understandings.
In sparse Bayesian learning (SBL), Gaussian scale mixtures (GSMs) have been used to model sparsity-inducing priors that realize a class of concave penalty functions for the regression task in real-valued signal models. Motivated by the relative scarcity of formal tools for SBL in complex-valued models, this paper propo…
Derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous.
problem Smoothness of sub-Riemannian geodesics
method Proving Lp-Hölder continuity of derivatives result Derivatives of sub-Riemannian geodesics are Lp-Hölder continuous Study phase transitions in noisy transformer dynamics on spheres.
problem Understanding phase transitions in noisy transformer dynamics on spheres.
method Sharp Beckner--Onofri/logarithmic HLS inequality, Funk--Hecke/Bessel coefficients, degree-two quartic obstruction.
result Sharp global-minimizer dichotomy and phase transitions in noisy transformer dynamics in arbitrary dimension.
This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with research of Yor's in 1992, these questions about exponential functionals of Brownian motion have been studied in terms of …
Proposes a new birth-death process for better modeling of population dynamics.
problem Models of population or opinion dynamics with spurious long-range memory.
method Introduces Bessel-like birth-death process to address the spurious long-range memory.
result Derives equations for the burst and inter-burst duration of the new process.
Paper proves existence and uniqueness of solutions to PIDEs in Bessel spaces for option pricing.
problem Existence and uniqueness of solutions to PIDEs in Bessel spaces.
method Abstract semilinear parabolic equations and Bessel potential spaces.
result Proves existence and uniqueness of solutions in Bessel potential spaces.
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …
Solves a long-standing problem on step-two groups with exact formulas.
problem Long-standing Gaveau--Brockett open problem on step-two groups.
method Combining Varadhan's formulas, heat kernel, and operator convexity.
result Exact formula for Carnot--Carathéodory distance on step-two groups.
This paper extends barrier option pricing to CIR and CEV models using semi-closed form solutions.
problem Pricing barrier options in time-dependent CEV and CIR models.
method Developed two new methods: Bessel potentials and generalized integral transform, both applied to Bessel processes.
result The methods provide more accurate and stable pricing compared to finite difference methods, especially for small and large maturities.
It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in Rn, continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the s…
The paper defines function spaces on manifolds with bounded or singular geometries.
problem Defining function spaces on manifolds with various geometries.
method Introduces and analyzes Sobolev, Besov, and Bessel potential spaces on uniformly regular and singular Riemannian manifolds.
result Demonstrates maximal regularity for a linear parabolic problem on singular manifolds.
A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…
The paper examines how the angle between inputs in ReLU networks decreases with depth, impacting training.
problem Depth degeneracy in neural networks, leading to constant function behavior on initialization.
method Combinatorial expansions and Monte Carlo experiments to analyze the angle between inputs in ReLU networks of increasing depth.
result The angle between inputs in ReLU networks decreases exponentially with depth, leading to constant function behavior on initialization.
Model predicts stock market volatility, leading to successful trading.
problem Real-time risk management in stock markets.
method Algebraic theory of news impact, Bessel and hypergeometric functions, ML procedures.
result Trading system proved successful in historical and real-time experiments.
This work extends Tweedie's formulae to non-Gaussian processes for better diffusion model generation.
problem Limited exploration of non-Gaussian diffusion models and corresponding Tweedie's formulae.
method Extended Tweedie's formulae to geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes.
result Demonstrated potential of non-Gaussian models in image and financial time series generation.
The paper introduces new methods for Asian option pricing using Laguerre quadrature.
problem Developing accurate pricing models for Asian options.
method Utilizes Laguerre quadrature and diffusion kernel approach.
result Demonstrates new techniques to solve complex Asian option pricing equations.
Paper proves smoothness for variational inference, giving convergence guarantees.
problem Proving convergence guarantees for black-box variational inference.
method Describes gradients in an inner-product space, using Bessel's inequality.
result Objective is M-Lipschitz smooth if target is, excluding entropy.
Paper analyzes EM algorithm's trajectory in 2MLR, revealing cycloid behavior.
problem Understanding the convergence and trajectory of EM algorithm in 2MLR.
method Explicit closed-form expressions for EM updates, recurrence relation derivation at population level.
result EM iterations lie on a cycloid trajectory, leading to theoretical estimate of convergence exponent.
Unique solutions found for diffusive martingale problems.
problem Finding unique solutions to Cauchy problems for diffusive real-valued strict local martingales.
method Provided sets of smooth functions under local Hölder and Engelbert-Schmidt conditions for unique classical and weak solutions.
result Unique solutions found for specific martingale models.
Model financial market with fundraiser and stock, derive option prices.
problem Derive option prices in a market with a fundraiser and multiple solutions to the Black-Scholes equation.
method Model financial market with two types of agents, use Pitman's theorem for Bessel process, derive option prices using numerical scheme.
result Derive option prices for European options and call options in a market with a bubble.
Paper solves PDEs for optimal investment strategies in volatile markets.
problem Finding optimal investment strategies in volatile markets.
method Numerical methods using time-changed Bessel bridges.
result Solves PDEs for relative arbitrage opportunities in volatility-stabilized markets.
DCFNet decomposes CNN filters into learned coefficients with bases, reducing parameters and computation.
problem Reduction of model parameters and computation in CNNs.
method DCFNet decomposes convolutional filters into a truncated expansion with pre-fixed bases, learning only the coefficients.
result DCFNet maintains accuracy for image classification tasks with significantly fewer parameters, including with random bases.
In this paper, we present a Bayesian channel estimation algorithm for multicarrier receivers based on pilot symbol observations. The inherent sparse nature of wireless multipath channels is exploited by modeling the prior distribution of multipath components' gains with a hierarchical representation of the Bessel K pro…
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
New method for pricing barrier options in time-dependent λ-SABR model.
problem Pricing barrier options in the time-dependent λ-SABR model.
method Modified integral transform method and Fourier-Bessel series solution.
result Semi-analytical solution for barrier options in λ-SABR model.
We provide an affirmative answer to a question posed by Tod \cite{Tod:1995b}, and construct all four-dimensional Kahler metrics with vanishing scalar curvature which are invariant under the conformal action of Bianchi V group. The construction is based on the combination of twistor theory and the isomonodromic problem …
In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional Lévy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete…
We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of…
In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the W1,2 Sobolev inequality along the Ricci flow …
At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vor…
We propose two main applications of Gyöngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous Itô processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility mod…
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
Unified framework for solving first passage times of diffusion processes.
problem Solving first passage times of time-homogeneous diffusion processes.
method Unified framework based on killed version potential theory and perturbation theory.
result Closed-form solutions for probability densities of level crossing problems.
We consider random vectors drawn from a multivariate normal distribution and compute the sample statistics in the presence of non-stationary correlations. For this purpose, we construct an ensemble of random correlation matrices and average the normal distribution over this ensemble. The resulting distribution contains…