Real analytic maps can be unstable even if infinitesimal changes are stable.
problem Unstability of real analytic maps despite infinitesimal stability.
method Used a relative version of Whitney's Analytic Approximation Theorem and H. Cartan's Theorems A and B.
result Infinitesimal Cω stability does not imply Cω stability. Introduces stability conditions for polarized varieties, linking to K-stability.
problem Stability conditions for polarized varieties.
method Analogue of Bridgeland's stability for polarized varieties, Z-stability, Z-critical Kähler metrics.
result Polarized varieties with certain stability conditions admit Z-critical Kähler metrics.
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.
Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for ge…
We obtain stability estimates and derive analytic expansions for local solutions of multi-dimensional quadratic BSDEs. We apply these results to a financial model where the prices of risky assets are quoted by a representative dealer in such a way that it is optimal to meet an exogenous demand. We show that the prices …
Classifies real-analytic SL(n,R) actions on closed manifolds.
problem Classifying real-analytic SL(n,R) actions on closed manifolds.
method Analytic and smooth classification methods.
result Extensions and classifications of Fisher--Melnick's work.
This paper introduces TDA and TSI for better business analytics.
problem Nonlinear, multi-scale business datasets under-represented by traditional tools.
method Topological Data Analysis (TDA) and Topological Stability Index (TSI).
result TSI reveals structural variability in business data.
Proves existence of Hermitian-Einstein metrics on stable bundles.
problem Analytic stability of vector bundles and instantons/monopoles.
method Analyzes curvature decay and proves existence of Hermitian-Einstein metrics.
result Existence of Hermitian-Einstein metrics on analytically stable bundles.
An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite…
The paper proves stability of certain singularities in integrable systems.
problem Stability of singularities in integrable systems under perturbations.
method Analytic and smooth perturbations of completely integrable systems, connectedness condition.
result Non-degenerate singular fibers are structurally stable under small perturbations.
Abstract: Generalizes stability theories over toric varieties and Novikov type rings.
problem Stability theories over toric varieties and Novikov type base.
method Generalization of stability theories to families over toric varieties and their analytic analogues.
result Established properness of moduli of Calabi-Yau cones and Kahler-Ricci solitons.
AI threatens financial stability through misuse and stealth adoption.
problem Misuse and stealth adoption of AI in financial regulations.
method Analysis of AI's potential risks and criteria for AI suitability.
result AI will likely become widely used by stealth, affecting high-level financial functions.
Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
The paper shows how to destabilize unstable Fano varieties using stability thresholds.
problem Optimally destabilizing K-unstable Fano varieties.
method Using divisorial valuations and special test configurations to induce stability thresholds.
result Fano varieties degenerate to uniquely determined twisted K-polystable varieties.
We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group…
The paper studies stability of CR structures on compact manifolds.
problem Stability of CR structures on compact manifolds.
method Smooth deformations and projective embeddings.
result Nearby structures still admit projective CR embeddings.
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
We show that every finite volume hyperbolic manifold of dimension greater or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Note that we do not need to make any decay assumptions on this perturbation. It will turn ou…
Study on stability of interpretable models in machine learning.
problem Bias in data and model construction affects accountability of interpretable models.
method Experimental study on feature, instance, and model selection.
result Interpretable models are unstable and require stability impact assessment.
This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.
problem Understanding growth and stability conditions for real-analytic functions over vector bundles.
method Outline theory of functionals and variational problems over vector bundles, explore applications to real-analytic functionals.
result Describes the energy functional on $S^{n-1$ as a functional over a vector bundle.
The paper generalizes SKT and HS properties to arbitrary p and studies their deformation stability.
problem Studying deformation stability of p-SKT and p-HS manifolds.
method Introducing p-hermitian-symplectic and p-pluriclosed manifolds, proving equivalence and deformation openness on ∂∂-manifolds.
result The cones of Aeppli cohomology classes of strictly weakly positive (p,p)-forms are equal on the limit fibre under certain assumptions.
As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently by Baird--Danielo and Lott: nongradient homogeneous expanding Ricci solitons on…
Extends fractional Lp uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional Lp settings. method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
New algorithms learn stability certificates from data, avoiding complex dynamics.
problem Synthesizing stability certificates from complex dynamical systems.
method Developed algorithms to learn certificate functions from trajectory data, establishing generalization error bounds.
result Efficiently learned certificates can be used for adaptive control.
In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second pa…
Develops a semi-analytic resampling method for Lasso regression.
problem Statistical fluctuations and high computational cost in Lasso resampling.
method Semi-analytic message passing algorithm based on state evolution analysis.
result Significant reduction in computational time and improved inference accuracy.
We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a re…
Paper proves families of singularities can be topologically trivialized.
problem Understanding behavior of singularities under small perturbations.
method Establishes sufficient conditions for embedded topological trivialization.
result New instances of topological stability, including μ-constant deformations. In this paper, simple mathematical models from Control Theory are applied to three very important economic paradigms, namely (a) minimum wages in self-regulating markets, (b) market-versus-true values and currency rates, and (c) government spending and taxation levels. Analytical solutions are provided in all three par…
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.
A new Shapley value approach for neural networks interpretable and stable.
problem Neural networks' interpretability and training stability issues.
method Shapley value approximation for ReLU activation, globally continuous Shapley gradient, Shapley Activation function.
result SA consistently outperforms ReLU in training convergence, accuracy, and stability.
Analyzes how many people can receive stable income in a pooled annuity fund.
problem Quantifying the trade-off between income stability and the number of members in a pooled annuity fund.
method Investment returns held constant, systematic longevity risk omitted. Derived an analytical expression for income stability.
result The number of fund members who receive stable income is independent of the mortality model.
The paper explores hidden torus symmetries in integrable systems and their stability.
problem Structural stability of singularities in integrable systems.
method Use of hidden torus actions near singular orbits and integrable perturbations.
result Persistence of toric symmetries and structural stability of Kalashnikov's parabolic orbits.
A new method for non-rigid point set registration reduces computational complexity.
problem Efficiently registering non-rigid point sets with large numbers of points.
method Structured Analytic Coherent Point Drift (Analytic-CPD) reformulates CPD for structured analytic mappings.
result Analytic-CPD reduces computational complexity by controlling the deformation model's dimensionality.
New bound on partition function proves Kähler-Einstein stability.
problem Proving Kähler-Einstein metrics on complex manifolds.
method Quantitative bound on partition function, connecting probabilistic and quantization approaches.
result Direct analytic proof of Kähler-Einstein stability for uniformly Gibbs stable manifolds.
Study unipotent group actions on Kähler manifolds using moment maps.
problem Analyzing unipotent group actions on compact Kähler manifolds.
method Moment map techniques, stability conditions, symplectic reduction.
result Geometric quotients with compactifiable Kähler structures.
New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.
problem Stabilizing quantum ergodicity for complex systems.
method Combines mixed quantization techniques with stable ergodicity results for partially hyperbolic systems.
result Establishes stable quantum ergodicity for spin Hamiltonians.
Develops stability conditions for estimating affine jump-diffusions.
problem Ergodicity and consistency of parameter estimation for affine jump-diffusions.
method Establishes stochastic stability conditions and ergodicity under specific conditions.
result Proves strong laws of large numbers and functional central limit theorems for additive functionals.
Following work of Colding-Minicozzi, we define a notion of entropy for connections over Rn which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stabilit…
We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polari…
In this note we identify the leading terms of the (reduced) K-energy map with a universal linear combination of the principal and subdominant coefficients of the weight of the mth Hilbert point. This shows that the weight F1(λ;X) introduced by Donaldson in [SKD02] is just the weight of the CM-polarisation.The eq…
The stability of money value is an important requisite for a functioning economy, yet it critically depends on the actions of participants in the market themselves. Here we model the value of money as a dynamical variable that results from trading between agents. The basic trading scenario can be recast into an Ising t…
Instability and variability of Deep Reinforcement Learning (DRL) algorithms tend to adversely affect their performance. Averaged-DQN is a simple extension to the DQN algorithm, based on averaging previously learned Q-values estimates, which leads to a more stable training procedure and improved performance by reducing …
We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface …
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
New stability theorems for H-type Carnot groups established.
problem Characterize H-type Carnot groups using stability theorems.
method Introduced H-type deviation, computed for families, established new characterizations.
result H-type Carnot groups are uniquely characterized by specific conditions.
We consider a non-Gaussian option pricing model, into which the underlying log-price is assumed to be driven by an α-stable distribution. We remove the a priori divergence of the model by introducing a Mellin regularization for the Lévy propagator. Using distributional and Cn tools, we derive an analytic …
Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.