For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue…
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Researchers found explicit solutions to a complex equation in advanced geometry.
problem Critical Yamabe type equation in sub-Finsler geometry.
method Computed a two-parameter family of explicit positive solutions.
result Explicit solutions to a critical equation in sub-Finsler geometry.
Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.
problem Investigating metrics maximizing eigenvalues of Paneitz operator.
method Critical points of eigenvalues of Paneitz operator on Riemannian metrics with fixed volume.
result Critical metrics associated with extrinsic conformal-harmonic maps into round spheres.
Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.
problem Analyzing Q-curvatures and Paneitz operators for hypersurfaces.
method Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.
result Explicit formulas for the extrinsic Paneitz operators P_4 and extrinsic Q-curvatures for totally umbilic hypersurfaces in any dimension.
We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.
problem Foliation of an asymptotically flat end by critical hypersurfaces.
method Constructing hypersurfaces as critical points of a functional, solving an over-determined boundary value problem.
result Solutions to the Laplace-Beltrami operator over a foliation of critical spheres.
The paper proves conditions under which critical point metrics are Einstein.
problem Proving that critical point metrics are Einstein under specific curvature constraints.
method Analyzing the traceless Ricci operator and scalar curvature constraints.
result The conjecture that critical point metrics are Einstein is proven under certain curvature conditions.
Essential self-adjointness and spectrum of CR GJMS operator proved.
problem Characterizing the spectrum of CR GJMS operator.
method Proving essential self-adjointness and closed range, analyzing spectrum.
result CR GJMS operator has discrete spectrum with finite-dimensional eigenspaces.
New framework for studying eigenvalue functionals of metrics.
problem Understanding critical points of eigenvalue functionals.
method Clarke subdifferential theory to unify previous research.
result Unified understanding of critical metrics and new examples.
Critical metrics for the quadratic curvature functional on complete four-dimensional manifoldsmath.DG
Critical metrics on four-dimensional manifolds are either Einstein or product of two-dimensional manifolds.
problem Classifying critical metrics of a curvature functional on complete four-dimensional manifolds.
method Analyzing the curvature operator and energy condition to prove metric properties.
result Complete four-dimensional manifolds with finite energy are either Einstein or product of two-dimensional manifolds.
On the cohomology of sl(m+1,R) acting on differential operators and sl(m+1,R)-equivariant symbolmath.DG
One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his cla…
Study on critical Lagrangian phase singularities in mean curvature flow.
problem Analyzing singularities in the Lagrangian mean curvature flow at the critical phase.
method Developed new method to prove estimates by using concave operators.
result Established interior estimates for critical Lagrangian phase singularities.
We prove the existence and uniqueness of a *projectively equivariant symbol map*, which is an isomorphism between the space of bidifferential operators acting on tensor densities over and that of their symbols, when both are considered as modules over an imbedding of into polynomial vector fields. Th…
For a pinched Hadamard manifold and a discrete group of isometries of , the critical exponent is the exponential growth rate of the orbit of a point in under the action of . We show that the critical exponent for any family of normal subgroups of has the same coarse behaviour…
In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our argumen…
A methodology is developed to identify, as units of study, each decrease in the value of a stock from a given maximum price level. A critical level in the amount of price declines is found to separate a segment operating under a random walk from a segment operating under a power law. This level is interpreted as a poin…
Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR structuresmath.DG
This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…
Symmetry proven for positive solutions of a weighted p-Laplace operator inequality.
problem Proving symmetry of positive solutions to a specific type of inequality.
method Analyzing positive critical points of Caffarelli-Kohn-Nirenberg inequalities with a weighted p-Laplace operator.
result Complete classification and symmetry result for positive solutions in a range of parameters.
In this paper, we study some fourth order singular critical equations of Lichnerowicz type involving the Paneitz-Branson operator, and we prove existence and non existence results under given assumptions.
Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.
This paper critiques the Standardized Measurement Approach (SMA) for operational risk and recommends maintaining Advanced Measurement Approach (AMA).
problem Weaknesses and failures of the Standardized Measurement Approach (SMA) in operational risk.
method Critical review and analysis of SMA and AMA approaches.
result SMA is unstable, insensitive to risk, and implicitly related to systemic risk in the banking sector.
We present a new Q-function operator for temporal difference (TD) learning methods that explicitly encodes robustness against significant rare events (SRE) in critical domains. The operator, which we call the -operator, allows to learn a robust policy in a model-based fashion without actually observing the SRE. We i…
This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.
In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We fin…
In this paper we investigate the properties of a semi-linear problem on a spin manifold involving the Dirac operator, through the construction of Rabinowitz-Floer homology groups. We give several existence results for sub-critical and critical non-linearities as application of the computation of the different homologie…
We look at several problems in even dimensional conformal geometry based around the de Rham complex. A leading and motivating problem is to find a conformally invariant replacement for the usual de Rham harmonics. An obviously related problem is to find, for each order of differential form bundle, a ``gauge'' operator …
Stock markets are complex systems exhibiting collective phenomena and particular features such as synchronization, fluctuations distributed as power-laws, non-random structures and similarity to neural networks. Such specific properties suggest that markets operate at a very special point. Financial markets are believe…
Study critical quasilinear equations on Riemannian manifolds with curvature constraints.
problem Investigate critical quasilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature.
method Utilize a new nonlinear Kato inequality and Cheng-Yau type gradient estimates for positive solutions.
result Classify positive solutions to the critical -Laplace equation and show rigidity concerning the ambient manifold.
Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.
problem Solving PDEs for varying parameters is computationally expensive.
method Physics-informed neural networks and neural operators learn solution mappings across parameter spaces.
result Neural operators achieve computational speedups of 10^3 to 10^5 times faster than traditional methods.
Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.
problem Computing the Morse index of the critical catenoid
method Physics-Informed Neural Network (PINN) enforces parity and eigenvalue as trainable parameters
result Returns eigenvalues within to of exact values
CR-harmonic mapsmath.DG
We develop the notion of renormalized energy in CR geometry, for maps from a strictly pseudoconvex pseudohermitian manifold to a Riemannian manifold. This energy is a CR invariant functional, whose critical points, which we call CR-harmonic maps, satisfy a CR covariant subelliptic partial differential equation. The cor…
Unsupervised detection and open-set classification of fast-ramped flexibility activation eventseess.SY
Automates detection of fast-ramped flexibility events for DSOs.
problem Monitoring and supervising flexibility activations in power systems.
method Unsupervised detection and open-set classification.
result Automatically identifies critical flexibility activations for early intervention.
CMCO provides robust uncertainty estimates for neural operators without retraining.
problem Uncertainty quantification in deep learning for real-time virtual sensing.
method Unified Monte Carlo dropout and split conformal prediction in DeepONet.
result Near-nominal empirical coverage in diverse applications.
Random subgroups in hyperbolic spaces have full limit sets and bounded critical exponents.
problem Understanding stationary random subgroups in hyperbolic spaces.
method Analyzing limit sets and critical exponents of random subgroups.
result Random subgroups have full limit sets and bounded critical exponents.
The paper finds sign-changing solutions for a specific type of elliptic equation.
problem Existence of sign-changing solutions for a Yamabe type equation.
method Investigates a critical elliptic equation with a Yamabe type operator on a compact manifold with boundary.
result Existence of sign-changing solutions assured under certain geometric conditions.
Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…
AgraSSt assesses graph generators using Stein operators and kernel discrepancies.
problem Assessing the quality of graph generators that are implicit or not in explicit form.
method AgraSSt uses Stein operators and kernel discrepancies to assess graph generators, providing interpretable criticisms.
result Theoretical guarantees and empirical validation for various graph models.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.
problem Existence and rigidity of critical Z2 eigenvalues on sphere configurations.
method Algebraic identities and finite group representation theory.
result Construction of infinitely many configurations admitting critical eigensections and proof of deformation rigidity of Taubes-Wu tetrahedral eigensections.
Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.
problem Finding optimal bounds for Dirac eigenvalues on spin surfaces.
method Minimization problem within a fixed conformal class, focusing on surfaces.
result Derive isoperimetric inequalities for the Dirac operator on the sphere, complete conformal spectrum characterization.
We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator on each regular level set of a fixed Morse function defining this cobordism. We show that as we approac…
Single-timescale actor-critic finds globally optimal policy.
problem Finding globally optimal policy in reinforcement learning.
method Simultaneous actor and critic updates with linear or deep neural network approximations.
result Actor sequence converges to globally optimal policy at rate.
CyPhERS provides real-time event info for CPSs, avoiding downtime.
problem Real-time event identification in CPSs is challenging due to complex interdependencies and rare events.
method CyPhERS integrates cyber and physical components, generating event signatures for known and unknown events.
result Event signatures provide relevant and inferable information on both known and unknown event types.
HOFLON automates process start-ups and grade-changes using offline RL and online optimization.
problem Manual operation of start-ups and grade-changes by experts is declining, leaving plant owners without the necessary tacit know-how.
method HOFLON combines offline RL to learn a latent manifold and long-horizon Q-critic, and online optimization to maximize Q-critic while penalizing deviations and excessive variable changes.
result HOFLON outperforms standard offline RL in industrial case studies, delivering better cumulative rewards than historical data.
Defines a new energy for submanifolds, comparing to Willmore energy.
problem Finding new conformally invariant energies for submanifolds.
method Coupling tractor connection to GJMS operators for higher-dimensional analogues.
result Shows comparison with existing energy in 4D.
New spinorial functional connects Perelman's W- and F-functionals.
problem Unifying Perelman's functionals for spin manifolds.
method Introduced a new energy functional on spin manifolds, computed its first variation, and established a gradient flow.
result Critical points of the functional are twisted Ricci solitons and eigen-spinsors.
We discuss the solution theory of operators of the form , acting on smooth sections of a vector bundle with connection over a manifold , where is a vector field having a critical point with positive linearization at some point . As an operator on a suitable space of smooth section…
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.