We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…
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48 results for “set operations”
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…
Self-ONNs adapt nodal operators during training for higher diversity and efficiency.
problem Limited network heterogeneity and high computational demand in ONNs.
method Self-organized ONNs with generative neurons that adapt nodal operators during training.
result Self-ONNs achieve utmost heterogeneity and computational efficiency.
The singular sets of degenerate and nonlocal elliptic equations on Poincaré-Einstein manifoldsmath.DG
The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield independent characteristic numbers mod 2, which generalize the Akbulut-K…
Elite ONNs learn better with synaptic plasticity, improving performance over CNNs.
problem Limited heterogeneity in ONNs due to fixed operator sets.
method Synaptic plasticity-based search for optimal operator sets.
result Elite ONNs achieve superior learning performance compared to conventional methods.
We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form if and only if it arises as the noncontact set of an obstacle problem involving the …
A softmax operator applied to a set of values acts somewhat like the maximization function and somewhat like an average. In sequential decision making, softmax is often used in settings where it is necessary to maximize utility but also to hedge against problems that arise from putting all of one's weight behind a sing…
Decision-calibrated prediction sets improve power system operations by reducing unnecessary costs.
problem Balancing operating costs and reliability in power systems with renewable uncertainty.
method Learn conditional prediction sets as sub-level sets of norm-based score functions, calibrate uncertainty sets based on reliability of downstream decisions.
result Decision-calibrated sets lead to more efficient operations with smaller uncertainty sets and lower costs compared to standard coverage-based calibration.
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…
Online learning of linear operators between infinite-dimensional spaces is possible but with limitations.
problem Learning linear operators between infinite-dimensional Hilbert spaces in an online setting.
method Online learning approach for linear operators with bounded -Schatten norm, proving impossibility for operator norm.
result Separation between online learnability and uniform convergence for bounded linear operators.
Calderón projector extended to fibred cusp operators.
problem Extending Calderón projector to non-compact manifolds with special fibred structure.
method φ-pseudodifferential calculus by Mazzeo and Melrose.
result Calderón projector for fibred cusp operators is a pseudodifferential operator.
New method for operating envelope identifies key performance indicators without arbitrary binning.
problem Accurate identification of operating envelope for optimal KPIs.
method Regularized GA algorithm with interpretability and implementability constraints.
result Validated through simulations and real-world application in mining.
We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.
Mixtures of neural operators reduce active complexity in operator learning.
problem Reduction of active complexity in operator learning models.
method Constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction.
result Every scalar uniformly continuous nonlinear operator can be approximated by a MoNO whose active expert has smaller depth, width, and rank scaling.
Study magnetic Schrödinger operators in Euclidean space.
problem Semiclassical spectral analysis of magnetic Schrödinger operators.
method Spectral problems for Bochner-Schrödinger operator on manifolds.
result Survey and describe ideas of proofs for magnetic Schrödinger operators.
Study solves inverse problems for real principal type operators using unique data sets and ray transforms.
problem Determining coefficients in real principal type equations from boundary data.
method Unique data sets, bicharacteristic ray transforms, and propagation of singularities.
result Global uniqueness results for determining coefficients in nonlinear real principal type equations.
Maps sets to probability distributions to minimize information loss.
problem Learning to map sets to probability distributions to preserve information.
method Relates set operations to probability distribution interpolations and demonstrates a preliminary solution.
result Experimental results show the effectiveness of the set embedding approach.
Study proves boundedness of operators in variable exponent Morrey spaces.
problem Boundedness of operators in global Morrey-type spaces with variable exponents.
method Analysis of Hardy-Littlewood maximal operator and potential type operator in variable exponent Morrey spaces.
result Boundedness of the Hardy-Littlewood maximal operator and potential type operator in global Morrey-type spaces with variable exponents.
We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-D…
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.
problem Deriving formulas for eigenvalues of elliptic operators on compact manifolds.
method Variational methods applied to elliptic operators on compact Riemannian manifolds.
result Generic subsets of metrics yield simple spectra of elliptic operators.
LSCI provides locally adaptive prediction sets for operator models with tighter coverage.
problem Generating robust, calibrated uncertainty quantification for operator models.
method Local Sliced Conformal Inference (LSCI) for operator models.
result LSCI yields tighter prediction sets with stronger adaptivity compared to conformal baselines.
Study on removing sets and uniqueness of diffusion operators on various spaces.
problem Determining the effect of removing small sets on the self-adjointness and uniqueness of diffusion operators.
method Analyzes symmetric diffusion operators on metric measure spaces, proving a truncation result for potentials.
result Characterizes the critical size of removed sets and their effect on operator properties.
New algorithm learns Koopman operator online, with complexity control and convergence guarantees.
problem Online learning of Koopman operator for general nonlinear systems.
method Sparse online learning via stochastic approximation, RKHS action, CME operator.
result Provably convergent algorithm with finite-time guarantees in mis-specified setting.
We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal growth of positive solutions of Lw = 0 towards the singular set is a stable propert…
Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method…
Study on magnetic Dirac operators and their spectrum.
problem Understanding the spectrum of magnetic Dirac operators.
method Analysis of magnetic Dirac operators over complete Riemannian manifolds.
result Find sufficient conditions for maximal or discrete spectrum.
Study differential operators over maps and their applications in supermanifolds.
problem Understanding differential operators over smooth maps and their applications.
method Recall and study differential operators, formal -differential operators, pullbacks by thick morphisms, and quantization of symplectic micromorphisms.
result Developed constructions and examples of differential operators over maps.
We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the cur…
Paper tackles conditional expectation estimation using compactification operators.
problem Estimating conditional expectations from product of two random variables.
method Operator theoretic approach using kernel integral operators in reproducing kernel Hilbert space.
result Solutions allow numerical approximation and convergence of data-driven implementations.
Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. This survey article studies when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of definition. It deals with results for the Jacobi operator, the higher order Jacobi opera…
ICON learns differential equation operators from examples, revealing probabilistic inference.
problem Learning operators for differential equations from limited examples.
method Probabilistic operator learning using ICON architectures trained on diverse datasets.
result ICON implicitly performs Bayesian inference on solution operators.
Proves surjectivity of certain smooth maps with non-properness sets.
problem Surjectivity of linear operators and global diffeomorphisms of semialgebraic maps.
method Analytic proof involving semialgebraic local diffeomorphisms and linear partial differential operators.
result A new analytic conjecture for polynomial local diffeomorphisms of implies known results.
We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…
Locally isotropic pseudo-Riemannian manifolds are known to be locally symmetric; this result is due to Wolf. In the Riemannian setting one proof, due to Szabó, uses spectral properties of the so-called Szabó operator. In this paper we extend Szabó's method to the pseudo-Riemannian setting, obtaining results comparable …
Study Dirac operators on finite warped cylinders with gauge fields.
problem Characterize spectral flow on finite warped cylinders with gauge fields.
method Identify endpoint operators, derive determinant characterization, introduce regularized APS conditions.
result Regularized APS conditions admit a spectral-flow framework, matching zero-mode sets.
Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are…
A note on lattice knotsmath.GT
One type of switch simplifies operations on lattice knots.
problem Operations on lattice knots are complex.
method Reduced operations to one type of local switch.
result Simplified set of operations on lattice knots.
Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.
problem Analyzing Dirac operator on non-compact spacetimes with non-compact Cauchy hypersurface.
method Building on previous works, extends Fredholm result to non-compact Lorentzian spaces, using von Neumann algebras and Galois coverings.
result Γ-Fredholmness of the Dirac operator under APS boundary conditions.
In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators(we will call it the "algebraic" version in this article). In 2012, Przytycki defined another homology theory for pre-Yang-Baxter operators which has a nice graphic visualization(we will call it the "graphic" version i…
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.
Framework extends neural operators to handle functions outside training set.
problem Robust handling of functions beyond the training set.
method Kernel approximation techniques and Reproducing Kernel Hilbert Spaces (RKHSs) theory.
result Theoretical framework and empirical validation for reliable function extension.
Study optimal holomorphic extensions on complex manifolds with transitivity property.
problem Optimal holomorphic extensions on complex manifolds with transitivity property.
method Use Toeplitz operators and transitivity property for optimal holomorphic extensions.
result Transitivity property of optimal holomorphic extensions with small defect.
Study geometric isomorphisms between spacetime solutions using paracausal metrics.
problem Geometric isomorphisms between solutions of normally hyperbolic operators over different spacetimes.
method Introduce paracausal relation to define isomorphisms between spacetime metrics and use Møller operators.
result Møller operators preserve causal propagators and natural symplectic forms on initial data.
Absolute index theorem for warped product manifolds.
problem Equivariant index computation for manifolds with warped product structures.
method Warped product structure, Fredholm operator, Atiyah-Segal-Singer index theorem.
result Equivariant relative index theorem for manifolds with warped product structures.
The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testin…