New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.
problem Understanding the asymptotic behavior of finite energy SU(2) monopoles on AC 3-manifolds.
method Analysis of critical points of the SU(2) Yang--Mills--Higgs energy on asymptotically conical 3-manifolds.
result Proves integrality of the monopole number and quadratic decay of curvature, among other findings.
Many practical machine learning tasks employ very deep convolutional neural networks. Such large depths pose formidable computational challenges in training and operating the network. It is therefore important to understand how fast the energy contained in the propagated signals (a.k.a. feature maps) decays across laye…
This paper tackles class imbalance in high energy physics experiments.
problem Extracting a signal from a large background in high energy physics experiments.
method Overview of class imbalance techniques and case studies.
result Demonstrates the effectiveness of class imbalance techniques in high energy physics experiments.
We give conditions on a general stress-energy tensor T_{αβ} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay p…
Removes singularities for Yang-Mills-Higgs fields in higher dimensions.
problem Yang-Mills-Higgs fields with isolated singularities.
method Establishes decay estimates and conformally invariant energy bounds.
result Removable singularity theorem for Yang-Mills-Higgs fields.
New proof removes decay assumptions for spacetime positive mass theorem.
problem Proving the rigidity of the spacetime positive mass theorem without additional decay assumptions.
method Uses spacetime harmonic functions and Liouville's theorem, and an alternative proof based on Killing development.
result Removes additional decay assumptions for the spacetime positive mass theorem.
This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_gψ = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general ψ or (II2) |a| < M, ψ axisymmetric. In either case, we prove a version of 'integrated local energy decay', specificall…
Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.
problem Understanding Willmore surfaces and their properties.
method Applying the 3-circle theorem to analyze the second fundamental form of Willmore surfaces.
result Proves a decay estimate of the second fundamental form along the neck region.
Second part of series studying charged scalar fields on Reissner--Nordström spacetimes.
problem Analyzing late-time behavior and stability of charged scalar fields on black hole backgrounds.
method Purely physical-space based methods, energy estimates, inverse-power laws.
result First pointwise decay estimates for charged scalar fields on black hole backgrounds.
The fundamental properties of J-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a J-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …
Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.
problem Analyzing decay and non-decay rates of solutions to the massless Vlasov equation on Reissner-Nordström spacetimes.
method Quantitative analysis of geodesic flow and comparison to wave equation instability results.
result Exponential decay rates in subextremal cases and polynomial rates in extremal cases, with non-decay of transversal derivatives in extremal cases.
We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain …
We consider solutions to the linear wave equation □gφ=0 on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may con…
We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the ∂ˉ-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then…
The paper studies decay near singularities of 3d Yang-Mills-Higgs fields.
problem Understanding isolated singularities of 3d Yang-Mills-Higgs fields.
method Derives decay estimates and applies removable singularity theorems.
result Generalizes removable singularity theorems for 3d Yang-Mills-Higgs fields.
New energy functional bounds Ricci flows on ancient spaces.
problem Bounding Ricci flows on ancient spaces.
method Introducing a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows.
result Provides an upper bound for the ordinary λ-functional.
The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.
problem Investigating the asymptotic behavior of G2-monopoles on nonparabolic G2-manifolds.
method Proof of finite mass for monopoles with bounded curvature on nonparabolic G2-manifolds; sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
result Sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
We establish that finite-time singularities do not occur in four-dimensional Yang-Mills flow, confirming the conjecture of Schlatter, Struwe, and Tahvildar-Zadeh. The proof relies on a weighted energy identity and sharp decay estimates in the neck region.
Bayesian analysis predicts properties of proton-emitting nuclei beyond the proton drip line.
problem Predicting properties of unstable nuclei in the proton-rich region.
method Bayesian Gaussian processes and mass models corrected with statistical emulators.
result Quantified predictions for separation energies and probabilities of proton emission.
The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fa…
We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 …
In recent work, we have proven uniform decay bounds for solutions of the wave equation □gφ=0 on a Schwarzschild exterior, in particular, the uniform pointwise estimate ∣φ∣≤Cv+−1, which holds throughout the domain of outer communications, where v is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…
Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.
problem Approximating the Willmore functional using nonlocal methods.
method Gamma-convergence and fractional Laplacian analysis in Fermi coordinates.
result Proves Γ-limsup estimate for the proposed nonlocal approximation. This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial da…
New magnetic memory effects found in gravitational waves and memory.
problem Understanding new effects in gravitational waves and memory.
method Analyzing asymptotically-flat spacetimes with slow decay.
result Diverging magnetic memory sourced by curvature and neutrino cloud.
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
Survey on stability of Minkowski spacetime in relativity.
problem Nonlinear stability of Minkowski spacetime in general relativity.
method Decay assumptions, geometric foliations, energy identities, and gauge choices.
result Understanding of decay, dispersion, and geometry-analysis interplay.
Future stability of FLRW solutions in expanding 3D space is shown for compact perturbations.
problem Future stability of expanding FLRW solutions with spatial topology R^3.
method Nonlinear stability analysis of spherically symmetric perturbations.
result Decay rates of energy momentum tensor components compared to Minkowski space.
This paper reduces the complexity of deep CNNs by optimizing their topology.
problem Reduces computational complexity and storage in deep CNNs for resource-constrained platforms.
method Analyzes the impact of CNN topology (depth and width) on feature extraction, focusing on scattering networks.
result Designs networks of fixed depth to retain a significant portion of input signal energy in the feature vector.
Iterative subtraction method outperforms other feature ranking techniques in high-energy physics.
problem Determining the most important features for classification in high-energy physics experiments.
method Comparison of feature ranking methods including Iterative Addition, Iterative Removal, and BDT Selection Frequency.
result Iterative Removal method is the most efficient for feature ranking in classification tasks.
We propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds derived from the extended musculo-skeletal configuration manifold. The …
Finite energy solutions classified for Seiberg-Witten equations on complex plane and Riemann surface.
problem Classifying solutions to Seiberg-Witten equations with finite energy.
method Established a classification theorem for solutions on X=CimesΣ with finite analytic energy. result Finite energy solutions correspond to polynomial maps from C to H0(Σ,L+,∂ˉ). New insights into Bartnik mass from improvability of dominant energy scalar.
problem Characterizing Bartnik mass minimizing initial data sets.
method Introducing improvability concept, proving non-improvability consequences, and analyzing pp-wave counterexamples.
result Bartnik mass minimizing initial data sets are characterized, advancing conjectures.
We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in R2 introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…
Physics-informed learning framework for pH systems and EB-PBC control.
problem Control of port-Hamiltonian systems from trajectory data.
method Co-learning of pH system model and EB-PBC through alternating optimization.
result Proven stability and robustness of the learned controller.
Machine learning identifies jet substructure from boosted Higgs decays.
problem Distinguishing jets from boosted heavy particles from QCD jets.
method Spectral analysis using neural networks on angular scale.
result ANN of angular spectrum input performs similarly to existing taggers.
We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension n=3 to dimensions 3≤n<8. This requires us to address several technical difficulties that are not present when n=3. The regularity and decay assumptions for the initial data sets to which our argume…
Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.
problem Proving growth of spinors in GSW equations on R4 and R3. method Unified framework of GSW equations, averaged L2-norm, curvature decay assumption, Yang-Mills-Higgs energy. result Growth of spinors in GSW equations on R4 and R3 faster than a power of the radius under suitable curvature decay. Study on helix curves and their Möbius energy asymptotics.
problem Understanding the asymptotic behavior of Möbius energy for helix curves.
method Investigation of helix curves with fixed radius, focusing on energy decay and blow-up.
result Proven asymptotics for both uncoiling and coiling helix curves, revealing distinct strategies for each.
Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.
problem Understanding the behavior and stability of charged scalar fields on near-extremal Reissner-Nordström spacetimes.
method Global integrated energy decay and boundedness estimates for solutions to the charged scalar field equation.
result Established global, weighted integrated energy decay and boundedness estimates for solutions on (near-)extremal Reissner-Nordström(--de Sitter) spacetimes.
New energy definition for expanding de Sitter spacetime with umbilic boundaries.
problem Defining energy for spacetimes with expanding de Sitter background and umbilic boundaries.
method Adapting Liu-Yau energy to a quasi-local setting in expanding de Sitter spacetime.
result Positivity of the defined energy for certain values of the cosmological constant.
New method for LVEBMs using saddle-point optimization and Langevin updates.
problem Expressive generative modeling of latent variables with hidden structure.
method Reformulate LVEBM training as a saddle problem, using Langevin updates and gradient flows.
result Proves existence and convergence of the algorithm under standard assumptions, with improved ELBO bounds.
Paper proves rigidity theorems for AE Q-singular spaces.
problem Analyzing Q-curvature on AE manifolds. method Introducing a fourth order energy and rewriting it in terms of a fourth order Ricci-like tensor.
result Yamabe positive J-flat AE manifolds are isometric to Euclidean space. The study proves constraints on the structure of compact half-conformally flat manifolds.
problem Analyzing the structure of compact half-conformally flat manifolds.
method Analyzes manifolds with bounded L2 energy, scalar curvature, and non-collapsing assumption. result Proves all Betti numbers are bounded for certain manifolds.
We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting…
Study axisymmetric waves on extremal Kerr spacetime using physical-space estimates.
problem Obtain integrated local energy decay estimates for axisymmetric waves on extremal Kerr backgrounds.
method Use physical-space analysis and a method introduced by Stogin, simplifying Aretakis' derivation.
result Extend Morawetz estimates to extremal Kerr spacetime using purely classical currents.