Estimates for special Lagrangian curvature equations in critical and convex cases.
problem Interior estimates for special Lagrangian curvature equations.
method Establishes a priori interior curvature and gradient estimates.
result Proves interior curvature and gradient estimates for special Lagrangian curvature equations.
Characterizes critical points in convex double and triple bubbles.
problem Critical points of double and triple bubbles in convex shapes.
method Characterization through stationary varifolds in Rn and R3. result Characterization of critical points in convex shapes.
This paper identifies critical cases for evaluating PV investment impacts on MV networks efficiently.
problem Challenges in maintaining and controlling voltages in MV distribution networks due to increasing PV generation.
method Clustering MV nodes based on electrical adjacency and time blocks, identifying critical cases for further study.
result A scalable method to time efficiently identify critical cases for PV investment evaluation.
The paper proves new rigidity results for critical metrics of quadratic curvature functionals.
problem Proving rigidity of critical metrics for specific quadratic curvature functionals.
method Rigidity results for conformal vector fields, ODE argument, and new pointwise and integral estimates.
result Critical metrics are rigid under specific conditions.
Actor-critic converges globally in LQR with ergodic cost.
problem Theoretical understanding of actor-critic algorithm's global convergence.
method Nonasymptotic convergence analysis of actor-critic in linear quadratic regulator (LQR) setting.
result Actor-critic finds globally optimal policy and value function at a linear rate.
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.
Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. It …
Geodesic distance vanishes for critical Sobolev norms on diffeomorphism groups.
problem Analyzing geodesic distance in diffeomorphism groups for critical Sobolev norms.
method Combining techniques from [JM19] and [BHP18]
result Geodesic distance vanishes for Ws,n/s norms when s∈(0,1) and sp≤n. We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak …
We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…
When f : R power n to R power p, is a surjective real analytic map with isolated critical value, we prove that the (m)-regularity condition (in a sense we define) ensures that f ||f|| is a fibration on small spheres, f induces a fibration on the tubes and both fibrations are equivalent. In particular, we make the state…
This paper studies symplectic critical surfaces in Hermite surfaces.
problem Generalizing results about Kähler angle to the general case.
method Focuses on symplectic critical surfaces in Hermite surfaces.
result Provides a definition of symplectic critical surfaces in Hermite surfaces.
Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.
problem Existence of least energy solutions for nonlinear p-Laplacian problems with critical exponent.
method Proving existence of solutions through critical point theory and variational methods.
result Significant difference in existence results between p-Laplacian and Laplacian cases.
Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e. the induced metric g on M satisfies the equation:−(Δgλ)g+∇g2λ−λRic=g, where λ is a smooth function on M. At first, for the case wher…
Solves critical LYZ equation in Kähler geometry.
problem Solvability of LYZ equation at critical phase.
method Establishes existence of smooth solutions.
result Solves critical case of LYZ equation.
Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
problem Solving Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
method Uses interior C2 estimate. result Result is sharp, showing existence of singular solutions in subcritical phase.
We investigate fourth order Paneitz equations of critical growth in the case of n-dimensional closed conformally flat manifolds, n≥5. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the Q-cu…
Solves local minima problems on smooth manifolds.
problem Local minima issues on smooth manifolds.
method Introducing valley functions and applying Morse's lemma.
result Eliminates critical points and reduces to 1D.
We compute the minimum number of critical points of a small codimension smooth map between two manifolds. We give as well some partial results for the case of higher codimension when the manifolds are spheres.
This paper studies properties of weak reducing pairs in critical Heegaard splittings.
problem Characterize weak reducing pairs in critical Heegaard splittings.
method Analyze the properties of weak reducing pairs in critical Heegaard splittings.
result Provide a necessary condition for a Heegaard surface to be critical.
Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.
problem Minimizing the Euler-Plateau energy with elastic modulus.
method Analyzing the energy functional and its minimizers, considering different boundary conditions and topological constraints.
result Potential minimizers are highly dependent on physical rigidity parameters, and the area of critical surfaces can be computed from boundary data.
We present the complete analytical classification of the atoms arising at the critical points of rank 1 of the Kowalevski-Yehia gyrostat. To classify the Smale-Fomenko diagrams, all separating values of the gyrostatic momentum are found. We present a kind of constructor of the Fomenko graphs; its application gives the …
Proposes a deep reinforcement learning framework for dynamic multichannel access.
problem Efficient use of limited spectral resources in dynamic multichannel access.
method Deep actor-critic reinforcement learning framework for both single-user and multi-user scenarios.
result Demonstrates improved performance and adaptive ability compared to existing methods.
The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold M with smooth boundary ∂M. Here, we will give the complete classification for an n-dimensional, n=3 or 4, weakly Einstein critical metric of the volume functional with nonnegative scalar …
We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, c…
Proves planar Lipschitz critical points of area functional are smooth.
problem Lawson-Osserman conjecture about smoothness of critical points.
method Outer variations to prove smoothness of critical points.
result Proves conjecture for planar case.
The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
In this paper we study the energy function associated to fourth order equations of critical growth on smooth compact conformally flat manifolds of dimension greater or equal than 5.
Actor-critic methods solve reinforcement learning problems by updating a parameterized policy known as an actor in a direction that increases an estimate of the expected return known as a critic. However, existing actor-critic methods only use values or gradients of the critic to update the policy parameter. In this pa…
The paper studies a flow equation on even-dimensional manifolds, proving convergence under critical conditions.
problem Proving convergence of the prescribed Q-curvature flow equation in critical cases. method Analyzes the flow equation on arbitrary even-dimensional closed Riemannian manifolds, proving convergence under specific geometric hypotheses.
result Proves convergence of the flow equation when the integral of Q equals (n−1)!Vol(Sn), extending previous results. We develop a topology data analysis-based method to detect early signs for critical transitions in financial data. From the time-series of multiple stock prices, we build time-dependent correlation networks, which exhibit topological structures. We compute the persistent homology associated to these structures in order…
We prove sharp pointwise decay estimates for critical Dirac equations on Rn with n≥2. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited state…
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.
The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.
problem The rigidity and positivity of the Hawking energy on specific surfaces in general relativity.
method Evaluation of the Hawking energy on area-constrained critical surfaces under the dominant energy condition.
result The Hawking energy is nonnegative and rigid on area-constrained surfaces, including charged and cosmological constant variants.
New method improves off-policy critic evaluation in reinforcement learning.
problem High variance and instability in off-policy policy evaluation.
method Doubly robust estimators applied to actor-critic algorithms.
result Doubly robust estimation significantly improves performance in continuous control tasks.
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or…
The paper classifies cosymplectic manifolds with critical metrics in dimension 3.
problem Classifying cosymplectic manifolds with critical metrics.
method Study of Chern-Hamilton energy functional on compact cosymplectic manifolds.
result Classification of manifolds admitting critical compatible metrics in dimension 3.
The Brasselet number helps calculate function germs with one-dimensional critical sets.
problem Calculating topological information of function germs with nonisolated singularities.
method Using the Brasselet number, the paper presents formulas for function germs with a one-dimensional critical locus.
result Formulas for function germs with a one-dimensional critical locus.
New foliations found for critical surfaces of Hawking energy, resolving discrepancies.
problem Finding consistent critical surfaces for the Hawking energy in non-totally geodesic spacelike hypersurfaces.
method Constructing a unique local foliation of area constrained critical surfaces of the Hawking energy in the general case of non-totally geodesic spacelike hypersurfaces.
result Discrepancy found in the small sphere limit of the Hawking energy, explained and resolved.
Stability of biharmonic maps in critical dimension proven.
problem Stability of biharmonic maps between manifolds in critical dimension.
method Generalization of Morse stability theory to biharmonic maps, development of strong energy quantization method.
result Strong energy quantization in a wide class of problems in geometric analysis.
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. Analyticity of critical points for O'Hara's knot energies proved.
problem Analyzing the regularity of critical points for O'Hara's knot energies.
method Cauchy's method of majorants and a Möbius energy-inspired gradient decomposition.
result Smooth critical points of O'Hara's knot energies are analytic.
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 O(K−1/2) rate. In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question becomes easy to answer.
The paper explores the critical point equation on Kenmotsu and almost Kenmotsu manifolds.
problem Investigating the critical point equation on specific types of manifolds.
method Analyzing Kenmotsu and almost Kenmotsu manifolds with nullity conditions.
result Complete Kenmotsu metrics satisfying the CPE are Einstein and locally isometric to H2n+1.
Accelerated method finds critical points faster on manifolds.
problem Optimization on non-convex manifolds.
method Accelerated gradient methods on Riemannian manifolds.
result Find approximate first-order critical points faster than regular gradient descent.
Study of surfaces meeting a plane orthogonally or contained in a line, deriving weak forms and proving regularity.
problem Critical points of the Willmore functional with boundary constraints.
method Weak forms of free boundary conditions derived by reflection.
result Proved regularity of the surfaces.
Machine learning predicts critical points for directed percolation models.
problem Determining critical points for directed percolation models.
method Supervised and unsupervised machine learning algorithms (CNN and DBSCAN) were used.
result Machine learning accurately predicts critical points for both models.