Upper bounds for Lagrangian capacities of Liouville domains
problem Lagrangian capacity of Liouville domains
method Using -equivariant techniques
result Extremal Lagrangian torus on the boundary of ellipsoid
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48 results for “Lagrangian capacity”
The codisc radius capacitymath.SG
We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…
We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of . Namely, if is a Lagrangian submanifold with weakly harmonic Lagrangian phase then must be smooth. In the process we also discuss a local version of the equation, which is a nonline…
The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.
problem Understanding ECH capacities and their relation to Anosov flows.
method Relating ECH capacities to Anosov flows dynamics, proving infinite capacities and obstructions.
result ECH capacities are infinite for many symplectic 4-manifolds, including cotangent disk bundles over surfaces of genus at least two.
Improved non-squeezing theorem for calibrated geometries proved.
problem Proving an improved non-squeezing theorem for calibrated geometries.
method Two proofs: direct and reduction to classical case.
result Established an improved non-squeezing theorem for calibrated geometries.
New methods prove non-squeezing in locally conformal symplectic geometry.
problem Non-squeezing theorem in locally conformal symplectic geometry.
method Generating functions and spectral selectors for lcs Hamiltonian diffeomorphisms.
result Proves a non-squeezing theorem in .
This work extends the randomized shortest paths (RSP) model by investigating the net flow RSP and adding capacity constraints on edge flows. The standard RSP is a model of movement, or spread, through a network interpolating between a random-walk and a shortest-path behavior [30, 42, 49]. The framework assumes a unit f…
Defines a distance function on a manifold using symplectic embeddings and recovers the metric.
problem Recovering a Riemannian metric from symplectic embeddings in cotangent bundles.
method Defines a distance-like function using symplectic embeddings and recovers the metric when is the unit disc-cotangent bundle.
result The distance function recovers the Riemannian metric when is the unit disc-cotangent bundle.
We seek to improve the data efficiency of neural networks and present novel implementations of parameterized piece-wise polynomial activation functions. The parameters are the y-coordinates of n+1 Chebyshev nodes per hidden unit and Lagrangian interpolation between the nodes produces the polynomial on [-1, 1]. We show …
Fisher GANcs.LG
Generative Adversarial Networks (GANs) are powerful models for learning complex distributions. Stable training of GANs has been addressed in many recent works which explore different metrics between distributions. In this paper we introduce Fisher GAN which fits within the Integral Probability Metrics (IPM) framework f…
We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the -Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the esta…
Graph neural networks optimize radio resource management policies for wireless networks.
problem Optimizing user selection and power control in wireless networks with fairness constraints.
method Formulated as a Lagrangian dual problem, RRM policies are parameterized by a GNN architecture trained on channel conditions.
result The method achieves superior tradeoff between average and 5th percentile rates, demonstrating fairness.
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.
problem Characterize measures generated by electrostatic p-capacity.
method Solves the discrete logarithmic Minkowski problem for 1 < p < n.
result Solves the discrete logarithmic Minkowski problem for measures in general position.
CapOptix uses options theory to price capacity in electricity markets.
problem Traditional capacity market designs fail to account for risk and price shocks.
method Interprets capacity commitments as reliability options and uses Markov Regime Switching Process.
result CapOptix provides more accurate pricing of capacity premia compared to existing mechanisms.
In this article, we propose the notion of the general -affine capacity and prove some basic properties for the general -affine capacity, such as affine invariance and monotonicity. The newly proposed general -affine capacity is compared with several classical geometric quantities, e.g., the volume, the -var…
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of sympl…
Study excess capacity in neural networks using Rademacher complexity.
problem Understanding how much capacity deep networks have beyond what's needed for classification.
method Unified Rademacher complexity bounds for function composition and convolutional layers, considering Lipschitz constants and initialization norms.
result There is substantial excess capacity per task, and capacity can be kept similar across different tasks.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
Constructing translating solitons from Lagrangian Grim Reapers.
problem Creating Lagrangian translating solitons from intersections of Grim Reapers.
method Desingularizing intersections with special Lagrangian Lawlor necks.
result Constructing Lagrangian translating solitons with multiple ends and loops.
Study binary perceptrons' capacity using random duality theory.
problem Characterize the capacity of binary perceptrons with general thresholds.
method Utilized fully lifted random duality theory (fl RDT) to characterize the capacity.
result Characterizations match replica symmetry breaking predictions and uncover the capacity for zero-threshold scenario.
Study capacity constraints in continual learning with a simple model.
problem Understanding optimal resource allocation for agents with limited memory and compute resources.
method Analyzes a capacity-constrained linear-quadratic-Gaussian (LQG) sequential prediction problem and demonstrates optimal capacity allocation strategies.
result Derives a solution to the capacity-constrained LQG sequential prediction problem and shows how to optimally allocate capacity across sub-problems in the steady state.
In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…
New complete panel dataset for LMICs helps analyze innovation and development.
problem Lack of complete data for empirical analyses in LMICs.
method Predictive Mean Matching multiple imputation technique.
result Created a large dataset of 47 variables for 82 LMICs from 2005-2019.
Memory capacity of DAM scales exponentially with feature separation, unaffected by correlations.
problem Understanding how feature correlations impact DAM's capacity.
method Developed an empirical framework to analyze DAM's capacity under varying feature correlations and pattern separations.
result Memory capacity scales exponentially with feature separation, unaffected by correlations.
Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.
problem Finding local maximizers for higher Ekeland-Hofer capacities in specific domains.
method Analogous to 4D local Viterbo conjecture, proving maximizers for rational ellipsoids.
result Local maximizers of the k-th Ekeland-Hofer capacities are symplectomorphic to rational ellipsoids.
Develops a theory for mth order p-affine capacity for convex bodies containing the origin.
problem Defines and studies the mth order p-affine capacity for convex bodies containing the origin.
method Provides equivalent definitions, proves properties, and establishes inequalities.
result Establishes inequalities comparing to other geometric measures.
Derives an empirical capacity model for self-attention neural networks.
problem Theoretical capacity of large transformer models is not fully utilized by current optimization algorithms.
method Analyzes memory capacity of transformers using synthetic training data and common training algorithms.
result Derives an empirical capacity model (ECM) for a generic transformer.
Improves online learning algorithms for functional models with capacity assumptions.
problem Convergence rates of online stochastic gradient descent algorithms for functional linear models.
method Characterizations of slope function regularity, kernel space capacity, and sampling process covariance operator.
result Capacity assumptions can alleviate saturation of convergence rates as function regularity increases.
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacitie…
Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problemmath.DG
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth sp…
Study relates symplectic homology capacity to periodic orbits in Liouville domains.
problem Relating symplectic homology capacity to periodic orbits in Liouville domains.
method Uses positive symplectic homology and Hofer-Zehnder capacity to establish bounds and existence of periodic points.
result Non-zero positive symplectic homology implies finite upper bound for Hofer-Zehnder capacity relative to skeleton and Hamiltonian diffeomorphisms.
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian …
Learning capacity measures model complexity, correlating with test loss and sample size.
problem Understanding model complexity and its relation to test performance.
method Formal correspondence between thermodynamics and inference; learning capacity as a measure of effective dimensionality.
result Learning capacity correlates with test loss and is a small fraction of model parameters.
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
Study compares Monge-Ampère capacities on Kähler manifolds.
problem Comparing Monge-Ampère capacities on compact Kähler manifolds.
method Proved all capacities comparable, used Xia's integration by parts formula.
result All Monge-Ampère capacities are comparable.
Generalizes memory and forecasting capacities for nonlinear recurrent networks with dependent inputs.
problem Understanding memory and forecasting capabilities in networks with dependent inputs.
method Formulated bounds for memory and forecasting capacities in terms of network size and input properties.
result Proved that memory capacity for linear recurrent networks with independent inputs is given by the rank of the controllability matrix.
The Affine BV-capacitymath.DG
This paper is devoted to a geometric-measure-theoretic study of the brand new affine BV-capacity which is essentially different from the classic BV-capacity in dimension greater than one.
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
For any Lie group , we construct a -equivariant analogue of symplectic capacities and give examples when , in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic -…
Study proves inequalities for mass-capacity on curved spaces.
problem Proving nonnegativity and positive lower bounds of mass on curved spaces.
method Applying mass-capacity inequalities from \cite{M22} to manifolds with nonnegative scalar curvature.
result Sufficient conditions for nonnegativity and positive lower bounds of mass.
Study online learning with delays and capacity constraints, achieving optimal regret bounds.
problem Online learning with delays and capacity constraints.
method Novel scheduling and preemptive techniques, matching upper and lower bounds.
result Achieves optimal regret bounds across all capacity levels.
In this paper, we investigate the common scenario where every candidate item for recommendation is characterized by a maximum capacity, i.e., number of seats in a Point-of-Interest (POI) or size of an item's inventory. Despite the prevalence of the task of recommending items under capacity constraints in a variety of s…
New method fills cluster seeds with exact Lagrangian structures.
problem Surjectivity of map from Lagrangian fillings to cluster seeds.
method Construction of quiver with potential and Lagrangian disk surgeries.
result Trivial deformation space of quiver allows manipulation of fillings.
Quantifies closeness of special Lagrangians under Floer conditions.
problem Estimating closeness of special Lagrangians.
method Floer theoretic conditions leading to quantitative estimates.
result Strong-weak uniqueness theorem for special Lagrangians.
In this paper, we prove that any Lagrangian translating soliton is Lagrangian -stable.
Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
Ancient solutions and translators identified for Lagrangian flow.
problem Characterizing ancient solutions and translators of Lagrangian mean curvature flow.
method Analyzing almost calibrated, exact, ancient solutions with specific geometric properties.
result All ancient solutions with entropy less than 3 are special Lagrangian, planes, or translators in \(\mathbb{C}^2\).