The noncompact Yamabe flow can lead to incomplete metrics over infinite time.
problem Incompleteness of noncompact Yamabe flow solutions over infinite time.
method Analysis of long-time behavior of the noncompact Yamabe flow.
result Existence of a long-time solution that is complete for each time but converges to an incomplete metric.
Infinite rank groups found in 3-manifolds with infinite fundamental groups.
problem Understanding the structure of diffeomorphism and homeomorphism groups of 3-manifolds with infinite fundamental groups.
method Analyzing actions of barbell diffeomorphisms on spaces of embedded arcs and configuration spaces.
result Groups of diffeomorphisms and homeomorphisms have infinite rank.
Project infinite time series graphs to finite marginal models using number theory.
problem Handling infinite time series graphs for causal inference.
method Projection method using number theory to find common ancestors in infinite graphs.
result Developed algorithm to project infinite graphs to finite marginal models.
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.
problem Infinite-time singularities in Lagrangian mean curvature flow.
method Constructing solutions by gluing special Lagrangian 'Lawlor necks' and analyzing dynamics of neck size.
result The flow decomposes initial data into a union of special Lagrangians intersecting at one point.
Study on Yang-Mills heat flow on R4 bundles, showing infinite time bubbling.
problem Understanding the long-time behavior of Yang-Mills heat flow on R4 bundles. method Construction of initial data and globally defined solutions, proof of existence of bubble-tower solutions.
result Demonstrates infinite time bubbling for Yang-Mills heat flow on R4 bundles. Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
We aim to construct the optimal solutions to the undiscounted continuous-time infinite horizon optimization problems, the objective functionals of which may be unbounded. We identify the condition under which the limit of the solutions to the finite horizon problems is optimal for the infinite horizon problems under th…
The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.
problem Establishing convergence and long-time existence of the combinatorial p-th Calabi flow.
method Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns.
result Sharp criterion for convergence in finite case and long-time existence in infinite case for p≥2. We prove the existence of nonperiodic, properly embedded minimal surfaces in R2×S1 with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).
The Conway knot can't be smoothly tied to any other knot an infinite number of times.
problem Understanding the concordance order of knots.
method Rasmussen invariant, satellite operations, null-homologous twists.
result The Conway knot has infinite concordance order.
New smooth structures found on certain 4D spaces.
problem Finding distinct smooth structures on specific 4D spaces.
method Constructing specific 4-manifolds with infinite fundamental group.
result Infinitely many non-diffeomorphic structures found.
This paper conditions non-linear infinite-dimensional diffusion processes.
problem Conditioning non-linear and infinite-dimensional diffusion processes.
method Infinite-dimensional Girsanov's theorem to condition function-valued stochastic processes.
result Conditioning of non-linear infinite-dimensional diffusion processes is achieved.
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.
Infinite Hidden Markov Models (iHMM's) are an attractive, nonparametric generalization of the classical Hidden Markov Model which can automatically infer the number of hidden states in the system. However, due to the infinite-dimensional nature of transition dynamics performing inference in the iHMM is difficult. In th…
On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.
Most existing approaches to clustering gene expression time course data treat the different time points as independent dimensions and are invariant to permutations, such as reversal, of the experimental time course. Approaches utilizing HMMs have been shown to be helpful in this regard, but are hampered by having to ch…
A manifold has infinitely many sphere fibrations over a sphere.
problem Infinitely many fibrations over a sphere.
method Examining X=S2imesS3 and finding fibrations over B=S2 for lens spaces. result Found infinitely many fibrations from L(p,1) to X over B. The Yamabe flow can blow up in infinite time with small perturbations.
problem Understanding the behavior of the Yamabe flow under small perturbations.
method Constructive proof using solutions of the Yamabe problem on the unit sphere as blow-up profiles.
result The Yamabe flow can blow up at multiple points on a Riemannian manifold in infinite time with small perturbations.
Deep neural nets approximate random dynamical system trajectories uniformly in time.
problem Approximating trajectories of random dynamical systems over infinite time horizons.
method Recurrent neural networks with simple feedback structures.
result Certain random trajectories can be approximated uniformly in time to any desired accuracy.
Infinite type surfaces can be perfectly divided into triangles.
problem Triangulating surfaces of infinite type.
method Showed arcs can be completed into triangulations if they intersect curves a finite number of times.
result Any surface of infinite type admits an ideal triangulation.
Study maps 4-manifolds with 1-handles, revealing infinite rank centers.
problem Understanding mapping class groups of 4-manifolds with 1-handles.
method Developed a generalized framework for W3 invariant. result Center of mapping class group is of infinite rank for certain 4-manifolds.
Improved algorithm for optimal stopping problems reduces runtime.
problem Optimal stopping problems with infinite time horizon and random discounting.
method Flexible forward improvement iteration with a variable look-ahead distance.
result The new algorithm converges and can significantly reduce runtime.
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Enhances understanding of Kähler-Ricci flow singularities.
problem Understanding singularities in Kähler-Ricci flow.
method Relates to classic Kähler-Ricci flow and degenerate complex Monge-Ampère equation.
result Improves understanding of finite and infinite time singularities.
We derive asymptotic expansions for option data to detect infinite variation volatility.
problem Detecting infinite variation volatility in high-frequency option data.
method Nonparametric higher-order asymptotic expansions for small-time changes of characteristic functions of Itô semimartingales.
result Evidence of infinite variation volatility in high-frequency option data.
For the Kähler-Ricci flow on a compact Kähler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some existing classification results on infinite-time singularity type of the Kähler-Ricci fl…
Study combinatorial Yamabe flow on infinite triangulated surfaces.
problem Solve discrete Yamabe problem on noncompact surfaces.
method Introduced and analyzed combinatorial Yamabe flow with short-time and long-time existence proofs.
result Established short-time and long-time existence of the flow, and proved convergence in hexagonal triangulations.
We construct a contact 5-manifold supported by infinitely many distinct open books with the identity monodromy and pairwise exotic Stein pages (i.e. pages are pairwise homeomorphic but non-diffeomorphic Stein fillings of a fixed contact 3-manifold), moreover we describe a process of generating infinitely many such exam…
This paper proposes new get-rich-quick schemes that involve trading in a financial security with a non-degenerate price path. For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes non-constant (if it ever does). If i…
We study a moduli space of ASD connections over S3×R. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the …
We consider an arbitrage-free, discrete time and frictionless market. We prove that an investor maximising the expected utility of her terminal wealth can always find an optimal investment strategy provided that her dissatisfaction of infinite losses is infinite and her utility function is non-decreasing, continuous an…
The infinite Viterbi alignment is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. For models on state-space Rd satisfying a new ``decay-convexity'' condition, we develop an approach to existence of the infinite Viterbi ali…
Time series models generalize ARMA and ARFIMA with non-Gaussian dependence.
problem Modeling non-Gaussian serial dependence in time series data.
method Infinite-order partial copula dependence in s-vine processes.
result Rich class of models that generalize linear processes.
Study hot spots on warped product manifolds and infinite cones.
problem Analyzing hot spots on specific geometric structures.
method Examining solutions to the heat equation on warped product manifolds and infinite cones.
result Hot spots behavior on warped product manifolds and infinite cones determined.
Developed LQ MFG theory with common noise, proving existence and uniqueness.
problem Linear-quadratic mean field games with common noise.
method Coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces.
result Existence and uniqueness of solutions for small and arbitrary finite time horizons.
Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding …
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type;…
We construct a non-abelian extension Γ of S1 by $\cy 3 \times \cy 3$, and prove that Γ acts freely and smoothly on S5×S5. This gives new actions on S5×S5 for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_…
We construct an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold S2×S2.
We establish a number of results about smooth and topological concordance of knots in S1×S2. The winding number of a knot in S1×S2 is defined to be its class in H1(S1×S2;Z)≅Z. We show that there is a unique smooth concordance class of knots with winding number one. …
The paper examines how deep linear neural networks behave as they become infinitely wide.
problem Understanding the behavior of deep linear neural networks as they approach infinite width.
method Analyzes the infinite-width limit of deep linear neural networks, proving convergence to deterministic models and providing precise laws for random weights.
result The training dynamics of deep linear neural networks converge to those of a deterministic model, and the weights' behavior is precisely described.
We construct infinite-dimensional families of non-singular static space times, solutions of the vacuum Einstein-Maxwell equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity.
Study shows automorphism groups of certain hyperbolic manifolds are infinitely generated.
problem Understanding the homotopy groups of automorphism groups of hyperbolic manifolds.
method Analyzing the homotopy groups of specific topological groups and applying these results to automorphism groups of hyperbolic manifolds.
result The (n-4)-th homotopy group of the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (n >= 4) is infinitely generated.
Derives time-averaged active inference from control principles.
problem Finite-horizon or discounted-surprise problems in active inference.
method Derives infinite-horizon, average-surprise active inference from optimal control principles.
result Unified objective functional for sensorimotor control.
The completeness of a bond market model with infinite number of sources of randomness on a finite time interval in the Heath-Jarrow-Morton framework is studied. It is proved that the market is not complete. A construction of a bounded contingent claim, which can not be replicated, is provided.
Paper introduces infinite-dimensional generative models using Doob's h-transform.
problem Defining generative models in infinite dimensions.
method Using Doob's h-transform to force a reference diffusion towards a target distribution.
result The forced process can be approximated by minimising a score-matching objective.
Study evaluates initialization strategies for infinite hidden Markov models.
problem Limited attention to initialization in infinite hidden Markov models.
method Systematically evaluated distance-based clustering, model-based, and uniform initializations.
result Distance-based clustering initializations consistently outperform other methods.