Study of Hamilton's Ricci flow on Finsler spaces, proving existence and convergence.
problem Existence and convergence of Hamilton's Ricci flow on Finsler spaces.
method Defined and proved existence of Finslerian Ricci-DeTurck flow, used to find Hamilton's Ricci flow solution.
result Existence of short time solution to Hamilton's Ricci flow on Finsler spaces.
Study proves smooth solutions for fractional mean curvature flow within short time.
problem Short-time existence of smooth solutions for fractional mean curvature flow.
method Established using short-time existence theorem for bounded, C^{1,1}-regular initial sets.
result Smooth solutions exist for both fractional mean curvature flow and volume preserving flow.
The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.
problem Existence and uniqueness of Ricci flow solutions on Finsler manifolds.
method Investigation of short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds.
result Theorems demonstrating the short-time existence of the flow solution for n-dimensional Finsler manifolds and the uniqueness of the solution for isotropic Finsler manifolds.
Study the Ricci flow on Finsler surfaces and find solutions.
problem Existence and uniqueness of solutions to the Ricci flow on Finsler surfaces.
method First, study the Finslerian Ricci-DeTurck flow and find a unique short time solution. Then, use this to find a solution to the original Ricci flow.
result Find solutions to the Ricci flow on Finsler surfaces.
Ricci flow preserves bounded curvature for a short time.
problem Bounding the Weyl curvature tensor under Ricci flow with bounded initial curvature.
method First-order energy quantity to prove uniqueness of Ricci flow solutions.
result Weyl curvature tensor remains bounded for a short time if initial curvature is bounded.
Study on combinatorial Yamabe flow on hyperbolic surfaces, proving existence and uniqueness.
problem Existence and uniqueness of solutions to combinatorial Yamabe flow on hyperbolic surfaces.
method Introduced combinatorial Yamabe flow and extended flow with generalized curvature to address potential degeneration of triangles.
result Established existence and uniqueness of solutions to the extended flow under certain conditions.
A flow is introduced to study G2-structures with divergence-free torsion.
problem Analyzing G2-structures with specific properties. method Introduced a flow of G2-structures with a divergence-free torsion condition. result Short-time existence and uniqueness of the flow solution.
Study proves flow of curves with special shapes exists for short time.
problem Curvature flow of curves with special shapes.
method Proves existence of solutions to anisotropic and crystalline curvature flow.
result Short-time existence of φ-regular solutions proved.
We prove short time existence and uniqueness of solutions to the Laplacian flow for closed G2 structures on a compact manifold M7. The result was claimed in \cite{BryantG2}, but its proof has never appeared.
The Ricci flow is shown to exist for short time on specific 3-manifolds.
problem Existence of Ricci flow on 3-manifolds with bounded curvature and volume.
method Proof of existence for short time using specific curvature and volume bounds.
result Ricci flow exists for short time on 3-manifolds with given curvature and volume constraints.
Proves short-time existence of Ricci flows on certain manifolds.
problem Proving short-time existence of Ricci flows on specific manifolds.
method Uses Perelman's pseudolocality theorem to study short-time behavior of solutions.
result Establishes short-time existence of Ricci flows on manifolds with specific curvature and isoperimetric conditions.
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
Given a compact three-manifold together with a Riemannian metric, we prove the short-time existence of a solution to the renormalization group flow, truncated at the second order term, under a suitable hypothesis on the sectional curvature of the initial metric.
Proves short-term existence of heat flow for Dirac-harmonic maps on closed manifolds.
problem Existence of Dirac-harmonic maps on closed manifolds.
method Coupling heat flow for harmonic maps to a spinor.
result Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.
This paper models short rates with jumps using PDEs.
problem Capturing jumps and spikes in interest rates.
method PDE approach for pricing interest rate derivatives.
result Established Feynman-Kač representation and derived solutions.
We modify the Laplacian coflow of co-closed G2-structures - dtdψ=Δψ where ψ is the closed dual 4-form of a G2-structure φ. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…
Theory of Ricci flow for Courant algebroids, preserving isometry group.
problem Analytic theory of geometric flows for Courant algebroids.
method Develops Ricci flow theory for Courant algebroids, proving existence and uniqueness.
result Shows scalar curvature monotonicity and convergence for certain solutions.
The (α,β)-Ricci-Yamabe flow exists on closed manifolds.
problem Existence of solutions to the (α,β)-Ricci-Yamabe flow. method Showed short time existence and established long time existence theorems.
result Existence of smooth solutions to the (α,β)-Ricci-Yamabe flow on closed manifolds. We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving "closed" sections of a vector bundle, generalizing a method used recently by Bryant and Xu for studying the Laplacian flow in G_2-geometry. We apply this theorem in balanced geometry introducin…
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.
Solves short-term solutions for complex nonlinear systems on manifolds.
problem Solving the Cauchy problem for fully nonlinear parabolic systems on manifolds.
method Linearization procedure and fixed-point argument, using Schauder estimates.
result Short-time existence and uniqueness of solutions for arbitrary even-order systems.
We discuss a simple extension of the Ho and Lee model with generic time-dependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the sho…
In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum…
Paper solves MV portfolio selection in jump-diffusion models with no-shorting constraint.
problem Mean-variance portfolio selection in jump-diffusion model with no-shorting constraint.
method Reduces problem to LQ control and finding a maximal point of a function, constructs viscosity solution.
result Explicit viscosity solution to Hamilton-Jacobi-Bellman equation, optimal controls derived.
We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently…
Study proves a criterion for curve diffusion flow blow-up.
problem Analyzing curve diffusion flow with contact angle constraints.
method Contradiction proof using compactness and short time existence.
result Proves blow-up criterion for L2 curvature bound. Novel LSTM network predicts pulsar timing residuals with few-shot data.
problem Predicting pulsar timing residuals with limited data.
method Long Short-Term Memory (LSTM) network optimized with model-agnostic meta-learning and particle swarm optimization.
result Robust generalization and accurate predictions across high-frequency test domains with minimal data.
Paper proves short-time existence for network flow, providing detailed insights.
problem Short-time existence for the flow of a network of curves in the plane.
method Direct PDE approach, handling singularities at vertices using self-similar expanding solutions.
result Substantially more detailed information about network resolution into a regular one.
Model optimal growth strategy in a market with short-lived assets.
problem Investment market with short-lived assets and endogenous prices.
method Formulate stochastic equation for wealth processes and prove existence of optimal strategy.
result Existence of a submartingale strategy ensuring investor's wealth growth asymptotically.
We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial …
Study hexagonal network evolution under curvature flow.
problem Understanding hexagonal network evolution under curvature flow.
method Proved local existence of classical solutions and classified homothetically shrinking solutions.
result Provided an example of network shrinking to a segment with multiplicity two.
Simple uniqueness proof for McKean-Vlasov systems with weak feedback.
problem Global and short-time uniqueness for McKean-Vlasov systems with small feedback.
method Simple probabilistic comparison argument robust to solution regularity.
result Global uniqueness for a broad class of McKean-Vlasov problems in the weak feedback regime.
Short proof shows only circles contract under curve shortening flow.
problem Characterizing contracting self-similar solutions of the curve shortening flow.
method Intuitive geometric proof using Gage's idea.
result Only circles contract under curve shortening flow.
Smoothness of graphs evolving by fractional mean curvature is proven.
problem Evolution of graphs by fractional mean curvature.
method Analytic semigroup approach to nonlocal quasilinear evolution equation.
result Short time existence, uniqueness, and optimal Hölder regularity of classical solutions.
Study optimal trading times for mean-reverting prices with deadlines.
problem Optimal timing strategies for mean-reverting price processes with deadlines.
method Solve optimal double stopping problems with sequential deadlines using local time-space calculus.
result Derive optimal trading boundaries for long-short, short-long, and chooser strategies.
We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas…
The paper proposes a fast method to predict tactical solutions to operational problems under imperfect information.
problem Predicting tactical solutions to operational planning problems under imperfect information.
method Formulated as a two-stage optimal prediction stochastic program, solved with a supervised machine learning algorithm using training data from deterministic problems.
result Deep learning algorithms produce highly accurate predictions in very short computing time (milliseconds or less).
Scheme minimizes p-elastic energy of curves over time.
problem Minimizing p-elastic energy of curves over time. method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
In this paper, we use the DeTurck trick to study the short-time existence of solutions to the Dirichlet and Newmann boundary problems of the cross curvature flow on 3-manifolds with boundary.
New models for short rates show longer periods at higher rates.
problem Modeling longer periods of higher interest rates.
method Developed a class of time-homogeneous one-factor Markov diffusion models with specific boundary conditions.
result Explicit expressions for bond prices and transition densities in new probability measure.
We consider the Cauchy problem associated with a general parabolic partial differential equation in d dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provi…
The paper explores bond pricing in short rate models using numerical and analytical methods.
problem Bond pricing in short rate convergence models of interest rates.
method Numerical and analytical methods for obtaining approximate solutions to partial differential equations.
result Approximations of bond prices in short rate convergence models.
In complex systems such as turbulent flows and financial markets, the dynamics in long and short time-lags, signaled by Gaussian and fat-tailed statistics, respectively, calls for a unified description. To address this issue we analyze a real dataset, namely, price fluctuations, in a wide range of temporal scales to em…
In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.
This paper studies the normalized Ricci flow on surfaces with conical singularities. It's proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear …
We analyse the dynamics of the Warsaw Stock Exchange index WIG at a daily time horizon before and after its well defined local maxima of the cusp-like shape decorated with oscillations. The rising and falling paths of the index peaks can be described by the Mittag-Leffler function superposed with various types of oscil…
Proves existence of Lagrangian mean curvature flow solutions.
problem Desingularizing transverse intersection points of immersed Lagrangians.
method Direct PDE approach using manifolds with corners and a-corners.
result Existence of Lagrangian mean curvature flow solutions with stronger convergence.
PPINN uses parareal method to speed up long-time PDE solutions.
problem Efficiently solving long-time PDEs with physics-informed neural networks.
method Parareal method applied to physics-informed neural networks (PINNs).
result Significant speedup for long-time PDE solutions.