Constructs explicit solutions to Spin(7)-structures gradient flow.
problem Finding explicit solutions to Spin(7)-structures gradient flow.
method Expressed Spin(7)-torsion tensor and gradient flow in terms of torsion forms; used these formulae to find solutions.
result Found explicit solutions including a shrinking soliton on SU(3) and another on a T7-bundle over S1. Paper proposes an efficient method to optimize neural networks without backpropagation.
problem Computational inefficiency and scalability issues in neural network optimization.
method Derives explicit solutions to optimize neural networks, reducing computational costs.
result Explicit solutions achieve near-optimality and can discover better optima than backpropagation.
Method finds explicit solutions to certain PDEs.
problem Finding solutions to specific types of PDEs.
method Exploiting solvable structures to find explicit solutions.
result Effectiveness demonstrated on several examples.
Researchers found explicit solutions to a complex equation in advanced geometry.
problem Critical Yamabe type equation in sub-Finsler geometry.
method Computed a two-parameter family of explicit positive solutions.
result Explicit solutions to a critical equation in sub-Finsler geometry.
Explicit formulas found for ancient solutions of heat equation.
problem Finding explicit formulas for ancient solutions of the heat equation.
method Explicit representation formulas for positive ancient solutions in Euclidean and Riemannian cases.
result Ancient solutions are the Laplace transform of positive solutions of a family of elliptic operators.
Two self-similar solutions found for time-like hypersurfaces in Minkowski spacetime.
problem Finding self-similar solutions for time-like extremal hypersurfaces in Minkowski spacetime.
method Explicit construction of two self-similar solutions.
result An untable eigenvalue found in the linearized equation around the solutions.
Some special solutions to the multidimensional Lamé and Bourlet type equations are constructed in an explicit form.
Researchers find explicit solutions to complex Monge-Ampère equation.
problem Solving complex Monge-Ampère equation with constant right-hand side.
method Explicit pluripotential and viscosity solutions.
result Presented solutions lie in Wloc1,2∩Wloc2,1 and are not Dini continuous. Explicit solutions found for mean field equations on hyperelliptic curves of genus two.
problem Finding explicit solutions to mean field equations on complex hyperelliptic curves.
method Analyzing the Gaussian curvature function of the canonical metric on hyperelliptic curves of genus two.
result Explicit solutions to mean field equations are determined by the Gaussian curvature function.
Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving …
Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, tr…
An explicit solution found for maximizing/minimizing agreement in a 2x2 table.
problem Maximizing or minimizing agreement between clusterings with given marginals.
method Formal framework for several agreement measures, explicit solution for 2x2 table.
result An explicit solution for the 2x2 case.
Developed a simulation method for 3/2 stochastic volatility model.
problem Pricing options in the 3/2 stochastic volatility model.
method Explicit weak solution for the 3/2 model, using inverse CIR process property.
result Simulation algorithm performance comparable to other methods.
The paper solves a complex portfolio selection problem with nonlinear wealth equations.
problem Continuous time mean-variance portfolio selection with nonlinear wealth equations.
method Invoking the HJB equation and providing an explicit viscosity solution.
result Explicit efficient portfolio strategy and efficient frontier obtained.
Constructs solutions for gravitational instantons from minimal surfaces.
problem Finding solutions for gravitational instantons.
method Using correspondence between minimal surfaces and gravitational instantons, derived explicit maximal surface solutions.
result Explicit solutions for maximal surface with zero mean curvature.
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.
The study examines VIX-linked fees for GMWBs using explicit solution simulation methods.
problem Decreasing the sensitivity of insurer's liability to volatility risk in GMWBs.
method Explicit weak solution for VA account value and Monte Carlo simulations.
result VIX-linked fees decrease the sensitivity of the insurer's liability to volatility risk.
Solves infinite horizon portfolio problem with path-dependent labor income.
problem Infinite horizon portfolio choice with path-dependent labor income.
method Solves an infinite dimensional stochastic optimal control problem using explicit solutions to the HJB equation.
result Explicit solutions to the optimal controls in feedback form are found.
Develops a new framework to analyze gradient flow regimes and derive explicit solutions.
problem Analyzing scaling regimes and deriving explicit analytic solutions for gradient flow in large learning problems.
method Formal power series expansion of the loss evolution with coefficients encoded by diagrams.
result Reveals different learning phases and obtains explicit solutions in some cases.
Several models for the pricing of derivative securities in illiquid markets are discussed. A typical type of nonlinear partial differential equations arising from these investigation is studied. The scaling properties of these equations are discussed. Explicit solutions for one of the models are obtained and studied.
We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …
We obtain explicit solutions of the mean curvature flow in some submanifolds of the Euclidean space. We give particularly an explicit solution of the flow of a hypersurface in the Lagrangian self-expander L which is constructed in the article of Joyce, Lee and Tsui and show that it converge to a minimal one.
Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
Efficient neural network optimization reduces costs and improves model performance.
problem High computational costs in optimizing neural networks, especially at scale.
method Introduces self-attentive feed-forward neural units (SAFFU) for efficient optimization.
result Explicit solutions outperform models optimized by backpropagation alone, and further training with backpropagation leads to better optima from smaller data sets.
Improved simulation for path-dependent options without matrix inversion.
problem Evaluating path-dependent options efficiently and accurately.
method Stochastic approximation, explicit solutions to Heston model, importance sampling.
result Up to two orders of magnitude speed improvement over standard Monte Carlo methods.
We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to simila…
Study prescribed curvature tensor in locally conformally flat manifolds.
problem Solving the Prescribed Curvature Tensor problem in locally conformally flat manifolds.
method Explicit solutions provided for special cases of the tensor R, including complete metrics on Rn.
result Explicit examples of metrics g and conformal metrics g that solve the problem are exhibited.
In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resultin…
Develops methods for constructing exact, non-stationary solutions to Euler equations.
problem Constructing exact, non-stationary solutions to the incompressible Euler equations.
method Arnold's geometric framework with a generalized Coriolis force.
result Explicit, smooth, global-in-time solutions on curved surfaces and three-dimensional manifolds.
In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß ℘-functi…
Solves optimal control for trading multiple mean-reverting assets.
problem How to construct a portfolio from mean-reverting assets.
method Optimal control problem for power utility agent.
result Nearly explicit solution with properties of optimal solution.
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…
Solves geodesic equations on specific metrics types.
problem Finding explicit solutions for geodesic equations on Eguchi-Hanson metrics.
method Analyzes geodesic equations under different constant conditions.
result Explicit solutions not yet available for geodesic equations when only ψ is constant. Global study of capillarity equation solutions.
problem Understanding solutions to capillarity equation globally.
method Global analysis without boundary conditions.
result Characterization of solutions' constraints.
Study consumption-investment problem in markets with rank-based returns.
problem Consumption-investment problem in markets with rank-based returns.
method Derives an HJB equation with Neumann boundary conditions for the value function and proves a corresponding verification theorem.
result Explicit solutions for unconstrained, open market constraints, and fully invested cases.
We construct some explicit quasihomogeneous algebraic solutions to the associativity (WDVV) equations by using analytical methods of the finite gap integration theory. These solutions are expanded in the uniform way to non-semisimple Frobenius manifolds.
We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms the τ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.
New representation of field equations for QMAG, presenting new solutions.
problem Finding new explicit solutions for quadratic metric-affine gravity.
method Developed a new explicit representation of field equations without assumptions on torsion properties.
result Presented two conjectures on new types of solutions of QMAG.
Paper finds infinitely many smooth solutions to complex geometry problem.
problem Finding smooth solutions to the Strominger system on specific geometric manifolds.
method Constructing solutions on generalized Calabi-Gray manifolds.
result Explicit construction of infinitely many solutions.
The paper solves quadratic equations in free groups and provides explicit solutions for certain cases.
problem Solving quadratic equations in free groups with specific constraints.
method Analyzes and provides explicit solutions for orientable quadratic equations in free groups.
result Explicit solutions for certain quadratic equations in free groups, providing conditions for the minimal number of unknowns.
We present an explicit example of a fast decaying solution to the modified Novikov--Veselov equation with a one-point singularity in the space-time. It is constructed by using the geometrical interpretation of the Moutard transformation of solutions to this equation and the Enneper minimal surface.
The paper solves Seiberg-Witten equations in dimensions 5, 6, and 8.
problem Solving Seiberg-Witten equations in higher dimensions.
method Elliptic system of equations, constructing explicit solutions, relating to vortices.
result Explicit solutions in dimensions 5, 6, and 8, linking to vortices.
This paper solves the dynamic portfolio choice problem. Using an explicit solution with a power utility, we construct a bridge between a continuous and discrete VAR model to assess portfolio sensitivities. We find, from a well analyzed example that the optimal allocation to stocks is particularly sensitive to Sharpe ra…
Geometrically constructs solutions to 11D supergravity.
problem Finding supersymmetric solutions to 11D supergravity.
method Warped product manifolds with non-vanishing flux.
result Explicit 5-parameter moduli space of solutions.
We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature con…
This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.
problem Understanding magnetic geodesics on the Heisenberg group with invariant Lorentz force.
method Analyzing the Heisenberg Lie group with a non-commutative product, deriving magnetic equations, identifying symmetries, and solving variational problems.
result Magnetic trajectories are solutions to a variational problem, providing explicit examples of Lagrangians.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
In this article we study an optimal stopping/optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corr…