The paper proves a formula for complex Monge-Ampère equations on manifolds.
problem Proving a mean value formula for complex Monge-Ampère equations.
method Using subharmonic Hermitian matrix valued functions and Liouville type theorems.
result Obtained a Liouville theorem for complex Monge-Ampère equations.
We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation …
The paper establishes a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.
problem Analyzing bounded pluriharmonic functions on Teichmüller space.
method Establishing a Poisson integral formula.
result A Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.
A new formula predicts stock prices using median instead of mean for skewed distributions.
problem Erroneous predictions from expected value in skewed stock price distributions.
method Uses geometric mean or median for log-normal distribution, especially for long-term outcomes.
result More realistic prediction for heavy-tailed distributions of stock price variations.
We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local mo…
New risk measure includes VaR, TVaR, and Entropic Risk Measure.
problem Risk management with specific focus on tail risk.
method Generalized Quasi-Linear Means restricted to the tail of the risk distribution.
result Unified measure for VaR, TVaR, and Entropic Risk Measure.
A new formula reveals symmetries between mean excess and ES functions.
problem Optimizing risk measures in financial models.
method Established a reverse ES optimization formula.
result Reveals elegant symmetries and relationships between mean excess and ES functions.
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
The paper analyzes Reliability Options in electricity markets, deriving pricing formulas and simulating real market scenarios.
problem Determining the value of Reliability Options in electricity markets under various price regimes.
method The paper derives closed-form pricing formulae and simulates real market scenarios using data from the Italian power market. Sensitivity analyses are performed to highlight the impact of different parameters.
result The value of Reliability Options is influenced by the level and volatility of power and strike prices, mean reversion speeds, and correlation coefficients.
The paper values perpetual callable American volatility options using a mean-reverting volatility model.
problem Valuation of callable American volatility put options.
method Modeling volatility dynamics as a mean-reverting 3/2 process and proposing a pricing formula.
result The value of perpetual callable American volatility put options is discussed under given conditions.
Paper proves inequality for capillary hypersurfaces with new proof.
problem Proving a Heintze-Karcher type inequality for hypersurfaces with capillary boundary.
method Using a mixed boundary value problem in Reilly type formula to establish the inequality.
result New proof of Alexandrov type theorem for capillary hypersurfaces.
The paper simplifies calculus for semimartingales using multiplicative compensation.
problem Developing a formula for complex-valued semimartingales to simplify stochastic calculus.
method Multiplicative compensation for complex-valued semimartingales.
result The stochastic exponential of complex-valued semimartingales becomes a true martingale after compensation.
The transformation formula of the Berezin integral holds, in the non-compact case, only up to boundary integrals, which have recently been quantified by Alldridge-Hilgert-Palzer. We establish divergence theorems in semi-Riemannian supergeometry by means of the flow of vector fields and these boundary integrals, and sho…
Using a Levy process we generalize formulas in Bo et al.(2010) for the Esscher transform parameters for the log-normal distribution which ensure the martingale condition holds for the discounted foreign exchange rate. Using these values of the parameters we find a risk-neural measure and provide new formulas for the di…
Derives formula for present value of future consumer goods multiplier.
problem Evaluating the present value of future consumer goods investments.
method Derives a formula based on geometric sequence and investigates macroeconomic implications.
result The present value of the future consumer goods multiplier is close to one.
Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type inva…
We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typ…
Formula extends mean curvature to convex sets.
problem Extending mean curvature to convex sets.
method Derived a special formula for ovaloids.
result Proved integral formula for ovaloids.
Entropy of Angenent torus estimated to be about 1.85122.
problem Estimating the entropy of the Angenent torus.
method Numerical estimation using discrete Euler-Lagrange equations.
result Entropy of Angenent torus is approximately 1.85122.
We study U(N|M) character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We deri…
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
New method proves achiral Lefschetz fibrations exist using Riemannian geometry.
problem Existence of achiral Lefschetz fibrations.
method Riemannian geometry and Smith signature formula extension.
result Existence of achiral Lefschetz fibrations proven.
For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivari…
The study classifies second fundamental forms on complex projective manifolds.
problem Characterizing the second fundamental form of complex projective manifolds.
method Deriving a formula for the mean value of the squared length of the second fundamental form and classifying them.
result Classifies the second fundamental forms under certain conditions.
The paper analyzes optimal investment strategies for life insurance contracts using mean-variance optimization.
problem Optimal portfolio choice for equity holders in life insurance contracts.
method Mean-variance optimization, explicit formulas, Hamilton-Jacobi-Bellman equations, numerical analysis.
result Equity holders increase investment in risky assets during economic downturns.
Paper proposes a novel method to test differences in spatial point patterns.
problem Detecting differences in the first-order structures of spatial point patterns.
method Kernel mean embedding with approximate version tailored for spatial point processes, reducing comparison to Euclidean space t-tests.
result The proposed method is powerful and well-calibrated, demonstrated on real-world data.
Unified formula for surfaces in Euclidean or Lorentzian 3-space.
problem Describe surfaces in Euclidean or Lorentzian 3-space.
method Unified Kenmotsu-type formula for surfaces in Euclidean or Lorentzian 3-space.
result Unified single equation for Kenmotsu-type formulas in Euclidean and Lorentzian 3-space.
Study on surfaces with unbounded mean curvature and singularities.
problem Understanding surfaces with unbounded mean curvature and singularities.
method Analyzing surfaces given by Kenmotsu-type formula.
result Characterization of singularities in surfaces with unbounded mean curvature.
We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…
Improved portfolio optimization using VaR and CVaR with NMVM models.
problem Optimizing portfolios with VaR and CVaR under NMVM distributions.
method Transformed mean-CVaR-skewness problems into quadratic optimization with closed-form solutions for NMVM models.
result Approximate closed-form expressions for VaR and CVaR of NMVM portfolios.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
We analyse derivative securities whose value is NOT a deterministic function of an underlying which means presence of a basis risk at any time. The key object of our analysis is conditional probability distribution at a given underlying value and moment of time. We consider time evolution of this probability distributi…
In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.
Study on convex ordering in stochastic control for swing contracts, proving value function convexity.
problem Pricing of swing contracts under stochastic dynamics.
method Discrete-time stochastic optimal control problem, convexity propagation, Brownian diffusion model, Stein's formula.
result Value function is convex in underlying asset price, relaxation of convexity assumption for semi-convexity.
We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. As an application, we obtain a version of the Duistermaat-Heck…
New formulas compare total mean curvatures of nested hypersurfaces.
problem Computing total mean curvatures of nested hypersurfaces.
method Developed differential forms based on Chern's work to compare curvatures.
result Quicker proof of recent result on total mean curvatures.
Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae…
New approach for prudent risk evaluation using model aggregation.
problem Risk evaluation and optimization under uncertainty.
method Model Aggregation (MA) approach based on stochastic dominance.
result Produces robust risk evaluation and distributional models.
Formula for complex SVD backpropagation developed.
problem No specific problem stated; focuses on complex SVD.
method Back propagation formula for complex SVD developed.
result Back propagation formula for complex SVD created.
Develops a new reinforcement learning framework for complex control problems.
problem Continuous-time extended mean field control with deterministic policies.
method Model-free sensitivity formula, deterministic policy gradient, local value and advantage-rate representations.
result Demonstrates efficiency, stability, and robustness in solving complex control problems.
Let Σbe a minimal submanifold of \R^{n+m} that can be represented as the graph of a smooth map f:\R^n-->\R^m. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which Σmust be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in…
Formula for spacelike submanifolds in warped products.
problem Finding formulas for spacelike submanifolds in semi-Riemannian warped products.
method Extending Simons' type formulas for spacelike submanifolds in semi-Riemannian warped products.
result Compact spacelike hypersurfaces with parallel mean curvature and non-negative sectional curvature are isoparametric hypersurfaces.
We derived similar to Bo et al. (2010) results but in the case when the dynamics of the FX rate is driven by a general Merton jump-diffusion process. The main results of our paper are as follows: 1) formulas for the Esscher transform parameters which ensure that the martingale condition for the discounted foreign excha…
Study applies Huisken formula to mean curvature flow in Ricci soliton background.
problem Analyzing mean curvature flow in Ricci soliton backgrounds.
method Applies Huisken's monotonicity formula to a shrinking self-similar solution of the extended Ricci flow.
result Establishes new results and solves noncompact case under natural geometric assumptions.
The paper uses moment matching method for pricing spread options under Lévy models.
problem Pricing spread options under Lévy models with mean-variance mixture.
method Moment matching method applied to Lévy models with mean-variance mixture.
result Obtains semi-closed form formulas for spread option prices.
Explicit formula found for certain curved surfaces.
problem Finding surfaces with specific curvature properties.
method Provided an explicit formula for singular surfaces of revolution.
result Conditions for singularities of surfaces are given.
A new, computationally friendly formula for a class of risk-averse preferences.
problem Characterizing a class of risk-averse preferences called uniformly weighted divergence preferences.
method Introducing a new formula that characterizes UWDP as the translation-invariant hull of state-independent expected utility.
result UWDP are the translation-invariant hull of state-independent expected utility over L0. For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula co…