Study shows flows with critical forcing term satisfy area change formula.
problem Analyzing flows with critical forcing terms and their area change.
method Identifying minimal conditions for Brakke flows to satisfy area change formula.
result Generalized BV flows with critical forcing terms have a lower bound for extinction time.
Researchers derive expressions for metric perturbations of extremal surfaces.
problem Understanding changes in extremal surfaces under metric perturbations.
method Derived explicit expressions for position and surface area changes.
result Found an expansion of surface area involving multiple integrals of geometric quantities.
Simple proof for Cauchy's surface area formula.
problem Proving Cauchy's surface area formula.
method Short and simple proof.
result Average projection area equals surface area up to a constant.
Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.
problem Finding surface area of arbitrary submanifolds in R^n.
method Defining natural projected areas and volumes, deriving a recursive formula.
result Derived a new surface area formula that coincides with Crofton's and De Jong's formulas.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
New formulas for geometric measures in vector spaces.
problem Local additive kinematic formulas for vector spaces.
method Introducing dual area measures and proving their convolution product.
result Local additive kinematic formulas in hermitian vector spaces.
We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. As applications, we give a short, "euclidean" proof of the anisotropic Sobolev inequality and describe an anisotropic tube formula for hypersurfaces in mathbbRn. A discussion about the first variation of the anisotropic…
Study area and coarea formulas for graphs and submanifolds in Carnot groups.
problem Understanding geometric properties of submanifolds in Carnot groups.
method Developed area and coarea formulas for CH1 intrinsic graphs and submanifolds. result Deduced density properties for Hausdorff measures and coarea formula for Carnot groups.
New formulae for minimal submanifolds improve area bounds.
problem Finding sharp area bounds for minimal submanifolds.
method Moving-centre monotonicity formulae involving asymptotic analysis and divergence theorem.
result Sharp area bounds for minimal submanifolds when the prescribed point is not the centre of the ball.
New Steiner formula for Lp affine surface area in Minkowski theory.
problem Developing a new Steiner formula for Lp affine surface area. method Proving a new Steiner formula for the Lp affine surface area of a Minkowski outer parallel body. result New curvature measures with properties not previously seen in literature.
New geometric measure simplifies complex analysis.
problem Complex geometric analysis challenges.
method Geometric integration and convergence methods.
result Smallest measure satisfying Area Formula.
A formula for triangle area in Deep Sets form.
problem Finding a polynomial formula for triangle area in Deep Sets form.
method Expressing area as a permutation-invariant function and finding a suitable Deep Sets form.
result Explicit polynomial formula for triangle area in Deep Sets form.
The coarea formula is proven for Heisenberg group maps, addressing open questions.
problem Proving the coarea formula for Lipschitz maps from the Heisenberg group to Euclidean space.
method Introducing a new integral to define symplectic area of curves and proving convergence conditions.
result The coarea formula is established for CH1 maps from the Heisenberg group to R2n. Minimum area formula for convex projective surfaces.
problem Calculating the minimum area of convex projective surfaces.
method Using Fock-Goncharov coordinates and triangle invariants.
result Minimum area formula derived for surfaces of genus g≥2.
The Basel liquidity formula is extended for elliptical risk-factor changes.
problem Extending the Basel liquidity formula for non-Gaussian risk-factor changes.
method Generalized the formula for multivariate elliptical risk-factor changes, using Fourier approach for expected shortfall calculation.
result The formula tends to be conservative for heavier-tailed distributions.
Paper derives formulas for static Einstein spaces, linking Neumann data to stability.
problem Stability of conformally compact static spaces.
method First and second variation formulas for renormalized area.
result Negativity of Neumann data implies instability.
Formula for renormalized area of hypersurfaces in hyperbolic spaces.
problem Calculating the renormalized area of asymptotically minimal hypersurfaces in hyperbolic spaces.
method Combining Chen's conformal invariant quantity and Chern-Gauss-Bonnet formulas.
result Extension of renormalized area formulas to higher dimensions and non-minimal cases.
The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…
We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…
New Lp-Steiner quermassintegrals defined from Steiner formula.
problem Defining new Lp-Steiner quermassintegrals. method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
Revisits contact measures in isotropic spaces, linking kinematic formulas to curved spaces.
problem Understanding contact measures in isotropic spaces.
method Using the theory of valuations on manifolds and integral geometry.
result Explicit kinematic formula for surface area measure in hermitian space.
Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic Q-curvature and application to renormalized area. result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmi…
The paper proves a statement about surfaces diffeomorphic to annuli.
problem Proving a statement about surfaces diffeomorphic to annuli in Perelman's paper.
method Uses extrinsic techniques, co-area formula, and is potentially generalizable.
result Potential generalizability to higher dimensions.
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical t…
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.
Derives a new formula for optimal stopping problems with exploding derivatives.
problem Optimal stopping problems with complex boundary conditions.
method Develops a change of variable formula for functions with exploding derivatives near a surface.
result Derives a formula similar to Itô's but with less restrictive conditions.
The paper proves formulas for capillary surfaces and applies them to inequalities and area estimates.
problem Understanding capillary surfaces and their properties.
method Established monotonicity formulas for capillary surfaces in half-space and unit ball.
result Extended Li-Yau-type inequalities and optimal area estimates for capillary surfaces.
The paper develops a new formula for financial pricing under multiple interest rates and collateralization.
problem Financial pricing under multiple interest rates and collateralization.
method Derives a change of measure formula for recursive conditional expectations in a jump-diffusion setting.
result Generalizes the change of numéraire technique for multiple interest rates and collateralization.
K-area is an invariant for Riemannian manifolds introduced by Gromov as an obstruction to the existence of positive scalar curvature. However in general it is difficult to determine whether K-area is finite or not. though the definition of K-area is quite natural. In this paper, we study how the invariant changes under…
Study geodesic triangles in compact Hermitian symmetric spaces, proving symplectic area formulas.
problem Define and calculate the symplectic area of geodesic triangles in compact Hermitian symmetric spaces.
method Analyze conditions for geodesic triangle definition, compute integral over filling surfaces.
result Explicit formula for symplectic area of geodesic triangles in compact Hermitian symmetric spaces.
Ancient formula connects volume forms and infinitesimal square volumes in manifolds.
problem Relating volume forms and infinitesimal square volumes in Riemannian manifolds.
method Uses Heron's formula to link these concepts.
result Established a connection between volume forms and infinitesimal square volumes.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
Graph-based weather prediction adapted for local models.
problem Applying neural weather prediction to limited area modeling.
method Adapting graph-based Neural Weather Prediction approach to local models.
result Validation of multi-scale hierarchical model extension for Nordic region.
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
This paper studies rectifiability in Carnot groups and proves geometric area formulas.
problem The study of rectifiability in Carnot groups and related geometric properties.
method Analysis of rectifiable measures in Carnot groups, geometric area formulas, and rectifiability of geodesic spheres.
result Geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in Carnot groups.
Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a…
Study proves Chern-Osserman type equality for surfaces in Euclidean space.
problem Characterize properties of complete surfaces in Euclidean space.
method Used Chern-Osserman type equality and monotonicity formula.
result Proved area growth for noncompact surfaces with finite mean curvature.
Consider a random smooth Gaussian field G(x):F→R, where F is a compact in Rd. We derive a formula for average area of a surface generated by the equation G(x)=0 and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \e…
Novel equations for nonsmooth level sets on Heisenberg group.
problem Parametrizing level sets of irregular maps on the Heisenberg group.
method Rough path theory equations for sub-Riemannian geometry.
result Well-posedness and calculus on nonsmooth level sets.
Researchers relax the CVF's smoothness requirement to create more flexible flow models.
problem Challenges in constructing flexible density models due to the CVF's smoothness requirement.
method Introduce L-diffeomorphisms as generalized transformations that may violate smoothness on zero Lebesgue-measure sets. result The relaxation allows for the use of non-smooth activation functions like ReLU in residual flows.
We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…
Study shows increased precipitation variability in Paris area over years.
problem Evaluating the evolution of precipitation variability over time.
method Shape-based Dynamic Time Warping (IMS-DTW) for clustering rainfall time series.
result Precipitation variability increased in Paris area over years.