The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.
problem Finding minimal hypersurfaces in wedge-shaped manifolds with boundary.
method Developed a min-max theory for locally wedge-shaped manifolds with boundary.
result Proved existence of smooth free boundary minimal hypersurfaces in wedge-shaped manifolds.
Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
problem Estimating curvature of stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
method Compactness theorem and Schoen-Simon-Yau estimates.
result Curvature estimate for free boundary minimal hypersurfaces in wedge-shaped manifolds.
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
Flat minimal hypersurfaces found in wedge-shaped domains.
problem Finding minimal surfaces in wedge-shaped domains.
method Proving stability and flatness of C1,1-to-edge minimal hypersurfaces. result Stable minimal hypersurfaces are flat in wedge-shaped domains.
Paper proves inequality for capillary hypersurfaces in a wedge.
problem Proving a best version of Heintze-Karcher inequality for capillary hypersurfaces.
method Utilized Heintze-Karcher method and modified parallel hypersurfaces.
result Classified capillary constant mean curvature hypersurfaces hitting the edge in a wedge.
Study heat kernel and spectrum on manifolds with wedge singularities.
problem Analyzing the spectrum and heat kernel on manifolds with wedge singularities.
method Uniform constructions of the resolvent and heat kernel on manifolds with corners, using spectral gap and analytic torsion.
result Obtain a Cheeger-Müller theorem relating analytic torsion and Reidemeister torsion.
Study on scalar curvature in wedge spaces with existence and obstruction results.
problem Existence and obstructions of scalar curvature in wedge spaces.
method Utilized established tools for wedge spaces including Yamabe, elliptic, and index theories.
result Provided existence and obstruction results for scalar curvature under suitable positivity assumptions.
Heat trace analysis reveals singularity properties on wedge-shaped manifolds.
problem Analyzing heat trace asymptotics on manifolds with wedge-like singularities.
method Heat trace expansion and asymptotic analysis on manifolds with wedge-like singularities.
result Two terms in the heat trace expansion contain information about the singularity.
Paper equates torsions on wedge singularities.
problem Analyzing spaces with wedge singularities.
method Equating analytic and intersection Reidemeister torsions.
result Equation of torsions in specific wedge singularities.
Consider a financial market in which an agent trades with utility-induced restrictions on wealth. By introducing a general convex-analytic framework which includes the class of umbrella wedges in certain Riesz spaces and faces of convex sets (consisting of probability measures), together with a duality theory for polar…
Geometrodynamics derived from Riemannian manifolds using geospin matrix.
problem Formulating dynamics on Riemannian manifolds using Cartan structural equations.
method Introducing four real dynamical variables and applying them to Cartan structural equations.
result Rewritten Cartan structural equations in a real geometrodynamical form.
New insights into Khovanov homology complexity and topological structure.
problem Complexity of computing Khovanov homology for closed braids.
method Analysis of independence simplicial complexes and polynomial time algorithms.
result Independence simplicial complexes associated to 4-braid diagrams are homotopy equivalent to wedges of spheres.
A fake wedge is a diagram of spaces K <- A -> C whose double mapping cylinder is contractible. The terminology stems from the special case A = K v C with maps given by the projections. In this paper, we study the homotopy type of the moduli space D(K,C) of fake wedges on K and C. We formulate two conjectures concerning…
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
Let Σ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in Rn+1. Suppose that Σ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if ∂Σ is embedded for n=2, or if ∂Σ is convex…
Examines how irreducibility and rigidity affect digital images.
problem Understanding interactions between irreducibility and rigidity in digital images.
method Analyzes Cartesian products, wedges, and cold and freezing sets.
result Interactions between irreducibility and rigidity in digital images.
Wedge Sampling improves tensor completion with nearly-linear sample complexity.
problem Efficiently completing low-rank tensors from a subset of entries.
method Non-adaptive wedge sampling to promote structured connections in tensor completion.
result Polynomial-time algorithms achieve weak and exact recovery with nearly linear sample complexity.
Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.
problem Understanding capillary surfaces with anisotropic forces.
method Generalized Minkowski norm on the unit sphere, new Minkowski formulae, Heintze-Karcher inequality.
result Proved an Alexandrov-type theorem in the anisotropic setting.
Algorithm finds characteristic maps over complex shapes.
problem Finding non-singular characteristic maps over wedged simplicial complexes.
method Constructive puzzle algorithm based on linear algebra.
result Algorithm performs well compared to other methods.
We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this approach with explicit examples. In particular, we describe a short program to determin…
We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions. These correlation functions a…
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus g with one boundary component to ∧3H, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…
Decomposes elements in multiplicative multivectors and relates to Lie algebroid cohomology.
problem Understanding the structure of multiplicative multivectors on Lie groupoids.
method Proves canonical decomposition formula and establishes relations between quotient spaces.
result Canonical decomposition formula and key relation between reduced spaces.
New internal symmetry found for Lie pair algebra.
problem Understanding Lie pair structures and their associated algebras.
method Introduced a Lie algebra action by Der(L) on the L_{≤3} algebra.
result Found internal symmetry of the L_{≤3} algebra.
CR embeddings in complex spaces for specific Lie groups.
problem Embedding specific Lie groups in complex spaces.
method Using integrable complex structures on subbundles of tangent bundles.
result CR embeddings possible as the edge of wedges in complex domains.
Study shows physical drift affects put-call parity enforcement, not just option payoffs.
problem Inconsistency between quoted put-call parity and actual market behavior.
method Examined SPX and RUT index options, used drift-preserving GBM term to improve fit.
result Physical drift enters the enforcement of risk-neutral parity, not just option payoffs.
We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dol…
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
Given a pair of (real or complex) Lie algebroid structures on a vector bundle A (over M) and its dual A∗, and a line bundle $\module$ such that $\module\otimes\module=(\wedge^{\TOP} A^*\otimes\wedge^{\TOP} T^*M)$, there exist two canonically defined differential operators $\bdees$ and $\bdel$ on $\sections{\wedg…
Study of Ricci-flat metrics on C^2 with cone singularities.
problem Ricci-flat metrics with cone singularities on C^2.
method Gibbons-Hawking ansatz over wedges in R^3.
result Asymptotic behavior and energy computation of constructed metrics.
Proves Khovanov homology has no torsion for bipartite circle graphs.
problem Proving properties of Khovanov homology for bipartite circle graphs.
method Proved homotopy equivalence of independence complexes to wedges of spheres.
result Extreme Khovanov homology has no torsion.
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM⊕∧nT∗M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n+1)-vector fi…
New integral defined for Hölder continuous functions, characterizing distributional volume forms.
problem Defining and characterizing a new integral for Hölder continuous functions.
method Constructing a distribution from Hölder continuous functions and using integral properties.
result Characterizes the Hölder regularity of the constructed distribution.
New geometric structures defined in contact metric geometry.
problem Introducing new geometric structures in almost contact metric geometry.
method Defining locally conformal almost generalized f-cosymplectic manifolds and deriving integrability conditions. result Dimensional dichotomy: transverse components in dimension 3, proportional to η in higher dimensions.
We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. …
Constructs Dirac generating operators for split Courant algebroids.
problem Defines Dirac generating operators for split Courant algebroids.
method Explicit construction of Dirac generating operators.
result Square of Dirac generating operator yields invariant.
Given a smooth bounded domain Ø⊆R2, we consider the equation $\D v = 2 v_x \wedge v_y$ in Ø, where v:Ø→R3. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple…
This paper introduces modal epistemic tools for risk management.
problem Identifying and certifying risk claims when institutions lack the necessary epistemic stance.
method Develops crisp and fuzzy modal semantics for assurance and working commitment, distinguishing between object-level risk claims and meta-level epistemic diagnostics.
result Risk governance should model evidential incompleteness and failures of escalation, not just hazards and losses.
In \cite{KOT:MORITA}, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in $\ds \HGF{7}{2}{}{8}$ is decomposed as a product η∧ω of some leaf cohomology class η and a transverse symplectic class ω. In other words, the Kontsevich homomorphism $\dsω\wedge :\HGF{5}{2}{0}{10} \rightarrow\HGF{7}{2}…
The paper solves a problem related to exterior multiplication with singularities.
problem Finding sections of a p-exterior product given sections of a vector bundle.
method Analyzes the sufficiency of a condition involving regular singularities and depth of ideals.
result Shows that the condition is sufficient for existence of a continuous right inverse in smooth, real analytic, and holomorphic categories.
We study pseudo-differential operators on a wedge with continuous and variable discrete branching asymptotics.
Study modular geodesics and wedge domains in non-compactly causal symmetric spaces.
problem Understanding the geometric implementation of modular group in symmetric spaces.
method Analyzing the flow generated by Euler elements and their geometric properties.
result The wedge region W is connected and coincides with the observer domain under certain conditions.
Mess showed that the genus 2 Torelli group T2 is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta func…
A polynomial curve of degree 5, α, is a helix, if and only if both $||α^'||$ and $||α^'\wedge α^{''}||$ are polynomial functions.
The paper linearizes higher Courant algebroids using jet and differential operators.
problem Understanding the structure of higher Courant algebroids.
method Isomorphic spaces of sections and linearization of jet and differential operator bundles.
result Higher Courant algebroids can be linearized as pseudo-linearization and Weinstein-linearization.
Let (Mn,g) be a closed, connected, oriented, C∞, Riemannian, n-manifold with a transversely oriented foliation $\boldkey F$. We show that if {X,Y} are basic vector fields, the leaf component of [X,Y], $\Cal{V}[X,Y]$, has vanishing leaf divergence whenever $κ\wedge χ_{\boldkey F}$ is a c…
We prove that on a compact Sasakian manifold (M,η,g) of dimension 2n+1, for any 0≤p≤n the wedge product with η∧(dη)p defines an isomorphism between the spaces of harmonic forms ΩΔn−p(M) and ΩΔn+p+1(M). Therefore it induces an isomorphism between the de Rham cohomology spaces $H^{n-p…