The paper studies special Lagrangian manifolds using algebraic topology.
problem Understanding special Lagrangian submanifolds in calibrated geometries.
method Algebraic topology of Grassmannian spaces, focusing on cohomology rings.
result Various results on the topology of Grassmannian spaces and special Lagrangian embeddings.
A new method represents rod shapes as paths in special Euclidean algebra.
problem Representing the shapes of rods and framed curves for mechanical analysis.
method Representing shapes as paths in the special Euclidean algebra.
result The method avoids expensive reconstruction and interpolation in rod mechanics.
Researchers prove an equivariant index theorem on Euclidean space.
problem Calculating the equivariant index of the Bott-Dirac operator on R2n. method Continuous field of C∗-algebras and equivariant index theorem. result Explicit calculation of the equivariant index of the Bott-Dirac operator on R2n. The abstract proves the non-existence of certain real algebraic surfaces.
problem The existence of real polynomial functions with specific properties.
method Algebraic solution to a problem proposed by D. A. Panov.
result There does not exist a real polynomial function with the specified properties.
We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanif…
New method for special generic maps into 3D space.
problem Understanding special generic maps into 3D space.
method Constructing pseudo quotient maps onto 2D polyhedra.
result Explicit construction of special generic maps into 3D space.
Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…
New method classifies special Vinberg cones of rank 4.
problem Classifying special Vinberg cones of rank 4.
method Using Clifford Nil-algebras and directed acyclic graphs.
result Explicit classification of rank 4 special Vinberg cones.
We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K…
The affine Grassmannian generalizes Euclidean and linear subspaces with rich geometric properties.
problem Formulating machine learning and statistical problems on the affine Grassmannian.
method Showed the affine Grassmannian has multiple structures and affords an analogue of Schubert calculus.
result The affine Grassmannian serves as a concrete computational platform for various machine learning and statistical problems.
The study characterizes flat pseudo-Euclidean Lie algebras and their properties.
problem Characterizing flat pseudo-Euclidean Lie algebras and their properties.
method Using the double extension process and analyzing the center of the Lie algebras.
result All flat pseudo-Euclidean nilpotent Lie algebras of signature (2,n−2) can be obtained by the double extension process from flat Lorentzian nilpotent Lie algebras. Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the ∇u=f(u) Poisson's equation, which has a subalgebra isomorphic to the 3−dimensional special Euclidean group SE(3) or group of rigid motions of R3. Looking the adjoint representation of ${\rm SE}(3)…
No algebraic 3rd degree hypersurfaces in Euclidean spaces have constant mean curvature.
problem Existence of algebraic hypersurfaces with constant mean curvature.
method Analytical proof.
result No such hypersurfaces exist.
The paper adapts differential signatures to algebraic curves under group actions.
problem Equivalence problem for complex plane algebraic curves under group actions.
method Adapting differential signature construction to algebraic curves, using classifying invariants.
result Explicit sets of rational classifying invariants and formulas for signature curve degree.
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce t…
Euclidean volumes of hyperbolic knots are algebraic numbers.
problem Understanding the algebraic nature of Euclidean volumes in hyperbolic knots.
method Deforming hyperbolic structures into Euclidean structures and analyzing the normalised Euclidean volumes.
result Normalised Euclidean volumes of hyperbolic knots are always algebraic numbers.
New, algebraic surfaces found in curved spaces.
problem Finding new types of surfaces in curved spaces.
method Analyzing zero-mean-curvature hypersurfaces in pseudo-Euclidean spaces.
result Three new classes of algebraic surfaces discovered.
Construct special Lagrangian pair of pants in n dimensions.
problem Constructing special Lagrangian pair of pants in general dimensions.
method Inside the cotangent bundle of Tn with the Euclidean structure. result Special Lagrangian pair of pants constructed in general dimensions.
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
The duality principle connects algebraic curvature tensors in pseudo-Euclidean spaces.
problem Understanding algebraic curvature tensors in pseudo-Euclidean spaces.
method Proving equivalence between the Jordan-Osserman condition and the Rakić duality principle.
result The Osserman condition and the duality principle are equivalent in the diagonalisable case.
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special ca…
Geometrically convex return risk measures on AM-algebras
problem Quantifying risk in time series analysis
method Extending return risk measures to general ordered vector spaces
result Establishing results on finiteness, continuity, separability, and dual and aggregation-based representations
New invariant distinguishes real algebraic surfaces.
problem Distinguishing real algebraic surfaces up to birational diffeomorphism.
method Introducing real (logarithmic)-Kodaira dimension.
result Constructs infinite families of non-birationally diffeomorphic surfaces.
We classify superintegrable systems in the Euclidean plane using algebraic geometry.
problem Classifying superintegrable systems in the Euclidean plane.
method Derived and solved a system of algebraic equations to classify systems.
result Associated a unique line triple arrangement to each superintegrable system.
No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.
problem Existence of regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces.
method Analyzing polynomials defining hypersurfaces of various degrees and shapes.
result Hyperspheres and round cylinders are the only such hypersurfaces defined by polynomials of degree ≤3.
We consider non-degenerate graph immersions into affine space An+1 whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs (J,γ), where J is an n-dimensional real Jordan algebra and γ is a no…
The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.
problem Whether rational homology n-spheres admit special generic maps into Rp for p<n. method Stein factorization technique to derive a necessary homological condition.
result New results on the (non-)existence of special generic maps for specific rational homology spheres.
We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our exampl…
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
Proves regularity of harmonic maps into Euclidean buildings and applies to superrigidity of algebraic groups.
problem Regularity of harmonic maps into Euclidean buildings.
method Analyzes singular sets and applies geometric settings.
result Proves singular sets of Hausdorff codimension 2 for harmonic maps.
Study infinite Euclidean distance discriminants of algebraic varieties.
problem Understanding the structure of data points with infinitely many critical points in Euclidean distance correspondence.
method Developed computer code to compute discriminants and proved properties of fibers.
result Infinite Euclidean distance discriminants contain all data points with infinitely many critical points for the nearest-point problem.
In this paper, we study the special curves and ruled surfaces on helix hypersurface whose tangent planes make a constant angle with a fixed direction in Euclidean n-space Besides, we observe some special ruled surfaces in and give requirement of being developable of the ruled surface. Also, we investigate the helix sur…
Investigates fundamental groups and path lifting for algebraic varieties.
problem Understand fundamental groups and path lifting properties of algebraic varieties.
method Analyzes three fundamental questions about fundamental groups of algebraic varieties.
result Identifies conditions for surjectivity on fundamental groups and path lifting properties.
Tropical curves match to special Lagrangian shapes.
problem Connecting tropical geometry to special Lagrangian shapes.
method Gluing construction that matches tropical local models to Lagrangian shapes.
result Locally planar tropical curves can be realized as special Lagrangian limits.
Derives Kerr metric from two commuting complex structures.
problem Deriving the Kerr metric from complex geometry.
method Observation of two commuting complex structures and two Killing vector fields leads to an ansatz for the metrics.
result Derives the Kerr metric using linear algebra.
The paper describes metrics on left Leibniz algebras, linking them to quadratic Lie algebras.
problem Understanding metrics on left Leibniz algebras and their connections to quadratic Lie algebras.
method Analyzing left multiplications, right multiplications, and bilinear forms on left Leibniz algebras.
result Left Leibniz algebras with associative metrics can be derived from their underlying quadratic Lie algebras.
Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.
problem Identifying metric lines in the Special Euclidean group on the plane.
method Alternative proof using Hamilton-Jacobi theory.
result Metric lines in SE(2) are identified.
Non-Euclidean number rings have non-integral Steinberg modules.
problem Characterizing when Steinberg modules are generated by integral elements.
method Analyzing special linear groups over non-Euclidean imaginary number rings.
result Steinberg modules are not generated by integral elements in non-Euclidean rings.
New class of maps restricts manifolds strongly in algebraic topology.
problem Restricting manifolds in algebraic topology.
method Proposed a class of generalized special generic maps.
result Extended fundamental results on structures and algebraic topological properties.
We investigate in detail the class of Euclidean affine Kac-Moody symmetric spaces and their orthogonal symmetric affine Kac-Moody algebras (OSAKAs). These spaces are the only class of Kac-Moody symmetric spaces, that is not directly derived from affine Kac-Moody algebras in the classical sense.
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
The study classifies meridian surfaces with constant mean curvature in a special pseudo-Euclidean space.
problem Classifying meridian surfaces with constant mean curvature in a pseudo-Euclidean space.
method Analyzing one-parameter systems of meridians of rotational hypersurfaces in a four-dimensional pseudo-Euclidean space with neutral metric.
result Complete classification of meridian surfaces with constant mean curvature, including minimal and quasi-minimal surfaces.
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
Classifies surfaces with a special direction in 4D space.
problem Classifying surfaces with a canonical principal direction.
method Complete classification of surfaces in Euclidean 4-space.
result Obtained complete classification of CPD surfaces.
The study restricts manifolds with certain explicit SGL maps and constructs them.
problem Restrictions on manifolds admitting specific SGL maps.
method Generalization of Morse functions and canonical projections to construct SGL maps.
result Manifolds admitting certain explicit SGL maps are strongly topologically restricted.
The paper calculates period matrices for specific algebraic curves.
problem Calculating period matrices for a class of algebraic curves.
method Constructing algebraic curves from Euclidean polygons and using symplectic bases.
result Symplectic bases are derived from Euclidean polygons for the constructed curves.
Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants. In the present paper we give the complete classification of Chen…
Study special affine connections on symmetric spaces and their products.
problem Characterize special affine connections on symmetric spaces.
method Analyze canonical affine connections, introduce special products, and study holonomy Lie algebras.
result Established a correspondence between special affine connections and special products on the Lie algebra.