Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
New findings on non-congruent curves with identical signatures.
problem Identifying congruence of non-degenerate curves with non-simple signatures.
method Associated directed graphs to signatures and used paths to reflect global and local symmetries.
result Non-congruent, non-degenerate curves can have identical signatures.
given two minimal surfaces embedded in §3 of genus g we prove the existence of a sequence of non-congruent compact minimal surfaces embedded in §3 of genus g that converges in C2,α to a compact embedded minimal surface provided some conditions are satisfied. These conditions also imply that, if any of th…
Constructs hyperbolic affine spheres and Calabi-Yau metrics.
problem Constructing non-isometric Calabi-Yau metrics.
method Non-Abelian Hodge theory for parabolic Higgs bundles.
result Answers a question about Calabi-Yau metrics.
The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
problem Classifying minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
method General construction of homogeneous minimal flat n-tori in spheres, detailed investigations of shortest vectors in lattices.
result There exists a 2-parameter family of non-congruent λ1-minimal flat 4-tori.
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
We introduce a class of surfaces in euclidean space motivated by a problem posed by Élie Cartan. This class furnishes what seems to be the first examples of pairs of non-congruent surfaces in euclidean space such that, under a diffeomorphism Φ, lines of curvatures are preserved and principal curvatures are switched. …
We prove the existence of multiparameter isospectral deformations of metrics on SO(n) (n=9 and n≥11), SU(n) (n≥8), and Sp(n) (n≥4). For these examples, we follow a metric construction developed by Schueth who had given one-parameter families of isospectral metrics on orthogonal and unitary group…
A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vecto…
Letting C be a compact Cω-curve embedded in R3 (Cω means real analyticity), we consider a Cω-cuspidal edge f along C. When C is non-closed, in the authors' previous works, the local existence of three distinct cuspidal edges along C whose first fundamental forms coincide with that of $…
A new neural network model identifies hysteresis universally.
problem Inability of existing models to simulate hysteresis universally.
method Inspired by the Preisach model, an Extended Preisach Neural Network (EPNN) is introduced with two hidden layers and a hybrid training algorithm.
result EPNN successfully identifies various hysteresis phenomena from different fields.
Four distinct curved foldings with a given crease and crease pattern are discovered.
problem Identifying all possible curved foldings with a given crease and crease pattern.
method Analyzing the singular set of curved foldings and the initially given plane curve.
result Four distinct non-congruent curved foldings are found.
New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
We construct and analyze minimal disc stackings with bounds on their Morse index.
problem Constructing and analyzing minimal free boundary disc stackings.
method Constructing minimal free boundary disc stackings in a three-dimensional Euclidean unit ball, proving bounds on their Morse index.
result Uniform, linear bounds on the Morse index of all such surfaces.
The paper explores the existence of multiple curved foldings with a common crease pattern.
problem Determining the number of distinct curved foldings with a given crease and crease pattern.
method Analyzing origami maps and their singular sets, focusing on the crease and crease pattern.
result For a non-closed simple arc crease, there are exactly 4 distinct non-congruent curved foldings.