EasiCS improves neck function assessment by objectively classifying cervical spondylosis.
problem Subjective and coarse-grained neck function assessment methods.
method Developed clustering algorithms on sEMG data to objectively classify cervical spondylosis.
result EasiCS outperforms existing seven algorithms overall.
Paper proposes a smart neck-band for detecting neck postures using integrated kinematic and kinetic data.
problem Improper neck postures lead to musculoskeletal disorders requiring therapy and rehabilitation.
method Integrated use of kinematic and kinetic data with machine learning algorithms.
result 100% accuracy in predicting neck postures using the proposed platform.
Lipid necks, large curvature bridges, are shown to be metastable.
problem Understanding the energetically prohibitive yet ubiquitous lipid necks in cell membranes.
method Geometric triality approach to demonstrate metastability.
result Lipid necks can exist for finite but potentially long times without stabilizing mechanisms.
Study on harmonic maps from surfaces with energy bounds and neck domains.
problem Behavior of harmonic maps with bounded energy on complex domains.
method Analysis of a sequence of harmonic maps in generalized neck domains.
result Upper bound of energy density and study of nullity and index limits.
The paper extends energy identities and neck existence for ε-harmonic maps.
problem Understanding the energy identity and neck formation for ε-harmonic maps.
method Finding analogues of energy identities and neck existence results for ε-harmonic maps.
result Specific quantities determine energy identity and neck formation for ε-harmonic maps.
Paper analyzes harmonic maps blow-up neck regions.
problem Understanding blow-up regions in harmonic maps.
method Proved refined estimates in neck regions.
result New estimate reveals neck shape and inequality about nullity and index.
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Paper solves long neck problem on odd-dimensional spin manifolds.
problem Long neck problem on odd-dimensional spin manifolds.
method Spectral flow of Callias operators.
result Complete answer to Gromov's long neck problem.
Paper proves energy identity and no-neck property for special harmonic maps.
problem Analyzing special harmonic maps with homogeneous targets.
method Introduced equivariant embedding for ε-harmonic case. result Energy identity and no-neck property established for ε- and α-harmonic maps. A multi-neck spacetime wormhole is constructed with a simple metric tensor.
problem Existence of multi-neck spacetime wormholes.
method Spherical inversion of a 3-torus to create a 3-neck spacetime wormhole.
result Exact solution of Einstein's field equations for a multi-neck spacetime wormhole.
Derives generalizations of the long neck principle and spectral width inequality.
problem Understanding the spectral width of geodesic collar neighborhoods.
method Spinorial Callias operator approach and relative Gromov-Lawson pair.
result Generalizations of the long neck principle and spectral width inequality.
In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.
Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.
problem Geodesics behavior on neck-degenerate manifolds with cuspidal singularities.
method Detailed multiscale analysis, blow-up techniques.
result Geodesics exhibit focussing and winding behavior as the neck degenerates.
The shape equation and linking conditions for a vesicle with two-phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. \textbf{70}, 2964 (1993); Phys. Rev. E \textbf{53}, 2670 (1996)], and then we us…
We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
Researchers create non-degenerate harmonic functions on n-dimensional space.
problem Creating non-degenerate Z2-harmonic functions on Rn. method Using a variant of ellipsoidal coordinates, the construction is explicit and involves Lawlor's necks in Cn. result First known family of non-degenerate Z2-harmonic 1-forms with compact branching sets. Over half a million individuals are diagnosed with head and neck cancer each year worldwide. Radiotherapy is an important curative treatment for this disease, but it requires manual time consuming delineation of radio-sensitive organs at risk (OARs). This planning process can delay treatment, while also introducing int…
In this paper we are dealing with mean curvature flow with surgeries of two-convex hypersurfaces. The main focus is to expand on the discussion in Section 3 of Mean Curvature Flow with Surgeries of Two-Convex Hypersurfaces by Huisken and Sinestrari. Firstly we wish to establish how the neck detection lemma allows us …
Classifies 85 tie knots into mathematical categories.
problem Classifying and understanding the mathematical properties of tie knots.
method Formal language and sequence of moves to describe tie knots, classification based on knot theory.
result Proves that any tie knot is prime and alternating.
We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.
Study controls curvature in Ricci flows using necks.
problem Controlling curvature in Ricci flows.
method Introducing necks of maximal symmetry and decomposing curvature into uniform bounds.
result Established L1-bounds on Riemann curvature tensor. Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
The paper proves the existence of maxfaces with multiple swallowtails and planar ends.
problem Existence of maxfaces with specific geometric properties.
method Analyzes 1-parameter infinite genus families of maxfaces with swallowtails and planar ends.
result Existence of maxfaces with infinitely many swallowtails and planar ends.
Proves principles and estimates for initial data sets in Einstein equations.
problem Understanding initial data sets in Einstein equations.
method Spinorial Callias operator approach.
result Proves long neck principle and width estimates.
The study proves all limit flows are self-similar under specific conditions.
problem The selfsimilarity of limit flows in mean curvature flow.
method Proof of selfsimilarity conditions for mean convex surfaces and neck singularities.
result All limit flows are self-similar if and only if there are finitely many spherical singularities.
We study moduli spaces of Seiberg-Witten monopoles over spin^c Riemannian 4-manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg-Witten invariants.
Constructs minimal immersions with singularities.
problem Minimal immersions with singularities in metric spaces.
method Constructs minimal immersions with catenoidal necks or floating disks converging to a singular point.
result Constructs minimal immersions with singularities.
We establish a gluing theorem for monopoles over 4--manifolds containing long necks. The theorem is stated in terms of an ungluing map defined explicitly in terms of data that appear naturally in applications. Orientations of moduli spaces are handled using Benevieri--Furi's concept of orientations of Fredholm operator…
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …
The paper studies neck-pinching of CP1-structures on surfaces, describing their limits.
problem Characterizing the degeneration of CP1-structures on surfaces. method Analyzing a path of CP1-structures leaving every compact subset, converging holonomy in the PSL(2, C)-character variety. result The limit of the path Ct is described in terms of developing maps, holomorphic quadratic differentials, and pleated surfaces. 3D Ricci flows have bounded diameter before Type I singularities.
problem Bounding the diameter of 3D Ricci flows before Type I singularities.
method Introduced a neck-region concept and proved packing measure Ahlfors regularity.
result Uniformly bounded diameter up to Type I singular time.
Researchers create minimal surfaces with Scherk ends and find catenoid limits.
problem Constructing minimal surfaces with specific end types and understanding their limits.
method Constructing families of embedded, singly periodic minimal surfaces with Scherk-type ends and analyzing their limits.
result The limit of the constructed surfaces are catenoid necks connecting planes, determined by Stieltjes polynomials.
Constructing translating solitons from Lagrangian Grim Reapers.
problem Creating Lagrangian translating solitons from intersections of Grim Reapers.
method Desingularizing intersections with special Lagrangian Lawlor necks.
result Constructing Lagrangian translating solitons with multiple ends and loops.
Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.
problem Verifying Joyce's conjectures for specific Lagrangian surfaces.
method Continuation of Lagrangian mean curvature flow through finite time neck pinches.
result Flow converges to a chain of special Lagrangians, verifying conjectures.
The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.
problem Stabilizing Z/2-harmonic 1-forms on closed 3-manifolds.
method Explicit construction of harmonic 1-forms by modifying metrics near links.
result Construction of harmonic 1-forms degenerating to manifolds with cylindrical ends.
CAST models time-varying treatment effects in cancer patients.
problem Estimating treatment effects at fixed time points limits understanding of dynamic changes over time.
method CAST combines parametric and non-parametric methods to model continuous time-varying treatment effects.
result CAST reveals how treatment effects rise, peak, and decline over the follow-up period.
The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.
problem Existence of non-trivial harmonic maps into higher-dimensional target manifolds.
method Perturbative argument, refined neck-analysis, energy identity, min-max problems.
result Construction of an infinite family of new null-homotopic n-harmonic n-spheres. New proof classifies ancient flows in 3D space.
problem Classifying ancient noncollapsed flows in R3. method Combining neck theorem and Harnack inequality rigidity.
result Directly establishes self-similarity of flows.
Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.
problem Understanding Willmore surfaces and their properties.
method Applying the 3-circle theorem to analyze the second fundamental form of Willmore surfaces.
result Proves a decay estimate of the second fundamental form along the neck region.
Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.
Real Lagrangian tori in S2imesS2 are Hamiltonian isotopic to the Clifford torus.
problem Unknottedness of real Lagrangian tori in S2imesS2. method Neck-stretching argument, Gromov's foliation theorem, Cieliebak-Schwingenheuer criterion.
result Real Lagrangian tori in S2imesS2 are Hamiltonian isotopic to the Clifford torus. In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on S3, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point bet…
The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.
problem Stability and energy identity of Yang-Mills-Higgs pairs on vector bundles.
method Bubble-neck decomposition and analysis of weakly stable pairs.
result A sequence of Yang-Mills-Higgs pairs converges to a Yang-Mills-Higgs pair with uniformly bounded energy.
Classifies ancient noncollapsed flows in 4D space.
problem Classify all noncollapsed singularities of the mean curvature flow in R^4.
method Proves differential neck theorem, introduces new ideas like switch and differential Merle-Zaag dynamics.
result Classifies all ancient noncollapsed solutions in R^4.
Study investigates singularity formation in α-Yang-Mills-Higgs fields on spheres.
problem Singularity formation in α-Yang-Mills-Higgs fields on spheres. method Established α-energy identity, no-neck property through Hodge decomposition and new conservation law. result Unified and quantitative framework for singularity formation in variational gauge theories.
The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.
problem Infinite-time singularities in Lagrangian mean curvature flow.
method Constructing solutions by gluing special Lagrangian 'Lawlor necks' and analyzing dynamics of neck size.
result The flow decomposes initial data into a union of special Lagrangians intersecting at one point.
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.