We give a detailed construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.
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48 results for “smooth counterexample”
The study refines known counterexamples in 4D to satisfy certain inequalities.
problem Addressing counterexamples in Gromov's and Rosenberg's conjectures.
method Analyzing simply connected and non-simply connected four manifolds up to homeomorphism.
result Gromov's and Rosenberg's conjectures hold for simply connected four manifolds up to homeomorphism.
The study finds counterexamples to curvature estimates for minimizing surfaces.
problem Curvature estimates for minimizing surfaces in metric convergence.
method Constructing sequences of smooth minimizing surfaces in metrics converging to Euclidean.
result Found counterexamples with diverging norm of second fundamental form.
Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based…
Disproves a conjecture about knots in 4D space.
problem Non-orientable slice genus of torus knots
method Found a specific knot that bounds a Möbius band in 4-ball
result Counterexample to Batson's conjecture
We outline the construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.
Compactness fails for curvature equations in high dimensions.
problem Compactness of solutions to curvature equations fails in high dimensions.
method Chen and Wu constructed a smooth counterexample.
result Compactness fails in dimensions not less than 35.
Counterexamples found for knot conjectures.
problem Knot conjectures regarding bounding Möbius bands in 4-ball.
method Examined torus knots T(2,5) and T(2,9) in 3D space.
result Found counterexamples to Allen's conjectures.
The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.
problem The Double Soul Conjecture for non-simply connected manifolds.
method Analysis of double disk bundles and counterexamples.
result Infinitely many counterexamples to the generalized Double Soul Conjecture.
In this letter a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely---taking into account that the conjecture lacks any precise formulation yet---first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically st…
Researchers find counterexamples to inverse problems for wave equations.
problem Inverse problems for wave equations on domains and Lorentzian manifolds.
method Constructing non-isometric Lorentzian metrics leading to same partial data measurements.
result Non-isometric Lorentzian metrics can produce identical partial data measurements.
For applications in computing, Bezier curves are pervasive and are defined by a piecewise linear curve L which is embedded in R^3 and yields a smooth polynomial curve C embedded in R^3. It is of interest to understand when L and C have the same embeddings. One class of counterexamples is shown for L being unknotted, wh…
We construct a new aperiodic symplectic plug and hence new smooth counterexamples to the Hamiltonian Seifert conjecture in R^{2n} for n>2. In other words, we develop an alternative procedure, to those of V. L. Ginzburg and M. Herman, for constructing smooth Hamiltonian flows, on the standard symplectic R^{2n} for n>2, …
Counterexample disproves log canonical Beauville--Bogomolov decomposition.
problem Disproving the log canonical Beauville--Bogomolov decomposition.
method Constructing a specific log canonical, K-trivial variety with non-birational fibers.
result Provides a counterexample to the Beauville--Bogomolov decomposition in the log canonical setting.
Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equationmath.DG
Study shows non-uniqueness and failure of compactness in constant curvature equations.
problem Non-uniqueness and failure of compactness in constant curvature equations.
method Warped product manifold construction and smooth counterexample creation.
result Compactness of solutions fails for dimensions ≥ 62.
A smooth counterexample to the Hamiltonian Seifert conjecture for six-dimensional symplectic manifolds is found. In particular, we construct a smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4, such that one of its non-singular level sets carries no periodic orbits of the Hamiltonian flow. Th…
Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.
New 4D shapes found that defy smoothness rules.
problem Smoothness rules in 4D shapes don't always hold.
method Applied reflection group trick to exotic 4-manifolds.
result Found counterexamples to smooth Borel conjecture.
Three counterexamples show SLCE's limitations in 2D and 3D.
problem Demonstrate limitations of the special Lagrangian curvature equation.
method Construct counterexamples in 2D and 3D, showing non- solutions and unbounded curvature.
result SLCE's a priori estimates are sharp and require convexity and critical phase.
In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.
We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C^3) locally isometrically embedded in R^3. Moreover, the Gauss curvature of the metric can be made negative except for one point.
A counterexample is given for the Knaster-like conjecture of Makeev for functions on . Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively.
The paper constructs manifolds without smooth psc metrics but with -metrics that are psc outside singular points.
problem Constructing manifolds without smooth positive scalar curvature metrics.
method Constructing manifolds with point singularities and -metrics that are psc outside the singular set.
result Examples of manifolds with point singularities that do not admit smooth psc metrics but do admit -metrics that are psc outside the singular set.
Two different spacetimes can mimic each other's boundary measurements.
problem Lorentzian Calderón problem
method Counterexample construction
result Non-isometric spacetimes can have identical boundary measurements
Counterexamples found for Torelli groups of Kähler and hyper-Kähler manifolds.
problem Torelli groups of Kähler and hyper-Kähler manifolds are not always finite.
method Construction of homomorphisms and analysis of diffeomorphisms' actions.
result Found counterexamples showing Torelli groups are not always finite.
Counterexamples show failure of uniform laws of large numbers for subdifferentials.
problem Failure of uniform laws of large numbers for subdifferentials under natural assumptions.
method Univariate and bivariate random Lipschitz and convex functions with smooth pieces.
result Counterexamples demonstrate failure of uniform laws of large numbers for subdifferentials.
Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as `flows' on hypersurface geometries are considered. It is shown that these hypersurface 'flows' are not generally well-posed for smooth initial data and counterexamples…
New non-perturbative counterexamples to Min-Oo's Conjecture are created.
problem Min-Oo's Conjecture on positive curvature and mass.
method Quantitative Gromov-Lawson Schoen-Yau surgery.
result Construction of new non-perturbative counterexamples with more complex topology.
Counterexample disproves conjecture about 3-manifold modules.
problem Disproving a conjecture about 3-manifold modules.
method Provided a specific counterexample.
result Disproved Marché's conjecture about 3-manifold modules.
Counterexample shows state-constrained optimal control problems can have Young measure gaps.
problem Existence of Young measure gaps in state-constrained optimal control problems.
method Provided a counterexample for smooth controllable systems state-constrained to the unit ball.
result Gap occurs in a regular setting with non-convex Lagrangian density.
Counterexample disproves recent Penrose conjecture variant.
problem A variant of the Penrose conjecture.
method Provided a counterexample.
result The conjectured variant of the Penrose inequality is false.
Four counterexamples in surface homology.
problem Understanding surface homology and minimal homology bases.
method Presenting four specific counterexamples.
result Minimal homology bases can be challenging to work with.
If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Gene…
After reconsidering the Dasbach-Hougardy counterexample to the Kauffman Conjecture on alternating knots, we reformulate the conjecture and consider Dasbach-Hougardy counterexample and similar counterexamples in the light of the reformulated conjecture.
Numerical study confirms Brennan's conjecture for a counterexample to Thurston's conjecture.
problem Thurston's conjecture and Brennan's conjecture in planar domains.
method Numerical analysis of a specific counterexample to Thurston's conjecture.
result The counterexample does not contradict Brennan's conjecture.
Counterexample disproves HK-conjecture for flat manifolds of dimension 9 and higher.
problem Disproving the HK-conjecture for flat manifolds of various dimensions.
method Using flat manifolds of dimension 9 and higher, constructing counterexamples.
result Minimal dimension 9 is required for counterexamples, and higher dimensions are possible.
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
The paper disproves some implications in convex projective geometry.
problem Geometrical finiteness in round convex projective geometry.
method Construction of counterexamples and description of invariant domains.
result Existence of a counterexample with infinite Hilbert volume.
Calegari's 4-spheres from fibered knots are proven standard.
problem Proving Calegari's 4-spheres from fibered knots are standard.
method 5-dimensional handlebody techniques and mapping class groups of 3-dimensional handlebodies.
result All Calegari's homotopy 4-spheres from fibered knots are diffeomorphic to the standard 4-sphere.
Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as and admits neither 1- nor 3-handles, by using rational blow-downs and K…
A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this ar…
We present a novel framework for augmenting data sets for machine learning based on counterexamples. Counterexamples are misclassified examples that have important properties for retraining and improving the model. Key components of our framework include a counterexample generator, which produces data items that are mi…
Counterexample disproves conjectures about log canonical thresholds.
problem Conjectures about log canonical thresholds were disproved.
method Provided a counterexample to both conjectures.
result Conjectures about log canonical thresholds are false.
Completes classification of high-dimensional manifolds up to exotic sums.
problem Classifying high-dimensional manifolds up to exotic sums.
method Algebraic refinement of intersection forms and Witten genus.
result All but finitely many dimensions are resolved, completing the classification.
New method finds large counterexamples by selectively exploring triangulations.
problem Finding small counterexamples in -manifold triangulations.
method Selective enumeration of triangulations using heuristics.
result Found counterexamples to three conjectures about vertex triangulations.
Construction of counterexamples to the jet determination Chern-Moser Theorem in higher codimensionmath.CV
Researchers found counterexamples to a 2-jet determination theorem in higher codimension.
problem Counterexample construction to the 2-jet determination Chern-Moser Theorem in higher codimension.
method Constructed counterexamples of quadratic submanifolds with specific properties.
result Generated counterexamples to the 2-jet determination Chern-Moser Theorem in higher codimension.
Proves Girth Alternative for some HNN extensions, finds counterexamples.
problem Girth Alternative for HNN extensions of finitely generated groups.
method Proves Girth Alternative for a specific class of HNN extensions.
result Girth Alternative holds for some HNN extensions but fails in general.
We found an infinite family of counterexamples to Batson's conjecture.
problem Batson's conjecture about nonorientable 4-ball genus of torus knots is false.
method We identified an infinite family of counterexamples to Batson's conjecture.
result We found an infinite family of counterexamples to Batson's conjecture.