Using direct analysis of the motional Stark effect (MSE) signals, an explicit measurement of the 'missing' bootstrap current density around the island location of a neoclassical tearing mode (NTM) is made for the first time. When the NTM is suppressed using co-electron cyclotron current drive, the measured changes in the current profile that restore the bootstrap current are also directly found from the MSE measurements. Additionally, direct analysis of helical perturbations in the MSE signals during slowly rotating 'quasi-stationary' modes shows the first explicit measurement of the deficit in the toroidal current density in the island O-point.
Abstract. We study oriented closed manifolds M n possessing the following Universal Realisation of Cycles (URC ) Property: For each topological space X and each homology class z ∈ H n (X, Z), there exist a finite-sheeted covering M n → M n and a continuous mapping f : M n → X such that f * [ M n ] = kz for a non-zero integer k. We find wide class of examples of such manifolds M n among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we obtain that for each 4-dimensional oriented closed manifold N 4 , there exists a mapping of non-zero degree of a hyperbolic manifold M 4 to N 4 . This was conjectured by Kotschick and Löh.
In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a
3-dimensional Euclidean space is a root of certain polynomial with coefficients
depending on the combinatorial type and on edge lengths of the polyhedron only.
Moreover, the coefficients of this polynomial are polynomials in edge lengths
of the polyhedron. This result implies that the volume of a simplicial
polyhedron with fixed combinatorial type and edge lengths can take only
finitely many values. In particular, this yields that the volume of a flexible
polyhedron in a 3-dimensional Euclidean space is constant. Until now it has
been unknown whether these results can be obtained in dimensions greater than
3. In this paper we prove that all these results hold for polyhedra in a
4-dimensional Euclidean space.Comment: 23 pages; misprints corrected, Lemma 6.1 slightly rewritten, title
change
In 1996 Sabitov proved that the volume of an arbitrary simplicial polyhedron P in the 3-dimensional Euclidean space R 3 satisfies a monic (with respect to V ) polynomial relation F (V, ℓ) = 0, where ℓ denotes the set of the squares of edge lengths of P . In 2011 the author proved the same assertion for polyhedra in R 4 . In this paper, we prove that the same result is true in arbitrary dimension n ≥ 3. Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular 2-faces. As a corollary, we obtain the proof in arbitrary dimension of the well-known Bellows Conjecture posed by Connelly in 1978. This conjecture claims that the volume of any flexible polyhedron is constant. Moreover, we obtain the following stronger result. If Pt, t ∈ [0, 1], is a continuous deformation of a polyhedron such that the combinatorial type of Pt does not change and every 2-face of Pt remains congruent to the corresponding face of P 0 , then the volume of Pt is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in C n from their orthogonal edge lengths.
Abstract. We construct self-intersected flexible cross-polytopes in the spaces of constant curvature, that is, Euclidean spaces E n , spheres S n , and Lobachevsky spaces Λ n of all dimensions n. In dimensions n ≥ 5, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces E n , S n , and Λ n . For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.
We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, unlike in the spheres, in the Euclidean spaces and the Lobachevsky spaces of dimensions 4 and higher, still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are nonconstant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was built only in dimension 3 (Alexandrov, 1997), and was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type among which there are our counterexamples to the usual Bellows Conjecture. By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write relations on the volumes of their faces of codimensions 1 and 2. 4
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