252 research outputs found
On Thurston's Euler class one conjecture
In 1976, Thurston proved that taut foliations on closed hyperbolic
3-manifolds have Euler class of norm at most one, and conjectured that
conversely, any integral second cohomology class with norm equal to one is the
Euler class of a taut foliation. This is the first from a series of two papers
that together give a negative answer to Thurston's conjecture. Here
counterexamples have been constructed conditional on the fully marked surface
theorem. In the second paper, joint with David Gabai, a proof of the fully
marked surface theorem is given.Comment: 42 pages, 21 figures. The paper is split into two parts, and the
appendix is appearing as a separate article joint with David Gabai. The
results on taut foliations on sutured solid tori are generalised. A section
on relative Euler class is added to address a possible oversight in the
literature. Exposition is improved, and new open questions are raised. Final
version to appear in Acta Mathematic
Lower bound for the Perron-Frobenius degrees of Perron numbers
Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius
degree of a Perron number that is not totally-real. As an application, we prove
that there are cubic Perron numbers whose Perron-Frobenius degrees are
arbitrary large; a result known to Lind, McMullen and Thurston. A similar
result is proved for biPerron numbers.Comment: To appear in Ergodic Theory and Dynamical Systems, 15 pages, 4
figure
Non-negative integral matrices with given spectral radius and controlled dimension
A celebrated theorem of Lind states that a positive real number is equal to
the spectral radius of some integral primitive matrix, if and only if, it is a
Perron algebraic integer. Given a Perron number , we prove that there is an
integral irreducible matrix with spectral radius , and with dimension
bounded above in terms of the algebraic degree, the ratio of the first two
largest Galois conjugates, and arithmetic information about the ring of
integers of its number field. This arithmetic information can be taken to be
either the discriminant or the minimal Hermite-like thickness. Equivalently,
given a Perron number , there is an irreducible shift of finite type with
entropy defined as an edge shift on a graph whose number of vertices
is bounded above in terms of the aforementioned data.Comment: The referee's suggestions are incorporated; in particular an upper
bound for the Perron--Frobenius degree is derived from the main result. See
Theorem 1.6. To appear in Ergodic Theory and Dynamical System
Pseudo-Anosov maps with small stretch factors on punctured surfaces
Consider the problem of estimating the minimum entropy of pseudo-Anosov maps
on a surface of genus with punctures. We determine the behaviour of
this minimum number for a certain large subset of the plane, up to a
multiplicative constant. In particular it has been shown that for fixed ,
this minimum value behaves as , proving what Penner speculated in
1991.Comment: To appear in Algebraic & Geometric Topology, 26 pages, 10 figure
LATE DEVONIAN-CARBONIFEROUS CONODONTS FROM EASTERN IRAN
Conodont data from acid-leaching 110 samples from two Late Devonian-Carboniferous areas in the Shotori Range (Tabas region) of eastern Iran are presented. At Howz-e-Dorah, a section (88 samples) commencing high in the Bahram Formation (Givetian-early Frasnian) extended through the Shishtu Formation (Frasnian, Early hassi Zone or older, to latest Tournaisian, anchoralis-latus Zone) and the Sardar Formation (earliest Visean, texanus Zone, to late Namurian, sinuatus-corrugatus-sulcatus Zone) and into the Jamal Formation (Permian). Four less exhaustively sampled sections (22 samples) show the Kale Sardar area to be tectonically more complicated than the Howz-e-Dorah area. Useful marker horizons in the Howz-e-Dorah section, well constrained by conodont data, are: the early Frasnian (no older than Early hassi Zone) biostromal beds of the Shishtu Formation, an early Famennian (Late triangularis to Early crepida) interval of oolitic limestone, a cyclothem sequence straddling the Early Carboniferous-Late Carboniferous boundary, and an Early Permian interval of siliceous sand ("the white quartzite" of previous authors). Additionally, several iron-rich horizons, readily traceable from locality to locality, are well constrained by conodont ages. Eighty-five conodont species/subspecies are documented representing 24 genera.. Two new species, Polygnathus capollocki and Polygnathus ratebi and one new subspecies, Icriodus alternatus mawsonae are described. 
The fully marked surface theorem
In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a
foliation has Euler characteristic equal, up to sign, to the Euler class of the
foliation evaluated on [S], the homology class represented by S. The main
result of this paper is a converse for taut foliations: if the Euler class of a
taut foliation evaluated on [S] equals up to sign the Euler
characteristic of S and the underlying manifold is hyperbolic, then there
exists another taut foliation such that is homologous to a
union of leaves and such that the plane field of is homotopic to
that of . In particular, and have the
same Euler class.
In the same paper Thurston proved that taut foliations on closed hyperbolic
3-manifolds have Euler class of norm at most one, and conjectured that,
conversely, any integral cohomology class with norm equal to one is the Euler
class of a taut foliation. This is the second of two papers that together give
a negative answer to Thurston's conjecture. In the first paper, counterexamples
were constructed assuming the main result of this paper.Comment: 36 pages, 16 figures. Portions of this work previously appeared as an
appendix to arXiv:1603.03822, but has evolved into its own work and has been
accepted for publication separately. Final version to appear in Acta
Mathematic
Thurston norm and Euler classes of tight contact structures
Bill Thurston proved that taut foliations of hyperbolic 3-manifolds have
Euler classes of norm at most one, and conjectured that any integral second
cohomology class of norm equal to one is realised as the Euler class of some
taut foliation. Recent work of the second author, joint with David Gabai, has
produced counterexamples to this conjecture. Since tight contact structures
exist whenever taut foliations do and their Euler classes also have norm at
most one, it is natural to ask whether the Euler class one conjecture might
still be true for tight contact structures. In this short note, we show that
the previously constructed counterexamples for Euler classes of taut foliations
in [Yaz20] are in fact realised as Euler classes of tight contact structures.
This shows that the Euler class one conjecture has a chance of being true for
tight contact structures.Comment: 12 pages, 4 figure
A New Amplification Regime for Traveling Wave Tubes with Third Order Modal Degeneracy
Engineering of the eigenmode dispersion of slow-wave structures (SWSs) to
achieve desired modal characteristics, is an effective approach to enhance the
performance of high power traveling wave tube (TWT) amplifiers or oscillators.
We investigate here for the first time a new synchronization regime in TWTs
based on SWSs operating near a third order degeneracy condition in their
dispersion. This special three-eigenmode synchronization is associated with a
stationary inflection point (SIP) that is manifested by the coalescence of
three Floquet-Bloch eigenmodes in the SWS. We demonstrate the special features
of "cold" (without electron beam) periodic SWSs with SIP modeled as coupled
transmission lines (CTLs) and investigate resonances of SWSs of finite length.
We also show that by tuning parameters of a periodic SWS one can achieve an SIP
with nearly ideal flat dispersion relationship with zero group velocity or a
slightly slanted one with a very small (positive or negative) group velocity
leading to different operating schemes. When the SIP structure is synchronized
with the electron beam potential benefits for amplification include (i) gain
enhancement, (ii) gain-bandwidth product improvement, and (iii) higher power
efficiency, when compared to conventional Pierce-like TWTs. The proposed theory
paves the way for a new approach for potential improvements in gain, power
efficiency and gain-bandwidth product in high power microwave amplifiers.Comment: 15 pages, 11 figure
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