scholarly journals Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

2021 ◽  
Vol -1 (-1) ◽  
Author(s):  
Jun-Muk HWANG
Keyword(s):  
Rational Curves ◽  
Contact Structures

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

scholarly journals Contact structures and reducible surgeries

2015 ◽  
Vol 152 (1) ◽  
pp. 152-186 ◽  
Author(s):  
Tye Lidman ◽  
Steven Sivek
Keyword(s):  
Contact Structures ◽  
Surgery Theory ◽  
Contact Topology ◽  
Exceptional Surgery ◽  
Genus 2 ◽  
Cabling Conjecture

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus$g$must have slope$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

scholarly journals Non Kählerian surfaces with a cycle of rational curves

2021 ◽  
Vol 8 (1) ◽  
pp. 208-222
Author(s):  
Georges Dloussky
Keyword(s):  
Deformation Theory ◽  
Complex Surface ◽  
Rational Curves ◽  
Intersection Form ◽  
Connected Component ◽  
Hirzebruch Surface ◽  
Compact Complex Surface ◽  
A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

scholarly journals Microfluidic Optical Shutter Flexibly x-y Actuated via Electrowetting-on-Dielectrics with <20 ms Response Time

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Henning Fouckhardt ◽  
Johannes Strassner ◽  
Carina Heisel ◽  
Dominic Palm ◽  
Christoph Doering
Keyword(s):  
Optical Properties ◽  
Response Time ◽  
Visible Spectral Range ◽  
Contact Structures ◽  
Light Path ◽  
Cell Type ◽  
Electrically Conductive ◽  
Droplet Movement ◽  
In Series ◽  
Optical Shutter

Tunable microoptics deals with devices of which the optical properties can be changed during operation without mechanically moving solid parts. Often a droplet is actuated instead, and thus tunable microoptics is closely related to microfluidics. One such device/module/cell type is an optical shutter, which is moved in or out of the path of the light. In our case the transmitting part comprises a moving transparent and electrically conductive water droplet, embedded in a nonconductive blackened oil, that is, an opaque emulsion with attenuation of 30 dB at 570 nm wavelength over the 250 μm long light path inside the fluid (15 dB averaged over the visible spectral range). The insertion loss of the cell is 1.5 dB in the “open shutter” state. The actuation is achieved via electrowetting-on-dielectrics (EWOD) with rectangular AC voltage pulses of 2·90 V peak-to-peak at 1 kHz. To flexibly allow for horizontal, vertical, and diagonal droplet movement in the upright x-y plane, the contact structures are prepared such that four possible stationary droplet positions exist. The cell is configured as two capacitors in series (along the z axis), such that EWOD forces act symmetrically in the front and back of the 60 nl droplet with a response time of <20 ms.

scholarly journals Tight contact structures on some small Seifert fibered 3-manifolds

2007 ◽  
Vol 129 (5) ◽  
pp. 1403-1447 ◽  
Author(s):  
Paolo. Ghiggini ◽  
Paolo. Lisca ◽  
András. Stipsicz
Keyword(s):  
Contact Structures ◽  
Tight Contact
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