Smooth dependence on data for the SPDE: the non-Lipschitz case (II)

2007 ◽  
pp. 205-220
Keyword(s):  
Smooth Dependence

scholarly journals Helix-Free Ferroelectric Liquid Crystals: Electro Optics and Possible Applications

2018 ◽  
Vol 8 (12) ◽  
pp. 2429 ◽  
Author(s):  
Alexander Andreev ◽  
Tatiana Andreeva ◽  
Igor Kompanets ◽  
Nikolay Zalyapin
Keyword(s):  
Liquid Crystals ◽  
Low Voltage ◽  
Light Transmission ◽  
Optical Response ◽  
Phase Delay ◽  
High Rate ◽  
Ferroelectric Liquid Crystals ◽  
Light Modulators ◽  
Director Reorientation ◽  
Smooth Dependence

This is a review of results from studying ferroelectric liquid crystals (FLCs) of a new type developed for fast low-voltage displays and light modulators. These materials are helix-free FLCs, which are characterized by spatially periodic deformation of smectic layers and a small value of spontaneous polarization (less than 50 nC/cm2). The FLC director is reoriented due to the motion of solitons at the transition to the Maxwellian mechanism of energy dissipation. A theoretical model is proposed for describing the FLC deformation and director reorientation. The frequency and field dependences of the optical response time are studied experimentally for modulation of light transmission, scattering, and phase delay with a high rate. The hysteresis-free nature and smooth dependence of the optical response on the external electric field in the frequency range up to 6 kHz is demonstrated, as well as bistable light scattering with memorization of an optical state for a time exceeding the switching time by up to 6 orders of magnitude. Due to the spatially inhomogeneous light phase delay, the ability of a laser beam to cause interference is effectively suppressed. The fastest FLCs under study are compatible with 3D, FLC on Silicon (FLCoS), and Field Sequential Colors (FSC) technologies.

scholarly journals Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters

2020 ◽  
Author(s):  
Arkadiusz Lewandowski
Keyword(s):  
Precise Formulation ◽  
Smooth Dependence

Abstract We prove that the mappings obtained in Forstnerič splitting lemma vary in a $$\mathcal {C}^{\lfloor {\frac{l-1}{2}}\rfloor }$$ C ⌊ l - 1 2 ⌋ -continuous way if only the input family of biholomorphic mappings close to Id (and their domains) is $$\mathcal {C}^l$$ C l -continuous (see Theorem 1.3 for a precise formulation).

scholarly journals N-Player Games and Mean-Field Games With Smooth Dependence on Past Absorptions

2019 ◽  
Author(s):  
Luciano Campi ◽  
Maddalena Ghio ◽  
Giulia Livieri
Keyword(s):  
Mean Field ◽  
Mean Field Games ◽  
Smooth Dependence

Mg2+-dependent electrical control of flagellar activity in Euglena

1977 ◽  
Vol 23 (1) ◽  
pp. 211-225
Author(s):  
K.M. Nichols ◽  
R. Rikmenspoel
Keyword(s):  
High Speed ◽  
External Medium ◽  
Average Frequency ◽  
Current Injection ◽  
Initial Activity ◽  
Free Medium ◽  
Electrical Control ◽  
Negative Current ◽  
Smooth Dependence ◽  
Injection Cycle

When a Euglena gracilis in a Ca2+-containing medium is impaled with a microelectrode, the flagellum is instantly ejected. In a Ca2+-free medium to which 1 mM EGTA has been added, the flagellum remains attached to the organism, but it loses activity upon impalement. Externally added ATP at a concentration of 10 mM will sustain normal flagellar activity (at approximately 20 Hz) of an impaled Euglena. If negative direct current of several tenths of a microamp is injected through the impaling microelectrode, the flagellar activity is stopped or much reduced. When the current injection is turned off the flagellum returns to its initial activity. This cycle can be repeated many times on the same animal, independent of whether Mg2+ is present in the external medium or not. If 1 micrometer of gramicidin is added to Ca2+-free medium containing 1 mM EGTA and 10 mM ATP, the flagellar activity becomes dependent on external Mg2+. Without external Mg2+ no flagellar activity is present after one or two current injection cycles as described above. With 1 mM Mg2+ present in the external medium many cycles (up to 10) can be produced. This Mg2+-dependent flagellar activity shows a smooth dependence on the amount of current injected. Observations taken by high speed cinemicrography show that in the third injection cycle the average frequency of the flagellar motion is 16-3 Hz at 0 muA, is 8 Hz at 0-2 muA, and is approximately 0 at 0-6 muA of negative current. The injection of positive current results in an increase in flagellar frequency dependent on the amount of current injected. The data indicate that the control of motility of Euglena flagella is dependent on an electrically activated Mg2+ pump.

On perturbations of a translationally-invariant differential equation

1988 ◽  
Vol 110 (1-2) ◽  
pp. 1-25 ◽  
Author(s):  
R.J. Magnus
Keyword(s):  
Differential Equation ◽  
Perturbation Parameter ◽  
Dimensional Euclidean Space ◽  
Existence Of A Solution ◽  
Smooth Perturbation ◽  
Invariant Differential ◽  
Smooth Dependence ◽  
Space Conditions ◽  
Perturbed Equation ◽  
Unperturbed Solution

SynopsisWe study certain perturbations of the differential equation Δu − u + up = 0 on all of n-dimensional Euclidean space. Conditions are obtained which ensure the existence of a solution to the perturbed equation near a given solution to the unperturbed equation. We have to overcome degeneracy of the unperturbed solution and lack of smooth dependence on the perturbation parameter. An abstract version of the argument is sketched in a functional-analytic setting related toequivariant bifurcation theory. We consider also a smooth perturbation with several parameters and study the singularities of the mapping which maps each solution to its associated parameters.

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