937 resultados para Periodic cohomology
Resumo:
The bifurcation and nonlinear stability properties of the Meinhardt-Gierer model for biochemical pattern formation are studied. Analyses are carried out in parameter ranges where the linearized system about a trivial solution loses stability through one to three eigenfunctions, yielding both time independent and periodic final states. Solution branches are obtained that exhibit secondary bifurcation and imperfection sensitivity and that appear, disappear, or detach themselves from other branches.
Resumo:
Six topics in incompressible, inviscid fluid flow involving vortex motion are presented. The stability of the unsteady flow field due to the vortex filament expanding under the influence of an axial compression is examined in the first chapter as a possible model of the vortex bursting observed in aircraft contrails. The filament with a stagnant core is found to be unstable to axisymmetric disturbances. For initial disturbances with the form of axisymmetric Kelvin waves, the filament with a uniformly rotating core is neutrally stable, but the compression causes the disturbance to undergo a rapid increase in amplitude. The time at which the increase occurs is, however, later than the observed bursting times, indicating the bursting phenomenon is not caused by this type of instability.
In the second and third chapters the stability of a steady vortex filament deformed by two-dimensional strain and shear flows, respectively, is examined. The steady deformations are in the plane of the vortex cross-section. Disturbances which deform the filament centerline into a wave which does not propagate along the filament are shown to be unstable and a method is described to calculate the wave number and corresponding growth rate of the amplified waves for a general distribution of vorticity in the vortex core.
In Chapter Four exact solutions are constructed for two-dimensional potential flow over a wing with a free ideal vortex standing over the wing. The loci of positions of the free vortex are found and the lift is calculated. It is found that the lift on the wing can be significantly increased by the free vortex.
The two-dimensional trajectories of an ideal vortex pair near an orifice are calculated in Chapter Five. Three geometries are examined, and the criteria for the vortices to travel away from the orifice are determined.
Finally, Chapter Six reproduces completely the paper, "Structure of a linear array of hollow vortices of finite cross-section," co-authored with G. R. Baker and P. G. Saffman. Free streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored vortices. If each vortex has area A and the separation is L, then there are two possible shapes if A^(1/2)/L is less than 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable, while the less deformed shape is stable.
Resumo:
A study is made of solutions of the macroscopic Maxwell equations in nonlinear media. Both nonlinear and dispersive terms are responsible for effects that are not taken into account in the geometrical optics approximation. The nonlinear terms can, depending on the nature of the nonlinearity, cause plane waves to focus when the amplitude varies across the wavefront. The dispersive terms prevent the singularities that nonlinearity alone would produce. Solutions are found which de scribe periodic plane waves in fully nonlinear media. Equations describing the evolution of the amplitude, frequency and wave number are generated by means of averaged Lagrangian techniques. The equations are solved for near linear media to produce the form of focusing waves which develop a singularity at the focal point. When higher dispersion is included nonlinear and dispersive effects can balance and one finds amplitude profiles that propagate with straight rays.
Resumo:
In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.
Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.
Resumo:
Research has proven that Shoreline Erosion is caused by excess water contained within the shore face. This Research presents an opportunity to control erosion by managing the near shore water table. Our Research on Bogue Banks North Carolina suggests that our buildings and other impervious surfaces collect and concentrate water from storm rain runoff into the surface water table and within the critical beach front water exit point. Presently our Potable Fresh Water is supplied from deep wells located beneath an impervious layer of Marl. After our use, the Waste water is drained into the Surface Aquifer, the combined waste and storm rain water raises the Surface Aquifer water table and produces Erosion. The Deep Aquifers presently supplying our Potable Water have an unknown recharge rate, with increasing reports of Salt Water intrusion. We believe our Vital Fresh water supply system should be modified to supply Reverse Osmosis treatment plants from shallow wells. This will lower the Surface Water Table. These Shallow wells, either horizontal or vertical, might be located within the beach front, adjacent to high erosion risk properties. Beach Drains and Reverse Osmosis Water systems are new and proven technologies. By combining these technologies we can reduce or reverse Shore Erosion, ensure a safe Potable Water supply, reduce requirements for periodic beach nourishment, reduce taxes and protect our property well into the Future. (PDF contains 5 pages)
Resumo:
How is climate change affecting our coastal environment? How can coastal communities adapt to sea level rise and increased storm risk? These questions have garnered tremendous interest from scientists and policy makers alike, as the dynamic coastal environment is particularly vulnerable to the impacts of climate change. Over half the world population lives and works in a coastal zone less than 120 miles wide, thereby being continuously affected by the changes in the coastal environment [6]. Housing markets are directly influenced by the physical processes that govern coastal systems. Beach towns like Oak Island in North Carolina (NC) face severe erosion, and the tax assesed value of one coastal property fell by 93% in 2007 [9]. With almost ninety percent of the sandy beaches in the US facing moderate to severe erosion [8], coastal communities often intervene to stabilize the shoreline and hold back the sea in order to protect coastal property and infrastructure. Beach nourishment, which is the process of rebuilding a beach by periodically replacing an eroding section of the beach with sand dredged from another location, is a policy for erosion control in many parts of the US Atlantic and Pacific coasts [3]. Beach nourishment projects in the United States are primarily federally funded and implemented by the Army Corps of Engineers (ACE) after a benefit-cost analysis. Benefits from beach nourishment include reduction in storm damage and recreational benefits from a wider beach. Costs would include the expected cost of construction, present value of periodic maintenance, and any external cost such as the environmental cost associated with a nourishment project (NOAA). Federal appropriations for nourishment totaled $787 million from 1995 to 2002 [10]. Human interventions to stabilize shorelines and physical coastal dynamics are strongly coupled. The value of the beach, in the form of storm protection and recreation amenities, is at least partly capitalized into property values. These beach values ultimately influence the benefit-cost analysis in support of shoreline stabilization policy, which, in turn, affects the shoreline dynamics. This paper explores the policy implications of this circularity. With a better understanding of the physical-economic feedbacks, policy makers can more effectively design climate change adaptation strategies. (PDF contains 4 pages)
Resumo:
Granular crystals are compact periodic assemblies of elastic particles in Hertzian contact whose dynamic response can be tuned from strongly nonlinear to linear by the addition of a static precompression force. This unique feature allows for a wide range of studies that include the investigation of new fundamental nonlinear phenomena in discrete systems such as solitary waves, shock waves, discrete breathers and other defect modes. In the absence of precompression, a particularly interesting property of these systems is their ability to support the formation and propagation of spatially localized soliton-like waves with highly tunable properties. The wealth of parameters one can modify (particle size, geometry and material properties, periodicity of the crystal, presence of a static force, type of excitation, etc.) makes them ideal candidates for the design of new materials for practical applications. This thesis describes several ways to optimally control and tailor the propagation of stress waves in granular crystals through the use of heterogeneities (interstitial defect particles and material heterogeneities) in otherwise perfectly ordered systems. We focus on uncompressed two-dimensional granular crystals with interstitial spherical intruders and composite hexagonal packings and study their dynamic response using a combination of experimental, numerical and analytical techniques. We first investigate the interaction of defect particles with a solitary wave and utilize this fundamental knowledge in the optimal design of novel composite wave guides, shock or vibration absorbers obtained using gradient-based optimization methods.
Resumo:
Freshwater fish of the genus Apteronotus (family Gymnotidae) generate a weak, high frequency electric field (< 100 mV/cm, 0.5-10 kHz) which permeates their local environment. These nocturnal fish are acutely sensitive to perturbations in their electric field caused by other electric fish, and nearby objects whose impedance is different from the surrounding water. This thesis presents high temporal and spatial resolution maps of the electric potential and field on and near Apteronotus. The fish's electric field is a complicated and highly stable function of space and time. Its characteristics, such as spectral composition, timing, and rate of attenuation, are examined in terms of physical constraints, and their possible functional roles in electroreception.
Temporal jitter of the periodic field is less than 1 µsec. However, electrocyte activity is not globally synchronous along the fish 's electric organ. The propagation of electrocyte activation down the fish's body produces a rotation of the electric field vector in the caudal part of the fish. This may assist the fish in identifying nonsymmetrical objects, and could also confuse electrosensory predators that try to locate Apteronotus by following its fieldlines. The propagation also results in a complex spatiotemporal pattern of the EOD potential near the fish. Visualizing the potential on the same and different fish over timescales of several months suggests that it is stable and could serve as a unique signature for individual fish.
Measurements of the electric field were used to calculate the effects of simple objects on the fish's electric field. The shape of the perturbation or "electric image" on the fish's skin is relatively independent of a simple object's size, conductivity, and rostrocaudal location, and therefore could unambiguously determine object distance. The range of electrolocation may depend on both the size of objects and their rostrocaudal location. Only objects with very large dielectric constants cause appreciable phase shifts, and these are strongly dependent on the water conductivity.
Resumo:
Electron acceleration in a tightly focused ultra-intensity linear polarized laser beam is investigated numerically. It has been found that the acceleration is strong phase dependent and is periodic to the variety of the initial laser field phase. When optimal initial parameters are chosen, the electron can be accelerated effectively. The accelerated electrons are emitted in pulses of which the full width is less than the half period of the laser field.
Resumo:
Electron acceleration using a tightly focused ultraintensity laser beam is investigated numerically and strong phase dependence is found. The acceleration is periodic to the variety of the initial laser field phase, and the accelerated electrons are emitted in pulses of which the full width is the half period of the laser field. When a 10 PW intense laser beam is used, the electron with energy less than 1 Mev can be accelerated up to energies about 1.4 GeV. The optimal initial condition for electron acceleration is found. (C) 2005 American Institute of Physics.
Resumo:
Home to hundreds of millions of souls and land of excessiveness, the Himalaya is also the locus of a unique seismicity whose scope and peculiarities still remain to this day somewhat mysterious. Having claimed the lives of kings, or turned ancient timeworn cities into heaps of rubbles and ruins, earthquakes eerily inhabit Nepalese folk tales with the fatalistic message that nothing lasts forever. From a scientific point of view as much as from a human perspective, solving the mysteries of Himalayan seismicity thus represents a challenge of prime importance. Documenting geodetic strain across the Nepal Himalaya with various GPS and leveling data, we show that unlike other subduction zones that exhibit a heterogeneous and patchy coupling pattern along strike, the last hundred kilometers of the Main Himalayan Thrust fault, or MHT, appear to be uniformly locked, devoid of any of the “creeping barriers” that traditionally ward off the propagation of large events. The approximately 20 mm/yr of reckoned convergence across the Himalaya matching previously established estimates of the secular deformation at the front of the arc, the slip accumulated at depth has to somehow elastically propagate all the way to the surface at some point. And yet, neither large events from the past nor currently recorded microseismicity nearly compensate for the massive moment deficit that quietly builds up under the giant mountains. Along with this large unbalanced moment deficit, the uncommonly homogeneous coupling pattern on the MHT raises the question of whether or not the locked portion of the MHT can rupture all at once in a giant earthquake. Univocally answering this question appears contingent on the still elusive estimate of the magnitude of the largest possible earthquake in the Himalaya, and requires tight constraints on local fault properties. What makes the Himalaya enigmatic also makes it the potential source of an incredible wealth of information, and we exploit some of the oddities of Himalayan seismicity in an effort to improve the understanding of earthquake physics and cipher out the properties of the MHT. Thanks to the Himalaya, the Indo-Gangetic plain is deluged each year under a tremendous amount of water during the annual summer monsoon that collects and bears down on the Indian plate enough to pull it away from the Eurasian plate slightly, temporarily relieving a small portion of the stress mounting on the MHT. As the rainwater evaporates in the dry winter season, the plate rebounds and tension is increased back on the fault. Interestingly, the mild waggle of stress induced by the monsoon rains is about the same size as that from solid-Earth tides which gently tug at the planets solid layers, but whereas changes in earthquake frequency correspond with the annually occurring monsoon, there is no such correlation with Earth tides, which oscillate back-and-forth twice a day. We therefore investigate the general response of the creeping and seismogenic parts of MHT to periodic stresses in order to link these observations to physical parameters. First, the response of the creeping part of the MHT is analyzed with a simple spring-and-slider system bearing rate-strengthening rheology, and we show that at the transition with the locked zone, where the friction becomes near velocity neutral, the response of the slip rate may be amplified at some periods, which values are analytically related to the physical parameters of the problem. Such predictions therefore hold the potential of constraining fault properties on the MHT, but still await observational counterparts to be applied, as nothing indicates that the variations of seismicity rate on the locked part of the MHT are the direct expressions of variations of the slip rate on its creeping part, and no variations of the slip rate have been singled out from the GPS measurements to this day. When shifting to the locked seismogenic part of the MHT, spring-and-slider models with rate-weakening rheology are insufficient to explain the contrasted responses of the seismicity to the periodic loads that tides and monsoon both place on the MHT. Instead, we resort to numerical simulations using the Boundary Integral CYCLes of Earthquakes algorithm and examine the response of a 2D finite fault embedded with a rate-weakening patch to harmonic stress perturbations of various periods. We show that such simulations are able to reproduce results consistent with a gradual amplification of sensitivity as the perturbing period get larger, up to a critical period corresponding to the characteristic time of evolution of the seismicity in response to a step-like perturbation of stress. This increase of sensitivity was not reproduced by simple 1D-spring-slider systems, probably because of the complexity of the nucleation process, reproduced only by 2D-fault models. When the nucleation zone is close to its critical unstable size, its growth becomes highly sensitive to any external perturbations and the timings of produced events may therefore find themselves highly affected. A fully analytical framework has yet to be developed and further work is needed to fully describe the behavior of the fault in terms of physical parameters, which will likely provide the keys to deduce constitutive properties of the MHT from seismological observations.
Resumo:
What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderly, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive.
The KAM theory is mute about the disrupted tori, but, for two-dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori," tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.
Resumo:
This paper reports self-organized nanostructures observed on the surface of ZnO crystal after irradiation by a focused beam of a femtosecond Ti:sapphire laser with a repetition rate of 250 kHz. For a linearly polarized femtosecond laser, the periodic nanograting structure on the ablation crater surface was promoted. The period of self-organization structures is about 180 nm. The grating orientation is adjusted by the laser polarization direction. A long range Bragg-like grating is formed by moving the sample at a speed of 10 mu m/s. For a circularly polarized laser beam, uniform spherical nanoparticles were formed as a result of Coulomb explosion during the interaction of near-infrared laser with ZnO crystal.
Resumo:
The development of Ring Opening Metathesis Polymerization has allowed the world of block copolymers to expand into brush block copolymers. Brush block copolymers consist of a polymer backbone with polymeric side chains, forcing the backbone to hold a stretched conformation and giving it a worm-like shape. These brush block copolymers have a number of advantages over tradition block copolymers, including faster self-assembly behavior, larger domain sizes, and much less entanglement. This makes them an ideal candidate in the development of a bottom-up approach to forming photonic crystals. Photonic crystals are periodic nanostructures that transmit and reflect only certain wavelengths of light, forming a band gap. These are used in a number of coatings and other optical uses. One and two dimensional photonic crystals are commercially available, though are often expensive and difficult to manufacture. Previous work has focused on the creation of one dimensional photonic crystals from brush block copolymers. In this thesis, I will focus on the synthesis and characterization of asymmetric brush block copolymers for self-assembly into two and three dimensional photonic crystals. Three series of brush block copolymers were made and characterized by Gel Permeation Chromatography and Nuclear Magnetic Resonance spectroscopy. They were then made into films through compressive thermal annealing and characterized by UV-Vis Spectroscopy and Scanning Electron Microscopy. Evidence of non-lamellar structures were seen, indicating the first reported creation of two or three dimensional photonic crystals from brush block copolymers.
Resumo:
Chapter 1
Cyclobutanediyl has been studied in both its singlet and triplet states by ab initio electronic structure theory. The triplet, which is the ground state of the molecule, exists in both C_(2h) and C_(2v) forms, which interconvert via a C_s transition state. For the singlet, only a C_(2h) form is found. It passes, via a C_s transition state, onto the C_(2v) surface on which bicyclobutane is the only minimum. The ring-flipping (inversion) process in bicyclobutane includes the singlet biradical as an intermediate, and involves a novel, nonleast motion pathway. Semiclassical periodic orbit theory indicates that the various minima on both the singlet and triplet surfaces can interconvert via quantum mechanical tunneling.
Chapter 2
The dimethylenepolycyclobutadienes (n) are the non-Kekulé analogues of the classical acenes. Application of a variety of theoretical methods reveals several novel features of such structures. Most interesting is the emergence of a parity rule. When n is even, n is predicted to be a singlet, with n disjoint NBMOs. When n is odd, theory predicts a triplet ground state with (n+1) NBMOs that are not fully disjoint.
Chapter 3
Bi(cyclobutadienyl) (2), the cyclobutadiene analogue of biphenyl, and its homologues tri- (3) and tetra(cyclobutadienyl) (4) have been studied using electronic structure theory. Ab initio calculations on 2 reveal that the central bond is a true double bond, and that the structure is best thought of as two allyl radicals plus an ethylene. The singlet and triplet states are essentially degenerate. Trimer 3 is two allyls plus a dimethylenecyclobutanediyl, while 4 is two coplanar bi(cyclobutadienyl) units connected by a single bond. For both 3 and 4, the quintet, triplet, and singlet states are essentially degenerate, indicating that they are tetraradicals. The infinite polymer, polycyclobutadiene, has been studied by HMO, EHCO, and VEH methods. Several geometries based on the structures of 3 and 4 have been studied, and the band structures are quite intriguing. A novel crossing between the valence and conduction bands produces a small band gap and a high density of states at the Fermi level.
Chapter 4
At the level of Hückel theory, polyfulvene has a HOCO-LUCO degeneracy much like that seen in polyacetylene. Higher levels of theory remove the degeneracy, but the band gap (E_g) is predicted to be significantly smaller than analogous structures such as polythiophene and polypyrrole at the fulvenoid geometry. An alternative geometry, which we have termed quinoid, is also conceivable for polyfulvene, and it is predicted to have a much larger E_g. The effects of benzannelation to produce analogues of polyisothianaphthene have been evaluated. We propose a new model for such structures based on conventional orbital mixing arguments. Several of the proposed structures have quite interesting properties, which suggest that they are excellent candidates for conducting polymers.
Chapter 5
Theoretical studies of polydimethylenecyclobutene and polydiisopropylidene- cyclobutene reveal that, because of steric crowding, they cannot achieve a planar, fully conjugated structure in either their undoped or doped states. Rather, the structure consists of essentially orthogonal hexatriene units. Such a structure is incompatible with conventional conduction mechanisms involving polarons and bipolarons.