A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

The hardnesses of poly(methylmethacrylate)

The paper describes an experimental study of the normal and scratch hardnesses of a poly(methylmethacrylate). The deformations have been introduced using hard steel cones of a range of included cone angles. The influence of the state of interfacial lubrication is examined and rationalized. The observed time dependence of the two types of computed hardness data is compared and the nature of the correlations between these data is evaluated. It is observed that when the imposed strains are modest, say less than 0.2, the scratch hardness and normal hardness deformations produce self consistent data using first order and rather indiscriminate analyses for both types of deformations. At higher levels of imposed strain, a more critical appraisal of the nature of the deformation produced in the two cases is necessary in order to provide mutually consistent hardness values and hence unequivocal rheological characteristics for this polymer.

Shock tunnel study of spiked aerodynamic bodies flying at hypersonic Mach numbers

A three-component accelerometer balance system is used to study the drag reduction effect of an aerodisc on large angle blunt cones flying at hypersonic Mach numbers. Measurements in a hypersonic shock tunnel at a freestream Mach number of 5.75 indicate more than 50% reduction in the drag coefficient for a 120degrees apex angle blunt cone with a forward facing aerospike having a flat faced aerodisc at moderate angles of attack. Enhancement of drag has been observed for higher angles of attack due to the impingement of the flow separation shock on the windward side of the cone. The flowfields around the large angle blunt cone with aerospike assembly flying at hypersonic Mach numbers are also simulated numerically using a commercial CFD code. The pressure and density levels on the model surface, which is under the aerodynamic shadow of the flat disc tipped spike, are found very low and a drag reduction of 64.34% has been deduced numerically.

On an oscillatory point force in a rotating stratified fluid

The forced oscillations due to a point forcing effect in an infinite or contained, inviscid, incompressible, rotating, stratified fluid are investigated taking into account the density variation in the inertia terms in the linearized equations of motion. The solutions are obtained in closed form using generalized Fourier transforms. Solutions are presented for a medium bounded by a finite cylinder when the oscillatory forcing effect is acting at a point on the axis of the cylinder. In both the unbounded and bounded case, there exist characteristic cones emanating from the point of application of the force on which either the pressure or its derivatives are discontinuous. The perfect resonance existing at certain frequencies in an unbounded or bounded homogeneous fluid is avoided in the case of a confined stratified fluid.

Linearization of Drucker-Prager yield criterion for axisymmetric problems: implementation in lower-bound limit analysis

The linearization of the Drucker-Prager yield criterion associated with an axisymmetric problem has been achieved by simulating a sphere with the truncated icosahedron with 32 faces and 60 vertices. On this basis, a numerical formulation has been proposed for solving an axisymmetric stability problem with the usage of the lower-bound limit analysis, finite elements, and linear optimization. To compare the results, the linearization of the Mohr-Coulomb yield criterion, by replacing the three cones with interior polyhedron, as proposed earlier by Pastor and Turgeman for an axisymmetric problem, has also been implemented. The two formulations have been applied for determining the collapse loads for a circular footing resting on a cohesive-friction material with nonzero unit weight. The computational results are found to be quite convincing. (C) 2013 American Society of Civil Engineers.

Lower bounds on Ricci flow invariant curvatures and geometric applications

We consider Ricci flow invariant cones C in the space of curvature operators lying between the cones ``nonnegative Ricci curvature'' and ``nonnegative curvature operator''. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies R + epsilon I is an element of C at the initial time, then it satisfies R + epsilon I is an element of C on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I is an element of C along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in C. Finally, we study the case where C is contained in the cone of operators whose sectional curvature is nonnegative. This allows us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.

Laser microstructuring of Si surfaces for low-threshold field-electron emission

Dense arrays of high aspect ratio Si micro-pyramids have been formed by cumulative high intensity laser irradiation of doped Si wafers in an SF6 environment. A comparative study using nanosecond (XeCl, 308 nm) and femtosecond (Ti: Sapphire, 800 nm and KrF, 248 nm) laser pulses has been performed in this work. The influence of pulse duration and ambient gas pressure (SF6) is also presented. Scanning electron microscopy has shown that upon laser irradiation conical features appear on the Si surface in a rather homogenous distribution and with a spontaneous self alignment into arrays. Their lowest tip diameter is 800 nm; while their height reaches up to 90 mum. Secondary tip decoration appears on the surface of the formed spikes. Areas of 2 X 2 mm(2) covered with Si cones have been tested as cold cathode field emitters. After several conditioning cycles, the field emission threshold for the studied Si tips is as low as 2 V/mum, with an emission current of 10(-3) A/cm(2) at 4 V/mum. Even though these structures have smaller aspect ratios than good quality carbon nanotubes, their field emission properties are similar. The simple and direct formation of field emission Si arrays over small pre-selected areas by laser irradiation could lead to a novel approach for the development of electron sources. (C) 2003 Elsevier B.V. All rights reserved.

The Effect of Conical Flowfields on the Performance of Waveriders at Mach 6

The performance of 23 kinds of waveriders, derived from different conical flowfields, is analyzed by the numerical computation under the conditions of fight speed of Mach 6, attack angle of 0° and flight altitude of 30 km. These results indicate that the performance is influenced by the shapes and the width to height ratios (W/H ) of generating cones. The geometrical parameter and the lift coefficient are proportional to W/H, while the drag coefficient and the lift to drag ratio (L/D ) have extreme values. Considering the base drag and the computation errors, the waverider with the highest L/D is cut from the elliptical cone’s flowfield (W/H = 1.5―1.618), and the configuration with the lowest drag can also be obtained at W/H = 1:1.5. Accordingly, good suggestions are proposed for practical design based on these computational results.

A possible alchemist apparatus from the early islamic period excavated at Ramla, Israel

Abstract: During extensive salvage excavations carried out during the years 2004-2008 in a large Early Islamic industrial area at the vicinities of Ramla, in Central Israel, an unparalleled industrial device was unearthed. The star-shaped, soil embedded installation, whose lower part was preserved, consisted of a central pottery jar surrounded by five minor jars, linked by ceramic pipes. Evidence of heat was observed mainly around the central vessel, and metal hollow cones perforated in the tip were found inside the surrounding jars. Although the manufacturing procedures and operation techniques of the installation are not completely clear, it is proposed that the installation is part of an industrial workshop or an alchemy laboratory. Both industry and alchemy were well-developed during the Early Islamic period and very often closely related, to the point that sometimes it is difficult to distinguish between them. The identification proposed is based on comparisons with tools described in literary sources, and somewhat later drawings and etchings. Circumstantial ceramic evidence was found, as well as the proximity of a bathhouse whose guests could have been the consumers of perfumes and unguents seem to reinforce this possibility. Due to the poor state of preservation of the device and the lack of available comparisons, the identification proposed here is tentative, and future research coupled with eventual new discoveries is needed in order to clarify this matter.

Waverider configurations derived from general conical flowfields

A method based on the computational fluid dynamics (CFD) is presented for a flexible waverider's design. The generating bodies of this method could be any cones. In addition, either the leading edge or the profile of the scramjet's inlet is used as the waverider's definition curve, parameterized by the quadric function, the sigmoid function or the B-spline function. Furthermore, several numerical examples are carried out to validate the method and the relevant codes. The CFD results of the configurations show that all the designs are successful. Moreover, primary suggestions are proposed for practical design by comparing the geometrical and aerodynamic performances of the cone-derived waveriders at Mach 6.

A method for age determination of dolphins.

The use of growth layers in teeth as an indicator of age in odnotocetes and pinnipeds was suggested by Laws (1954) and since then the method has been used extensively in both marine and non-marine mammals. Dentinal growth layers are groups (growth layer groups) of repetitive alternating bands which in cross-section are similar to growth rings in trees. The most commonly used methods for counting growth layer groups (GLGs) are by undecalcified longitudinal thin sections (150 um) or decalcified and stained thin sections (10-30 um). In longitudinal sections viewed with light microscopy, GLGs appear as opaque and translucent cones nestled one inside another, with the oldest dentine Iying adjacent to the enamel, and the newest layer borderinq the pulp cavity.

Evaluation of salmon and steelhead spawning habitat quality in the South Fork Trinity River Basin, 1997

Sediment sampling was used to evaluate chinook salmon (Oncorhynchus tshawytscha) and steelhead (O. mykiss) spawning habitat quality in the South Fork Trinity River (SFTR) basin. Sediment samples were collected using a McNeil-type sampler and wet sieved through a series of Tyler screens (25.00 mm, 12.50 mm, 6.30 mm, 3.35 mm, 1.00 mm, and 0.85 mm). Fines (particles < 0.85 mm) were determined after a l0-minute settling period in Imhoff cones. Thirteen stations were sampled in the SFTR basin: five stations were located in mainstem SFTR between rk 2.1 and 118.5, 2 stations each were located in EF of the SFTR, Grouse Creek, and Madden Creek, and one station each was located in Eltapom and Hayfork Creeks. Sample means for fines(particles < 0.85 mm) fer SFTR stations ranged between 14.4 and 19.4%; tributary station sample mean fines ranged between 3.4 and 19.4%. Decreased egg survival would be expected at 4 of 5 mainstem SFTR stations and at one station in EF of SFTR and Grouse Creek where fines content exceed 15%. Small gravel/sand content measured at all stations were high, and exceed levels associated with reduced sac fry emergence rates. Reduction of egg survival or sac fry emergence due to sedimentation in spawning gravels could lead to reduced juvenile production from the South Fork Trinity River. (PDF contains 18 pages.)

Effect of pressure on mechanical behavior of filled elastomers

The Young's modulus, stress-strain curves, and failure properties of glass bead-filled EPDM vulcanizates were studied under superposed hydrostatic pressure. The glass bead-filled EPDM was employed as a representation of composite systems, and the hydrostatic pressure controls the filler-elastomer separation under deformation. This separation shows up as a volume change of the system, and its infuence is reflected in the mechanical behavior as a reinforcing effect of variable degree.

The strain energy stored in the composite system in simple tension was calculated by introducing a model which is described as a cylindrical block of elastomer with two half spheres of filler on each end with their centers on the axis of the cylinder. In the derivation of the strain energy, assumptions were made to obtain the strain distribution in the model, and strain energy-strain relation for the elastomer was also assumed. The derivation was carried out for the case of no filler-elastomer separation and was modified to include the case of filler-elastomer separation.

The resulting strain energy, as a function of stretch ratio and volume of the system, was used to obtain stress-strain curves and volume change-strain curves of composite systems under superposed hydrostatic pressure.

Changes in the force and the lateral dimension of a ring specimen were measured as it was stretched axially under a superposed hydrostatic pressure in order to calculate the mechanical properties mentioned above. A tensile tester was used which is capable of sealing the whole system to carry out a measurement under pressure. A thickness measuring device, based on the Hall effect, was built for the measurement of changes in the lateral dimension of a specimen.

The theoretical and experimental results of Young's modulus and stress-strain curves were compared and showed fairly good agreement.

The failure data were discussed in terms of failure surfaces, and it was concluded that a failure surface of the glass-bead-filled EPDM consists of two cones.

An investigation of detached shock waves

This investigation demonstrates an application of a flexible wall nozzle for testing in a supersonic wind tunnel. It is conservative to say that the versatility of this nozzle is such that it warrants the expenditure of time to carefully engineer a nozzle and incorporate it in the wind tunnel as a permanent part of the system. The gradients in the test section were kept within one percent of the calibrated Mach number, however, the gradients occurring over the bodies tested were only ± 0.2 percent in Mach number.

The conditions existing on a finite cone with a vertex angle of 75° were investigated by considering the pressure distribution on the cone and the shape of the shock wave. The pressure distribution on the surface of the 75° cone when based on upstream conditions does not show any discontinuities at the theoretical attachment Mach number.

Both the angle of the shock wave and the pressure distribution of the 75° cone are in very close agreement with the theoretical values given in the Kopal report, (Ref. 3).

The location of the intersection of the sonic line with the surface of the cone and with the shock wave are given for the cone. The blocking characteristics of the GALCIT supersonic wind tunnel were investigated with a series of 60° cones.

Stability of Hypervelocity Boundary Layers

The early stage of laminar-turbulent transition in a hypervelocity boundary layer is studied using a combination of modal linear stability analysis, transient growth analysis, and direct numerical simulation. Modal stability analysis is used to clarify the behavior of first and second mode instabilities on flat plates and sharp cones for a wide range of high enthalpy flow conditions relevant to experiments in impulse facilities. Vibrational nonequilibrium is included in this analysis, its influence on the stability properties is investigated, and simple models for predicting when it is important are described.

Transient growth analysis is used to determine the optimal initial conditions that lead to the largest possible energy amplification within the flow. Such analysis is performed for both spatially and temporally evolving disturbances. The analysis again targets flows that have large stagnation enthalpy, such as those found in shock tunnels, expansion tubes, and atmospheric flight at high Mach numbers, and clarifies the effects of Mach number and wall temperature on the amplification achieved. Direct comparisons between modal and non-modal growth are made to determine the relative importance of these mechanisms under different flow regimes.

Conventional stability analysis employs the assumption that disturbances evolve with either a fixed frequency (spatial analysis) or a fixed wavenumber (temporal analysis). Direct numerical simulations are employed to relax these assumptions and investigate the downstream propagation of wave packets that are localized in space and time, and hence contain a distribution of frequencies and wavenumbers. Such wave packets are commonly observed in experiments and hence their amplification is highly relevant to boundary layer transition prediction. It is demonstrated that such localized wave packets experience much less growth than is predicted by spatial stability analysis, and therefore it is essential that the bandwidth of localized noise sources that excite the instability be taken into account in making transition estimates. A simple model based on linear stability theory is also developed which yields comparable results with an enormous reduction in computational expense. This enables the amplification of finite-width wave packets to be taken into account in transition prediction.