Development of AMREM: a Maxwell's equations solver with Adaptive Mesh Refinement

El principal objetivo de la presente tesis es el de desarrollar y probar un código capaz de resolver las ecuaciones de Maxwell en el dominio del tiempo con Malla Refinada Adaptativa (AMR por sus siglas en inglés). AMR es una técnica de cálculo basada en dividir el dominio físico del problema en distintas mallas rectangulares paralelas a las direcciones cartesianas. Cada una de las mallas tendrá distinta resolución y aquellas con mayor resolución se sitúan allí dónde las ondas electromagnéticas se propagan o interaccionan con los materiales, es decir, dónde mayor precisión es requerida. Como las ondas van desplazándose por todo el dominio, las mayas deberán seguirlas. El principal problema al utilizar esta metodología se puede encontrar en las fronteras internas, dónde las distintas mallas se unen. Ya que el método más corrientemente utilizado para resolver las ecuaciones de Maxwell es el de las diferencias finitas en el dominio del tiempo (FDTD por sus siglas en inglés) , el trabajo comenzó tratando de adaptar AMR a FDTD. Tras descubrirse que esta interacción resultaba en problemas de inestabilidades en las fronteras internas antes citadas, se decidió cambiar a un método basado en volúmenes finitos en el dominio del tiempo (FVTD por sus siglas en inglés). Este se basa en considerar la forma en ecuaciones de conservación de las ecuaciones de Maxwell y aplicar a su resolución un esquema de Godunov. Se ha probado que es clave para el correcto funcionamiento del código la elección de un limitador de flujo que proteja los extremos de la onda de la disipación típica de los métodos de este tipo. Otro problema clásico a la hora de resolver las ecuaciones de Maxwell es el de tratar con las condiciones de frontera física cuando se simulan dominios no acotados, es decir, dónde las ondas deben salir del sistema sin producir ninguna reflexión. Normalmente la solución es la de disponer una banda absorbente en las fronteras físicas. En AMREM se ha desarrollado un nuevo método basado en los campos característicos que con menor requisito de CPU funcina suficientemente bien incluso en los casos más desfaborables. El código ha sido contrastado con soluciones analíticas de diferentes problemas y también su velocidad ha sido comparada con la de Meep, uno de los programas más conocidos del ámbito. También algunas aplicaciones han sido simuladas con el fin de demostrar el amplio espectro de campos en los que AMREM puede funcionar como una útil herramienta.

Overtopping of harbour breakwaters: a comparison of semi-empirical equations, neural networks, and physical model tests

This paper reports extensive tests of empirical equations developed by different authors for harbour breakwater overtopping. First, the existing equations are compiled and evaluated as tools for estimating the overtopping rates on sloping and vertical breakwaters. These equations are then tested using the data obtained in a number of laboratory studies performed in the Centre for Harbours and Coastal Studies of the CEDEX, Spain. It was found that the recommended application ranges of the empirical equations typically deviate from those revealed in the experimental tests. In addition, a neural network model developed within the European CLASH Project is tested. The wind effects on overtopping are also assessed using a reduced scale physical model

Ricci flows on surfaces related to the Einstein weyl and Abelian vortex equations

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Hyers–Ulam stability of functional equations with a square-symmetric operation

The stability of the functional equation f(x ○ y) = H(f(x), f(y)) (x, y ∈ S) is investigated, where H is a homogeneous function and ○ is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.

Reduction in levels of the cyclin-dependent kinase inhibitor p27kip-1 coupled with transforming growth factor β neutralization induces cell-cycle entry and increases retroviral transduction of primitive human hematopoietic cells

Successful gene therapy depends on stable transduction of hematopoietic stem cells. Target cells must cycle to allow integration of Moloney-based retroviral vectors, yet hematopoietic stem cells are quiescent. Cells can be held in quiescence by intracellular cyclin-dependent kinase inhibitors. The cyclin-dependent kinase inhibitor p15INK4B blocks association of cyclin-dependent kinase (CDK)4/cyclin D and p27kip-1 blocks activity of CDK2/cyclin A and CDK2/cyclin E, complexes that are mandatory for cell-cycle progression. Antibody neutralization of β transforming growth factor (TGFβ) in serum-free medium decreased levels of p15INK4B and increased colony formation and retroviral-mediated transduction of primary human CD34+ cells. Although TGFβ neutralization increased colony formation from more primitive, noncycling hematopoietic progenitors, no increase in M-phase-dependent, retroviral-mediated transduction was observed. Transduction of the primitive cells was augmented by culture in the presence of antisense oligonucleotides to p27kip-1 coupled with TGFβ-neutralizing antibodies. The transduced cells engrafted immune-deficient mice with no alteration in human hematopoietic lineage development. We conclude that neutralization of TGFβ, plus reduction in levels of the cyclin-dependent kinase inhibitor p27, allows transduction of primitive and quiescent hematopoietic progenitor populations.

Cloning, expression, and characterization of a cDNA encoding a novel human growth factor for primitive hematopoietic progenitor cells

Multiple growth factors synergistically stimulate proliferation of primitive hematopoietic progenitor cells. A human myeloid cell line, KPB-M15, constitutively produces a novel hematopoietic cytokine, termed stem cell growth factor (SCGF), possessing species-specific proliferative activities. Here we report the molecular cloning, expression, and characterization of a cDNA encoding human SCGF using a newly developed λSHDM vector that is more efficient for differential and expression cloning. cDNA for SCGF encodes a 29-kDa polypeptide without N-linked glycosylation. SCGF transiently produced by COS-1 cells supports growth of hematopoietic progenitor cells through a short-term liquid culture of bone marrow cells and exhibits promoting activities on erythroid and granulocyte/macrophage progenitor cells in primary semisolid culture with erythropoietin and granulocyte/macrophage colony-stimulating factor, respectively. Expression of SCGF mRNA is restricted to myeloid cells and fibroblasts, suggesting that SCGF is a growth factor functioning within the hematopoietic microenvironment. SCGF could disclose some human-specific mechanisms as yet unidentified from studies on the murine hematopoietic system.

On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Purification of primitive human hematopoietic cells capable of repopulating immune-deficient mice

The purification of primitive human hematopoietic stem cells has been impaired by the absence of repopulation assays. By using a stringent two-step strategy involving depletion of lineage-positive cells followed by fluorescence-activated cell sorting, we have purified a cell population that is highly enriched for cells capable of multilineage repopulation in nonobese diabetic/severe combined immunodeficient (NOD/SCID) recipients. These SCID-repopulating cells (SRCs) were exclusively found in a cell fraction that expressed high levels of CD34 and no CD38. Through limiting dilution analysis using Poisson statistics, we calculated a frequency of 1 SRC in 617 CD34+ CD38− cells. The highly purified SRC were capable of extensive proliferation in NOD/SCID mice. Mice transplanted with 1 SRC (at limiting cell doses) were able to produce approximately 400,000 progeny 6 weeks after the transplant. Detailed flow cytometric analysis of the marrow of highly engrafted mice demonstrated both lymphoid and myeloid differentiation, as well as the retention of a significant fraction of CD34+ CD38− cells. These highly purified fractions should be useful for identification of the cellular and molecular mechanisms that regulate primitive human hematopoietic cells. Moreover, the ability to detect and purify primitive cells provides a means to develop conditions for maintaining and/or expanding these cells during in vitro culture.

Chimeric Cytokine Receptor Can Transduce Expansion Signals in Interleukin 6 Receptor α (IL-6Rα)-, IL-11Rα-, and gp130-low to -negative Primitive Hematopoietic Progenitors

We generated transgenic mice expressing chimeric receptors, which comprise extracellular domains of the human granulocyte–macrophage colony-stimulating factor (hGM-CSF) receptor and transmembrane and cytoplasmic domains of the mouse leukemia inhibitory factor receptor. In suspension cultures of lineage-negative (Lin−), 5-fluorouracil-resistant bone marrow cells of the transgenic mice, a combination of hGM-CSF and stem cell factor (SCF) induced exponential expansions of mixed colony-forming unit. The combination of hGM-CSF and SCF was effective on enriched, Lin−Sca-1+c-kit+ progenitors and increased either mixed colony-forming unit or cobblestone area–forming cells. In case of stimulation with hGM-CSF and SCF, interleukin-6 (IL-6) and SCF, or IL-11 and SCF, the most efficient expansion was achieved with hGM-CSF and SCF. When Lin−Sca-1+c-kit+CD34− further enriched progenitors were clone sorted and individually incubated in the presence of SCF, hGM-CSF stimulated a larger number of cells than did IL-6, IL-6 and soluble IL-6 receptor (IL-6R), or IL-11. These data suggest the presence of IL-6Rα-, IL-11Rα-, and gp130-low to -negative primitive hematopoietic progenitors. Such primitive progenitors are equipped with signal transduction molecules and can expand when these chimeric receptors are genetically introduced into the cells and stimulated with hGM-CSF in the presence of SCF.

In vitro and in vivo differentiation into B cells, T cells, and myeloid cells of primitive yolk sac hematopoietic precursor cells expanded >100-fold by coculture with a clonal yolk sac endothelial cell line

The yolk sac, first site of hematopoiesis during mammalian development, contains not only hematopoietic stem cells but also the earliest precursors of endothelial cells. We have previously shown that a nonadherent yolk sac cell population (WGA+, density <1.077, AA4.1+) can give rise to B cells, T cells, and myeloid cells both in vitro and in vivo. We now report on the ability of a yolk sac-derived cloned endothelial cell line (C166) to provide a suitable microenvironment for expansion of these early precursor cells. Single day 10 embryonic mouse yolk sac hematopoietic stem cells were expanded >100 fold within 8 days by coculture with irradiated C166 cells. Colony-forming ability was retained for at least three passages in vitro, with retention of the ability to differentiate into T-cell, B-cell, and myeloid lineages. Stem cell properties were maintained by a significant fraction of nonadherent cells in the third passage, although these stem cells expressed a somewhat more mature cell surface phenotype than the initial yolk sac stem cells. When reintroduced into adult allogeneic immunocompromised (scid) hosts, they were able to give rise to all of the leukocyte lineages, including T cells, B cells, and myeloid cells. We conclude that yolk sac endothelial cells can support the stable proliferation of multipotential hematopoietic stem cells, thus generating adequate numbers of cells for study of the mechanisms involved in their subsequent development and differentiation, for in vivo hematopoietic restitution, and for potential use as a vehicle for gene transfer.

Behavior of several two-dimensional fluid equations in singular scenarios

We give conditions that rule out formation of sharp fronts for certain two-dimensional incompressible flows. We show that a necessary condition of having a sharp front is that the flow has to have uncontrolled velocity growth. In the case of the quasi-geostrophic equation and two-dimensional Euler equation, we obtain estimates on the formation of semi-uniform fronts.

Self-renewal of primitive human hematopoietic cells (long-term-culture-initiating cells) in vitro and their expansion in defined medium.

A major goal of experimental and clinical hematology is the identification of mechanisms and conditions that support the expansion of transplantable hematopoietic stem cells. In normal marrow, such cells appear to be identical to (or represent a subset of) a population referred to as long-term-culture-initiating cells (LTC-ICs) so-named because of their ability to produce colony-forming cell (CFC) progeny for > or = 5 weeks when cocultured with stromal fibroblasts. Some expansion of LTC-ICs in vitro has recently been described, but identification of the factors required and whether LTC-IC self-renewal divisions are involved have remained unresolved issues. To address these issues, we examined the maintenance and/or generation of LTC-ICs from single CD34+ CD38- cells cultured for variable periods under different culture conditions. Analysis of the progeny obtained from cultures containing a feeder layer of murine fibroblasts engineered to produce steel factor, interleukin (IL)-3, and granulocyte colony-stimulating factor showed that approximately 20% of the input LTC-ICs (representing approximately 2% of the original CD34+ CD38- cells) executed self-renewal divisions within a 6-week period. Incubation of the same CD34+ CD38- starting populations as single cells in a defined (serum free) liquid medium supplemented with Flt-3 ligand, steel factor, IL-3, IL-6, granulocyte colony-stimulating factor, and nerve growth factor resulted in the proliferation of initial cells to produce clones of from 4 to 1000 cells within 10 days, approximately 40% of which included > or = 1 LTC-IC. In contrast, in similar cultures containing methylcellulose, input LTC-ICs appeared to persist but not divide. Overall the LTC-IC expansion in the liquid cultures was 30-fold in the first 10 days and 50-fold by the end of another 1-3 weeks. Documentation of human LTC-IC self-renewal in vitro and identification of defined conditions that permit their extensive and rapid amplification should facilitate analysis of the molecular mechanisms underlying these processes and their exploitation for a variety of therapeutic applications.