Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure

Combustion of liquid sprays at high pressures

The combustion of pressure atomized fuel sprays in high pressure stagnant air was studied. Measurements were made of flame and spray boundaries at pressures in the range 0.1-9 MPa for methanol and n-pentane. At the higher test pressure levels, critical phenomena are important. The experiments are compared with theoretical predictions based on a locally homogeneous two-phase flow model. The theory correctly predicted the trends of the data, but underestimates flame and spray boundaries by 30-50 percent, indicating that slip is still important for the present experiments (Sauter mean diameters of 30 microns at atmospheric pressure under cold flow conditions). Since the sprays are shorter at high pressures, slip effects are still important even though the density ratio of the phases approach one another as the droplets heat up. The model indicates the presence of a region where condensed water is present within the spray and provides a convenient means of treating supercritical phenomena

CELLULAR DIFFERENTIATION OF THE IMMUNE SYSTEM OF MICE : III. SEPARATE ANTIGEN-SENSITIVE UNITS FOR DIFFERENT TYPES OF ANTI-SHEEP IMMUNOCYTES FORMED BY MARROW-THYMUS CELL MIXTURES

Marrow cell suspensions of unprimed donor mice have been transplanted into X-irradiated syngeneic hosts. 5–46 days later, bone cavities and spleens contained regenerated cells of the immune system which required interaction with thymocytes (from intact donors) and antigen (SRBC) to form antigen-sensitive units (ASU) and to generate mature immunocytes. These cells were capable of differentiating either into direct or indirect hemolytic plaque-forming cells (PFC). The precursors of PFC regenerated earlier than the other cell type necessary for immunocompetence, the antigen-reactive cell (ARC). The latter was not found until 10 or more days after transplantation. Availability of ARC was inferred from PFC responses elicited by grafted mice challenged with SRBC at varying intervals. In a second series of experiments, graded numbers of marrow cells (ranging from 107 to 5 x 107) were transplanted with 5 x 107 or 108 thymocytes into irradiated mice, and SRBC were given 18 hr later. After 9–12 days the recipient spleens contained all or some of the following immunocytes: direct and indirect PFC, and hemagglutinating cluster-forming cells. The frequency of each immune response varied independently of the others, but in relation to the number of marrow cells grafted. This was interpreted to indicate that ASU formed in irradiated mice by interaction of marrow and thymus cells were similar to those of intact mice. In particular, they were specialized for the molecular class (IgM or IgG) and function (lysis or agglutination) of the antibody to be secreted by their descendent immunocytes. Hence, class-differentiation appeared to be conferred upon ASU by their marrow-derived components

DISTINCT EVENTS IN THE IMMUNE RESPONSE ELICITED BY TRANSFERRED MARROW AND THYMUS CELLS : I. ANTIGEN REQUIREMENTS AND PROLIFERATION OF THYMIC ANTIGEN-REACTIVE CELLS

Marrow cells and thymocytes of unprimed donor mice were transplanted separately into X-irradiated syngeneic hosts, with or without sheep erythrocytes (SRBC). Antigen-dependent changes in number or function of potentially immunocompetent cells were assessed by retransplantation of thymus-derived cells with fresh bone marrow cells and SRBC; of marrow-derived cells with fresh thymocytes and SRBC; and of thymus-derived with marrow-derived cells and SRBC. Plaque-forming cells (PFC) of the direct (IgM) and indirect (IgG) classes were enumerated in spleens of secondary host mice at the time of peak responses. By using this two-step design, it was shown (a) that thymus, but not bone marrow, contained antigen-reactive cells (ARC) capable of initiating the immune response to SRBC (first step), and (b) that the same antigen complex that activated thymic ARC was required for the subsequent interaction between thymus-derived and marrow cells and/or for PFC production (second step). Thymic ARC separated from marrow cells but exposed to SRBC proliferated and generated specific inducer cells. These were the cells that interacted with marrow precursors of PFC to form the elementary units for plaque responses to SRBC, i.e. the class- and specificity-restricted antigen-sensitive units. It was estimated that each ARC generated 80–800 inducer cells in 4 days by way of a minimum of 6–10 cell divisions. On the basis of the available evidence, a simple model was outlined for cellular events in the immune response to SRBC

Bulk commodities and the Liverpool and London markets of the mid-19th century

We study British prices and the degree of commodity market integration between Liverpool, the bulk commodity port of mid-19th century, and London. A new wholesale commodity price index is presented for Liverpool and this is compared with the Klovland-Sauerbeck index. Next, we examine the relationship between Liverpool and London markets in specific bulk commodities. Our data consist of price indices for identically described goods in both Liverpool and London: three commodity groups (metal products, wood products, and processed foods), and the specific commodities of wheat and flour. Tests for cointegration reveal convergence among the six price pairs. We also find that the markets were highly integrated in the short-run because three of the commodity group pairs (processed foods, wheat, and flour) shared common features or cycles. A common cycle implies that transitory price shocks in Liverpool had the same impact on prices in London and vice versa. The importance of the London and Liverpool common cycle to a shock is brief. Its shock explains less than 20 percent of the variation in the relevant price levels after twelve months, on average.

Investigation of Critical Burning of Fuel Droplets

An earlier analysis for the combustion response of a liquid monopropellant strand (hydrazine) was extended to consider individual droplets and sprays. While small drops gave low or negative response, large droplets provided response near unity at low frequencies, with the response declining at frequencies greater than the characteristic liquid phase frequency. Temperature gradients in the liquid phase resulted in response peaks greater than unity. A second response peak was found for large drops which corresponded to gas phase transient effects. Spray response was generally reduced from the response of the largest injected droplet, however, even a small percentage of large droplets can yield appreciable response. An apparatus was designed and fabricated to allow observation of bipropellant fuel spray combustion at elevated pressures. A locally homogeneous model was developed to describe this combustion process which allows for high pressure phenomena associated with the thermodynamic critical point

CONTRIBUTION OF BONE MARROW CELLS AND LACK OF EXPRESSION OF THYMOCYTES IN GENETIC CONTROLS OF IMMUNE RESPONSES FOR TWO IMMUNOPOTENT REGIONS WITHIN POLY-(PHE,GLU)-POLY-PRO--POLY-LYS IN INBRED MOUSE STRAINS

Previous cellular studies on the genetic regulation of immunological responsiveness for two immunopotent regions within the branched chain synthetic polypeptide (Phe, G)-Pro--L demonstrated a direct correlation between the number of detectable immunocompetent splenic precursor cells and the response patterns of SJL, DBA/1, and F1 mice (21). In order to establish the cellular origin(s) of the genetic defect, the present study first demonstrated that thymus and bone marrow cell cooperation was required for (Phe, G)- and Pro--L-specific immune responses. Secondly, limiting dilution experiments, in which several graded and limiting inocula of marrow cells were mixed with a non-limiting number of 108 thymocytes and injected into irradiated, syngeneic recipients, indicated that the low responsiveness of the SJL and DBA/1 strains to the (Phe, G) and Pro--L specificities, respectively, could be attributed to a reduced number of precursor cells found in bone marrow. About five times more marrow precursors were detected in SJL mice for Pro--L than for (Phe, G), whereas about five times as many precursor cells were estimated for (Phe, G) as for Pro--L in the DBA/1 strain. These differences are similar to those obtained using spleen cells from unimmunized SJL and DBA/1 donors (21), and indicate that these genetically determined variations in responsiveness can be accounted for by differences in the frequencies of monospecific populations of immunocompetent cells present in bone marrow. In contrast, limiting dilution transfers of thymocytes or thymus-derived cells with an excess of syngeneic marrow cells resulted in equally frequent (Phe, G) and Pro--L responses for both SJL ad DBA/1 strains. This finding in conjunction with the observation that the generation of (Phe, G)- and Pro--L-specific responses were associated in individual recipients injected with limiting inocula of thymocytes indicated that a single population of thymocytes was stimulated by (Phe,G)-Pro--L. Therefore, it is improbable that the thymic population of immunocompetent cells contributes to expression of these genetically controlled defects

Unpulsed UBV Optical Emission from the Crab Pulsar

Based on observations of the Crab pulsar using the TRIFFID high speed imaging photometer in the UBV bands using the Special Astrophysical Observatory's 6m telescope in the Russian Caucasus, we report the detection of pronounced emission during the so-called `off' phase of emission. Following de-extinction, this unpulsed component of emission is shown to be consistent with a power law with an exponent of alpha = -0.60 +/- 0.37, the uncertainty being dominated by the error associated with the independent CCD photometry used to reference the TRIFFID data. This suggests a steeper power law form than that reported elsewhere in the literature for the total integrated spectrum, which is essentially flat with alpha ~ 0.1, although the difference in this case is only significant at the ~ 2 sigma level. Deeper reference integrated and TRIFFID phase-resolved photometry in these bands in conjunction with further observations in the UV and R region would constrain this fit further.Comment: 26 pages, 2 figures, uses aasms4.sty, accepted for publication in the Astrophysical Journa

CELLULAR DIFFERENTIATION OF THE IMMUNE SYSTEM OF MICE : VI. STRAIN DIFFERENCES IN CLASS DIFFERENTIATION AND OTHER PROPERTIES OF MARROW CELLS

Marrow cells and 5 x 107 thymocytes of unprimed (C57BL/6 x DBA/2)F1, (C57BL/10 x WB)F1 and (C3H x C57BL)F1 donor mice were mixed in vitro and transplanted into X-irradiated syngeneic hosts. Upon injection of sheep erythrocytes, splenic plaque-forming cells (PFC) secreting IgM (direct PFC or IgG (indirect PFC) hemolytic antibody were enumerated at the time of peak responses. By grading the numbers of marrow cells, inocula were found that contained few immunocompetent cells reaching the recipient spleens, interacting with thymocytes or other accessory cells (or both), and generating PFC. The frequency of responses in BDF1 mice conformed to Poisson statistics, indicating that immunocompetent marrow cells participated in a single-hit interaction limiting PFC responses. The marrow cells assayed were not restricted for the antibody class (IgM versus IgG) to be secreted by mature PFC. Unrestricted marrow cells could have been either the precursors of PFC or accessory cells. Different results were obtained in BWF1 and C3BF1 mice. The frequency of responses in relation to the number of marrow cells grafted did not follow Poisson statistics, and the limiting cells were restricted for antibody class. Presumably, immunocompetent cells of these strains were more heterogeneous than those of BDF1 mice and participated in a multiplicity of cell-to-cell interactions. The strain differences reflected inherent properties of marrow cells and not influences of the environment in which PFC were produced. The results confirmed for bone marrow the heterogeneity of immunocompetent cells reported by others for spleen, and suggested that genetic factors such as "immune response" genes regulate cellular differentiation also for functions other than those related to antibody specificity