Sonic Limit Singularities in the Hodograph Method

In the hodograph transformation, introduced to linerize the equations governing the two-dimensional inviscid potential flow of a compressible fluid, there may appear so-called limit-points and limit-lines at which the Jacobian J = ∂(x,y)/ ∂(q,θ) of the transformation vanishes. This thesis investigate these singularities when they occur at points or segments of arc of the sonic line (Mach number unity). Assuming the streamfunction to be regular in the hodograph variables, it is show that sonic limit points cannot be isolated but must lie on a supersonic limit line or form a sonic limit line [cf. H. Geiringer, Math. Zeitschr., 63, (1956), 514-524]. Using this dichotomy a classification of sonic limit points is set up and certain geometrical properties of the mapping in the neighborhood of the singularity are discussed. In particular the general sonic limit line is shown to be an equipotential and an isovel; an envelope of both families of characteristics; and the locus of cusps of the streamlines and the isoclines. Flows containing sonic limit lines may be constructed by forming suitable linear combinations of the Chaplygin product solutions for any value of the separation constant n ≥ 0. For n less than a certain value n0 and greater than zero (n = 0 corresponds to the well-known radial flow), these flows represent a compressible analogue of the incompressible corner flows and may be envisaged as taking place on a quadruply-sheeted surface. The sheets are joined at a super-sonic limit line and at the sonic limit line which has the shape of a hypocycloid (n >1), cycloid (n = 1), or epicycloid (n <1). To exemplify the general behavior, the flows are constructed explicitly for n = 1/2, 1, and 2. The shape of the sonic limit line is also discussed when solutions corresponding to different n are superposed, and it is shown how then the supersonic limit line can be eliminated so that an isolated sonic limit line is obtained. A flow containing such an isolated sonic limit line is presented. An appendix derives the asymptotic solution for large values of n which corresponds to the sonic limit solution. The above results have been published in part in Math. Zeitschr., 67, (1957), 229-237. Other portions of this thesis will appear in two papers in Archive Rational Mech. and Anal., 2, (1958)

Eighty years of Sommerfeld's radiation condition

AbstractIn 1912 Sommerfeld introduced his radiation condition to ensure the uniqueness of the solution of certain exterior boundary value problems in mathematical physics. In physical applications these problems generally describe wave propagation where an incident time-harmonic wave is scattered by an object, and the resulting diffracted or scattered waves need to be calculated. When formulated mathematically, these problems usually take the form of an exterior Dirichlet or Neumann problem for the Helmholtz partial differential equation. The Sommerfeld condition is applied at infinity and, when added to the statement of the boundary value problem, singles out only the solution which represents “outgoing” (rather than “incoming” or “standing”) waves in the physical applications. Since its introduction, the Sommerfeld radiation condition has become indispensable for these types of problems and has stimulated a considerable amount of mathematical research, especially in uniqueness theorems. The present note traces the motivation and reasoning that led Sommerfeld to the original formulation of his radiation condition and surveys the extensions and modifications this condition has undergone since then

Chronic lymphocytic leukemia disease progression is accelerated by APRIL-TACI interaction in the TCL1 transgenic mouse model

Although in vitro studies pointed to the tumor necrosis factor family member APRIL (a proliferation-inducing ligand) in mediating survival of chronic lymphocytic leukemia (CLL) cells, clear evidence for a role in leukemogenesis and progression in CLL is lacking. APRIL significantly prolonged in vitro survival of CD5(+)B220(dull) leukemic cells derived from the murine E mu-TCL1-Tg (TCL1-Tg [transgenic]) model for CLL. APRIL-TCL1 double-Tg mice showed a significantly earlier onset of leukemia and disruption of splenic architecture, and survival was significantly reduced. Interestingly, clonal evolution of CD5(+) B220(dull) cells (judged by BCR clonality) did not seem to be accelerated by APRIL; both mouse strains were oligoclonal at 4 months. Although APRIL binds different receptors, APRIL-mediated leukemic cell survival depended on tumor necrosis factor receptor superfamily member 13B (TACI) ligation. These findings indicate that APRIL has an important role in CLL and that the APRIL-TACI interaction might be a selective novel therapeutic target for human CLL. (Blood. 2013;122(24):3960-3963