Mathematics

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    Completions
    (1964) Nielsen, Robert Maurice; Brace, John W.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)
    This paper presents a new approach to the theory of completions. The treatment is based on the concept of convergence on filters and related topologies. For a given uniform Hausdorff space Xu and a collection S of Cauchy filters in Xu, the basic result is the construction of a uniform Hausdorff space. Xu having the properties that Xu is isomorphic to a dense subspace of Xu and every filter in S converges to a point in S. As a special case, the completion of Xu of Xu is obtained. The construction is so given as to prove the existence of the space Xu. The technique involves embedding the object X to be "completed" in a space of functions F which has as its domain a space of continuous functions C(X) defined on X. The procedure is analogous to the process of taking the bidual E" of a locally convex topological vector space. Indeed, E" is obtained as a special case. In the absence of sufficient structure on X, the Xu is obtained as the closure of X in F. In a locally convex space or an abelian topological group having enough character to separate points, Xu is obtained as a bidual or a second character group of the object X.
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    Sonic Limit Singularities in the Hodograph Method
    (1958) Schot, Steven H.; Ludford, Geoffrey S.S.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)
    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).
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    A COMBINATORIAL REPRESENTATION FOR ORIENTED POLYHEDRAL SURFACES
    (1960) Edmonds, John Robert Jr; Reinhart, Bruce; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)
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    The Axiom of Choice for Collections of Finite Sets
    (1969) Gauntt, Robert James; Karp, Carol R.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)
    Some implications among finite versions of the Axiom of Choice are considered. In the first of two chapters some theorems are proven concerning the dependence or independence of these implications on the theory ZFU, the modification of ZF which permits the existence of atoms. The second chapter outlines proofs of corresponding theorems with "ZFU" replaced by "ZF" . The independence proofs involve Mostowski type permutation models in the first chapter and Cohen forcing in the second chapter. The finite axioms considered are C^n , "Every collection of n-element sets has a choice function"; W^n, "Every well-orderable collection of n-element sets has a choice function"; D^n, "Every denumerable collection of n-element sets has a choice function"; and A^n (x), "Every collection Y of n-element sets, with Y ≈ X, has a choice function". The conjunction C^nl &...& C^nk is denoted by CZ where Z = {nl ,...,nk}. Corresponding conjunctions of other finite axioms are denoted similarly by Wz, Dz and Az (X). Theorem: The following are provable in ZFU: W^k1n1+...+krnr ➔ W^n1 v...v W^nr, D^k1n1+...+krnr ➔ D^n1 v...v D^nr, and C^k1n1+...+krnr ➔ C^n1 v W^n2 v...v W^nr
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    An Historical and Critical Development of the Theory of Legendre Polynomials Before 1900
    (1938) Laden, Hyman N.; Lancaster, O.E.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md)