# A Subprime Lending Market Primer

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Let f: M m â†’ â„ m+1 be an immersion of an orientable m-dimensional connected smooth manifold M without boundary and assume that Î¾ is a unit normal field for f. For a real number t the map f tÎ¾: M m â†’ â„ m+1 is defined as f tÎ¾( p) = f( p) + tÎ¾( p). It is known...

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We consider solutions of the class of ODEs yâ€³ = 6y2 - xÎ¼, which contains the first PainlevÃ© equation (PI) for Î¼ = 1. It is well known that PI has a unique real solution (called a tritronquÃ©e solution) asymptotic to $-\backslash sqrt\{x/6\}$ and decaying monotonically on the...

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In this paper we consider a problem of investigating the dependence of Ç P (Rz) - Î²P (rz)Çp on ÇP(z)Çp for every real or complex number Î² with âˆ£Î²âˆ£â‰¥â‰¤1, R > r â‰¥1, p > 0 and present certain compact generalizations which, besides yielding some...

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The article discusses the power slide, which was a name coined by the author as the graph of f(x) resembles that of a children's slide. To come up with a power slide, one has to think of a real number that can be made into a sequence, which can be considered as a sequence of functions that can...

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Let P(z) be a polynomial of degree n having no zeros in |z| < k where k â‰¥ 1. Then it is known that for every real or complex number Î± with |Î±| â‰¥ 1, max |z|=1 |DÎ±P(z)| â‰¤ n (|Î±| + k/1 + k) max |z|=1 |P(z)|, where DÎ±P(z) = nP(z) + (Î± - z)P' (z) denotes the polar...

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We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: 1. The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong...

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We introduce the notion of F-parametrizable model and prove some general results on elementary submodels of F-parametrizable models. Using this notion, we can uniformly characterize all elementary submodels for the field of real numbers and for the group of all permutations on natural numbers in...

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We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself.

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The article classifies the various types of real numbers. A figure is presented showing the proportions of real numbers including irrationals, algebraic and transcendentals. It provides illustrations across the field of real numbers showing the location of some well-known numbers and sums of...