Natural gas liquid (NGL), a mixture consisting primarily of ethane, propane, and butane, is an excellent 17 enhanced oil recovery (EOR) solvent. However, NGL is typically about ten times less viscous than the 18 crude oil within the carbonate or sandstone porous media, which causes the NGL to finger through the 19 rock toward production wells resulting in low volumetric sweep efficiency in five-spot patterns or during a linear drive displacement. The viscosity of candidate polymeric NGL thickeners is measured with a windowed, close-clearance falling ball viscometer, and an expression for the average shear rate associated with this type of viscometer is derived. High molecular weight polydimethyl siloxane

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Introduction. We study some aspects of the question of the rationality of finite-dimensional representations of the hyperalgebra of an affine algebraic group scheme G. This is a question of Verma's*, according to [1].We will look at non-rational representations of the hyperalgebras of the additive and multiplicative groups. On the other hand, we show that for simply connected, connected semisimple groups at least the simple representations of the hyperalgebra are rational (see Section 5 for the precise results). Let G be a reduced affine algebraic group scheme over the prime field Fp of characteristic p, represented by the Hopf algebra A. Let M be the augmentation ideal of A, and suppose that G is connected in the sense that nM(P n)A= (0), where MPpn) =f{aPna E M}. The family {A/M(P )A}n>0 of finite dimensional Hopf algebras, together with the quotient morphisms A/M(Pn)A-)A/M(P )A for e < n, is a projective family. Definition. A is the projective limithhm Mp -A/M(P )A, n in the category of Hopf algebras (h lim is used to indicate the category in which the projective limit lies). Let G be the group scheme represented by A. G plays a role in the representation theory of the hyperalgebra of G. The dual family {(A/M(Pn)A)*}n>0 of Hopf algebras is an inductive family on a directed set.

The Salerno model is a discrete variant of the celebrated nonlinear Schrödinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz-Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions even at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.