Physical Sciences and Mathematics Commons^™

Unique Pseudo-Expectations For C∗-Inclusions, David R. Pitts, Vrej Zarikian

Department of Mathematics: Faculty Publications

Given an inclusion D⊆C of unital C ∗ -algebras (with common unit), a unital completely positive linear map Φ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set PsExp(C,D) of all pseudo-expectations is a convex set, and when D is Abelian, we prove a Krein–Milman type theorem showing that PsExp(C,D) can be recovered from its set of extreme points. In general, PsExp(C,D) is not a singleton. However, there are large and natural classes …

Comparison Theorems And Asymptotic Behavior Of Solutions Of Discrete Fractional Equations, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

Consider the following n-th order nabla and delta fractional difference equations

rn r (a)x(t) = c(t)x(t), t 2 Na+1, x(a) > 0.

and

Va+v-1x(t) = c(t)x(t + v - 1), t 2 Na, x(a + n - 1) > 0

We establish comparison theorems by which we compare the solutions x(t) of (*) and (**) with the solutions of the equations rn r(a)x(t) = bx(t) and Dn a+v-1x(t) = bx(t + v -1), respectively, where b is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011), …

Some Relations Between The Caputo Fractional Difference Operators And Integer-Order Differences, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

In this article, we are concerned with the relationships between the sign of Caputo fractional differences and integer nabla differences. In particular, we show that if N -1 < v < N. f : Na -N + 1 -> R, va * f(t) > O, for t - Na +1 and N-1f(a) > 0, then N -1 f(t) > 0 for t- Na +1, then va* f(t) > 0, for each t - Na +1. As applications of these two results, we get that if 1 < vR, va*f(t) > 0 for t - Na +1 and f(a) > f(a-1), then f(t) is an increasing function for t- Na -1. Conversely if 0 < vR and f is an increasing function for t - Na, then va*f(t) > 0, for each t - Na +1. …

Clique Topology Reveals Intrinsic Geometric Structure In Neural Correlations, Chad Giusti, Eva Pastalkova, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a 2D environment, and confirmed that our method is able to detect geometric organization using …

Toric Varieties, Monoid Schemes And Cdh Descent, Guillermo Cortiñas, C. Haesemeyer, Mark E. Walker, Charles Weibel

Department of Mathematics: Faculty Publications

We give conditions for the Mayer–Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.

The Interplay Between Wnt Mediated Expansion And Negative Regulation Of Growth Promotes Robust Intestinal Crypt Structure And Homeostasis, Huijing Du, Qing Nie, William R. Holmes

Department of Mathematics: Faculty Publications

The epithelium of the small intestinal crypt, which has a vital role in protecting the underlying tissue from the harsh intestinal environment, is completely renewed every 4–5 days by a small pool of stem cells at the base of each crypt. How is this renewal controlled and homeostasis maintained, particularly given the rapid nature of this process? Here, based on the recent observations from in vitro “mini gut” studies, we use a hybrid stochastic model of the crypt to investigate how exogenous niche signaling (from Wnt and BMP) combines with auto-regulation to promote homeostasis. This model builds on the sub-cellular …