By: Howard S. Cohl, Roberto S. Costas-Santos

We derive double product representations of nonterminating basic
hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We also present some generating functions that arise from it in the $q$ and $q$-inverse Askey schemes. Using this $q$-Chaundy theorem which expresses a product of two nonterminating basic hypergeometric
series as a sum over a terminating basic hypergeometric series, we study gener... more

We derive double product representations of nonterminating basic
hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We also present some generating functions that arise from it in the $q$ and $q$-inverse Askey schemes. Using this $q$-Chaundy theorem which expresses a product of two nonterminating basic hypergeometric
series as a sum over a terminating basic hypergeometric series, we study generating functions for the symmetric families of orthogonal polynomials in the $q$ and $q$-inverse Askey scheme. By applying the $q$-Chaundy theorem to $q$-exponential generating functions due to Ismail, we are able to derive alternative expansions of these generating functions and from these, new representations for the continuous $q$-Hermite and $q$-inverse Hermite polynomials which are connected by a quadratic transformation for the terminating basic hypergeometric series representations.
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By: David E. Evans, Ulrich Pennig

We develop an equivariant Dixmier-Douady theory for locally trivial bundles
of $C^*$-algebras with fibre $D \otimes \mathbb{K}$ equipped with a fibrewise
$\mathbb{T}$-action, where $\mathbb{T}$ denotes the circle group and $D =
\operatorname{End}\left(V\right)^{\otimes \infty}$ for a
$\mathbb{T}$-representation $V$. In particular, we show that the group of
$\mathbb{T}$-equivariant $*$-automorphisms $\operatorname{Aut}_{\mathbb{T}}(D
\otimes... more

We develop an equivariant Dixmier-Douady theory for locally trivial bundles
of $C^*$-algebras with fibre $D \otimes \mathbb{K}$ equipped with a fibrewise
$\mathbb{T}$-action, where $\mathbb{T}$ denotes the circle group and $D =
\operatorname{End}\left(V\right)^{\otimes \infty}$ for a
$\mathbb{T}$-representation $V$. In particular, we show that the group of
$\mathbb{T}$-equivariant $*$-automorphisms $\operatorname{Aut}_{\mathbb{T}}(D
\otimes \mathbb{K})$ is an infinite loop space giving rise to a cohomology
theory $E^*_{D,\mathbb{T}}(X)$. Isomorphism classes of equivariant bundles then
form a group with respect to the fibrewise tensor product that is isomorphic to
$E^1_{D,\mathbb{T}}(X) \cong [X, B\operatorname{Aut}_{\mathbb{T}}(D \otimes
\mathbb{K})]$. We compute this group for tori and compare the case $D =
\mathbb{C}$ to the equivariant Brauer group for trivial actions on the base
space.
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By: Shady E Ahmed, Omer San, Sivaramakrishnan Lakshmivarahan, John M Lewis

The four-dimensional variational data assimilation methodology for
assimilating noisy observations into a deterministic model has been the
workhorse of forecasting centers for over three decades. While this method
provides a computationally efficient framework for dynamic data assimilation,
it is largely silent on the important question concerning the minimum number
and placement of observations. To answer this question, we demonstrate the ... more

The four-dimensional variational data assimilation methodology for
assimilating noisy observations into a deterministic model has been the
workhorse of forecasting centers for over three decades. While this method
provides a computationally efficient framework for dynamic data assimilation,
it is largely silent on the important question concerning the minimum number
and placement of observations. To answer this question, we demonstrate the dual
advantage of placing the observations where the square of the sensitivity of
the model solution with respect to the unknown control variables, called
forward sensitivities, attains its maximum. Therefore, we can force the
observability Gramian to be of full rank, which in turn guarantees efficient
recovery of the optimal values of the control variables, which is the first of
the two advantages of this strategy. We further show that the proposed strategy
of placing observations has another inherent optimality: the square of the
sensitivity of the optimal estimates of the control with respect to the
observations (used to obtain these estimates) attains its minimum value, a
second advantage that is a direct consequence of the above strategy for placing
observations. Our analytical framework and numerical experiments on linear and
nonlinear systems confirm the effectiveness of our proposed strategy.
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By: Howard S. Cohl, Roberto S. Costas-Santos

We study special values for the continuous $q$-Jacobi polynomials and present
applications of these special values which arise from bilinear generating
functions, and in particular the Poisson kernel for these polynomials.

We study special values for the continuous $q$-Jacobi polynomials and present
applications of these special values which arise from bilinear generating
functions, and in particular the Poisson kernel for these polynomials.
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