Wavelets and graph C∗C^*-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs

Twisted k-graph algebras associated to Bratteli diagrams

Given a system of coverings of k-graphs, we show that the cohomology of the resulting (k+1)-graph is isomorphic to that of any one of the k-graphs in the system. We then consider Bratteli diagrams of 2-graphs whose twisted C*-algebras are matrix algebras over noncommutative tori. For such systems we calculate the ordered K-theory and the gauge-invariant semifinite traces of the resulting 3-graph C*-algebras. We deduce that every simple C*-algebra of this form is Morita equivalent to the C*-algebra of a rank-2 Bratteli diagram in the sense of Pask-Raeburn-R{\o}rdam-Sims.Comment: 28 pages, pictures prepared using tik

Use of Dual-Polarization Radar Variables to Assess Low-Level Wind Shear in Severe Thunderstorm Near-storm Environments in the Tennessee Valley

The upgrade of the National Weather Service (NWS) network of S ]band dual-polarization radars is currently underway, and the incorporation of polarimetric information into the real ]time forecasting process will enhance the forecaster fs ability to assess thunderstorms and their near ]storm environments. Recent research has suggested that the combination of polarimetric variables differential reflectivity (ZDR) and specific differential phase (KDP) can be useful in the assessment of low level wind shear within a thunderstorm. In an environment with strong low ]level veering of the wind, ZDR values will be largest along the right inflow edge of the thunderstorm near a large gradient in horizontal reflectivity (indicative of large raindrops falling with a relative lack of smaller drops), and take the shape of an arc. Meanwhile, KDP values, which are proportional to liquid water content and indicative of a large number of smaller drops, are maximized deeper into the forward flank precipitation shield than the ZDR arc as the smaller drops are being advected further from the updraft core by the low level winds than the larger raindrops. Using findings from previous work, three severe weather events that occurred in North Alabama were examined in order to assess the utility of these signatures in determining the potential for tornadic activity. The first case is from October 26, 2010, where a large number of storms indicated tornadic potential from a standard reflectivity and velocity analysis but very few storms actually produced tornadoes. The second event is from February 28, 2011, where tornadic storms were present early on in the event, but as the day progressed, the tornado threat transitioned to a high wind threat. The third case is from April 27, 2011, where multiple rounds of tornadic storms ransacked the Tennessee Valley. This event provides a dataset including multiple modes of tornadic development, including QLCS and supercell structures. The overarching goal of examining these three events is to compare dual ]polarization features from this larger dataset to previous work and to determine if these signatures can be a useful indication of the potential for tornadic activity associated with the amount of low ]level wind shear in the near ]storm environment

Strong Shift Equivalence of C∗C^*-correspondences

We define a notion of strong shift equivalence for $C^*$-correspondences and show that strong shift equivalent $C^*$-correspondences have strongly Morita equivalent Cuntz-Pimsner algebras. Our analysis extends the fact that strong shift equivalent square matrices with non-negative integer entries give stably isomorphic Cuntz-Krieger algebras.Comment: 26 pages. Final version to appear in Israel Journal of Mathematic

Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact

Cartan subalgebras and the UCT problem, II

We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a suitable UHF algebra. These results rely on a new construction of Cartan subalgebras in certain inductive limits of Cartan pairs. We also provide a characterisation of the UCT problem in terms of finite order automorphisms, Cartan subalgebras and inverse semigroups of partial isometries of the Cuntz algebra $\mathcal{O}_2$. This generalizes earlier work of the authors.Comment: minor revisions; final version, accepted for publication in Math. Ann.; 26 page

Confronting the Challenge of Modeling Cloud and Precipitation Microphysics

In the atmosphere, microphysics refers to the microscale processes that affect cloud and precipitation particles and is a key linkage among the various components of Earth\u27s atmospheric water and energy cycles. The representation of microphysical processes in models continues to pose a major challenge leading to uncertainty in numerical weather forecasts and climate simulations. In this paper, the problem of treating microphysics in models is divided into two parts: (i) how to represent the population of cloud and precipitation particles, given the impossibility of simulating all particles individually within a cloud, and (ii) uncertainties in the microphysical process rates owing to fundamental gaps in knowledge of cloud physics. The recently developed Lagrangian particle‐based method is advocated as a way to address several conceptual and practical challenges of representing particle populations using traditional bulk and bin microphysics parameterization schemes. For addressing critical gaps in cloud physics knowledge, sustained investment for observational advances from laboratory experiments, new probe development, and next‐generation instruments in space is needed. Greater emphasis on laboratory work, which has apparently declined over the past several decades relative to other areas of cloud physics research, is argued to be an essential ingredient for improving process‐level understanding. More systematic use of natural cloud and precipitation observations to constrain microphysics schemes is also advocated. Because it is generally difficult to quantify individual microphysical process rates from these observations directly, this presents an inverse problem that can be viewed from the standpoint of Bayesian statistics. Following this idea, a probabilistic framework is proposed that combines elements from statistical and physical modeling. Besides providing rigorous constraint of schemes, there is an added benefit of quantifying uncertainty systematically. Finally, a broader hierarchical approach is proposed to accelerate improvements in microphysics schemes, leveraging the advances described in this paper related to process modeling (using Lagrangian particle‐based schemes), laboratory experimentation, cloud and precipitation observations, and statistical methods

Graded C*-algebras, graded K-theory, and twisted P-graph C*-algebras

We develop methods for computing the graded $K$ -theory of $C^*$ -algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner'\''s six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner--Voiculescu sequence. We introduce the notion of a twisted $P$ -graph $C^*$ -algebra and establish connections with graded $C^*$ -algebras. Specifically, we show how a functor from a $P$ -graph into the group of order two determines a grading of the associated $C^*$ -algebra. We apply our graded version of Pimsner'\''s exact sequence to compute the graded $K$ -theory of a graph $C^*$ -algebra carrying such a grading