Analogues of Lusztig's higher order relations for the q-Onsager algebra

Let $A,A^*$ be the generators of the $q-$Onsager algebra. Analogues of Lusztig's $r-th$ higher order relations are proposed. In a first part, based on the properties of tridiagonal pairs of $q-$Racah type which satisfy the defining relations of the $q-$Onsager algebra, higher order relations are derived for $r$ generic. The coefficients entering in the relations are determined from a two-variable polynomial generating function. In a second part, it is conjectured that $A,A^*$ satisfy the higher order relations previously obtained. The conjecture is proven for $r=2,3$. For $r$ generic, using an inductive argument recursive formulae for the coefficients are derived. The conjecture is checked for several values of $r\geq 4$. Consequences for coideal subalgebras and integrable systems with boundaries at $q$ a root of unity are pointed out.Comment: 19 pages. v2: Some basic material in subsections 2.1,2.2,2.3 of pages 3-4 (Definitions 2.1,2.2, Lemma 2.2, Theorem 1) from Terwilliger's and coauthors works (see also arXiv:1307.7410); Missprints corrected; Minor changes in the text; References adde

A voice for the voiceless: lessons from a Hmong community's approach to music and self-expression

Since the turn of the century, the world has witnessed a rise in violence promulgated on American soil. From terrorism to bullying, citizens across the United States are left wondering, "What’s next or what can I do about it?" I imagine that I am not alone in feeling powerlessness, out of control, and sometimes apathetic about the constant newsfeed heralding bad news both at home and abroad. With this kind ofuncertainty, it is no surprise that our students might feel just as overwhelmed and confused as we teachers. What, then, can music educators do to be a voice for and with students and how will their songs be a voice for those who will not or cannot sing songs of their own? This essay is an account of how I connect what I learned in a Hmong community of rappers and poets to music education, what critical pedagogy might mean for music educators, and how teachers can employ "voice for the voiceless" strategies with their ensemble or general music students

On the permanent of random Bernoulli matrices

We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero

Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials

The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of irreducibility is added; v3: version for Linear Algebra and its Applications, one assumption added in Appendix about eq. (A.2

Using a novel source-localized phase regressor technique for evaluation of the vascular contribution to semantic category area localization in BOLD fMRI.

Numerous studies have shown that gradient-echo blood oxygen level dependent (BOLD) fMRI is biased toward large draining veins. However, the impact of this large vein bias on the localization and characterization of semantic category areas has not been examined. Here we address this issue by comparing standard magnitude measures of BOLD activity in the Fusiform Face Area (FFA) and Parahippocampal Place Area (PPA) to those obtained using a novel method that suppresses the contribution of large draining veins: source-localized phase regressor (sPR). Unlike previous suppression methods that utilize the phase component of the BOLD signal, sPR yields robust and unbiased suppression of large draining veins even in voxels with no task-related phase changes. This is confirmed in ideal simulated data as well as in FFA/PPA localization data from four subjects. It was found that approximately 38% of right PPA, 14% of left PPA, 16% of right FFA, and 6% of left FFA voxels predominantly reflect signal from large draining veins. Surprisingly, with the contributions from large veins suppressed, semantic category representation in PPA actually tends to be lateralized to the left rather than the right hemisphere. Furthermore, semantic category areas larger in volume and higher in fSNR were found to have more contributions from large veins. These results suggest that previous studies using gradient-echo BOLD fMRI were biased toward semantic category areas that receive relatively greater contributions from large veins

Distributed Data Summarization in Well-Connected Networks

We study distributed algorithms for some fundamental problems in data summarization. Given a communication graph G of n nodes each of which may hold a value initially, we focus on computing sum_{i=1}^N g(f_i), where f_i is the number of occurrences of value i and g is some fixed function. This includes important statistics such as the number of distinct elements, frequency moments, and the empirical entropy of the data. In the CONGEST~ model, a simple adaptation from streaming lower bounds shows that it requires Omega~(D+ n) rounds, where D is the diameter of the graph, to compute some of these statistics exactly. However, these lower bounds do not hold for graphs that are well-connected. We give an algorithm that computes sum_{i=1}^{N} g(f_i) exactly in {tau_{G}} * 2^{O(sqrt{log n})} rounds where {tau_{G}} is the mixing time of G. This also has applications in computing the top k most frequent elements. We demonstrate that there is a high similarity between the GOSSIP~ model and the CONGEST~ model in well-connected graphs. In particular, we show that each round of the GOSSIP~ model can be simulated almost perfectly in O~({tau_{G}}) rounds of the CONGEST~ model. To this end, we develop a new algorithm for the GOSSIP~ model that 1 +/- epsilon approximates the p-th frequency moment F_p = sum_{i=1}^N f_i^p in O~(epsilon^{-2} n^{1-k/p}) roundsfor p >= 2, when the number of distinct elements F_0 is at most O(n^{1/(k-1)}). This result can be translated back to the CONGEST~ model with a factor O~({tau_{G}}) blow-up in the number of rounds