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Analogues of Lusztig's higher order relations for the q-Onsager algebra
Let be the generators of the Onsager algebra. Analogues of
Lusztig's higher order relations are proposed. In a first part, based on
the properties of tridiagonal pairs of Racah type which satisfy the
defining relations of the Onsager algebra, higher order relations are
derived for generic. The coefficients entering in the relations are
determined from a two-variable polynomial generating function. In a second
part, it is conjectured that satisfy the higher order relations
previously obtained. The conjecture is proven for . For generic,
using an inductive argument recursive formulae for the coefficients are
derived. The conjecture is checked for several values of .
Consequences for coideal subalgebras and integrable systems with boundaries at
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
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
SQPR: Stream Query Planning with Reuse
When users submit new queries to a distributed stream processing system (DSPS), a query planner must allocate physical resources, such as CPU cores, memory and network bandwidth, from a set of hosts to queries. Allocation decisions must provide the correct mix of resources required by queries, while achieving an efficient overall allocation to scale in the number of admitted queries. By exploiting overlap between queries and reusing partial results, a query planner can conserve resources but has to carry out more complex planning decisions. In this paper, we describe SQPR, a query planner that targets DSPSs in data centre environments with heterogeneous resources. SQPR models query admission, allocation and reuse as a single constrained optimisation problem and solves an approximate version to achieve scalability. It prevents individual resources from becoming bottlenecks by re-planning past allocation decisions and supports different allocation objectives. As our experimental evaluation in comparison with a state-of-the-art planner shows SQPR makes efficient resource allocation decisions, even with a high utilisation of resources, with acceptable overheads
A short proof of Kahn-Kalai conjecture
In a recent paper, Park and Pham famously proved Kahn-Kalai conjecture. In
this notes, we simplify their proof, using an induction to replace the original
analysis. This reduces the proof to one page and from the argument it is also
easy to read that one can set the constant in the conjecture be , which could be the best value under the current method. Our argument
also applies to the -version of Park-Pham result, studied by Bell
Outward Influence and Cascade Size Estimation in Billion-scale Networks
Estimating cascade size and nodes' influence is a fundamental task in social,
technological, and biological networks. Yet this task is extremely challenging
due to the sheer size and the structural heterogeneity of networks. We
investigate a new influence measure, termed outward influence (OI), defined as
the (expected) number of nodes that a subset of nodes will activate,
excluding the nodes in S. Thus, OI equals, the de facto standard measure,
influence spread of S minus |S|. OI is not only more informative for nodes with
small influence, but also, critical in designing new effective sampling and
statistical estimation methods.
Based on OI, we propose SIEA/SOIEA, novel methods to estimate influence
spread/outward influence at scale and with rigorous theoretical guarantees. The
proposed methods are built on two novel components 1) IICP an important
sampling method for outward influence, and 2) RSA, a robust mean estimation
method that minimize the number of samples through analyzing variance and range
of random variables. Compared to the state-of-the art for influence estimation,
SIEA is times faster in theory and up to several orders of
magnitude faster in practice. For the first time, influence of nodes in the
networks of billions of edges can be estimated with high accuracy within a few
minutes. Our comprehensive experiments on real-world networks also give
evidence against the popular practice of using a fixed number, e.g. 10K or 20K,
of samples to compute the "ground truth" for influence spread.Comment: 16 pages, SIGMETRICS 201
Editorial
We use a linear FROG technique based on electro-optic modulation to fully characterise for the first time pulses from a 1.06 ”m FP laser diode and design a grating to provide optimum pulse compression
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