Shuffle Invariance of the Super-RSK Algorithm

As in the $(k,l)$-RSK (Robinson-Schensted-Knuth) of [1], other super-RSK algorithms can be applied to sequences of variables from the set $\{t_1,...,t_k,u_1,...,u_l\}$, where $t_1<...<t_k$, and $u_1<...<u_l$. While the $(k,l)$-RSK of [1] is the case where $t_i<u_j$ for all $i$ and $j$, these other super-RSK's correspond to all the (\big{(}{{k+l}\atop{k}}\big{)} shuffles of the $t$'s and $u$'s satisfying the above restrictions that $t_1<...<t_k$ and $u_1<...<u_l$. We show that the shape of the tableaux produced by any such super-RSK is independent of the particular shuffle of the $t$'s and $u$'s.Comment: 22 page

Cyclage, catabolism, and the affine Hecke algebra

We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n of type A. The subalgebra \pH_n is a \u-analogue of the monoid algebra of \S_n \ltimes \ZZ_{\geq 0}^n and inherits a canonical basis from that of \eH_n. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term \emph{positive affine tableaux} (PAT). We then exhibit a cellular subquotient \R_{1^n} of \pH_n that is a \u-analogue of the ring of coinvariants \CC[y_1,...,y_n]/(e_1,...,e_n) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element \pi \in \pH_n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that \R_{1^n} has cellular quotients \R_\lambda that are \u-analogues of the Garsia-Procesi modules R_\lambda with left cells labeled by (a PAT version of) the \lambda-catabolizable tableaux. We give a conjectural description of a cellular filtration of \pH_n, the subquotients of which are isomorphic to dual versions of \R_\lambda under the perfect pairing on \R_{1^n}. We conjecture how this filtration relates to the combinatorics of the cells of \eH_n worked out by Shi, Lusztig, and Xi. We also conjecture that the k-atoms of Lascoux, Lapointe, and Morse and the R-catabolizable tableaux of Shimozono and Weyman have cellular counterparts in \pH_n. We extend the idea of atom copies of Lascoux, Lapoint, and Morse to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.Comment: 58 pages, youngtab.sty included for tableau

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Current Frontiers and Perspectives in Cell Biology

A numerous internationally renowned authors in the pages of this book present the views of the fields of cell biology and their own research results or review of current knowledge. Chapters are divided into five sections that are dedicated to cell structures and functions, genetic material, regulatory mechanisms, cellular biomedicine and new methods in cell biology. Multidisciplinary and often quite versatile approach by many authors have imposed restrictions of this classification, so it is certain that many chapters could belong to the other sections of this book. The current frontiers, on the manner in which they described in the book, can be a good inspiration to many readers for further improving, and perspectives which are highlighted can be seen in many areas of fundamental biology, biomedicine, biotechnology and other applications of knowledge of cell biology. The book will be very useful for beginners to gain insight into new area, as well as experts to find new facts and expanding horizons