Splittings in varieties of logic

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

Extensions of homomorphisms between localities

We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal {F}$ depends only on $\mathcal {F}$ as long as the object set of the linking system is $\mathrm {Aut}(\mathcal {F})$ -invariant. This was known to be true for linking systems in Oliver’s definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this article. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

The Half Cleaner Lemma: Constructing Efficient Interconnection Networks from Sorting Networks

In general, efficient non-blocking interconnection networks can be derived from sorting networks, and to this end, one may either follow the merge-based or the radix-based sorting paradigm. Both paradigms require special modifications to handle partial permutations. In this article, we present a general lemma about half cleaner modules that were introduced as building blocks in Batcher’s bitonic sorting network. This lemma is the key to prove the correctness of many known optimizations of interconnection networks. In particular, we first show how to use any ternary sorter and a half cleaner for implementing an efficient split module as required for radix-based sorting networks for partial permutations. Second, our lemma formally proves the correctness of another known optimization of the Batcher-Banyan network.

Insertion-tolerance and repetitiveness of random graphs

Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular laws as a corollary of a general lemma about countable Borel equivalence relations on first countable Hausdorff spaces.

Central extensions and groups locally of minimal type

This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar fact in the connected reductive case. The chapter then considers four phenomena that go beyond the quadratic case, along with a pseudo-reductive group of minimal type that is locally of minimal type. It shows that the pseudo-split absolutely pseudo-simple k-groups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. In this classification there is no effect if the “minimal type” hypothesis is relaxed to “locally of minimal type.”

Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids

International audience This paper considers the representation theory of towers of algebras of $\mathcal{J} -trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras $FQSym , PBT$ , and $NCSF$ respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions.

Arithmetic Progressions in Sumsets and Lp-Almost-Periodicity

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least \begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation} Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least \begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation}

Iterative processes with errors for nonlinear equations

In this paper we introduce and consider a class of multi-valued and single-valued operators of generalised monotone type. We proved a new general lemma on the convergence of real sequences and some new convergence theorems for the Ishikawa and Mann iteration processes with errors to the unique fixed point of such operators, which are not necessarily Lipschitz operators. Our results generalise, improve, and extend several recent results.

A general lemma for fixed-point theorems involving more than two maps inD-metric spaces with applications

A general procedural lemma for fixed-point theorems for three and four maps in aD-metric space is proved, and it is further applied for proving the common fixed-point theorems of three and four maps in aD-metric space satisfying certain contractive conditions.

OPTIMAL PROOFS OF DETERMINACY II

We present a general lemma which allows proving determinacy from Woodin cardinals. The lemma can be used in many different settings. As a particular application we prove the determinacy of sets in [Formula: see text], n ≥ 1. The assumption we use to prove [Formula: see text] determinacy is optimal in the base theory of [Formula: see text] determinacy.