We deal with the relation between the Mittag-Leffler functions and the sum of the hypervolume of generalized l(p) n-balls. We derive from this result some simple properties of the Mittag-Leffler functions. We also define some new Abelian groups from the sum and the multiplication as well as the Pontryagin transform associated to them. We relate this approach to the calculus on measure chains.

We study finite subsets of l(p) and show that, up to a nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains l(p)(n).

This note is a companion to the article On the mutually non isomorphic l(p)(l(q)) spaces published in this journal, in which P. Cembranos and J. Mendoza showed that {l(p)(l(q)) : 1 <=3D p, q <=3D infinity} is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non-locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class {l(p)(l(q)) : 0 < p, q <=3D infinity} are isomorphic. Our approach is novel in the sense that we reach the isomorphism obstructions from the perspective of bases techniques and the different convexities of the spaces, both methods being intrinsic to quasi-Banach spaces. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Two topics are presented: synchronization games and synchronization costs. In a synchronization game on a deterministic finite automaton, there are two players, Alice and Bob, whose moves alternate. Alice wants to synchronize the given automaton, while Bob aims to make her task as hard as possible. We answer a few natural questions related to such games. Speaking about synchronization costs, we consider deterministic automata in which each transition has a certain price. The problem is whether or not a given automaton can be synchronized within a given budget. We determine the complexity of this problem.

Let l and p be primes, let F/Q(p) be a finite extension with absolute Galois group G(F), let F be a finite field of characteristic l, and let (rho) over bar : G(F) -> GL(n) (F) be a continuous representation. Let R-square((rho) over bar) be the universal framed deformation ring for (rho) over bar. If l =3D p, then the Breuil-Mezard conjecture ( as recently formulated by Emerton and Gee) relates the mod l reduction of certain cycles in R-square((rho) over bar) to the mod l reduction of certain representations of GL(n) (O-F) . We state an analogue of the Breuil-Mezard conjecture when l not equal p, and we prove it whenever l > 2 using automorphy lifting theorems. We give a local proof when l is "quasibanal" for F and (rho) over bar is tamely ramified. We also analyze the reduction modulo l of the types sigma(tau) defined by Schneider and Zink.

The problem of c-Approximate Nearest Neighbor (c-ANN) search in high-dimensional space is fundamentally important in many applications, such as image database and data mining. Locality-Sensitive Hashing (LSH) and its variants are the well-known indexing schemes to tackle the c-ANN search problem. Traditionally, LSH functions are constructed in a query-oblivious manner, in the sense that buckets are partitioned before any query arrives. However, objects closer to a query may be partitioned into different buckets, which is undesirable. Due to the use of query-oblivious bucket partition, the state-of-the-art LSH schemes for external memory, namely C2LSH and LSB-Forest, only work with approximation ratio of integer . In this paper, we introduce a novel concept of query-aware bucket partition which uses a given query as the "anchor" for bucket partition. Accordingly, a query-aware LSH function under a specific norm with is a random projection coupled with query-aware bucket partition, which removes random shift required by traditional query-oblivious LSH functions. The query-aware bucket partitioning strategy can be easily implemented so that query performance is guaranteed. For each norm , based on the corresponding p-stable distribution, we propose a novel LSH scheme named query-aware LSH (QALSH) for c-ANN search over external memory. Our theoretical studies show that QALSH enjoys a guarantee on query quality. The use of query-aware LSH function enables QALSH to work with any approximation ratio . In addition, we propose a heuristic variant named QALSH to improve the scalability of QALSH. Extensive experiments show that QALSH and QALSH outperform the state-of-the-art schemes, especially in high-dimensional space. Specifically, by using a ratio , QALSH can achieve much better query quality.