The Multi-Cover Persistence of Euclidean Balls

Given a locally finite $$X \subseteq {{{\mathbb {R}}}}^d$$ X ⊆ R d and a radius $$r \ge 0$$ r ≥ 0 , the k-fold cover of X and r consists of all points in $${{{\mathbb {R}}}}^d$$ R d that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in $${{{\mathbb {R}}}}^{d+1}$$ R d + 1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.

Decomposition of pointwise finite-dimensional persistence modules

We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on images and kernels.

TO WARDS DATA PERSIST ENCY IN REAL-TIME ONLINE INTERACTIVE APPLIC ATIONS

The class of distributed Real-time O nline Interactive Applications (ROIA) includes such important applications as Massively Multiplayer Online Games (MMOGs), as well as interactive e-Learning a nd simulation systems. These applications usually work in a persistent environment (also called world) which continues to exist and evolve also while the user is offline and away from the application. The challenge is how to efficiently make the world and the player characters persistent in the system over time. In this paper, we deal with storing persistent data of real-time interactive applications in modern relational databases. We analyze the major requirements to a system for persistency and we describe a preliminary design of the Entity Persistence Module (EPM) middleware which liberates the application developer from writing and maintaining complex a nd error-prone code for persistent data management. EPM automatically performs the mapping operations to store/ retrieve the complex data to/from different types of relational databases, supports the management of persistent data in memory, and integrates it into the main loop of the ROIA client-server architecture.