arXiv subjects

Russell Brown

Publications and source records attributed to Russell Brown.

5 recordsLinked to original sources

arXiv2026

A framework to reason about consistency and atomicity guarantees in a sparsely-connected, partially-replicated peer-to-peer system

For an offline-first collaborative application to operate in true peer-to-peer fashion, its collaborative features must function even in environments where internet connectivity is limited or unavailable. Each peer may only be interested in a subset of the application data relevant to its workload, and this subset can overlap in different ways with those of other peers. Limitations imposed by access control and mesh network technologies often result in peers being sparsely connected. Reasoning about consistency in these systems is hard, especially when considering transactional updates that may alter different sets of data in the same transaction. We present \textsc{IntersectionAtomicity} and \textsc{IntersectionCC} as models to reason about offline-first collaborative applications that are sparsely-connected and rely on partially replicating different subsets of a broader set of data. We then use these models to propose a set of guidelines to help developers design their application with atomicity and consistency guarantees.

cs.DC↗

arXiv2016

Big(ger) Sets: decomposed delta CRDT Sets in Riak

CRDT[24] Sets as implemented in Riak[6] perform poorly for writes, both as cardinality grows, and for sets larger than 500KB[25]. Riak users wish to create high cardinality CRDT sets, and expect better than O(n) performance for individual insert and remove operations. By decomposing a CRDT set on disk, and employing delta-replication[2], we can achieve far better performance than just delta replication alone: relative to the size of causal metadata, not the cardinality of the set, and we can support sets that are 100s times the size of Riak sets, while still providing the same level of consistency. There is a trade-off in read performance but we expect it is mitigated by enabling queries on sets.

cs.DB↗

arXiv2013

Heat kernel for the elliptic system of linear elasticity with boundary conditions

We consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second Korn inequality holds. We construct heat kernel of the system subject to Dirichlet, Neumann, or mixed boundary condition under the assumption that weak solutions of the elliptic system are Hölder continuous in the interior. Moreover, we show that if weak solutions of the mixed problem are Hölder continuous up to the boundary, then the corresponding heat kernel has a Gaussian bound. In particular, if the domain is a two dimensional Lipschitz domain satisfying a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary condition, then we show that the heat kernel has a Gaussian bound. As an application, we construct Green's function for elliptic mixed problem in such a domain.

math.AP↗

arXiv2010

Estimates for a family of multi-linear forms

We consider a special class of the multi-linear forms studied by Brascamp and Lieb. For these forms, we are able to characterize the L^p spaces for which the form is bounded. We use this characterization to study a non-linear map that arises in scattering theory.

math.AP↗

arXiv2005

The mixed problem in L^p for some two-dimensional Lipschitz domains

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary and the Neumann data is in L^p(N). We find conditions on the domain and the sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p.

math.AP↗