27,552 research outputs found
A study of the communication cost of the FFT on torus multicomputers
The computation of a one-dimensional FFT on a c-dimensional torus multicomputer is analyzed. Different approaches are proposed which differ in the way they use the interconnection network. The first approach is based on the multidimensional index mapping technique for the FFT computation. The second approach starts from a hypercube algorithm and then embeds the hypercube onto the torus. The third approach reduces the communication cost of the hypercube algorithm by pipelining the communication operations. A novel methodology to pipeline the communication operations on a torus is proposed. Analytical models are presented to compare the different approaches. This comparison study shows that the best approach depends on the number of dimensions of the torus and the communication start-up and transfer times. The analytical models allow us to select the most efficient approach for the available machine.Peer ReviewedPostprint (published version
Morita Equivalence of Noncommutative Supertori
In this paper we study the extension of Morita equivalence of noncommutative
tori to the supersymmetric case. The structure of the symmetry group yielding
Morita equivalence appears to be intact but its parameter field becomes
supersymmetrized having both body and soul parts. Our result is mainly in the
two dimensional case in which noncommutative supertori have been constructed
recently: The group , where denotes Grassmann even
number whose body part belongs to , yields Morita equivalent
noncommutative supertori in two dimensions.Comment: LaTeX 18 pages, the version appeared in JM
Developing a Mathematical Model for Bobbin Lace
Bobbin lace is a fibre art form in which intricate and delicate patterns are
created by braiding together many threads. An overview of how bobbin lace is
made is presented and illustrated with a simple, traditional bookmark design.
Research on the topology of textiles and braid theory form a base for the
current work and is briefly summarized. We define a new mathematical model that
supports the enumeration and generation of bobbin lace patterns using an
intelligent combinatorial search. Results of this new approach are presented
and, by comparison to existing bobbin lace patterns, it is demonstrated that
this model reveals new patterns that have never been seen before. Finally, we
apply our new patterns to an original bookmark design and propose future areas
for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as
Computer Scientists and Mathematician
TorusE: Knowledge Graph Embedding on a Lie Group
Knowledge graphs are useful for many artificial intelligence (AI) tasks.
However, knowledge graphs often have missing facts. To populate the graphs,
knowledge graph embedding models have been developed. Knowledge graph embedding
models map entities and relations in a knowledge graph to a vector space and
predict unknown triples by scoring candidate triples. TransE is the first
translation-based method and it is well known because of its simplicity and
efficiency for knowledge graph completion. It employs the principle that the
differences between entity embeddings represent their relations. The principle
seems very simple, but it can effectively capture the rules of a knowledge
graph. However, TransE has a problem with its regularization. TransE forces
entity embeddings to be on a sphere in the embedding vector space. This
regularization warps the embeddings and makes it difficult for them to fulfill
the abovementioned principle. The regularization also affects adversely the
accuracies of the link predictions. On the other hand, regularization is
important because entity embeddings diverge by negative sampling without it.
This paper proposes a novel embedding model, TorusE, to solve the
regularization problem. The principle of TransE can be defined on any Lie
group. A torus, which is one of the compact Lie groups, can be chosen for the
embedding space to avoid regularization. To the best of our knowledge, TorusE
is the first model that embeds objects on other than a real or complex vector
space, and this paper is the first to formally discuss the problem of
regularization of TransE. Our approach outperforms other state-of-the-art
approaches such as TransE, DistMult and ComplEx on a standard link prediction
task. We show that TorusE is scalable to large-size knowledge graphs and is
faster than the original TransE.Comment: accepted for AAAI-1
Generalized Property R and the Schoenflies Conjecture
There is a relation between the generalized Property R Conjecture and the
Schoenflies Conjecture that suggests a new line of attack on the latter. The
approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices
to prove the genus 3 case, even in the absence of new progress on the
generalized Property R Conjecture.Comment: 29 pages, 8 figure
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