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Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.


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Journal Article
Yong Zeng1
TL;DR: This paper proposes a formal definition of design problem using the axiomatic theory of design modeling, which implies that both design problem and design solutions are included in a product system.
Abstract: This paper proposes a formal definition of design problem using the axiomatic theory of design modeling. Based on the concept of product system and an informal description of design problem, a design problem is formulated into three parts: environment, structural requirements, as well as performance requirements. On the basis of this formulation, both structural requirements and performance requirements are related to product environment, which is divided into natural, built, and human environments. As a result, product environment is identified as the source of product requirements. An environment-based formulation of design problem is thus developed. This formulation implies that both design problem and design solutions are included in a product system. The design governing equation is formulated to capture the interdependence relation between product problem and product descriptions along the design process. An example of rivet setting tool design is used to illustrate the mathematical formulations throughout the paper.

102 citations

Journal ArticleDOI
TL;DR: The direct product theorems imply a time-space tradeoff $T^2S=\Om{N^3}$ for sorting N items on a quantum computer, which is optimal up to polylog factors, and give several tight time- space and communication- space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.
Abstract: A strong direct product theorem says that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then our overall success probability will be exponentially small in $k$. We establish such theorems for the classical as well as quantum query complexity of the OR-function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing $k$ instances of the disjointness function. Our direct product theorems imply a time-space tradeoff $T^2S=\Om{N^3}$ for sorting $N$ items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.

101 citations

Journal ArticleDOI
TL;DR: In this paper, a graded version of the convolution product on the symmetric group was introduced, and it was shown that there is a degree-preserving ring isomorphism to the Hilbert scheme of points in the complex affine plane.
Abstract: Let $\mathscr {C}(S_n)$ be the $\mathbb {Z}$-module of integer-valued class functions on the symmetric group $S_n)$. We introduce a graded version of the convolution product on $\mathscr {C}(S_n)$, and we show that there is a degree-preserving ring isomorphism $\mathscr {C}(S_n)\longrightarrow H^\ast({\rm Hilb}^n(\mathbb {A_C}^2);\mathbb {Z})$ to the cohomology of the Hilbert scheme of points in the complex affine plane.

101 citations

Journal Article
TL;DR: In this article, the authors study product graphs and obtain some results, and indicate the scope of its applications in a variety of fields, such as computer vision, artificial intelligence, etc.
Abstract: In this paper we study product graphs and obtain some results. Wealso indicate the scope of its applications in a variety of fields.

101 citations

Journal ArticleDOI
TL;DR: A systematic study on several types of soft subsets and various soft equal relations induced by them and it is shown that soft L-equal relations are congruence relations over free soft algebras and the corresponding quotient structures form commutative semigroups.

101 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20251
20244
20239,015
202219,171
20212,013
20202,263