Topic
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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TL;DR: In this paper, the special point formulas generated from product Gauss-Chebyshev integration formulas for fcc and bcc lattices are compared with those resulting from the special treatment suggested by Monkhorst and Pack.
Abstract: The special-point formulas generated from product Gauss-Chebyshev integration formulas for fcc and bcc lattices are compared with those resulting from the special treatment suggested by Monkhorst and Pack. It is found that the Monkhorst-Pack special-point formulas are more efficient for formulas with few points and less efficient for formulas with many points.
100 citations
TL;DR: A test that can distinguish efficiently between product states of n quantum systems and states that are far from product is given, which can be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalization of classical linearity testing.
Abstract: We give a test that can distinguish efficiently between product states of n quantum systems and states that are far from product. If applied to a state |ψ〉 whose maximum overlap with a product state is 1 − e, the test passes with probability 1 − Θ(e), regardless of n or the local dimensions of the individual systems. The test uses two copies of |ψ〉. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarizing channel.A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k) = QMA(2) for k ≥ 2. Building on a previous result of Aaronson et al., this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(√n) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities.Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalization of classical linearity testing.
100 citations
Patent•
26 Jul 1995
TL;DR: A product for surgical use, in the form of an open, integral mesh of substantially uniform thickness, is presented in this paper, where the authors present an open integral mesh with uniform thickness.
Abstract: A product, for surgical use, in the form of an open, integral mesh of substantially uniform thickness.
100 citations
Book•
01 Jan 1978
TL;DR: The authors constructed and tested models incorporating this dependence and tested them on a set of product classes (e.g., magazines, stereo albums, liquor in a home bar) and found that the assumption that product choices are made independently of each other is implausible.
Abstract: By focusing on consumer preference for single items, researchers have implicitly assumed that product choices are made independently of each other. For many product classes (e.g., magazines, stereo albums, liquor in a home bar) that assumption is implausible. “Balancing” or “rounding out” a group of items implies dependence among selections. Models incorporating this dependence are constructed and tested.
99 citations
TL;DR: Two new and fast methods of determining product ratios for kinetic schemes leading to more than one product on which the Acree–Curtin–Hammett (ACH) principle is based are presented and are shown to be completely verifiable with conventional more tedious treatments based on rate law determinations.
Abstract: This paper presents two new and fast methods of determining product ratios for kinetic schemes leading to more than one product on which the Acree–Curtin–Hammett (ACH) principle is based. The metho...
99 citations