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In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot prod

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Further results on latent discourse models and word embeddings

Youssef Allouah

We discuss some properties of generative models for word embeddings. Namely, (Arora et al., 2016) proposed a latent discourse model implying the concentration of the partition function of the word vectors. This concentration phenomenon led to an asymptotic linear relation between the pointwise mutual information (PMI) of pairs of words and the scalar product of their vectors. Here, we first revisit this concentration phenomenon and prove it under slightly weaker assumptions, for a set of random vectors symmetrically distributed around the origin. Second, we empirically evaluate the relation between PMI and scalar products of word vectors satisfying the concentration property. Our empirical results indicate that, in practice, this relation does not hold with arbitrarily small error. This observation is further supported by two theoretical results: (i) the error cannot be exactly zero because the corresponding shifted PMI matrix cannot be positive semidefinite; (ii) under mild assumptions, there exist pairs of words for which the error cannot be close to zero. We deduce that either natural language does not follow the assumptions of the considered generative model, or the current word vector generation methods do not allow the construction of the hypothesized word embeddings.

MICROTOME PUBL2021

Surface parameterization and optimum design methodology for hydraulic turbines

This thesis presents a methodology for the design optimization of hydraulic runner blades. The originality of the methodology comes from the geometric definition of the blade shapes, which uses parametric surfaces instead of a set of profiles. The main advantage of using surfaces is the number of parameters required. The use of surfaces requires a different technique for the blade construction when compared to traditional approaches. NURBS surfaces are used for the geometric representation, the properties provided by such parametric formulation permit to reach the necessary flexibility and accuracy to be attained. Moreover, the surface approach can be seen as a way to liberate the blade design from traditional discrete sectional approaches. Actual blade optimization procedures are a compromise between the quality of the design, its performance analysis and the subsequent computational time-consumed effort. The improvements provided by the above mentioned geometric definition allow the use of more realistic analysis tools for the design evaluation. Thus, Navier-Stokes (k – ε) simulations are integrated in a simple and direct optimization process. The resulting methodology is not penalized by the time-effort required and becomes of interest for its use in industrial applications. Finally, we supply a number of examples to demonstrate the feasibility of the optimization proposal. These examples illustrate the application of the methodology at different levels of geometric complexity. They are interesting not only through the results obtained, but also because they become acceptable in terms of time consumed on daily and industrial applications.

EPFL2006

MnO2 nanosheets as an artificial enzyme to mimic oxidase for rapid and sensitive detection of glutathione

Paul Joseph Dyson, Zhaofu Fei, Xing Liu, Jing Liu

Nanozymes are increasingly used as components in assays and diagnostics. Here, we describe a rapid and highly sensitive colorimetric assay for the detection and quantification of glutathione (GSH) employing MnO2 nanosheets as an artificial oxidase. In the assay pale yellow 3,3',5,5'-tetramethylbenzidine (TMB) is oxidized to a blue product (oxTMB) under catalyzing of MnO2 nanosheets with a significant change in absorption at 650 nm. GSH selectively inhibits this reaction with a detection limit of 300 nM. The high specificity of inhibition by GSH allows this system to be used to determine the GSH concentrations in human serum samples. The MnO2 nanosheet-based assay is simple, rapid, sensitive and selective for the quantification of GSH and surpasses detection methods based on other MnO2 nanomaterials.

Elsevier2017

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scal

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In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direct

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