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In grammar, tense is a category that expresses time reference. Tenses are usually manifested by the use of specific forms of verbs, particularly in their conjugation patterns.
In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. For example, 30 is the product of 6 and 5, and is the product of and .
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors, dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
In mathematics, the tensor product of two vector spaces V and W is a vector space that can be thought of as the space of all tensors that can be built from vectors from its constituent spaces using an additional operation that can be considered as a generalization and abstraction of the outer product. Because of the connection with tensors, which are the elements of a tensor product, tensor products find uses in many areas of application including in physics and engineering, though the full theoretical mechanics of them described below may not be commonly cited there. For example, in general relativity, the gravitational field is described through the metric tensor, which is a field of tensors, one at each point in the space-time manifold, and each of which lives in the tensor self-product of tangent spaces at its point of residence on the manifold.
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.
In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface push against them in (Newtonian) reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Stress is frequently represented by a lowercase Greek letter sigma (σ).
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar and a vector, a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space.
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly correlated to the local strain rate—the rate of change of its deformation over time. That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.
In the general theory of relativity, the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.
The present tense is a grammatical tense whose principal function is to locate a situation or event in the present time. The present tense is used for actions which are happening now. In order to explain and understand present tense, it is useful to imagine time as a line on which the past tense, the present and the future tense are positioned. The term present tense is usually used in descriptions of specific languages to refer to a particular grammatical form or set of forms; these may have a variety of uses, not all of which will necessarily refer to present time. For example, in the English sentence "My train leaves tomorrow morning", the verb form leaves is said to be in the present tense, even though in this particular context it refers to an event in future time. Similarly, in the historical present, the present tense is used to narrate events that occurred in the past.
Suppose that φ : M → N is a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback, and is frequently denoted by φ∗. More generally, any covariant tensor field – in particular any differential form – on N may be pulled back to M using φ.
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.
The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
On July 19, 2015, in Cincinnati, Ohio, Samuel DuBose, an unarmed black man, was fatally shot by Ray Tensing, a white University of Cincinnati police officer, during a traffic stop for a missing front license plate and a suspended driver's license.
TensorFlow is a free and open-source software library for machine learning and artificial intelligence. It can be used across a range of tasks but has a particular focus on training and inference of deep neural networks.
Tensor Processing Unit (TPU) is an AI accelerator application-specific integrated circuit (ASIC) developed by Google specifically for neural network machine learning, particularly using Google's own TensorFlow software. Google began using TPUs internally in 2015, and in 2018 made them available for third party use, both as part of its cloud infrastructure and by offering a smaller version of the chip for sale.
Google AI is a division of Google dedicated to artificial intelligence. It was announced at Google I/O 2017 by CEO Sundar Pichai.
References
- ↑ What is nonfuture tense? 5 January 2004
- ↑ (Li 1996, 1997)
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