Tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Euclidean vectors, often used in physics and engineering applications, and scalars themselves are also tensors. A more sophisticated example is the Cauchy stress tensorT, which takes a direction v as input and produces the stress T(v) on the surface normal to this vector for output, thus expressing a relationship between these two vectors, shown in the figure (right).
In terms of a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map is represented by a matrix (a 2-dimensional array) in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. The coordinate independence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one. The precise form of the transformation law determines the type (or valence) of the tensor. The tensor type is a pair of natural numbers (n, m), where n is the number of contravariant indices and m is the number of covariant indices. The total order of a tensor is the sum of these two numbers.
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 multi-linear concept. Their well-known 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.
Given a finite set { V1, ..., Vn } of vector spaces over a common fieldF, one may form their tensor productV1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.
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Google launches Gemini2.0 with autonomous AI agents, native image generation, and multilingual capabilities, doubling performance while introducing ProjectsAstra, Mariner, and Jules for enhanced task automation and digital assistance.Read More ....
GooglePixel 11 series could offer greatly improved thermal and battery efficiency, according to a report. The company's Pixel phones come with Tensor chips, which provide advanced AI capabilities and... .
From my own experience with Google’s smartphones, especially those with custom-designed Tensor processors, heating has been a mainstay ... of a rough template for designing its own Tensor silicon.
(MENAFN - EIN Presswire) Dr. Greg Vigna Study shows DTI can differentiate between infants with periventricular white matter injury and those with spastic cerebral palsy, aiding early diagnosis ... .
GooglePixel 11a is expected to be launched in 2027 alongside the Pixel Tablet 3 and both of these devices will reportedly feature Google's Tensor G6 chip, but a new report suggests these processors... .
Leaked details hint Google might go with a custom wearable Tensor chip in 2026 ... Starting in 2026, Google might go in-house with a custom Tensor processor for the PixelWatch 5.
GooglePixel 10 and Pixel 11 will reportedly be equipped with Tensor G5 and Tensor G6 chips, respectively, offering improved efficiency in comparison to the Tensor G4 that powers the... .
“The Tensor inside Pixels is essentially a mirror of Samsung’s Exynos silicon, and so are the problems.” That’s a rough summation of Pixel user sentiments over the past few years.
GPUs in particular have seen a recent rise to prominence with the emergence of artificial intelligence (AI). You may have also heard of tensor processing units (TPUs), which are a Google creation and only available via their cloud services ... .
