Product (mathematics)

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

On the complexity of matrix product

Abstract: We prove a lower bound of Ω(m2 log m) for the size of any arithmetic circuit for the product of two matrices, over the real or complex numbers, as long as the circuit doesn't use products with field elements of absolute value larger than 1 (where mxm is the size of each matrix). That is, our lower bound is super-linear in the number of inputs and is applied for circuits that use addition gates, product gates and products with field elements of absolute value up to 1. More generally, for any c = c(m) ρ 1, we obtain a lower bound of Ω(m2 log2c m) for the size of any arithmetic circuit for the product of two matrices (over the real or complex numbers), as long as the circuit doesn't use products with field elements of absolute value larger than c. We also prove size-depth tradeoffs for such circuits.

Axiomatic characterization of fields by the product formula for valuations

Abstract: Introduction. The theorems of class field theory are known to hold for two kinds of fields: algebraic extensions of the rational field and algebraic extensions of a field of functions of one variable over a field of constants. We shall refer to these fields as number fields and function fields, respectively. For class field theory, the function fields must indeed be restricted to those with a Galois field as field of constants; however, we make this restriction only in §5, and until then consider fields with an arbitrary field of constants. In proving these theorems, the product formula for valuations plays an important rôle. This formula states that, for a suitable set of inequivalent valuations | | p,

Structure de la mycosporine (P 310)

Abstract: The structure 1 (2-methoxy-3-bis(hydroxymethyl)methylamino-5-hydroxy-5-hydroxymethyl-2-cyclohexen-1-one has been attributed to mycosporine (P. 310), a product isolated from Stereumhirsutum (Willd. ex Fr.) Fr.; this result is derived from the uv, ir, ms, and nmr of 1H and 13C spectral analysis of the natural product and its derivatives.

Automatic generation of 3D models from packaged goods product images

Abstract: A computer-implemented method and apparatus of 3D modeling accesses product data of a consumer good product. The product data includes 2D image data of the subject consumer good product and shape information of the subject consumer good product. An extractor extracts the shape information from the product data, including a 3D shape class of the subject consumer good product, and derives a detailed 3D shape graph. The detailed 3D shape graph represents an orthographically correct 3D outline shape of the subject consumer good product. As a function of the determined 3D shape class, a mesh generator generates a 3D mesh of the subject consumer good product from the detailed 3D shape graph. An image mapper projects images onto the 3D mesh, the images being of the subject consumer good product and from the 2D image data. A 3D model of the subject consumer good product results and is configured for interactive display on web pages and in other user-interactive environments.