Product (mathematics)

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

Generalized Iterative Scaling for Log-Linear Models

Abstract: Say that a probability distribution $\{p_i; i \in I\}$ over a finite set $I$ is in "product form" if (1) $p_i = \pi_i\mu \prod^d_{s=1} \mu_s^{b_si}$ where $\pi_i$ and $\{b_{si}\}$ are given constants and where $\mu$ and $\{\mu_s\}$ are determined from the equations (2) $\sum_{i \in I} b_{si} p_i = k_s, s = 1, 2, \cdots, d$; (3) $\sum_{i \in I} p_i = 1$. Probability distributions in product form arise from minimizing the discriminatory information $\sum_{i \in I} p_i \log p_i/\pi_i$ subject to (2) and (3) or from maximizing entropy or maximizing likelihood. The theory of the iterative scaling method of determining (1) subject to (2) and (3) has, until now, been limited to the case when $b_{si} = 0, 1$. In this paper the method is generalized to allow the $b_{si}$ to be any real numbers. This expands considerably the list of probability distributions in product form which it is possible to estimate by maximum likelihood.

Lectures on Algebraic Topology

Abstract: I Preliminaries on Categories, Abelian Groups, and Homotopy.- x1 Categories and Functors.- x2 Abelian Groups (Exactness, Direct Sums, Free Abelian Groups).- x3 Homotopy.- II Homology of Complexes.- x1 Complexes.- x2 Connecting Homomorphism, Exact Homology Sequence.- x3 Chain-Homotopy.- x4 Free Complexes.- III Singular Homology.- x1 Standard Simplices and Their Linear Maps.- x2 The Singular Complex.- x3 Singular Homology.- x4 Special Cases.- x5 Invariance under Homotopy.- x6 Barycentric Subdivision.- x7 Small Simplices. Excision.- x8 Mayer-Vietoris Sequences.- IV Applications to Euclidean Space.- x1 Standard Maps between Cells and Spheres.- x2 Homology of Cells and Spheres.- x3 Local Homology.- x4 The Degree of a Map.- x5 Local Degrees.- x6 Homology Properties of Neighborhood Retracts in ?n.- x7 Jordan Theorem, Invariance of Domain.- x8 Euclidean Neighborhood Retracts (ENRs).- V Cellular Decomposition and Cellular Homology.- x1 Cellular Spaces.- x2 CW-Spaces.- x3 Examples.- x4 Homology Properties of CW-Spaces.- x5 The Euler-Poincare Characteristic.- x6 Description of Cellular Chain Maps and of the Cellular Boundary Homomorphism.- x7 Simplicial Spaces.- x8 Simplicial Homology.- VI Functors of Complexes.- x1 Modules.- x2 Additive Functors.- x3 Derived Functors.- x4 Universal Coefficient Formula.- x5 Tensor and Torsion Products.- x6 Horn and Ext.- x7 Singular Homology and Cohomology with General Coefficient Groups.- x8 Tensorproduct and Bilinearity.- x9 Tensorproduct of Complexes. Kunneth Formula.- x10 Horn of Complexes. Homotopy Classification of Chain Maps.- x11 Acyclic Models.- x12 The Eilenberg-Zilber Theorem. Kunneth Formulas for Spaces.- VII Products.- x1 The Scalar Product.- x2 The Exterior Homology Product.- x 3 The Interior Homology Product (Pontijagin Product).- x 4 Intersection Numbers in ?n.- x5 The Fixed Point Index.- x6 The Lefschetz-Hopf Fixed Point Theorem.- x7 The Exterior Cohomology Product.- x 8 The Interior Cohomology Product (?-Product).- x 9 ?-Products in Projective Spaces. Hopf Maps and Hopf Invariant.- x10 Hopf Algebras.- x11 The Cohomology Slant Product.- x12 The Cap-Product (?-Product).- x 13 The Homology Slant Product, and the Pontijagin Slant Product.- VIII Manifolds.- x1 Elementary Properties of Manifolds.- x2 The Orientation Bundle of a Manifold.- x3 Homology of Dimensions ? n in n-Manifolds.- x4 Fundamental Class and Degree.- x5 Limits.- x6 ?ech Cohomology of Locally Compact Subsets of ?n.- x7 Poincare-Lefschetz Duality.- x8 Examples, Applications.- x9 Duality in ?-Manifolds.- x10 Transfer.- x11 Thom Class, Thom Isomorphism.- x12 The Gysin Sequence. Examples.- x13 Intersection of Homology Classes.- Appendix: Kan- and ?ech-Extensions of Functors.- x1 Limits of Functors.- x2 Polyhedrons under a Space, and Partitions of Unity.- x3 Extending Functors from Polyhedrons to More General Spaces.

Evaluation of Brand Extensions: The Role of Product Feature Similarity and Brand Concept

Abstract: This article examines two factors that differentiate between successful and unsuccessful brand extensions: product feature similarity and brand concept consistency. The results reveal that, in identifying brand extensions, consumers take into account not only information about the product-level feature similarity between the new product and the products already associated with the brand, but also the concept consistency between the brand concept and the extension. For both function-oriented and prestige-oriented brand names, the most favorable reactions occur when brand extensions are made with high brand concept consistency and high product feature similarity. In addition, the relative impact of these two factors differs to some extent, depending on the nature of the brand-name concept. When a brand's concept is consistent with those of its extension products, the prestige brand seems to have greater extendibility to products with low feature similarity than the functional brand does. Copyright 1991 by the University of Chicago.

Product common object

Abstract: Stored product management information in a first format for use by a first computerized system is transformed to readily make the stored product management information available for use in a second computerized system that utilizes a second format in a cost-efficient and time-efficient manner.

Harmonic Polylogarithms

Abstract: The harmonic polylogarithms (hpl's) are introduced They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t) The coefficients of their expansions and their Mellin transforms are harmonic sums