Showing papers on "Product (mathematics) published in 2006"

Fast Monte Carlo Algorithms for Matrices III: Computing a Compressed Approximate Matrix Decomposition

Abstract: In many applications, the data consist of (or may be naturally formulated as) an $m \times n$ matrix $A$ which may be stored on disk but which is too large to be read into random access memory (RAM) or to practically perform superlinear polynomial time computations on it. Two algorithms are presented which, when given an $m \times n$ matrix $A$, compute approximations to $A$ which are the product of three smaller matrices, $C$, $U$, and $R$, each of which may be computed rapidly. Let $A' = CUR$ be the computed approximate decomposition; both algorithms have provable bounds for the error matrix $A-A'$. In the first algorithm, $c$ columns of $A$ and $r$ rows of $A$ are randomly chosen. If the $m \times c$ matrix $C$ consists of those $c$ columns of $A$ (after appropriate rescaling) and the $r \times n$ matrix $R$ consists of those $r$ rows of $A$ (also after appropriate rescaling), then the $c \times r$ matrix $U$ may be calculated from $C$ and $R$. For any matrix $X$, let $\|X\|_F$ and $\|X\|_2$ denote its Frobenius norm and its spectral norm, respectively. It is proven that $$ \left\|A-A'\right\|_\xi \le \min_{D:\mathrm{rank}(D)\le k} \left\|A-D\right\|_\xi + poly(k,1/c) \left\|A\right\|_F $$ holds in expectation and with high probability for both $\xi = 2,F$ and for all $k=1,\ldots,\mbox{rank}(A)$; thus by appropriate choice of $k$ $$ \left\|A-A'\right\|_2 \le \epsilon \left\|A\right\|_F $$ also holds in expectation and with high probability. This algorithm may be implemented without storing the matrix $A$ in RAM, provided it can make two passes over the matrix stored in external memory and use $O(m+n)$ additional RAM (assuming that $c$ and $r$ are constants, independent of the size of the input). The second algorithm is similar except that it approximates the matrix $C$ by randomly sampling a constant number of rows of $C$. Thus, it has additional error but it can be implemented in three passes over the matrix using only constant additional RAM. To achieve an additional error (beyond the best rank-$k$ approximation) that is at most $\epsilon \|A\|_F$, both algorithms take time which is a low-degree polynomial in $k$, $1/\epsilon$, and $1/\delta$, where $\delta>0$ is a failure probability; the first takes time linear in $\mbox{max}(m,n)$ and the second takes time independent of $m$ and $n$. The proofs for the error bounds make important use of matrix perturbation theory and previous work on approximating matrix multiplication and computing low-rank approximations to a matrix. The probability distribution over columns and rows and the rescaling are crucial features of the algorithms and must be chosen judiciously.

Multipartite nonlocality without entanglement in many dimensions

Abstract: We present a generic method to construct a product basis exhibiting nonlocality without entanglement with $n$ parties each holding a system of dimension at least $n\ensuremath{-}1$. This basis is generated via a quantum circuit made of controlled discrete Fourier transform gates acting on the computational basis. The simplicity of our quantum circuit allows for an intuitive understanding of this new type of nonlocality. We also show how this circuit can be used to construct unextendible product bases and their associated bound entangled states. To our knowledge, this is the first method which, given a general Hilbert space $\mathcal{H}={\ensuremath{\bigotimes}}_{i=1}^{n}{\mathcal{H}}_{{d}_{i}}$ with ${d}_{i}\ensuremath{\leqslant}n\ensuremath{-}1$, makes it possible to construct (i) a basis exhibiting nonlocality without entanglement, (ii) an unextendible product basis, and (iii) a bound entangled state.

Text mining for product attribute extraction

Abstract: We describe our work on extracting attribute and value pairs from textual product descriptions. The goal is to augment databases of products by representing each product as a set of attribute-value pairs. Such a representation is beneficial for tasks where treating the product as a set of attribute-value pairs is more useful than as an atomic entity. Examples of such applications include demand forecasting, assortment optimization, product recommendations, and assortment comparison across retailers and manufacturers. We deal with both implicit and explicit attributes and formulate both kinds of extractions as classification problems. Using single-view and multi-view semi-supervised learning algorithms, we are able to exploit large amounts of unlabeled data present in this domain while reducing the need for initial labeled data that is expensive to obtain. We present promising results on apparel and sporting goods products and show that our system can accurately extract attribute-value pairs from product descriptions. We describe a variety of application that are built on top of the results obtained by the attribute extraction system.

Using neural networks to represent potential surfaces as sums of products.

Abstract: By using exponential activation functions with a neural network (NN) method we show that it is possible to fit potentials to a sum-of-products form. The sum-of-products form is desirable because it reduces the cost of doing the quadratures required for quantum dynamics calculations. It also greatly facilitates the use of the multiconfiguration time dependent Hartree method. Unlike potfit product representation algorithm, the new NN approach does not require using a grid of points. It also produces sum-of-products potentials with fewer terms. As the number of dimensions is increased, we expect the advantages of the exponential NN idea to become more significant.

Packaging machine and method

Abstract: A packaging method and apparatus wherein each product is packaged by enveloping the product in flexible packaging material. A programmed microprocessor calculates the length of flexible packaging material needed to package the product based on the physical dimensions of the product, calculates the weight of the flexible packaging material needed, and calculates a total package weight as the sum of the weight of the product and the calculated weight of the flexible packaging material. A printer prints information specific to the product that is being packaged onto a label that is then affixed to the flexible packaging material prior to the product being packaged. The information can be a function of the package weight, and the calculated total package weight can be communicated from the microprocessor to the printer. Finally, the product is packaged in the flexible packaging material having the label already affixed thereto.

From the Mahler conjecture to Gauss linking integrals

Abstract: We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimized (non-uniquely) when $K$ is an $n$-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain $K^\diamond \subseteq K \times K^\circ$ is minimized when $K$ is an ellipsoid. It implies the Mahler conjecture up to a factor of $(\pi/4)^n \gamma_n$, where $\gamma_n$ is a monotonic factor that begins at $4/\pi$ and converges to $\sqrt{2}$. This strengthens a result of Bourgain and Milman, who showed that there is a constant $c$ such that the Mahler conjecture is true up to a factor of $c^n$. The proof uses a version of the Gauss linking integral to obtain a constant lower bound on $\Vol K^\diamond$, with equality when $K$ is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere $S^{n-1}$ and in hyperbolic space $H^{n-1}$.

Software Product Line Engineering with the UML: Deriving Products

Abstract: Software product line engineering introduces two new dimensions into the traditional engineering of software-based systems: the variability modeling and the product derivation. The variability gathers characteristics that differ from one product to another, while the product derivation is defined as a complete process of building products from the product line. Software Product Line Engineering with the UML has received a lot of attention in recent years. However most of these works only concern variability modeling in UML static models and few works concern behavioral models . In addition, there is very little research on product derivation. This chapter investigates the product derivation in the context of the product line engineering with the UML. First, a set of extensions are proposed to model product line variability in two types of UML mo dels: class diagrams (t he static aspect) and sequence diagrams (the behavioral aspect). Then we formalize product derivation using a UML model transformation. An algorithm is give n to derive a static model for a product and an algebraic approach is proposed to derive product-specific statecharts from the sequence diagrams of the product line. Two simple case studies are presented, based on a Mercure product line and the banking product line, to illustrate the overall process, from the modeling of the product line to the product derivation.

One-and-a-half quantum de Finetti theorems

Abstract: We prove a new kind of quantum de Finetti theorem for representations of the unitary group U(d). Consider a pure state that lies in the irreducible representation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing out U_nu. We show that xi is close to a convex combination of states Uv, where U is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the symmetric representation, this yields the conventional quantum de Finetti theorem for symmetric states, and our method of proof gives near-optimal bounds for the approximation of xi by a convex combination of product states. For the class of symmetric Werner states, we give a second de Finetti-style theorem (our 'half' theorem); the de Finetti-approximation in this case takes a particularly simple form, involving only product states with a fixed spectrum. Our proof uses purely group theoretic methods, and makes a link with the shifted Schur functions. It also provides some useful examples, and gives some insight into the structure of the set of convex combinations of product states.

A feature-oriented approach to developing dynamically reconfigurable products in product line engineering

Abstract: Dynamic product reconfiguration refers to making changes to a deployed product configuration while a system is running. Recently, there have been increasing demands for dynamic product reconfiguration in various application areas (e.g., ubiquitous computing, self-healing systems, etc.); however, most product line engineering methods in the literature have focused on the development of reusable core assets for statically configured products. In this paper, we propose a feature-oriented approach to develop dynamically reconfigurable core assets. This approach takes feature binding analysis results as a key design driver for identifying and managing variation points of dynamically reconfigurable products. We also provide a conceptual model for a reconfigurator, which monitors and manages product reconfiguration at run time. The method is illustrated with a home service robot product line example.

Combining abstract interpreters

Abstract: We present a methodology for automatically combining abstract interpreters over given lattices to construct an abstract interpreter for the combination of those lattices. This lends modularity to the process of design and implementation of abstract interpreters.We define the notion of logical product of lattices. This kind of combination is more precise than the reduced product combination. We give algorithms to obtain the join operator and the existential quantification operator for the combined lattice from the corresponding operators of the individual lattices. We also give a bound on the number of steps required to reach a fixed point across loops during analysis over the combined lattice in terms of the corresponding bounds for the individual lattices. We prove that our combination methodology yields the most precise abstract interpretation operators over the logical product of lattices when the individual lattices are over theories that are convex, stably infinite, and disjoint.We also present an interesting application of logical product wherein some lattices can be reduced to combination of other (unrelated) lattices with known abstract interpreters.

Noncommutative Gravity

Abstract: We consider simple extensions of noncommutativity from flat to curved spacetime One possibility is to have a generalization of the Moyal product with a covariantly constant noncommutative tensor $\theta^{\mu u}$ In this case the spacetime symmetry is restricted to volume preserving diffeomorphisms which also preserve $\theta^{\mu u}$ Another possibility is an extension of the Kontsevich product to curved spacetime In both cases the noncommutative product is nonassociative We find the the order $\theta^2$ noncommutative correction to the Newtonian potential in the case of a covariantly constant $\theta^{\mu u}$ It is still of the form $1/r$ plus an angle dependent piece The coupling to matter gives rise to a propagator which is $\theta$ dependent

Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs

Abstract: A strong direct product theorem says that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then our overall success probability will be exponentially small in $k$. We establish such theorems for the classical as well as quantum query complexity of the OR-function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing $k$ instances of the disjointness function. Our direct product theorems imply a time-space tradeoff $T^2S=\Om{N^3}$ for sorting $N$ items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.

Non-existence of Warped Product Semi-Slant Submanifolds of Kaehler Manifolds

Abstract: In this paper, we show that there are no warped product semi-slant submanifolds of Kaehler manifolds. Contrary to this result,we provide an elementary example of a CR-warped product submanifold of a Kaehler manifold

System and Method for Creating a Personalized Consumer Product

Abstract: A system and method for creating a personalized consumer product are provided. The system and method of the present disclosure enables a user, e.g., a consumer, to customize products containing solids and/or fluids by allowing a server communicating over the global computer network, e.g., the Internet, to provide product preferences of a user to a product or a mixing device, e.g., a product or beverage dispenser. The method for creating a product according to a user's preferences over a network includes the steps identifying a product to a server over the network; identifying a user to the server over the network; retrieving the user's product preferences from a database at the server based on the product's identity and user's identity; transmitting the user's product preferences to the product over the network; and mixing at least one element contained within the product based on the user's product preferences.

Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

Abstract: Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory ${\mathcal M}(G,R)$ of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that ${\mathcal M}(G,R)$ is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in ${\mathcal M}(G,R).$ We prove that the Krull--Schmidt theorem holds in many cases.

Software Product Lines

Abstract: Product Line Management.- A Scenario-Based Method for Software Product Line Architecting.- Strategic Scenario-Based Valuation of Product Line Roadmaps.- Experiences and Expectations Regarding the Introduction of Systematic Reuse in Small- and Medium-Sized Companies.- Product Line Requirements Engineering.- Capturing Product Line Information from Legacy User Documentation.- Scenario-Based Application Requirements Engineering.- Consolidated Product Line Variability Modeling.- Product Line Architecture.- Dealing with Architectural Variation in Product Populations.- A Software Product Line Reference Architecture for Security.- Architecture Reasoning for Supporting Product Line Evolution: An Example on Security.- A Method for Predicting Reliability and Availability at the Architecture Level.- Product Line Testing.- Product Line Use Cases: Scenario-Based Specification and Testing of Requirements.- System Testing of Product Lines: From Requirements to Test Cases.- The ScenTED Method for Testing Software Product Lines.- Specific Product Line Engineering Issues.- Incremental Systems Integration within Multidisciplinary Product Line Engineering Using Configuration Item Evolution Diagrams.- Software Product Line Engineering with the UML: Deriving Products.- Evaluation Framework for Model-Driven Product Line Engineering Tools.

The classification of surfaces with p_g = q = 0 isogenous to a product of curves

Abstract: We classify all the surfaces with p_g = q = 0 which admit an unramified covering which is isomorphic to a product of curves. Beyond the trivial case \PP^1 x \PP^1 we find 17 families which we explicitly describe. We reduce the problem to a combinatorial description of certain generating systems for finite groups which we solve using also MAGMA's library of groups of small order.

Mixable shuffles, quasi-shuffles and Hopf algebras

Abstract: The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota---Baxter algebras.

Minimal surfaces of Riemann type in three-dimensional product manifolds

Abstract: We construct and classify minimal surfaces foliated by horizontal curves of constant curvature in H2×R, R2×R and S2×R. The main tool is the existence of a Shiffman Jacobi field; such fields characterize the property of being foliated by circles in these product manifolds.

Apparatus, system, and method for optical verification of product information

Abstract: An apparatus, system, and method are disclosed for verifying product information. The apparatus includes an identification module configured to obtain a first set of data associated with a product. A database, in communication with the identification module, is configured to store data associated with the product in a database. An image module is configured to capture at least one image of a product. A recognition module, in communication with the image module, is configured to obtain a second set of data from at least one captured image of the product. A determination module configured to determine whether a match exists between the second set of data and the first set of data by comparing the second set of data with data in the database.

Data mining for improvement of product quality

Abstract: The assemble-to-order strategy delays the final assembly operations of a product until a customer order is received. The modules used in the final assembly operation result in a large product diversity. This production strategy reduces the customer waiting time for the product. As the lead-time is short, any product rework may violate the delivery time. Since quality tests can be performed on the stocked modules without impacting the assembly schedule, the quality of the final assembly operations should be the focus. The data-mining approach presented in this paper uses the production data to determine the sequence of assemblies that minimizes the risk of producing faulty products. The extracted knowledge plays an important role in sequencing modules and forming product families that minimize the cost of production faults. The concepts introduced in the paper are illustrated with numerical examples.

Using pointwise mutual information to identify implicit features in customer reviews

Abstract: This paper is concerned with automatic identification of implicit product features expressed in product reviews in the context of opinion question answering. Utilizing a polarity lexicon, we map each adjectives in the lexicon to a set of predefined product features. According to the relationship between those opinion-oriented words and product features, we could identify what feature a review is regarding without the appearance of explicit feature nouns or phrases. The results of our experiments proved the validity of this method.

When to use features and aspects?: a case study

Abstract: Aspect-Oriented Programming (AOP) and Feature-Oriented Programming (FOP) are complementary technologies that can be combined to overcome their individual limitations. Aspectual Mixin Layers (AML) is a representative approach that unifies AOP and FOP. We use AML in a non-trivial case study to create a product line of overlay networks. We also present a set of guidelines to assist programmers in how and when to use AOP and FOP techniques for implementing product lines in a stepwise and generative manner.

Sparse finite element methods for operator equations with stochastic data

Abstract: Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed) An operator equation Au = f with stochastic data f is considered The goal of the computation is the mean field and higher moments $$\mathcal{M}^1 u \in V,\mathcal{M}^2 u \in V \otimes V,,\mathcal{M}^k u \in V \otimes \otimes V$$ of the solution We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment $$\mathcal{M}^k u$$ for k⩾1 The key tool in both algorithms is a “sparse tensor product” space for the approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom, instead of N k degrees of freedom for the full tensor product FEM space A sparse Monte-Carlo FEM with M samples (ie, deterministic solver) is proved to yield approximations to $$\mathcal{M}^k u$$ with a work of O(M N(log N) k−1) operations The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples The deterministic FEM is based on deterministic equations for $$\mathcal{M}^k u$$ in D k ⊂ ℝkd Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom converging at an optimal rate (up to logs) For nonlocal operators wavelet compression of the operators is used The linear systems are solved iteratively with multilevel preconditioning This yields an approximation for $$\mathcal{M}^k u$$ with at most O(N (log N) k+1) operations

Noncommutative symmetric functions III : D eformations of C auchy and convolution algebras

Abstract: This paper discusses various deformations of free associative algebras and of their convolution algebras Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q -analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics (the quon algebra, generalized Brownian motion)