Showing papers on "Product (mathematics) published in 2004"

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.

Improved simulation of stabilizer circuits

Abstract: The Gottesman-Knill theorem says that a stabilizer circuit\char22{}that is, a quantum circuit consisting solely of controlled-NOT (CNOT), Hadamard, and phase gates\char22{}can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor of 2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely available program called CHP (CNOT-Hadamard-phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class $\ensuremath{\bigoplus}\mathsf{L}$, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any $n$-qubit stabilizer circuit into a ``canonical form'' that requires at most $O({n}^{2}∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}n)$ gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.

Characterizing complex product architectures

Abstract: Due to the large-scale nature of complex product architectures, it is necessary to develop some form of abstraction in order to be able to describe and grasp the structure of the product, facilitating product modularization. In this paper we develop three methods for describing product architectures: (a) the Dependency Structure Matrix (DSM), (b) Molecular Diagrams (MD), and (c) Visibility-Dependency (VD) signature diagrams. Each method has its own language (and abstraction), which can be used to qualitatively or quantitatively characterize any given architecture spanning the modular-integrated continuum. A consequence of abstraction is the loss of some detail. So, it is important to choose the correct method (and resolution) to characterize the architecture in order to retain the salient details. The proposed methods are suited for describing architectures of varying levels of complexity and detail. The three methods are demonstrated using a sequence of illustrative simple examples and a case-study analysis of a complex product architecture for an industrial gas turbine. © 2003 Wiley Periodicals, Inc. Syst Eng 7: 35–60, 2004

Associativity of crossed products by partial actions, enveloping actions and partial representations

Abstract: Given a partial action a of a group G on an associative algebra A, we consider the crossed product A × α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of A × α G obtained in the context of C*-algebras. In particular, we prove that A × α G is associative, provided that A is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

Software Product Lines

Abstract: It’s a fact of life that organizations love to reorganize. Reorganizations have a profound effect on the way product lines are governed. We introduce the concept of the Responsibility, Authority, and Accountability (RAA) network. An RAA network assists in the governance process of product lines for internal information systems, even in the face of massive reorganization. Armour (“Reorg Cycle” [1]) describes the pressures of reorganization to balance the “dimensions of organization” (e.g., geography, customers, product technology); we apply polarity management to balance the dimensions. Armour describes the difficulty of applying hierarchical organization charts—“single dimension management structures”—to the above “multidimensional environments”; we apply lean RAA networks to span organization charts and provide the multidimensional view needed for product lines. Armour observes that network organization approaches “do not have a good track record”; our experience is that lean, resilient RAA networks document the product line’s governance architecture. These governance architecture patterns are applied repeatedly to the strata of products in the product line. We present the governance architect’s RAA to define, monitor, and sustain governance health using these tools: polarity maps, polarity networks, RAA maps, and RAA networks.

There are only finitely many Diophantine quintuples

Abstract: A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square Diophantus found a set of four positive rationals with the above property The first Diophantine quadruple was found by Fermat (the set {; ; 1, 3, 8, 120}; ; ) Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quintuple In this paper, we prove that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples

Data communication system with multi-dimensional error-correction product codes

Abstract: A data communication system has a combiner circuit that combines a set of information symbols with error correction codes and that generates a set of product codes that are at least three dimensional. A communication channel receives the set of product codes and provides the set of product codes with errors after a channel delay. A channel detector receives the set of product codes with the errors and generates a channel detector output. An error correction circuit receives the channel detector output and iteratively removes the errors to provide a set of reproduced information symbols with a reduced number of the errors.

Towards a UML profile for software product lines

Abstract: This paper proposes a UML profile for software product lines. This profile includes stereotypes, tagged values, and structural constraints and it makes possible to define PL models with variabilities. Product derivation consists in generating product models from PL models. The derivation should preserve and ensure a set of constraints which are specified using the OCL.

On the quantum product of Schubert classes

Abstract: We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Peterson's quantum version of Chevalley's formula, also for general G/P's.

Harmonic analysis on the infinite symmetric group

Abstract: The infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space l2(S(∞)) generates a type II1 von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification \(\mathfrak{S}\supset{S(\infty)}\), which we call the space of virtual permutations. Although \(\mathfrak{S}\) is no longer a group, it still admits a natural two–sided action of S(∞). Thus, \(\mathfrak{S}\) is a G–space, where G stands for the product of two copies of S(∞). On \(\mathfrak{S}\), there exists a unique G-invariant probability measure μ1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ1 into a family {μt: t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {Tz: z∈ℂ} of unitary representations of G, called generalized regular representations (each representation Tz with z≠=0 can be realized in the Hilbert space \(L^2(\mathfrak{S}, \mu_t)\), where t=|z|2). As |z|→∞, the generalized regular representations Tz approach, in a suitable sense, the “naive” two–sided regular representation of the group G in the space l2(S(∞)). In contrast with the latter representation, the generalized regular representations Tz are highly reducible and have a rich structure. We prove that any Tz admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z1, z2, the spectral types of the representations \(T_{z_1}\) and \(T_{z_2}\) are shown to be disjoint. In the case z∈ℤ, a complete description of the spectral type is obtained. Further work on the case z∈ℂ∖ℤ reveals a remarkable link with stochastic point processes and random matrix theory.

Product Variety Optimization Simultaneously Designing Module Combination and Module Attributes

Abstract: This paper proposes a product variety optimization method for both module combination and module attributes of multiple products. As manufacturing competition has become restricted with high profitability and external constraints, simultaneous design of multiple products, called product variety design etc., becomes an important strategy. The systems-based optimal design paradigm is essential for rationalizing such practices, since design for product variety is more complicated than for a single product. In such a direction, we configure an optimization method for simultaneously designing both module combination and module attributes across multiple products. The optimization method hybridizes a genetic algorithm, a mixed-integer programming method with a branch-and-bound technique, and a constrained nonlinear programming method, i.e., a successive quadratic programming method. In its optimization process, the first optimizes the combinatorial pattern of module commonality and similarity among different pr...

Online curriculum handling system including content assembly from structured storage of reusable components

Abstract: In a curricula system, courses are assembled from components where items are created by authors and/or authoring tools, items comprising questions, answers, narrative, media, interactions, or the like, and instructional designers design products that specify structures, strategies and the like for a product. A product, in the form of one or more course presentable to a student or other user, references reusable content items, such that a given item can be present in more than one unrelated product. A product can be represented with a hierarchical structure organized as one or more course each comprising one or more units, in turn comprising one or more modules, such as testing modules and learning modules, wherein each module in turn specifies items to be included. Products can take on one or more product template and one or more product class, wherein a product template specifies the “look-and-feel” of the product and the product class defines basic course structure and functionality. Items are atomic objects that are reusable across products, courses, units, etc. The atomic objects can be created in advance and can be stored in a non-product specific manner.

Multiple-view feature modelling for integral product development

Abstract: To allow a designer to focus on the information that is relevant for a particular product development phase, is an important aspect of integral product development. Unlike current modelling systems, multiple-view feature modelling can adequately support this, by providing an own view on a product for each phase. Each view contains a feature model of the product specific for the corresponding phase. An approach to multiple-view feature modelling is presented that supports conceptual design, assembly design, part detail design and part manufacturing planning. It does not only provide views with form features to model single parts, as previous approaches to multiple-view feature modelling did, but also a view with conceptual features, to model the product configuration with functional components and interfaces between these components, and a view with assembly features, to model the connections between components. The general concept of this multiple-view feature modelling approach, the functionality of the four views, and the way the views are kept consistent, are described.

Consumer product status monitoring

Abstract: Data collection and display representative of at least one characteristic relevant to a viability product contained in a sealed package, via monitoring at least one characteristic associated with a state condition of said product. A viability state condition may be related to probability of degradation of the product. The monitoring may include analyzing data associated with at least one said characteristic and, based on said analyzing, determining at least one critical date of said product which may be displayed. When current state of viability can be predicted, freezing said display with said signal representative of an expiration or maturity state condition, or dynamically altering a display accordingly. Options for displaying a history of said characteristic may be provided.

Quantum weights of dyons and of instantons with nontrivial holonomy

Abstract: We calculate exactly functional determinants for quantum oscillations about periodic instantons with a nontrivial value of the Polyakov line at spatial infinity. Hence, we find the weight or the probability with which calorons with nontrivial holonomy occur in the Yang-Mills partition function. The weight depends on the value of the holonomy, the temperature, ${\ensuremath{\Lambda}}_{\mathrm{QCD}},$ and the separation between the BPS monopoles (or dyons) that constitute the periodic instanton. At large separation between constituent dyons, the quantum measure factorizes into a product of individual dyon measures, times a definite interaction energy. We present an argument that at temperatures below a critical one related to ${\ensuremath{\Lambda}}_{\mathrm{QCD}},$ trivial holonomy is unstable, and that calorons ``ionize'' into separate dyons.

Patch un-installation

Abstract: Uninstalling a patch applied to a software product installed on a computer. A current state of a software product is identified. A desired state of the software product is determined. The desired state of the software product represents a state of the software product when a patch applied to the software product is removed from the software product. The patch is removed from the software product to transition the software product from the identified current state to the determined desired state.

On the product theory of singular integrals

Abstract: We establish Lp-boundedness for a class of product singular integral operators on spaces M = M1 x M2 x . . . x Mn. Each factor space Mi is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on Mi are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood- Paley theory on M. This in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor.

Computer-Aided Methods and Tools for Chemical Product Design

Abstract: Chemical product design involves a very large search space and evaluation of numerous alternatives. It also requires a number of tools and different sources of data for properties that define the needs of the chemical product. In this paper, a framework for computer-aided chemical product design is presented together with a review of methods and tools that may be useful in chemical product design. The objective of this framework is to provide a set of integrated methods and tools so that some of the chemical product design steps can be carried out faster, over a wider search space and using less resources. The requirements for such a framework and its resident methods and tools are that there exists a reliable set of models. Within this context, the differences between the process model, the property model and the chemical product model are highlighted together with a discussion on the roles of property models with respect to chemical product design. Several types of chemical product design problems are highlighted together with the corresponding methods and tools that can be used. Illustrative examples highlighting the use of the framework are also presented.

Environment-Based Formulation Of Design Problem

Abstract: This paper proposes a formal definition of design problem using the axiomatic theory of design modeling. Based on the concept of product system and an informal description of design problem, a design problem is formulated into three parts: environment, structural requirements, as well as performance requirements. On the basis of this formulation, both structural requirements and performance requirements are related to product environment, which is divided into natural, built, and human environments. As a result, product environment is identified as the source of product requirements. An environment-based formulation of design problem is thus developed. This formulation implies that both design problem and design solutions are included in a product system. The design governing equation is formulated to capture the interdependence relation between product problem and product descriptions along the design process. An example of rivet setting tool design is used to illustrate the mathematical formulations throughout the paper.

Secure product packaging system

Abstract: A product security system includes an RFID chip and antenna that are polymerized onto separable parts of a commercial product package. The RFID chip includes a unique serial number that can be interrogated by a wireless reader. A database of such unique serial numbers associated with particular manufacturing production runs is used in a method to detect counterfeiting. The RFID chip and antenna are embedded such that attempts to remove or transfer them will be obvious to an inspector.

On the maximal dimension of a completely entangled subspace for finite level quantum systems

Abstract: LetH ibe a finite dimensional complex Hilbert space of dimensiond i associated with a finite level quantum system Ai for i = 1, 2, ...,k. A subspaceS ⊂ $${\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k $$ is said to becompletely entangled if it has no non-zero product vector of the formu 1⊗u 2 ⊗ ... ⊗u k with ui inH i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that $$\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1$$ where e is the collection of all completely entangled subspaces. When $${\mathcal{H}} = {\mathcal{H}}_2 $$ andk = 2 an explicit orthonormal basis of a maximal completely entangled subspace of $${\mathcal{H}}_1 \otimes {\mathcal{H}}_2 $$ is given. We also introduce a more delicate notion of aperfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.

Fast backtrack-free product configuration using a precompiled solution space representation

Abstract: In this paper we describe a two-phase approach to interactive product configuration. In the first phase, a compressed symbolic representation of the set of valid configurations (the solution space) is compiled offline. In the second phase, this representation is embedded in an online configurator and utilized for fast, complete, and backtrack-free interactive product configuration. The main advantage of our approach compared to online search-based approaches is that we avoid searching for valid solutions in each iteration of the interactive configuration process. The computationally hard part of the problem is fully solved in the offline phase given that the produced symbolic representation is small. The employed symbolic representation is Binary Decision Diagrams (BDDs). More than a decade of research in formal verification has shown that BDDs often compactly encode formal models of systems encountered in practice. To our experience this is also the case for product models. Often the compiled BDD is small enough to be embedded directly in hardware. Our research has led to the establishment of a spin-off company called Configit Software A/S. Configit has developed software for writing product models in a strongly typed language and has patented a particularly efficient symbolic representation called Virtual Tables.

Systems and methods for planning demand for configurable products

Abstract: Systems and methods are disclosed for planning demand of a product, such as a configurable product. In one embodiment, a method is provided for planning demand for a configurable product in a managed supply chain. The method may comprise the steps of storing data relating to the possible combinations of characteristics defining configurable products, wherein product dependent characteristic data and product independent characteristic data are stored dependently with respect to one another, and converting a planning demand for a configurable product into a planning demand for an actual combination of characteristics defining a configured product.

Method of evaluating the performance of a product using a virtual environment

Abstract: A method of evaluating a product for use on a body used to develop a preferred product configuration using a computer-based virtual product development and testing system. A virtual wearer sub-model is created of the body and a virtual product sub-model is created of a product for use on the body. An environment sub-model is generated so that environmental factors affecting the product or the body are also used in designing or evaluating the product. Instructions defining how the wearer sub-model, the product sub-model and the environment sub-model interact are introduced in an interaction model. The sub-models and the interaction defined by the interaction model are then combined to create a virtual use model simulating the use of the virtual product sub-model by the virtual wearer sub-model. The use model determines the forces, deformations and stresses caused by movement and interaction between the virtual wearer sub-model and the virtual product sub-model using numerical method analysis. The results of the use model are analyzed to evaluate the performance of product features embodied in the virtual product sub-model such as when positioned on a virtual wearer and exposed to typical movements or forces. The analysis evaluates the performance of at least one product feature of the product and/or wearer body.

The Middle Product Algorithm I

Abstract: We present new algorithms for the inverse, division, and square root of power series. The key trick is a new algorithm – MiddleProduct or, for short, MP – computing the n middle coefficients of a (2n−1)×n full product in the same number of multiplications as a full n×n product. This improves previous work of Brent, Mulders, Karp and Markstein, Burnikel and Ziegler. These results apply both to series and polynomials.

Method for interfacing scanned product information with a source for the product over a global network

Abstract: A method for interfacing scanned product information with a source for the product over a global network. A method is provided for obtaining information regarding the source of a product from a remote information source location on a global communication network utilizing a product code associated with the product and unique thereto. The product code associated with the product is scanned with a scanner at a user location on the global communication network to extract the information contained in the unique product code therefrom. A unique scan ID code is associated with the scanning operation and a packet of information assembled that is comprised of the extracted product code and the unique scan ID code to provide a routing packet. The user location is then connected to the remote information source location utilizing the routing packet and in response to the step of scanning, wherein the routing packet is representative of the location of the remote information source location on the global communication network through an association with a routing table.

Product dispenser and carrier

Abstract: A product dispenser and carrier (10) for attachment to a surface such as an inner surface of a dryer includes a plate member (11) and a product carrier (21). The plate member (11) attaches to the surface and the product carrier (21) releasably attaches to the plate member (11). Product (31) is operatively connected to the product carrier (21).

A sensitivity-based commonality strategy for family products of mild variation, with application to automotive body structures

Abstract: Identification of the product platform is a key step in designing a family of products. This article presents a methodology for selecting the product plat- form by using information obtained from the individual optimization of the product variants. Under the assump- tion that the product variety requires only mild design changes, a performance deviation vector is derived by taking into consideration individual optimal designs and sensitivities of functional requirements. Commonality de- cisions are based on values of the performance deviation vector, and the product family is designed optimally with respect to the chosen platform. The proposed method- ology is applied to the design of a family of automotive body structures. Variants are defined by changing the functional requirements they need to satisfy and/or the geometry of the associated finite element models.