Showing papers on "Product (mathematics) published in 2003"

Deformation Quantization of Poisson Manifolds

Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

Abstract: We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.

A sum-product estimate in finite fields, and applications

Abstract: Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.

Entangled states and collective nonclassical effects in two-atom systems

Abstract: We propose a review of recent developments on entanglement and non-classical effects in collective two-atom systems and present a uniform physical picture of the many predicted phenomena. The collective effects have brought into sharp focus some of the most basic features of quantum theory, such as nonclassical states of light and entangled states of multiatom systems. The entangled states are linear superpositions of the internal states of the system which cannot be separated into product states of the individual atoms. This property is recognized as entirely quantum-mechanical effect and have played a crucial role in many discussions of the nature of quantum measurements and, in particular, in the developments of quantum communications. Much of the fundamental interest in entangled states is connected with its practical application ranging from quantum computation, information processing, cryptography, and interferometry to atomic spectroscopy.

Puzzles and (equivariant) cohomology of Grassmannians

Abstract: The product of two Schubert cohomology classes on a Grassmannian ${\rm Gr}_k (\mathbb{c}^n)$ has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (eg, the Littlewood-Richardson rule or the more symmetric puzzle rule from A Knutson, T Tao, and C Woodward [KTW]) Recently, W~Graham showed in [G], nonconstructively, that a similar positivity statement holds for {\em $T$-equivariant} cohomology (where the coefficients are polynomials) We give the first manifestly positive formula for these coefficients in terms of puzzles using an ``equivariant puzzle piece'' The proof of the formula is mostly combinatorial but requires no prior combinatorics and only a modicum of equivariant cohomology (which we include) As a by-product the argument gives a new proof of the puzzle (or Littlewood-Richardson) rule in the ordinary-cohomology case, but this proof requires the equivariant generalization in an essential way, as it inducts backwards from the ``most equivariant'' case This formula is closely related to the one in A Molev and B Sagan [MS] for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [G] We include a cohomological interpretation of their problem and a puzzle formulation for it

Methods and systems for object identification and interaction

Abstract: A method of facilitating an interaction between a user and a product item, the product item having an identity and the method including a number of steps. The first step involves providing the user with an interface surface associated with the product item and containing information relating to the product item. The interface surface includes coded data indicative of the identity of the product item and of a number of reference points of the interface surface. The next step involves receiving, in a computer system, indicating data from a sensing device regarding the identity of the product item and a position of the sensing device relative to the interface surface. The sensing device, when placed in an operative position relative to the interface surface, sensing the coded data in the vicinity of the sensing device and generating the indicating data using the sensed coded data. The next step involves facilitating, in the computer system and with reference to the indicating data, the interaction between the user and the product item.

Compact Einstein warped product spaces with nonpositive scalar curvature

Abstract: We study Einstein warped product spaces. As a result, we prove the following: if M is an Einstein warped product space with nonpositive scalar curvature and compact base, then M is simply a Riemannian product space.

Clifford group, stabilizer states, and linear and quadratic operations over GF(2)

Abstract: We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a $(2n+1)\ifmmode\times\else\texttimes\fi{}(2n+1)$ binary matrix product and binary quadratic forms. As an application we give two schemes to efficiently decompose Clifford group operations into one- and two-qubit operations. We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. Our results are useful for quantum error correction, entanglement distillation, and possibly quantum computing.

Content aggregation method and apparatus for an on-line product catalog

Abstract: The method comprises processing plural product information records from the product information sources into one or more groups based on which product information records are likely to correspond to the same product, correlating a unique product ID corresponding to the product associated with each of said groups to identify the product, comparing each identified product to categories of a taxonomy to determine a category for the identified products in the taxonomy, and determining attributes for each categorized product based on the product information records corresponding to each group, creating product specifications based on the determined attributes and storing the product specification in the corresponding determined categories of the taxonomy.

Determination of burn-in parameters and residual life for highly reliable products

Abstract: Today, many products are designed and manufactured to function for a long period of time before they fail. Determining product reliability is a great challenge to manufacturers of highly reliable products with only a relatively short period of time available for internal life testing. In particular, it may be difficult to determine optimal burn-in parameters and characterize the residual life distribution. A promising alternative is to use data on a quality characteristic (QC) whose degradation over time can be related to product failure. Typically, product failure corresponds to the first passage time of the degradation path beyond a critical value. If degradation paths can be modeled properly, one can predict failure time and determine the life distribution without actually observing failures. In this paper, we first use a Wiener process to describe the continuous degradation path of the quality characteristic of the product. A Wiener process allows nonconstant variance and nonzero correlation among data collected at different time points. We propose a decision rule for classifying a unit as normal or weak, and give an economic model for determining the optimal termination time and other parameters of a burn-in test. Next, we propose a method for assessing the product's lifetime distribution of the passed units. The proposed methodologies are all based only on the product's initial observed degradation data. Finally, an example of an electronic product, namely contact image scanner (CIS), is used to illustrate the proposed procedure. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003

Formal methods for the validation of automotive product configuration data

Abstract: In the automotive industry, the compilation and maintenance of correct product configuration data is a complex task. Our work shows how formal methods can be applied to the validation of such business critical data. Our consistency support tool BIS works on an existing database of Boolean constraints expressing valid configurations and their transformation into manufacturable products. Using a specially modified satisfiability checker with an explanation component, BIS can detect inconsistencies in the constraints set and thus help increase the quality of the product data. BIS also supports manufacturing decisions by calculating the implications of product or production environment changes on the set of required parts. In this paper, we give a comprehensive account of BIS: the formalization of the business processes underlying its construction, the modifications of satisfiability-checking technology we found necessary in this context, and the software technology used to package the product as a client–server information system.

Product dispensing controlled by RFID tags

Abstract: RFID tags attached to a product contain information regarding the product and communicate this information to a reader. This information can include the age of the product, the amount of product left and preparation or care instructions for the product. To record the age of a product, a timer is initiated in the RFID tag. Thereafter, the elapsed time on the timer can be read and displayed by the reader. For other products, the RFID tag can have contacts monitoring the level of remaining product. Once the contacts sense that the amount of product is low, this information can be transmitted and communicated by the reader. Preparation and care instructions are prerecorded on the tag. Instructions can also be communicated between a product and a base, such as a heater.

Products having rfid tags to provide information to product consumers

Abstract: RFID tags attached to a product contain information regarding the product and communicate this information to a reader. This information can include the age of the product, the amount of product left and preparation or care instructions for the product. To record the age of a product, a timer is initiated in the RFID tag. Thereafter, the elapsed time on the timer can be read and displayed by the reader. For other products, the RFID tag can have contacts monitoring the level of remaining product. Once the contacts sense that the amount of product is low, this information can be transmitted and communicated by the reader. Preparation and care instructions are prerecorded on the tag. Instructions can also be communicated between a product and a base, such as a heater.

Methods and systems for external localization

Abstract: Embodiments of the present invention relate to methods, systems and computer-readable media for external localization of a software product. This external localization comprises loading a base product having one or more calls to an operating system to load language specific data. A language specific localized product is generated, during loading of the base product, by converting the base product to a localized product using and at least one language map containing language specific data. The localized product is then executed in place of the base product.

The five independences as natural products

Abstract: Let be the class of all algebraic probability spaces. A "natural product" is, by definition, a map which is required to satisfy all the canonical axioms of Ben Ghorbal and Schurmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

Automated configuration system and method

Abstract: An automated configuration system (and method) is provided for facilitating the configuration of desired products, services, or other assemblages that require users to gather and assimilate disparate knowledge of makes, models, types, features, codes, and prices of the desired product/service to be configured. In accordance with a preferred embodiment, configuration is facilitated through interaction of a user with a frame engine that performs frame-based inferences to discern stored knowledge of a product (or the like), as supplemented by a rules-based inference system.

Product engineering of dispersed systems

Abstract: Product engineering is the design of products with desired properties using the methods of process engineering. To reduce the number of process parameters, necessary to obtain the desired properties, a characteristic feature of the respective product is helpful. For a dispersed system this characteristic feature is its complex structure on a micro-scale. This micro-structure is the link between the process and the properties of the final product. The principles of product engineering of dispersed systems are explained and demonstrated by means of different examples: Though there are limits to this concept, certain properties of dispersed systems can be obtained by determining and applying property and process functions.

Quantum Cohomology of Grassmannians

Abstract: We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.

Value distribution of the product of a meromorphic function and its derivative

Abstract: In the paper we discuss the value distribution of the product of a meromorphic function and its derivative and we improve a recent result of K.W.Yu.

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.

The massless two-loop two-point function

Abstract: We consider the massless two-loop two-point function with arbitrary powers of the propagators and derive a representation from which we can obtain the Laurent expansion to any desired order in the dimensional regularization parameter e. As a side product, we show that in the Laurent expansion of the two-loop integral only rational numbers and multiple zeta values occur. Our method of calculation obtains the two-loop integral as a convolution product of two primitive one-loop integrals. We comment on the generalization of this product structure to higher loop integrals.

A Genetic Algorithm for Developing Modular Product Architectures

Abstract: The architecture of a product is determined by both the elements that compose the product and the way in which they interact with each other. In this paper, we use the design structure matrix (DSM) as a tool to capture this architecture. Designing modular products can result in many benefits to both consumers and manufacturers. The development of modular products requires the identification of highly interactive groups of elements and arranging (i.e. clustering) them into modules. However, no rigorous DSM clustering technique can be found in product development literature. This paper presets a review of the basic DSM building blocks used in the identification of product modules. The DSM representation and building blocks are used to develop a new DSM clustering tool based on a genetic algorithm (GA) and the minimum description length (MDL) principle. The new tool is capable of partitioning the product architecture into an “optimal” set of modules or sub-systems. We demonstrate this new clustering method using an example of a complex product architecture for an industrial gas turbine.Copyright © 2003 by ASME

On the prime submodules of multiplication modules

Abstract: By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results.

Multi-tiered structure for storing and displaying product and process variants

Abstract: A computer based system is provided that is capable of displaying component and process structures for configurable product classes stored in a data structure. Product classes in the data structure are represented by product nodes. Components, which are abstractions of concrete items used in the product, are represented by component nodes. Component nodes are mapped to each product class with which the component is associated, and are associated with one or more component variants, which are concrete items used in the manufacture of a particular product variant. Process nodes, which are abstractions of actual steps required to produce a product variant, known as process variants, are represented by process nodes. The computer based system can display several alternative views of the data structure. For example, upon receiving a selection for a particular product class, the computer based system converts and can graphically display all component and process nodes associated with that product class. Alternatively, the system can receive input of a product variant and display all component variants used in its production, which is its bill of materials.

Servicing a component-based software product throughout the software product lifecycle

Abstract: Updating component-based software product with files stored in a service package. The service package also includes a plurality of instruction sets each corresponding to a state or operating context of a component in the software product. A component installer determines the state of the component and applies the instruction set corresponding to the determined state to install the files.

On the Complexity of Matrix Product

Abstract: Our main result is a lower bound of $\Omega(m^2 \log m)$ for the size of any arithmetic circuit for the product of two matrices, over the real or complex numbers, as long as the circuit does not use products with field elements of absolute value larger than 1 (where m × m is the size of each matrix). That is, our lower bound is superlinear 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. We also prove size-depth tradeoffs for such circuits: We show that if a circuit, as above, is of depth d, then its size is $\Omega(m^{2+ 1/O(d)})$.

Two Proofs of a Conjecture of Hori and Vafa

Abstract: We give two proofs of a conjecture of Hori and Vafa which expresses the J-function (i.e, the generating function for 1-point descendant Gromov-Witten invariants) of a Grassmannian in terms of the J-function of a product of projective spaces. Similar relations are obtained for two-point descendants, and three-point (primary) Gromov-Witten invariants. As an application we prove Givental's "R-Conjecture" - hence the Virasoro conjecture - for Grassmannians.

Use case-based testing of product lines

Abstract: This paper presents PLUTO, a simple and intuitive methodology to manage the testing process of product lines, described as Product Lines Use Cases (PLUCs). PLUCs are an extension of the well-known Cockburn's Use Cases, a notation based on natural language descriptions of requirements. The proposed test methodology is based on the Category Partition method, and can be used to derive a generic Test Specification for the product line, and a set of relevant test scenarios for a customer specific application.