Showing papers on "Product (mathematics) published in 2010"

Pseudo-Hermitian Representation of Quantum Mechanics

Abstract: A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as ${\mathcal{P}\mathcal{T}}$, the true meaning and significance of the so-called charge operators $\mathcal{C}$ and the ${\mathcal{C}\mathcal{P}\mathcal{T}}$-inner products,...

Delta-oriented programming of software product lines

Abstract: Feature-oriented programming (FOP) implements software product lines by composition of feature modules. It relies on the principles of stepwise development. Feature modules are intended to refer to exactly one product feature and can only extend existing implementations. To provide more flexibility for implementing software product lines, we propose delta-oriented programming (DOP) as a novel programming language approach. A product line is represented by a core module and a set of delta modules. The core module provides an implementation of a valid product that can be developed with well-established single application engineering techniques. Delta modules specify changes to be applied to the core module to implement further products by adding, modifying and removing code. Application conditions attached to delta modules allow handling combinations of features explicitly. A product implementation for a particular feature configuration is generated by applying incrementally all delta modules with valid application condition to the core module. In order to evaluate the potential of DOP, we compare it to FOP, both conceptually and empirically.

Digital point-of-sale analyzer

Abstract: A digital point-of-sale system for determining key performance indicators (KPIs) at a point-of-sale includes a product identification unit and a realogram creation unit. The product identification unit is configured to receive a captured image of a product display and to identify products in the captured image by comparing features determined from the captured image to features determined from products templates. The realogram creation unit is configured to create a realogram from the identified products and product templates. A product price KPI unit is configured to identify a product label proximally located to each identified product, and to recognize the product price on each product label. Each product price is compared to a predetermined range of prices to determine whether the product label proximally located to the identified product is a correct product label for the identified product.

Uniqueness of enhancement for triangulated categories

Abstract: The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded de- rived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de- rived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

Affine linear sieve, expanders, and sum-product

Abstract: Let $\mathcal{O}$ be an orbit in ℤ n of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (eg isomorphic to SL2) We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$ at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$ This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free

Algebraic approach to dynamics of multivalued networks

Abstract: Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is expressed as a multilinear...

Higher order decompositions of ordered operator exponentials

Abstract: We present a decomposition scheme based on Lie–Trotter–Suzuki product formulae to approximate an ordered operator exponential with a product of ordinary operator exponentials. We show, using a counterexample, that Lie–Trotter–Suzuki approximations may be of a lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. To address this problem, we present a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds on the approximation error. This work may shed light on why related product formulae fail to be as accurate as expected when applied to Coulomb potentials.

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

Abstract: We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira–Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.

Type safety for feature-oriented product lines

Abstract: A feature-oriented product line is a family of programs that share a common set of features. A feature implements a stakeholder's requirement and represents a design decision or configuration option. When added to a program, a feature involves the introduction of new structures, such as classes and methods, and the refinement of existing ones, such as extending methods. A feature-oriented decomposition enables a generator to create an executable program by composing feature code solely on the basis of the feature selection of a user--no other information needed. A key challenge of product line engineering is to guarantee that only well-typed programs are generated. As the number of valid feature combinations grows combinatorially with the number of features, it is not feasible to type check all programs individually. The only feasible approach is to have a type system check the entire code base of the feature-oriented product line. We have developed such a type system on the basis of a formal model of a feature-oriented Java-like language. The type system guaranties type safety for feature-oriented product lines. That is, it ensures that every valid program of a well-typed product line is well-typed. Our formal model including type system is sound and complete.

Abstract delta modeling

Abstract: Delta modeling is an approach to facilitate automated product derivation for software product lines. It is based on a set of deltas specifying modifications that are incrementally applied to a core product. The applicability of deltas depends on feature-dependent conditions. This paper presents abstract delta modeling, which explores delta modeling from an abstract, algebraic perspective.Compared to previous work, we take a more flexible approach with respect to conflicts between modifications and introduce the notion of conflict-resolving deltas. We present conditions on the structure of deltas to ensure unambiguous product generation.

Eigenvalues and singular values of products of rectangular gaussian random matrices.

Abstract: We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that they both have power--law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.

Computable representations for convex hulls of low-dimensional quadratic forms

Abstract: Let $${\mathcal{C}}$$ be the convex hull of points $${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$$. Representing or approximating $${\mathcal{C}}$$is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and $${\mathcal{F}}$$is a simplex, then $${\mathcal{C}}$$has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and $${\mathcal{F}}$$is a box, then $${\mathcal{C}}$$has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of $${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$when $${\mathcal{F}\subset\Re^2}$$is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of $${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$. When n = 3 and $${\mathcal{F}}$$is a box, we show that a representation for $${\mathcal{C}}$$can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.

A strong direct product theorem for disjointness

Abstract: A strong direct product theorem states that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then the overall success probability will be exponentially small in k. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication const• kn the success probability of solving k instances of size n can only be exponentially small in k. This solves an open problem of [KSW07, LSS08]. We also show that this bound even holds for $AM$-communication protocols with limited ambiguity. The main result implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a so-called Intersection Sampling Lemma that generalizes a result by Razborov [Raz92].

Incremental Test Generation for Software Product Lines

Abstract: Recent advances in mechanical techniques for systematic testing have increased our ability to automatically find subtle bugs, and hence, to deploy more dependable software. This paper builds on one such systematic technique, scope-bounded testing, to develop a novel specification-based approach for efficiently generating tests for products in a software product line. Given properties of features as first-order logic formulas in Alloy, our approach uses SAT-based analysis to automatically generate test inputs for each product in a product line. To ensure soundness of generation, we introduce an automatic technique for mapping a formula that specifies a feature into a transformation that defines incremental refinement of test suites. Our experimental results using different data structure product lines show that an incremental approach can provide an order of magnitude speedup over conventional techniques. We also present a further optimization using dedicated integer constraint solvers for feature properties that introduce integer constraints, and show how to use a combination of solvers in tandem for solving Alloy formulas.

Cluster aggregation model for discontinuous percolation transitions

Abstract: The evolution of the Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi (ER) network by adding edges is a basis model for irreversible kinetic aggregation phenomena. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel ${K}_{ij}\ensuremath{\sim}ij$, where $ij$ is the product of the sizes of two merging clusters. Here we study that when the giant cluster is discouraged to develop by a sublinear kernel ${K}_{ij}\ensuremath{\sim}{(ij)}^{\ensuremath{\omega}}$ with $0\ensuremath{\le}\ensuremath{\omega}l1/2$, the percolation transition (PT) is discontinuous. Such discontinuous PT can occur even when the ER dynamics evolves from proper initial conditions. The obtained evolutionary properties of the simple model sheds light on the origin of the discontinuous PT in other nonequilibrium kinetic systems.

Quasitoric manifolds over a product of simplices

Abstract: A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^{1})^{n}$ (resp. $(\mathbb{Z}_{2})^{n}$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\mathbb{F}$, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, is a sequence of projective bundles of the Whitney sum of $\mathbb{F}$-line bundles starting with a point. Each stage of the tower over $\mathbb{F}$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\mathbb{F}=\mathbb{C}$) and small covers (when $\mathbb{F}=\mathbb{R}$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\mathbb{Q}$ coefficients.

On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions

Abstract: We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Holder, and Young inequalities.

Sum-product Estimates in Finite Fields via Kloosterman Sums

Abstract: We establish improved sum-product bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.

Template-based recognition of food product information

Abstract: A method for template based recognition of food product information includes capturing an image of food product packaging and extracting an attribute from the image. The attribute is used to find a matching template in a template-database. The matching template is then used to extract food product information from the image.

Food safety indicator

Abstract: A food safety device for placement on a product is disclosed. The food safety device comprises one or more sensors that are configured to measure at least one condition of the product and/or its environment, one or more visual indicators that are configured to display a visual indication of freshness and/or safety of the product, an antenna that is configured to transmit and receive data regarding the at least one measured condition of the product and the freshness and/or safety of the product, and a logic module that is configured to execute programmable logic to determine the freshness and/or safety of the product from the at least one measured condition of the product, to cause the one or more visual indicators to display a visual indication of the freshness and/or safety it determines, and to transmit and receive data regarding the at least one measured condition of the product and the freshness and/or safety of the product via the antenna.

A probabilistic technique for finding almost-periods of convolutions

Abstract: We introduce a new probabilistic technique for finding ‘almost-periods’ of convolutions of subsets of groups. This gives results similar to the Bogolyubovtype estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth’s theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain–Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, . . . , N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov–Freiman–Halberstam–Ruzsa-type results of additive combinatorics, showing that product sets A 1 · A 2 · A 3 and A 2 · A −2 are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in A · B. Our results are ‘local’ in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they ‘interact nicely’ with some other set.

A tool to estimate materials and manufacturing energy for a product

Abstract: This study proposes an easy-to-use methodology to estimate the materials embodied energy and manufacturing energy for a product. The tool requires as input the product's Bill of Materials and the knowledge on how these materials are processed (or an educated guess); the resulting output represents the sum of all the energy inputs into a product system in the form of a value range on the energy requirements of the product during its beginning of life. This includes extraction of materials, processing, and manufacture of the final product.

Automatic method to generate product attributes based solely on product images

Abstract: Disclosed is a method to generate data describing a product. The method includes the steps of comparing a digitized query image of the product to digitized pre-existing product images in a pre-existing product database. The pre-existing product database is organized using a taxonomy and an ontology. The pre-existing product images are linked to a corresponding node in the taxonomy and are also linked to attribute data and attribute value data in the ontology. At least one pre-existing product image is then retrieved that most closely matches the query image based on at least one matching criterion selected in whole or in part by a user. From the pre-existing product database is extracted the node in the taxonomy, the attribute data, or the attribute value data linked to the pre-existing product image retrieved earlier. In this fashion, product data relevant to the item depicted in the query image can be generated automatically from existing product data.

Pure Spinors on Lie groups

Abstract: For any manifold M, the direct sum TM oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of emph{pure spinor}. In this paper, we study pure spinors and Dirac structures in the case when M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G semi-simple. The applications of our theory include the construction of distinguished volume forms on conjugacy classes in G, and a new approach to the theory of quasi-Hamiltonian G-spaces.

A probabilistic technique for finding almost-periods of convolutions

Abstract: We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, ..., N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A B C and A^2 A^{-2} are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in product sets A B. Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Abstract: Let Γ be a countable group and denote by \({\mathcal{S}}\) the equivalence relation induced by the Bernoulli action \({\Gamma\curvearrowright [0, 1]^{\Gamma}}\), where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation \({\mathcal{R}}\) of \({\mathcal{S}}\), there exists a partition {Xi}i≥0 of [0, 1]Γ into \({\mathcal{R}}\)-invariant measurable sets such that \({\mathcal{R}_{\vert X_{0}}}\) is hyperfinite and \({\mathcal{R}_{\vert X_{i}}}\) is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.