Showing papers on "Product (mathematics) published in 2009"

The Series Product and Its Application to Quantum Feedforward and Feedback Networks

Abstract: The purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks. These methods build on and generalize existing methods for series (or cascade) connections by allowing for more general interfaces, and by introducing an efficient algebraic tool, the series product. We also introduce another product, which we call the concatenation product, that is useful for assembling and representing systems without necessarily having connections. We show how the concatenation and series products can be used to describe feedforward and feedback networks. A selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology.

Lovelock black holes with a nonlinear Maxwell field

Abstract: We derive electrically charged black hole solutions of the Einstein-Gauss-Bonnet equations with a nonlinear electrodynamics source in $n(\ensuremath{\ge}5)$ dimensions. The spacetimes are given as a warped product ${\mathcal{M}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathcal{K}}^{n\ensuremath{-}2}$, where ${\mathcal{K}}^{n\ensuremath{-}2}$ is a $(n\ensuremath{-}2)$-dimensional constant curvature space. We establish a generalized Birkhoff's theorem by showing that it is the unique electrically charged solution with this isometry and for which the orbit of the warp factor on ${\mathcal{K}}^{n\ensuremath{-}2}$ is non-null. An extension of the analysis for full Lovelock gravity is also achieved with a particular attention to the Chern-Simons case.

Model Checking of Domain Artifacts in Product Line Engineering

Abstract: In product line engineering individual products are derived from the domain artifacts of the product line. The reuse of the domain artifacts is constraint by the product line variability. Since domain artifacts are reused in several products, product line engineering benefits from the verification of domain artifacts. For verifying development artifacts, model checking is a well-established technique in single system development. However, existing model checking approaches do not incorporate the product line variability and are hence of limited use for verifying domain artifacts. In this paper we present an extended model checking approach which takes the product line variability into account when verifying domain artifacts. Our approach is thus able to verify that every permissible product (specified with I/O-automata) which can be derived from the product line fulfills the specified properties (specified with CTL). Moreover, we use two examples to validate the applicability of our approach and report on the preliminary validation results.

Sparse Tensor Discretization of Elliptic sPDEs

Abstract: We propose and analyze sparse deterministic-stochastic tensor Galerkin finite element methods (sparse sGFEMs) for the numerical solution of elliptic partial differential equations (PDEs) with random coefficients in a physical domain $D\subset\mathbb{R}^d$. In tensor product sGFEMs, the variational solution to the boundary value problem is approximated in tensor product finite element spaces $V^\Gamma\otimes V^D$, where $V^\Gamma$ and $V^D$ denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces, respectively. These approaches lead to sGFEM algorithms of complexity $O(N_\Gamma N_D)$, where $N_\Gamma=\dim V^\Gamma$ and $N_D=\dim V^D$. In this work, we use hierarchic sequences $V^\Gamma_1\subset V^\Gamma_2\subset\ldots$ and $V^D_1\subset V^D_2\subset\ldots$ of finite dimensional spaces to approximate the law of the random solution. The hierarchies of approximation spaces allow us to define sparse tensor product spaces $V^\Gamma_\ell\hat{\otimes}V^D_\ell$, $\ell=1,2,\dots$, yielding algorithms of $O(N_\Gamma\log N_D+N_D\log N_\Gamma)$ work and memory. We estimate the convergence rate of sGFEM for algebraic decay of the input random field Karhunen-Loeve coefficients. We give an algorithm for an input adapted a-priori selection of deterministic and stochastic discretization spaces. The convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to Monte Carlo approximations. Numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach.

A characterization of higher rank symmetric spaces via bounded cohomology

Abstract: Let M be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group Γ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover \(\tilde{M}\) is a higher rank symmetric space iff \(H^{2}_{b} (M; {\mathbb{R}}) \rightarrow H^{2}(M;{\mathbb{R}})\) is injective (and otherwise the kernel is infinite dimensional). This is the converse of a theorem of Burger–Monod. The proof uses the celebrated Rank Rigidity Theorem, as well as a new construction of quasi-homomorphisms on groups that act on CAT(0) spaces and contain rank 1 elements.

Warped product submanifolds of Kaehler manifolds with a slant factor

Abstract: Recently, we showed that there exist no warped product semi-slant submanifolds in Kaehler manifolds. On the other hand, Carriazo introduced anti-slant submanifolds as a particular class of bi-slant submanifolds. In this paper, we study such submanifolds in detail and show that they are useful to define a new kind of warped product submanifolds of Kaehler manifolds. In this direction, we obtain the existence of warped product hemi-slant (anti-slant) submanifolds with examples. We give a characterization theorem and establish an inequality for the squared norm of the second fundamental form in terms of the warping function for such submanifolds. The equality case is also considered.

Variability Modelling for Model-Driven Development of Software Product Lines

Abstract: Model-driven development of software-intensive systems aims at designing systems by stepwise model refine- ment. In order to create software product lines by model-driven development, product variability has to be represented on every modelling level and preserved under model refinement. In this paper, we propose ∆-modelling as an generally applicable variability modelling concept that is orthogonal to model refinement. Products on each modelling level are represented by a core model and a set of ∆-models specifying changes to the core to incorporate product features. Core and ∆-models can be refined independently to obtain a more detailed model of the product line. Based on a formalization of ∆-modelling, we establish conditions that model refinement and model configuration commute resulting in an incremental model- driven development process.

Approximating strongly correlated wave functions with correlator product states

Abstract: We describe correlator product states, a class of numerically efficient many-body wave functions to describe strongly correlated wave functions in any dimension. Correlator product states introduce direct correlations between physical degrees of freedom in a simple way, yet provide the flexibility to describe a wide variety of systems. We show that many interesting wave functions can be mapped exactly onto correlator product states, including Laughlin’s quantum Hall wave function, Kitaev’s toric code states, and Huse and Elser’s frustrated spin states. We also outline the relationship between correlator product states and other common families of variational wave functions such as matrix product states, tensor product states, and resonating valence-bond states. Variational calculations for the Heisenberg and spinless Hubbard models demonstrate the promise of correlator product states for describing both two-dimensional and fermion correlations. Even in one-dimensional systems, correlator product states are competitive with matrix product states for a fixed number of variational parameters.

Nucleon form factors with 2 + 1 flavor dynamical domain-wall fermions

Abstract: We report our numerical lattice QCD calculations of the isovector nucleon form factors for the vector and axial-vector currents: the vector, induced tensor, axial-vector, and induced pseudoscalar form factors. The calculation is carried out with the gauge configurations generated with ${N}_{f}=2+1$ dynamical domain-wall fermions and Iwasaki gauge actions at $\ensuremath{\beta}=2.13$, corresponding to a cutoff ${a}^{\ensuremath{-}1}=1.73\text{ }\text{ }\mathrm{GeV}$, and a spatial volume of $(2.7\text{ }\text{ }\mathrm{fm}{)}^{3}$. The up and down-quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange quark mass is about 12% heavier than the physical one. We calculate the form factors in the range of momentum transfers, $0.2l{q}^{2}l0.75\text{ }\text{ }{\mathrm{GeV}}^{2}$. The vector and induced tensor form factors are well described by the conventional dipole forms and result in significant underestimation of the Dirac and Pauli mean-squared radii and the anomalous magnetic moment compared to the respective experimental values. We show that the axial-vector form factor is significantly affected by the finite spatial volume of the lattice. In particular in the axial charge, ${g}_{A}/{g}_{V}$, the finite-volume effect scales with a single dimensionless quantity, ${m}_{\ensuremath{\pi}}L$, the product of the calculated pion mass and the spatial lattice extent. Our results indicate that for this quantity, ${m}_{\ensuremath{\pi}}Lg6$ is required to ensure that finite-volume effects are below 1%.

Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds

Abstract: We study correlation functions of single-cycle chiral operators in ${\mathrm{Sym}}^{N}{T}^{4}$, the symmetric product orbifold of $N$ supersymmetric four-tori. Correlators of twist operators are evaluated on covering surfaces, generally of different genera, where fields are single valued. We compute some simple four-point functions and study how the sum over inequivalent branched covering maps splits under operator product expansions. We then discuss extremal $n$-point correlators, i.e. correlators of $n\ensuremath{-}1$ chiral and one antichiral operators. They obey simple recursion relations involving numbers obtained from counting branched covering maps with particular properties. In most cases we are able to solve explicitly the recursion relations. Remarkably, extremal correlators turn out to be equal to Hurwitz numbers.

Method And System For Simulated Product Evaluation Via Personalizing Advertisements Based On Portrait Images

Abstract: Advertising method and system for creating a personalized advertisement, and for recommending a product being advertised. An uploaded facial photograph is received, the facial photograph including a plurality of facial features. At least one facial feature of the plurality of facial features is detected. Personal information may be extracted by analyzing the at least one detected facial feature. A product being advertised may be recommended, wherein the recommended product applies to a treatment related to the extracted personal information.

Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations.

Abstract: A detailed comparison of the time-dependent wave packet method using the split operator propagator and recently introduced Chebyshev real wave packet approach for calculating reactive scattering processes is reported. As examples, the state-to-state differential cross sections of the H+H′D(v0=0,j0=1)→H′D+H/H′H+D reaction, the state-to-state reaction probabilities of the O16+O352 (v0=0,j0=0)→O17+O16O18/O18+O16O17 reaction, the H+O2→O+HO reaction, and the F+HD→HF+D reaction are calculated, using an efficient reactant-coordinate-based method on an L-shape grid which allows the extraction of the state-to-state information of the two product channels simultaneously. These four reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these two propagation methods for describing reactive scattering dynamics. The results indicate that the Chebyshev real wave packet method is typically more accurate, particularly for reactions dominated by long-lived resonances. However, the split operator approach is often more cost effective, making it a method of choice for fast reactions. In addition, our results demonstrate accuracy of the reactant-coordinate-based method for extracting state-to-state information.

Products of composition and differentiation operators on the weighted Bergman space

Abstract: Motivated by a recent paper by S. Ohno we calculate Hilbert-Schmidt norms of products of composition and differentiation operators on the Bergman space $A^2_\alpha,$ $\alpha>-1$ and the Hardy space $H^2$ on the unit disk. When the convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$ implies the convergence of product operators $C_{\varphi_n}D^k$ is also studied. Finally, the boundedness and compactness of the operator $C_{\varphi}D^k: A^2_\alpha\to A^2_\alpha$ are characterized in terms of the generalized Nevanlinna counting function.

Offer Reporting Apparatus and Method

Abstract: A computerized method for storing purchase information for a product in a price database to generate a purchase history includes receiving in a price database a set of sales information for a product, wherein the set of sales information includes a product identifier for a product and a price for the product; and storing in the price database a price entry, which include the set of sales information, wherein the price entry is editable to generate a purchase history.

Unbounded bivariant $K$-theory and correspondences in noncommutative geometry

Abstract: By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable $C^{*}$-module. The theory of operator spaces provides the required tools. Finally, the above mentioned $KK$-cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.

Small gaps between products of two primes

Abstract: Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 - qnle; 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ? is any positive integer, then (qn+1 - qn) ≤ e-γ(1 + o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms. © 2008 London Mathematical Society.

Fast amplification of QMA

Abstract: Given a verifier circuit for a problem in QMA, we show how to exponentially amplifythe gap between its acceptance probabilities in the 'yes' and 'no' cases, with a methodthat is quadratically faster than the procedure given by Marriott and Watrous [1]. Ourconstruction is natively quantum, based on the analogy of a product of two reflections anda quantum walk. Second, in some special cases we show how to amplify the acceptanceprobability for good witnesses to 1, making a step towards the proof that QMA withone-sided error (QMA1) is equal to QMA. Finally, we simplify the filter-state method tosearch for QMA witnesses by Poulin and Wocjan [2].

The Central value of the Rankin–Selberg L-Functions

Abstract: Let f be a Maass form for SL\((3, {\mathbb{Z}})\) which is fixed and uj be an orthonormal basis of even Maass forms for SL\((2, {\mathbb{Z}})\), we prove an asymptotic formula for the average of the product of the Rankin–Selberg L-function of f and uj and the L-function of uj at the central value 1/2. This implies simultaneous nonvanishing results of these L-functions at 1/2.

The hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free

Abstract: We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

Process similarity and developing new process models through migration

Abstract: An industrial process may operate over a range of conditions to produce different grades of product. With a data-based model, as conditions change, a different process model must be developed. Adapting existing process models can allow using fewer experiments for the development of a new process model, resulting in a saving of time, cost, and effort. Process similarity is defined and classified based on process representation. A model migration strategy is proposed for one type of process similarity, family similarity, which involves developing a new process model by taking advantage of an existing base model, and process attribute information. A model predicting melt-flow-length in injection molding is developed and tested as an example and shown to give satisfactory results.

The link between product data model and process model : the effect of changes in the product data model on the process model

Abstract: Following the ‘product based design’ approach, a process model can be generated directly from the product data model, i.e. a specification of the end product. Such end products, however, can be subject to change, and generating a new process model for every small change, can become a burden. Changes to the product data model affect the corresponding process model. In this paper these effects are investigated, to be able to alter the process model, based on changes in the product data model, without the need to regenerate the process model from scratch.

Multiparameter Riesz commutators

Abstract: It is shown that product BMO of S.-Y. A. Chang and R. Fefferman, defined on the space ${\Bbb R}^{d_1}\otimes \cdots \otimes {\Bbb R} ^{d_t}$, can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of R. Coifman, R. Rochberg, and G. Weiss, and at the same time extends the work of M. Lacey and S. Ferguson and M. Lacey and E. Terwilleger, on multiparameter commutators with Hilbert transforms.

The sum-product phenomenon in arbitrary rings

Abstract: The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense ``close'' to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\R$ and cyclic groups $\Z/q\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when $A$ encounters few zero-divisors of $R$. As applications we recover (and generalise) several sum-product theorems already in the literature.

Product of generalized topologies

Abstract: The definition of the product of topologies is generalized in such a way that topologies are replaced by generalized topologies in the sense of [3].