Showing papers on "Product (mathematics) published in 2005"

Finding Exponential Product Formulas of Higher Orders

Abstract: In the present article, we review a continual effort on generalization of the Trotter formula to higher-order exponential product formulas. The exponential product formula is a good and useful approximant, particularly because it conserves important symmetries of the system dynamics. We focuse on two algorithms of constructing higher-order exponential product formulas. The first is the fractal decomposition, where we construct higher-order formulas recursively. The second is to make use of the quantum analysis, where we compute higher-order correction terms directly. As interludes, we also have described the decomposition of symplectic integrators, the approximation of time-ordered exponentials, and the perturbational composition.

System, method and computer program product for prioritizing contacts

Abstract: A system, method and computer program product for prioritizing a message are described. In one embodiment, information about one or more characteristics of a message may be obtained. A score for the message may be calculated based on the obtained information. A priority based on the calculated score may then be assigned to the message.

Apparatus for dispensing and identifying product in washrooms

Abstract: An apparatus for the dispensing of product is provided. The apparatus includes a dispenser that is configured for dispensing product and a sensor in communication with an electrical circuit carried by the dispenser. The sensor is configured for detecting identification information about the product when the electrical circuit is completed by the product. Additional exemplary embodiments are also provided in which the sensor operates through optical detection, smell, physical contact with the product, or vibration instead of or in addition to the completion of an electrical circuit.

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

Abstract: We study finite dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type {\tt ADE}, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module. For the current algebra $\Lgc$ of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the $\Lgc$-module structure of the Kac-Moody algebra $\Lhg$-module $V(\ell\Lam_0)$ as a semi-infinite fusion product of finite dimensional $\Lgc$--modules.

General Framework for Consistent Sampling in Hilbert Spaces

Abstract: We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

Computationally Efficient Approximations of the Joint Spectral Radius

Abstract: The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed. For the special case of matrices with nonnegative entries we give elementary proofs of simple inequalities that we then use to obtain approximations of arbitrary high accuracy. From these inequalities it follows that the spectral radius of matrices with nonnegative entries is given by the simple expression $$ \rho(A_1, \ldots, A_m)= \lim_{k \rightarrow \infty} \rho^{1/k} ( A_1^{\otimes k} + \cdots + A_m^{\otimes k}), $$ where it is somewhat surprising to notice that the right-hand side does not directly involve any mixed product between the matrices. ($A^{\otimes k}$ denotes the $k$th Kronecker power of $A$.) For matrices with arbitrary entries (not necessarily nonnegative), we introduce an approximation procedure based on semidefinite liftings that can be implemented in a recursive way. For two matrices, even the first step of the procedure gives an approximation whose relative accuracy is at least ${1 / \sqrt{2}}$, that is, more than $70\%$. The subsequent steps improve the accuracy but also increase the dimension of the auxiliary problems from which the approximation can be found. Our approximation procedures provide approximations of relative accuracy $1-\epsilon$ in time polynomial in $n^{(\ln m) /\epsilon}$, where $m$ is the number of matrices and $n$ is their size. These bounds are close from optimality since we show that, unless P=NP, no approximation algorithm is possible that provides a relative accuracy of $1-\epsilon$ and runs in time polynomial in $n$ and $1/\epsilon$. As a by-product of our results we prove that a widely used approximation of the joint spectral radius based on common quadratic Lyapunov functions (or on ellipsoid norms) has relative accuracy $1/\sqrt m$, where $m$ is the number of matrices.

Product structures on four dimensional solvable lie algebras

Abstract: It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the three dimensional subalgebras.

Method, system and program product that utilize a hierarchical conceptual framework to model an environment containing a collection of items

Abstract: A method, system and program product for modeling an environment containing a collection of items are disclosed. In accordance with the method, an environmental hierarchy describing a model environment, a product catalog containing data describing a plurality of items that may be utilized to populate the modeled environment, and a configuration library containing data describing a spatial relationship between first and second items among the plurality of items in the product catalog are created. The modeled environment is then populated by storing, in a database, data representative of the spatial relationship between the environmental hierarchy and a collection of items including the first item.

Representations of Quantum Affinizations and Fusion Product

Abstract: In this paper we study general quantum affinizations $\mathcal{U}_q(\widehat{\mathfrak{g}})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Presley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), the combinatorics of q-characters give a ring structure * on the Grothendieck group $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ of the integrable representations that we classified. We propose a new construction of tensor products in a larger category by using the Drinfel'd new coproduct (it cannot directly be used for $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ because it involves infinite sums). In particular, we prove that * is a fusion product (a product of representations is a representation).

Character Formulae and Partition Functions in Higher Dimensional Conformal Field Theory

Abstract: A discussion of character formulae for positive energy unitary irreducible representations of the the conformal group is given, employing Verma modules and Weyl group reflections. Product formulae for various conformal group representations are found. These include generalisations of those found by Flato and Fronsdal for SO(3,2). In even dimensions the products for free representations split into two types depending on whether the dimension is divisible by four or not.

Fast parallel molecular algorithms for DNA-based computation: factoring integers

Abstract: The RSA public-key cryptosystem is an algorithm that converts input data to an unrecognizable encryption and converts the unrecognizable data back into its original decryption form. The security of the RSA public-key cryptosystem is based on the difficulty of factoring the product of two large prime numbers. This paper demonstrates to factor the product of two large prime numbers, and is a breakthrough in basic biological operations using a molecular computer. In order to achieve this, we propose three DNA-based algorithms for parallel subtractor, parallel comparator, and parallel modular arithmetic that formally verify our designed molecular solutions for factoring the product of two large prime numbers. Furthermore, this work indicates that the cryptosystems using public-key are perhaps insecure and also presents clear evidence of the ability of molecular computing to perform complicated mathematical operations.

Beauville surfaces without real structures

Abstract: Inspired by a construction by Arnaud Beauville of a surface of general type with K 2 = 8, p g = 0, the second author defined Beauville surfaces as the surfaces which are rigid, i.e., without nontrivial deformations, and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2.

Complex product structures on Lie algebras

Abstract: A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. If g is a Lie algebra with such a structure then its complexification has a hypercomplex structure. It is shown in addition that g splits into the sum of two left-symmetric subalgebras. Interpretations of these results are obtained that are relevant to the theory of both hypercomplex and hypersymplectic manifolds and their associated connections.

Configuration-oriented product modelling and knowledge management for made-to-order manufacturing enterprises

Abstract: In made-to-order (MTO) manufacturing enterprises (ME), product architectures are usually modularised and components standardised. Product configuration is a key technology for order realisation in MTO–ME and is a typical knowledge-based application. Through a configuration process, product modules or components are selected and assembled according to customer requirements. Product configuration relates to a great deal of knowledge that represents complexity relations among components or modules, such as configuration rules and assembly constraints. Traditional product modelling techniques are focused mainly on physical product modelling and geometric representation, which makes them insufficient to help in the product configuration process. This paper discusses configuration-oriented product modelling and knowledge management for MTO–ME. A general process of product configuration modelling is proposed. The configuration model represents a product family from which a specific configuration solution or product variant can be derived. Actually, configuration modelling is a process which captures and represents product knowledge. In this paper, product knowledge is organised and managed through a knowledge component (KCOM) that includes configuration rules and constraints. A KCOM-based product knowledge representation model is presented. Finally, a PDM system is extended to support product modelling and knowledge management for MTO configurable products .

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: Energy levels of coupled systems with as many as 16 coordinates

Abstract: In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.

Noncommutative Lp modules

Abstract: We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.

The exact law of large numbers via Fubini extension and characterization of insurable risks

Abstract: A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product. © 2005 Elsevier Inc. All rights reserved. JEL classification: C60; D51; D61; D80; E00

On the zappa-szép product

Abstract: The Zappa-Szep product of a pair of groups generalizes the semidirect product in that neither factor is assumed to be normal in the result. We extend the applicability of the Zappa-Szep product to multiplicative structures more general than groups with emphasis on categories and monoids. We also explore the preservation of various properties of the multiplication under the Zappa-Szep product.

Method, system, and program product for controlling chemical reactions in a digital microfluidic system

Abstract: The present invention provides, in a first aspect, a method, system, and program product for controlling chemical reactions in a digital microfluidic system that include logically partitioning cells of a digital microfluidic system array into a plurality of virtual components wherein at least one of the virtual components is capable of handling droplets of reactants associated with distinct chemical reactions concurrently. In a second aspect, a respective next cell is determined for each of a plurality of chemical droplets in the digital microfluidic system array, which may include droplets of reactants associated with distinct chemical reactions. In another aspect, a method, system, and program product for controlling chemical reactions in a digital microfluidic system in accordance with the present invention induce a chemical droplet of the plurality of chemical droplets in the digital microfluidic system array to move to the respective next cell determined for the chemical droplet.

Generating Series for Interconnected Analytic Nonlinear Systems

Abstract: Given two analytic nonlinear input-output systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection, and feedback connection. In each case, the corresponding generating series is produced and conditions for the convergence of the corresponding Fliess operator are given. In the process, an existing notion of a composition product for formal power series has its set of known properties significantly expanded. In addition, the notion of a feedback product for formal power series is shown to be well defined in a broad context, and its basic properties are characterized.

Modelling and solving engineering product configuration problems by constraint satisfaction

Abstract: A product configurator has been an effective software tool in successful implementation of mass customization strategy. It enables manufacturers to automatically generate product configuration information tailored to individual customers’ needs. However, current product configurator techniques are not adequate to solve an engineering product configuration problem, because the constraints in such a problem are often expressed by mathematical formulae and computable procedures. This type of constraint posts challenges on constraint modelling and solving within the constraint satisfaction paradigm. Constraints made up of mathematical formulae and computable procedures are not naturally supported with pre-defined constraint semantics in a constraint model. It is also difficult to achieve search efficiency for constraints over n-ary continuous variables. This paper presents an innovative approach to modelling and solving an engineering product configuration problem based on the constraint satisfaction paradigm...

Learning mixtures of product distributions over discrete domains

Abstract: We consider the problem of learning mixtures of product distributions over discrete domains in the distribution learning framework introduced by Kearns et al. (1994). We give a poly(n//spl epsi/) time algorithm for learning a mixture of k arbitrary product distributions over the n-dimensional Boolean cube {0, 1}/sup n/ to accuracy /spl epsi/, for any constant k. Previous poly(n)-time algorithms could only achieve this for k = 2 product distributions; our result answers an open question stated independently in M. Cryan (1999) and Y. Freund and Y. Mansour (1999). We further give evidence that no polynomial time algorithm can succeed when k is superconstant, by reduction from a notorious open problem in PAC learning. Finally, we generalize our poly(n//spl epsi/) time algorithm to learn any mixture of k = O(1) product distributions over {0, 1,... , b }/sup n/,for any b = O(1).

Every decision tree has an influential variable

Abstract: We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i(f), \] where $\delta_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is the influence of the $i$th variable on $f$. The variance, influence and probability are taken with respect to an arbitrary product measure on $\{-1,1\}^n$. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth $d$ has a variable with influence at least $\frac{1}{d}$. The only previous nontrivial lower bound known was $\Omega(d 2^{-d})$. Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least $\Omega(v^{4/3}/p^{1/3})$, where $v$ is the number of vertices and $p \leq \half$ is the critical threshold probability. This supersedes the milestone $\Omega(v^{4/3})$ bound of Hajnal and is sometimes superior to the best known lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.

System and method for modeling customer response using data observable from customer buying decisions

Abstract: A computer system models customer response using observable data. The observable data includes transaction, product, price, and promotion. The computer system receives data observable from customer responses. A set of factors including customer traffic within a store, selecting a product, and quantity of selected product is defined as expected values, each in terms of a set of parameters related to customer buying decision. A likelihood function is defined for each of the set of factors. The parameters are solved using the observable data and associated likelihood function. The customer response model is time series of unit sales defined by a product combination of the expected value of customer traffic and the expected value of selecting a product and the expected value of quantity of selected product. A linear relationship is given between different products which includes a constant of proportionality that determines affinity and cannibalization relationships between the products.

System and method for identifying a product

Abstract: A product that is uniquely identifiable according to a globally unique identifier (GUID) (100) includes a class identifier (CID) (110) that uniquely identifies at least one product class in class in which the product is categorized (along with a plurality of other products) within a class hierarchy of a global content directory (42). The product class defines one or more attributes of the products categorized in the class. The product also includes a product identifier (PID) (120) that uniquely identifies the particular product from among the plurality of products categorized in the product class uniquely identified by the CID (110). The CID (110) and PID (120) collectively provide the GUID (100), which may be specified or determined to facility a commercial transaction involving the product.

Sobolev active contours

Abstract: All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L2-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions. This is particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and region-based active contour models as Sobolev active contours and show the substantial improvements gained in segmentation and tracking applications.

Network-based system for selecting or purchasing hardware products

Abstract: System and method for receiving purchase information for a client system, e.g., a measurement system. A configuration diagram visually representing a current configuration of the client system is displayed. Multiple product icons are displayed representing products (hardware and/or programs) available for use in the client system. User input is received graphically associating at least one first product icon with the configuration diagram, where the first product icon represents a first product, and the user input indicates a desire to purchase the first product. An updated configuration diagram is displayed representing the configuration of the client system after receiving the user input, including the first product icon. Pricing information for the first product is displayed in response to receiving the user input. User input initiating purchase of the first product may be received in response to displaying the pricing information, and the product may be provided to the user.