Showing papers on "Product (mathematics) published in 1997"
Journal Article•
2,665 citations
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, which can be interpreted as correlators in topological open string theory.
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 ("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 use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.
2,223 citations
TL;DR: In this article, the authors consider families of {0, 1}-valued random variables indexed by the vertices of countable graphs with bounded degree and show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site, the probability of seeing a 1 at this site is at least a value $p$ which is large enough, then this random field dominates a product measure with positive density.
Abstract: 4 We consider families of {0, 1}-valued random variables indexed by the vertices of countable graphs with bounded degree. First we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site, the probability of seeing a 1 at this site is at least a value $p$ which is large enough, then this random field dominates a product measure with positive density. Moreover the density of this dominated product measure can be made arbitrarily close to 1, provided that $p$ is close enough to 1. Next we address the issue of obtaining the critical value of $p$, defined as the threshold above which the domination by positive-density product measures is assured. For the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than $k$ units, this critical value is shown to be $1 - k^k /(k + 1)^{k+1}$, and a discontinuous transition is shown to occur. Similar critical values of $p$ are found for other classes of probability measures on ${0, 1}^{\mathbb{Z}}$. For the class of $k$-dependent measures the critical value is again $1 - k^k /(k + 1)^{k+1}$, with a discontinuous transition. For the class of two-block factors the critical value is shown to be 1/2 and a continuous transition is shown to take place in this case. Thus both the critical value and the nature of the transition are different in the two-block factor and 1-dependent cases.
515 citations
TL;DR: In this article, the authors apply the separability criterion, local filtering, and Bennett et al. distillation protocol to show that any inseparable $2 ifmmode\times\else\texttimes\fi{}2$ system represents the entanglement which, however small, can be distilled to a singlet form.
Abstract: A quantum system is called inseparable if its density matrix cannot be written as a mixture of product states. In this Letter we apply the separability criterion, local filtering, and Bennett et al. distillation protocol [Phys. Rev. Lett. 76, 722 (1996)] to show that any inseparable $2\ifmmode\times\else\texttimes\fi{}2$ system represents the entanglement which, however small, can be distilled to a singlet form.
311 citations
TL;DR: In this paper, it is shown that the standard binning method utilized in the classical study of scattering processes dealing with more than three atoms, which consists in rounding the product vibrational actions (the x i's) to their nearest (the n i ''s), is not appropriate when the total energy disposal in the products is too low for the classical and quantum densities of vibrational states to be equal.
228 citations
TL;DR: In this paper, a linearization of the distributivity of product over sum is proposed to model Gentzen's cut rule and can be strengthened in two natural ways to generate full distributivity and ∗autonomous categories.
200 citations
TL;DR: A least-squares approximation to a first order system which involves a discrete inner product which is related to the inner product in H -1 (Ω) (the Sobolev space of order minus one on Ω) results in a method of approximation which is optimal with respect to the required regularity.
Abstract: The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in H -1 (Ω) (the Sobolev space of order minus one on Ω). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.
198 citations
TL;DR: In this article, the authors defined the notion of a star product of the Wick type on every Kahler manifold by a straightforward generalization of the corresponding star product in Cn: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions.
Abstract: In this Letter we compute some elementary properties of the Fedosov star product of Weyl type, such as symmetry and order of differentiation. Moreover, we define the notion of a star product of the Wick type on every Kahler manifold by a straightforward generalization of the corresponding star product in Cn: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions. By a Fedosov type procedure, we give an existence proof of such star products for any Kahler manifold.
161 citations
TL;DR: A semantic recognition and rule-oriented approach for developing a product design is proposed, using the development of an office chair and a number of referential products to quantify partial correlation coefficients.
160 citations
TL;DR: In this article, the authors dealt with the reaction between two solids A and B to form one or more product phases (A m B n ), during which the reaction has to be attributed to a transport of the reactants across phase boundaries and through reaction product.
Abstract: This chapter deals essentially with the reaction between two solids A and B to form one or more product phases (A m B n ). During the course of this heterogeneous solid-solid reaction A m B n separates the reactants spatially. Therefore the progress of the reaction has to be attributed to a transport of the reactants across phase boundaries and through the reaction product.
TL;DR: In this article, a dimension free lower bound for isoperimetric constants of product probability measures is derived, and some analytic inequalities are derived from this lower bound; see Section 2.1.
Abstract: A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived.
TL;DR: In this paper, the real-time order parameter correlations of a class of models in a transverse field at low temperatures on both sides of the quantum critical point were analyzed and compared with numerical studies on the nearest neighbor spin-1/2 model.
Abstract: We present asymptotically exact results for the real time order parameter correlations of a class of $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ Ising models in a transverse field at low temperatures ( $T$) on both sides of the quantum critical point. The correlations are a product of a $T$-independent factor determined by quantum effects, and a $T$-dependent relaxation function which comes from a classical theory. We confirm our predictions by a no-free-parameter comparison with numerical studies on the nearest neighbor spin-1/2 model.
TL;DR: In this article, the authors introduce the notion of normal ordering and Wick-like expansion of a product of fermionic creation/annihilation operators with respect to a multi-determinantal reference state ψ 0.
Patent•
11 Apr 1997
TL;DR: In this paper, execution control programming is used to determine whether or not an executable file will be allowed to be executed by a digital product, i.e., as would be provided by the operating system if loaded normally, and allows execution of the digital product under controlled conditions.
Abstract: Digital product execution control as disclosed contemplates production of a final version of a digital product and subsequently imposing execution control on that digital product. The manufacturer of the original digital product need not incorporate execution control features into the final version of the product. Execution control programming attaches to an executable file of the digital product to create a controlled executable file. The resulting operating environment when loaded is insufficient for the original executable file, and control programming determines whether or not execution will be allowed. If allowed, control programming creates the necessary operating environment for the digital product, i.e., as would be provided by the operating system if loaded normally, and allows execution of the digital product only under controlled conditions. Further security measures identify an executable portion of the digital product and a plurality of sub-portions of the executable portion are encrypted. An exception handler is established with the operating system and upon access to memory regions marked as protected the exception handler decrypts each sub-portion when accessed. A further security measure stores at a remote clearing house a key needed to decrypt the digital product for use. Upon execution, the controlled digital product interacts with the remote clearing house to obtain permission to execute and to obtain the key needed for execution.
TL;DR: In this article, it was shown that while Fubini's theorem is in general false if additivity is not granted, it is true when a certain type of function is being integrated, and the unique product thus obtained has an intuitive justification when the marginals are distributions induced by random correspondences.
Book•
01 Jan 1997
TL;DR: This chapter discusses extensional type theory in intensional type theory, which is an extension of TTI for which the interpretation in TTE is surjective.
Abstract: 1. Introduction.- 1.1 Definitional and propositional equality.- 1.2 Extensional constructs.- 1.3 Method.- 1.3.1 The use of categorical models.- 1.3.2 Syntactic models.- 1.4 Applications.- 1.4.1 Application to machine-assisted theorem proving.- 1.5 Overview.- 2. Syntax and semantics of dependent types.- 2.1 Syntax for a core calculus.- 2.1.1 Raw syntax.- 2.1.2 Judgements.- 2.1.3 Notation.- 2.1.4 Derived rules and meta-theoretic properties.- 2.2 High-level syntax.- 2.2.1 Telescopes.- 2.2.2 Elements of telescopes and context morphisms.- 2.2.3 Definitions and substitution.- 2.3 Further type formers.- 2.3.1 Unit type.- 2.3.2 ?-types.- 2.3.3 Function and cartesian product types.- 2.3.4 The Calculus of Constructions.- 2.3.5 Universes.- 2.3.6 Quotient types.- 2.4 Abstract semantics of type theory.- 2.4.1 Syntactic categories with attributes.- 2.4.2 Type constructors.- 2.5 Interpreting the syntax.- 2.5.1 Partial interpretation.- 2.5.2 Soundness of the interpretation.- 2.6 Discussion and related work.- 3. Syntactic properties of propositional equality.- 3.1 Intensional type theory.- 3.1.1 Substitution.- 3.1.2 Uniqueness of identity.- 3.1.3 Functional extensionality.- 3.2 Extensional type theory.- 3.2.1 Comparison with Troelstra's presentation.- 3.2.2 Undecidability of extensional type theory.- 3.2.3 Interpreting extensional type theory in intensional type theory.- 3.2.4 An extension of TTI for which the interpretation in TTE is surjective.- 3.2.5 Conservativity of TTE over TTI.- 3.2.6 Discussion and extensions.- 3.2.7 Conservativity of quotient types and functional extensionality.- 3.3 Related work.- 4. Proof irrelevance and subset types.- 4.1 The refinement approach.- 4.2 The deliverables approach.- 4.3 The deliverables model.- 4.3.1 Contexts.- 4.3.2 Families of specifications.- 4.3.3 Sections of specifications (deliverables).- 4.4 Model checking with Lego.- 4.4.1 Records in Lego.- 4.4.2 Deliverables in Lego.- 4.5 Type formers in the model D.- 4.5.1 Dependent products.- 4.5.2 Dependent sums.- 4.5.3 Natural numbers.- 4.5.4 The type of propositions.- 4.5.5 Proof irrelevance.- 4.5.6 Universes.- 4.6 Subset types.- 4.6.1 Subset types without impredicativity.- 4.6.2 A non-standard rule for subset types.- 4.7 Reinterpretation of the equality judgement.- 4.8 Related work.- 5. Extensionality and quotient types.- 5.1 The setoid model.- 5.1.1 Contexts of setoids.- 5.1.2 Implementing the setoid model S0 in Lego.- 5.1.3 Type formers in the setoid model.- 5.1.4 Propositions.- 5.1.5 Quotient types.- 5.1.6 Interpretation of quotient types in S0.- 5.1.7 A choice operator for quotient types.- 5.1.8 Type dependency and universes.- 5.2 The groupoid model.- 5.2.1 Groupoids.- 5.2.2 Interpretation of type formers.- 5.2.3 Uniqueness of identity.- 5.2.4 Propositional equality as isomorphism.- 5.3 A dependent setoid model.- 5.3.1 Families of setoids.- 5.3.2 Dependent product.- 5.3.3 The identity type.- 5.3.4 Inductive types.- 5.3.5 Quotient types.- 5.4 Discussion and related work.- 6. Applications.- 6.1 Tarski's fixpoint theorem.- 6.1.1 Discussion.- 6.2 Streams in type theory.- 6.3 Category theory in type theory.- 6.3.1 Categories in S0.- 6.3.2 Categories in S1.- 6.3.3 Discussion.- 6.4 Encoding of the coproduct type.- 6.4.1 Development in the setoid models.- 6.5 Some basic constructions with quotient types.- 6.5.1 Canonical factorisation of a function.- 6.5.2 Some categorical properties of S0.- 6.5.3 Subsets and quotients.- 6.5.4 Saturated subsets.- 6.5.5 Iterated quotients.- 6.5.6 Quotients and products.- 6.5.7 Quotients and function spaces.- 6.6 ? is co-continuous-intensionally.- 6.6.1 Parametrised limits of ?-chains.- 6.6.2 Development in TTE.- 6.6.3 Development in TTI.- 7. Conclusions and further work.- A.1 Extensionality axioms.- A.2 Quotient types.- A.3 Further axioms.- Appendix B. Syntax.- Appendix C. A glossary of type theories.- Appendix D. Index of symbols.
Patent•
20 May 1997TL;DR: In this article, a hand-held unit which contains a detector which identifies a product located nearby is used to provide product information to customers of a retail establishment, which can be accomplished through scanning a bar code affixed to the product.
Abstract: A device for providing product information to customers of a retail establishment. The device is a hand-held unit which contains a detector which identifies a product located nearby. The identification can be accomplished through scanning a bar code affixed to the product. Once the product is identified, the device retrieves descriptive information about the product from memory, and displays the information to the customer.
TL;DR: In this article, it was shown that any separable state on Hilbert space can be written as a convex combination of N pure product states with $N\leq (dim{cal H})^2}.
Abstract: It is shown that any separable state on Hilbert space ${\cal H}={\cal H}_1\otimes{\cal H}_2$, can be written as a convex combination of N pure product states with $N\leq (dim{\cal H})^2$. Then a new separability criterion for mixed states in terms of range of density matrix is obtained. It is used in construction of inseparable mixed states with positive partial transposition in the case of $3\times 3$ and $2\times 4$ systems. The states represent an entanglement which is hidden in a more subtle way than it has been known so far.
Posted Content•
TL;DR: In this paper, a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions is given, which is a special case of a more general rule which also gives the coefficients of a skew Schur function.
Abstract: We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Multiplication rules for the Capelli operators and quantum immanants are also given.
Patent•
11 Apr 1997
Abstract: Digital product execution control as disclosed contemplates production of a final version of a digital product and subsequently imposing execution control on that digital product. The manufacturer of the original digital product need not incorporate execution control features into the final version of the product. Execution control programming attaches to an executable file of the digital product to create a controlled executable file. Upon loading and execution of the controlled executable file, the resulting operating environment is insufficient for the original executable file, and control programming determines whether or not execution will be allowed. If allowed, control programming creates the necessary operating environment for the digital product, i.e., as would be provided by the operating system if loaded normally, and allows execution of the digital product only under controlled conditions, e.g., such as under an evaluation use of the digital product.
Posted Content•
TL;DR: In this paper, it was shown that the tensor product of a quasitriangular quasi-Hopf algebra can be used to construct a quasi-quantum double in terms of a Tannaka-Krein-like reconstruction procedure.
Abstract: Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is not a subalgebra of \D(\G) unless the coproduct on \G is strictly coassociative. It is shown that the category of finite dimensional representations of \D(\G) coincides with what has been called the double category of \G--modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi--quantum doubles in terms of a Tannaka--Krein--like reconstruction procedure. The whole construction is shown to generalize to weak quasi--Hopf algebras with \D(\G) now being linearly isomorphic to a subspace of \dG \otimes \G.
Patent•
03 Jun 1997
TL;DR: In this paper, the authors augment machine control with true position feedback from an independent measurement system to allow such inspection to occur with conventional machine tools rather than expensive, special purpose, limited versatility Coordinate Measuring Machines.
Abstract: Product acceptance requires accurate measurement of product features. We augment machine control with true position feedback from an independent measurement system to allow such inspection to occur with conventional machine tools rather than expensive, special purpose, limited versatility Coordinate Measuring Machines.
TL;DR: In this paper, Teubner et al. investigated the algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras and showed that when m = 3, V-b is a reducible variety with one component corresponding to V-t and another corresponding to the trifocal plane.
Abstract: In this paper the different algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, V-n, to work with is the image of P-3 in P-2 x P-2 x ... x P-2 under a corresponding product of projections, (A(1) x A(2) x ... x A(m)). Another descriptor, the variety V-b, is the one generated by all bilinear forms between pairs of views, which consists of all points in P-2 x P-2 x ... x P-2 where all bilinear forms vanish. Yet another descriptor, the variety V-t, is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, V-b is a reducible variety with one component corresponding to V-t and another corresponding to the trifocal plane. Furthermore, when m = 3, V-t is generated by the three bilinearities and one trilinearity, when m = 4, V-t is generated by the six bilinearities and when m greater than or equal to 4, V-t can be generated by the ((m)(2)) bilinearities. This shows that four images is the generic case in the algebraic setting, because V-t can be generated by just bilinearities. Furthermore, some of the bilinearities may be omitted when m greater than or equal to 5. (C) 1997 by B. G. Teubner Stuttgart - John Wiley & Sons Ltd. (Less)
TL;DR: In this paper, various deformations of free associative algebras and convolutional convolutions are discussed, such as non-commutative symmetric functions related to families of idempotents, and simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics.
Abstract: This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in mathematical physics (the quon algebra, generalized Brownian motion).
Patent•
23 Sep 1997TL;DR: In this paper, a centralized monitoring unit for an agricultural implement coupled with a work vehicle is described. The monitoring unit includes a product delivery apparatus which has at least one product channel for delivering the product to the row units, and an electronic product sensor which is coupled to the product delivery device and configured to generate a product signal representing an amount of the product which moves through the product channel.
Abstract: A centralized monitoring unit for an agricultural implement is disclosed herein. The monitoring unit is preferably used with a planting system including a planting implement coupled to a work vehicle. The implement includes a frame supporting row units which are configured to apply a product (e.g., seed, fertilizer, insecticide, herbicide) to the rows in a field. The monitoring unit includes a product delivery apparatus which has at least one product channel for delivering the product to the row units, and an electronic product sensor which is coupled to the product delivery apparatus and configured to generate a product signal representing an amount of the product which moves through the product channel. An electronic display located in the cab of the work vehicle is configured to generate an image in response to a display signal. A processor circuit is configured to monitor the product signal, calculate application progress statistics, calculate implement statistics, calculate product rate statistics and apply the display signal to the electronic display to generate the image on the electronic display representative of the application progress statistics, the implement statistics and the product rate statistics.
TL;DR: The authors develop a simple model of the product definition process that offers insights about how a firm should customize the process to suit its needs, and captures these insights as a managerial framework and illustrates them with industry examples involving the development process for laser printers and network test kits.
Abstract: The process of product definition, during which a firm establishes the product specifications, has a vital influence on the success of a product but has attracted little attention from researchers....