Showing papers on "Product (mathematics) published in 2015"

Partition functions of N = (2,2) gauge theories on S 2 and vortices

Abstract: We apply localization techniques to compute the partition function of a two-dimensional \({\mathcal{N}=(2,2)}\) R-symmetric theory of vector and chiral multiplets on S2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet–Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. For applications, we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

An introduction to semi-tensor product of matrices and its

Abstract: An introduction to semi-tensor product of matrices and its , An introduction to semi-tensor product of matrices and its , کتابخانه مرکزی دانشگاه علوم پزشکی تهران

Numerical approximation of fractional powers of elliptic operators

Abstract: We present and study a novel numerical algorithm to approximate the action of $T^\beta:=L^{-\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a representation formula for $T^{-\beta}$ in terms of Bochner integrals involving $(I+t^2L)^{-1}$ for $t\in(0,\infty)$. To develop an approximation to $T^\beta$, we introduce a finite element approximation $L_h$ to $L$ and base our approximation to $T^\beta$ on $T_h^\beta:= L_h^{-\beta}$. The direct evaluation of $T_h^{\beta}$ is extremely expensive as it involves expansion in the basis of eigenfunctions for $L_h$. The above mentioned representation formula holds for $T_h^{-\beta}$ and we propose three quadrature approximations denoted generically by $Q_h^\beta$. The two results of this paper bound the errors in the $H_0$ inner product of $T^\beta-T_h^\beta\pi_h$ and $T_h^\beta-Q_h^\beta$ where $\pi_h$ is the $H_0$ orthogonal projection into the finite element space. We note that the evaluation of $Q_h^\beta$ involves application of $(I+(t_i)^2L_h)^{-1}$ with $t_i$ being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error $T_h^\beta-Q_h^\beta$ and the finite element error $T^\beta-T_h^\beta\pi_h$.

Search for hidden-sector bosons in $B^0 \!\to K^{*0}\mu^+\mu^-$ decays

Abstract: A search is presented for hidden-sector bosons, $\chi$, produced in the decay ${B^0\!\to K^*(892)^0\chi}$, with $K^*(892)^0\!\to K^{+}\pi^{-}$ and $\chi\!\to\mu^+\mu^-$. The search is performed using $pp$-collision data corresponding to 3.0 fb$^{-1}$ collected with the LHCb detector. No significant signal is observed in the accessible mass range $214 \leq m({\chi}) \leq 4350$ MeV, and upper limits are placed on the branching fraction product $\mathcal{B}(B^0\!\to K^*(892)^0\chi)\times\mathcal{B}(\chi\!\to\mu^+\mu^-)$ as a function of the mass and lifetime of the $\chi$ boson. These limits are of the order of $10^{-9}$ for $\chi$ lifetimes less than 100 ps over most of the $m(\chi)$ range, and place the most stringent constraints to date on many theories that predict the existence of additional low-mass bosons.

Modified scattering for the cubic Schrödinger equation on product spaces and applications

Abstract: We consider the cubic nonlinear Schrodinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$ . We prove modified scattering and construct modified wave operators for small initial and final data respectively ( $1\leqslant d\leqslant 4$ ). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\geqslant 2$ . As a consequence, we obtain global strong solutions (for $d\geqslant 2$ ) with infinitely growing high Sobolev norms $H^{s}$ .

On Symmetric and Asymmetric LSHs for Inner Product Search

Abstract: We consider the problem of designing locality sensitive hashes (LSH) for inner product similarity, and of the power of asymmetric hashes in this context. Shrivastava and Li (2014a) argue that there is no symmetric LSH for the problem and propose an asymmetric LSH based on different mappings for query and database points. However, we show there does exist a simple symmetric LSH that enjoys stronger guarantees and better empirical performance than the asymmetric LSH they suggest. We also show a variant of the settings where asymmetry is in-fact needed, but there a different asymmetric LSH is required.

Interactive product placement system and method therefor

Abstract: A method for presenting advertisements for commercial products in video productions, whereby the commercial product is placed in the video production as an element of the video production A viewer is enabled to interact with the video production to select the product Information is then displayed about the selected product; and the viewer is enabled to purchase the selected product

Beyond the Tamm-Dancoff approximation for extended systems using exact diagonalization

Abstract: Linear optical properties can be accurately calculated using the Bethe-Salpeter equation. After introducing a suitable product basis for the electron-hole pairs, the Bethe-Salpeter equation is usually recast into a complex non-Hermitian eigenvalue problem that is difficult to solve using standard eigenvalue solvers. In solid-state physics, it is therefore common practice to neglect the problematic coupling between the positive- and negative-frequency branches, reducing the problem to a Hermitian eigenvalue problem [Tamm-Dancoff approximation (TDA)]. We use time-inversion symmetry to recast the full problem into a quadratic Hermitian eigenvalue problem, which can be solved routinely using standard eigenvalue solvers even at a finite wave vector $\mathbf{q}$. This allows us to access the importance of the coupling between the positive- and negative-frequency branch for prototypical solids. As a starting point for the Bethe-Salpeter calculations, we use self-consistent Green's-function methods ($\mathit{GW}$), making the present scheme entirely ab initio. We calculate the optical spectra of carbon (C), silicon (Si), lithium fluoride (LiF), and the cyclic dimer ${\mathrm{Li}}_{2}{\mathrm{F}}_{2}$ and discuss why the differences between the TDA and the full solution are tiny. However, at finite momentum transfer $\mathbf{q}$, significant differences between the TDA and our exact treatment are found. The origin of these differences is explained.

Function-Hiding Inner Product Encryption

Abstract: We extend the reach of functional encryption schemes that are provably secure under simple assumptions against unbounded collusion to include function-hiding inner product schemes. Our scheme is a private key functional encryption scheme, where ciphertexts correspond to vectors $$\vec {x}$$, secret keys correspond to vectors $$\vec {y}$$, and a decryptor learns $$\langle \vec {x}, \vec {y} \rangle $$. Our scheme employs asymmetric bilinear maps and relies only on the SXDH assumption to satisfy a natural indistinguishability-based security notion where arbitrarily many key and ciphertext vectors can be simultaneously changed as long as the key-ciphertext dot product relationships are all preserved.

Augmented reality virtual product for display

Abstract: A method includes receiving from a shopping device an image of a shopping target corresponding to a product for sale, identifying a virtual model of the product, obtaining the virtual model of the product, and displaying the virtual model of the product on the shopping device.

Nonlocality of orthogonal product-basis quantum states

Abstract: We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, where $d$ is odd, Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)] have constructed ${d}^{2}$ orthogonal product basis quantum states that are locally indistinguishable. We find a subset that contains $6d\ensuremath{-}9$ orthogonal product states that are still locally indistinguishable. We generalize our method to an arbitrary bipartite quantum system ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We present a small set with only $3(m+n)\ensuremath{-}9$ orthogonal product states and prove that these states are local operations and classical communication (LOCC) indistinguishable. Even though these $3(m+n)\ensuremath{-}9$ product states are LOCC indistinguishable, they can be distinguished by separable measurements. This shows that separable operations are strictly stronger than the local operations and classical communication.

Method of simulating a virtual out-0f-box experience of a packaged product

Abstract: A method of simulating a virtual out-of-box experience of a packaged product with an electronic device is disclosed. The packaging of the product has a computer-readable code which is scanned with a camera on the electronic device. A product demonstration application is retrieved for the product from the internet. Surfaces of the packaging are recognized by the application. An image of a virtual representation of the product is rendered on an image of the packaging to make a combined image, which is displayed to the user on the display of the electronic device wherein the combined image simulates X-Ray vision into the packaging, revealing the product. The method further updates the combined image as the user manipulates the product in front of the camera, revealing different view of the product, a three-dimensional view of the product may further be displayed outside the packaging, and features demonstrated of the virtual product.

Information augmented product guide

Abstract: A method includes receiving an image of a product, obtaining content relevant to using the product, and displaying the content in an augmented reality view of the product by overlaying the content on the image of the product on a display device.

Confirming product location using a subset of a product identifier

Abstract: A system and method include providing product location information to a user; prompting a user to provide check digits; receiving a subset of digits from a set of digits associated with a product that are used for a purpose different than for check digits; and comparing the subset of digits to product check digits to confirm the user has located the product.

Method, apparatus and computer program product for joining the displays of multiple devices

Abstract: A method is provided for enhancing a user interface with a mobile device by joining the displays of multiple devices together to function together with one another. In particular, example methods may provide for detecting a first touch or drag event, receiving a message indicating a second touch or drag event from a nearby device occurring during the first touch or drag event, creating a join event, and providing for transmission of the join event to the nearby device. The method may also include updating a registry of nearby devices in response to receiving a message regarding a touch or drag event from a nearby device. The method may further include providing for presentation of content on a display of a device and direction of content to be presented on a display of the nearby device, wherein the content to be presented on the display of the nearby device is related to the content on the display of the device.

Tensor Categories and Endomorphisms of von Neumann Algebras: with Applications to Quantum Field Theory

Abstract: Introduction.- Homomorphisms of von Neumann algebras.- Endomorphisms of infinite factors.- Homomorphisms and subfactors.- Non-factorial extensions.- Frobenius algebras, Q-systems and modules.- C* Frobenius algebras.- Q-systems and extensions.- The canonical Q-system.- Modules of Q-systems.- Induced Q-systems and Morita equivalence.- Bimodules.- Tensor product of bimodules.- Q-system calculus.- Reduced Q-systems.- Central decomposition of Q-systems.- Irreducible decomposition of Q-systems.- Intermediate Q-systems.- Q-systems in braided tensor categories.- a-induction.- Mirror Q-systems.- Centre of Q-systems.- Braided product of Q-systems.- The full centre.- Modular tensor categories.- The braided product of two full centres.- Applications in QFT.- Basics of algebraic quantum field theory.- Hard boundaries.- Transparent boundaries.- Further directions.- Conclusions.

Simplicity of heads and socles of tensor products

Abstract: We prove that, for simple modules $M$ and $N$ over a quantum affine algebra, their tensor product $M\otimes N$ has a simple head and a simple socle if $M\otimes M$ is simple. A similar result is proved for the convolution product of simple modules over quiver Hecke algebras.

A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^2$

Abstract: We propose a simple construction of the solution to the continuum parabolic Anderson model on $\mathbf{R}^2$ which does not rely on any elaborate arguments and makes extensive use of the linearity of the equation. A logarithmic renormalisation is required to counterbalance the divergent product appearing in the equation. Furthermore, we use time-dependent weights in our spaces of distributions in order to construct the solution on the unbounded space $\mathbf{R}^2$.

The Möbius function and distal flows

Abstract: We prove that the Mobius function is linearly disjoint from an analytic skew product on the 2 -torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of the Mobius function from various distal homogeneous flows.

Enabling quaternion derivatives: the generalized HR calculus.

Abstract: Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

Abstract: A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovasz (J. Comb. Theory Ser. A, 95:169–179, 2001), but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.

Algorithms for computing backbones of propositional formulae

Abstract: The problem of propositional satisfiability (SAT) has found a number of applications in both theoretical and practical computer science. In many applications, however, knowing a formula's satisfiability alone is insufficient. Often, some other properties of the formula need to be computed. This article focuses on one such property: the backbone of a formula, which is the set of literals that are true in all the formula's models. Backbones find theoretical applications in characterization of SAT problems and they also find practical applications in product configuration or fault localization. This article overviews existing algorithms for backbone computation and introduces two novel ones. Further, an extensive evaluation of the algorithms is presented. This evaluation demonstrates that one of the novel algorithms significantly outperforms the existing ones.

Type IIB supergravity solution for the T-dual of the eta-deformed AdS_5 x S^5 superstring

Abstract: We find an exact type IIB supergravity solution that represents a one-parameter deformation of the T-dual of the AdS_5 x S^5 background (with T-duality applied in all 6 abelian bosonic isometric directions). The non-trivial fields are the metric, dilaton and RR 5-form only. The latter has remarkably simple "undeformed" form when written in terms of a "deformation-rotated" vielbein basis. An unusual feature of this solution is that the dilaton contains a linear dependence on the isometric coordinates of the metric precluding a straightforward reversal of T-duality. If we still formally dualize back, we find exactly the metric, B-field and product of dilaton with RR field strengths as recently extracted from the eta-deformed AdS_5 x S^5 superstring action in arXiv:1507.04239. We also discuss similar solutions for deformed AdS_n x S^n backgrounds with n=2,3. In the eta -> i limit we demonstrate that all these backgrounds can be interpreted as special limits of gauged WZW models and are also related to (a limit of) the Pohlmeyer-reduced models of the AdS_n x S^n superstrings.

Exploring \( {\mathcal{W}}_{\infty } \) in the quadratic basis

Abstract: We study the operator product expansions in the chiral algebra $$ {\mathcal{W}}_{\infty } $$ , first using the associativity conditions in the basis of primary generating fields and then using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form expression for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part we verify the consistency with results derived previously by studying minimal models of $$ {\mathcal{W}}_{\infty } $$ and comparing them to known reductions of $$ {\mathcal{W}}_{\infty } $$ to $$ {\mathcal{W}}_N $$ . The results we obtain illustrate nicely the role of triality symmetry in the representation theory of $$ {\mathcal{W}}_{\infty } $$ .

Identifying the Discharge Product and Reaction Pathway for a Secondary Mg/O2 Battery

Abstract: Secondary Mg/O2 Battery Gulin Vardar,† Emily G. Nelson,‡ Jeffrey G. Smith, Junichi Naruse, Hidehiko Hiramatsu, Bart M. Bartlett,‡ Alice E. S. Sleightholme, Donald J. Siegel,*,†,§,∇ and Charles W. Monroe* †Materials Science and Engineering Department, ‡Chemistry Department, Mechanical Engineering Department, and Chemical Engineering Department, University of Michigan, Ann Arbor, Michigan 48109, United States North America Research & Development, DENSO International America, Inc., 24777 Denso Drive, Southfield, Michigan 48086, United States Research Laboratories, DENSO CORPORATION, 500-1, Minamiyama, Komenoki-cho, Nisshin 470-0111, Japan Department of Energy Conversion and Storage, Technical University of Denmark, Fysikvej, Building 309, 2800 Kgs Lyngby, Denmark Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom