Reflection Equation Research Articles

Integrating quantum groups over surfaces

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)-dimensional part of Crane–Yetter–Kauffman four-dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld–Jimbo quantum group U q ( g ) we obtain in this way an aspect of topologically twisted four-dimensional N = 4 super Yang–Mills theory, the setting introduced by Kapustin–Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of G-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to U q ( g ) , and from the punctured torus we recover the algebra of quantum differential operators associated to U q ( g ) . From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum D-modules.

Solutions of the reflection equations

We find the complete set of invertible solutions of the untwisted and twisted reflection equations for the Bazhanov–Jimbo R-matrix of type . We also show that all invertible solutions can be obtained by an appropriate affinization procedure from solutions of the constant untwisted and twisted reflection equations.

Square-Root Consider Filters with Hyperbolic Householder Reflections

Maintaining positive-definiteness of a filter’s error covariance estimate is of paramount importance to any stable and reliable estimation scheme. Two widely used methods for protecting a filter from loss of positive-definiteness are that of consider filtering and square-root filtering. Despite their widespread use in estimation applications, a filtering methodology that combines the two techniques has yet to be presented in a unified, streamlined architecture. This paper develops a new approach to square-root filtering using hyperbolic Householder reflections. A specialization of the method is considered to permit computationally efficient applications, and a numerical example pertaining to a ballistic reentry is presented to validate the proposed approaches.

Creation of qutrit and one-qubit gates in atom–cavity–laser systems by adiabatic passage

We derive an adiabatic technique that implements arbitrary qutrit and one-qubit rotation gates in a quantum system composed of two tripod atoms each of which interacting with two laser pulses and a cavity field. In this method, the atomic ground states playing the role of the qutrit and qubit subspace and losses due to atomic spontaneous emissions and the cavity decay are efficiently suppressed by employing adiabatic passage technique. In order to create qutrit gates, we exploit generalized quantum Householder reflection in which each Householder reflection is implemented by two-step adiabatic passage technique in a seven-state system with two dark states. With a similar method, an arbitrary rotation gate is constructed in this system. We also implement the Fourier transform in a qutrit as an example.

Two-stage and single-stage seismic-petrophysical inversions applied in the Nile Delta

We discuss two Bayesian seismic-petrophysical inversion techniques, the two-stage and single-stage approach, which estimate petrophysical properties from prestack seismic data. Both approaches are developed considering linear approximations and are applied for reservoir appraisal studies in the Nile Delta. The two-stage approach first uses an amplitude-variation-with-angle (AVA) inversion to infer P-wave velocity, S-wave velocity, and density. Then, a linear empirical rock-physics model, properly calibrated for the investigated area, is used to transform the estimated elastic parameters into the petrophysical properties of interest under the assumption of Gaussian mixture distributed petrophysical properties. The single-stage approach uses the linear rock-physics model to rewrite the time-continuous P-wave reflectivity equation as a function of the petrophysical contrasts instead of the elastic constants. This reformulation, together with the assumption of a Gaussian prior model, allows us to directly derive (in a single-stage inversion) the petrophysical properties from AVA data. Despite the differences in the forward-model parameterization and in the assumed a priori distribution for the petrophysical properties, the two inversion methods yield comparable predictions that are consistent with borehole data. For the investigated zone, this work demonstrates that, although neglecting the facies-dependent behavior of petrophysical properties, the a priori Gaussian assumption for the petrophysical properties can provide reliable predictions with a very limited computational effort. In addition, the good match between predicted and logged properties proves the suitability of the linear rock-physics model for reservoir characterization in the Nile Delta.

Matrix product solutions to the reflection equation from three dimensional integrability

We formulate a quantized reflection equation in which q-boson valued L and K matrices satisfy the reflection equation up to conjugation by a solution to the Isaev–Kulish 3D reflection equation. By forming its n-concatenation along the q-boson Fock space followed by suitable reductions, we construct families of solutions to the reflection equation in a matrix product form connected to the 3D integrability. They involve the quantum R matrices of the antisymmetric tensor representations of and the spin representations of , and .

Improvement of pressure-saturation changes estimations from time-lapse PP-AVO data by using non-linear optimization method

First-order approximations of seismic parameters in AVO intercept and gradient changes are not sufficient to estimate changes in saturation and effective stress accurately. Analysis from an unconsolidated reservoir and a compacting reservoir show that we need to consider higher order terms in seismic parameters to reduce the inaccuracy of the estimates. Furthermore, improper approximation of rock physics parameters increases the error in the estimates. Here, we implement a non-linear optimization method to estimate changes in saturation and pressure using reflectivity equation directly. We test the applicability of this non-linear optimization method on synthetic data for both the above-mentioned reservoir scenarios over a wide range of realistic saturation and pressure changes. We observe that the inversion results using the new method are reasonably good for both reservoir scenarios.

Solving infrastructural concerns through a market reorganization: A case study of a Danish smart grid demonstration

Following the rapid growth of wind power in Denmark in the past 20 years, energy infrastructure has become increasingly politicized. Fluctuating renewables not only contest the dominant ‘logic’ of operating the system, namely ‘supply-follows-demand’, but it also introduces new actors like aggregators and reconfigures existing market actors. In this paper, we study a case, EcoGrid 2.0 on the Danish island Bornholm, as a case of a ‘marketized’ solution to the infrastructural concerns emerging from the large share of fluctuating wind power in the system. The market design involves transforming ‘flexible consumption’ into an exchangeable good, as well as a transformation of households into ‘distributed energy resources’, making it possible to capitalize on the existing infrastructure in new ways. We end the paper with a discussion of the implications for infrastructure; when households become balancing entities and a digital and smart infrastructure is made indispensable to the operation of the system, the infrastructure grows significantly in terms of scope and complexity eventually introducing yet new challenges.

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

A two-step inversion approach for seismic-reservoir characterization and a comparison with a single-loop Markov-chain Monte Carlo algorithm

We have evaluated a two-step Bayesian algorithm for seismic-reservoir characterization, which, thanks to some simplifying assumptions, is computationally very efficient. The applicability and reliability of this method are assessed by comparison with a more sophisticated and computer-intensive Markov-chain Monte Carlo (MCMC) algorithm, which in a single loop directly estimates petrophysical properties and lithofluid facies from prestack data. The two-step method first combines a linear rock-physics model (RPM) with the analytical solution of a linearized amplitude versus angle (AVA) inversion, to directly estimate the petrophysical properties, and related uncertainties, from prestack data under the assumptions of a Gaussian prior model and weak elastic contrasts at the reflecting interface. In particular, we use an empirical, linear RPM, properly calibrated for the investigated area, to reparameterize the linear time-continuous P-wave reflectivity equation in terms of petrophysical contrasts instead of elastic constants. In the second step, a downward 1D Markov-chain prior model is used to infer the lithofluid classes from the outcomes of the first step. The single-loop (SL) MCMC algorithm uses a convolutional forward modeling based on the exact Zoeppritz equations, and it adopts a nonlinear RPM. Moreover, it assumes a more realistic Gaussian mixture distribution for the petrophysical properties. Both approaches are applied on an onshore 3D seismic data set for the characterization of a gas-bearing, clastic reservoir. Notwithstanding the differences in the forward-model parameterization, in the considered RPM, and in the assumed a priori probability density functions, the two methods yield maximum a posteriori solutions that are consistent with well-log data, although the Gaussian mixture assumption adopted by the SL method slightly improves the description of the multimodal behavior of the petrophysical parameters. However, in the considered reservoir, the main difference between the two approaches remains the very different computational times, the SL method being much more computationally intensive than the two-step approach.

True amplitude recovery in reverse time extrapolation of plane and spherical waves

A challenging outstanding problem in reverse time extrapolation is recovering accurate amplitudes at reflectors from the receiver wavefield. Various migrations have been developed to produce accurate image locations rather than correct amplitude information because of inadequate compensation of attenuation, dispersion, and transmission losses. We have evaluated the requirements, and determined the theoretical feasibility, of true amplitude recovery of 2D acoustic and elastic seismic data by using the analytic Zoeppritz equations for plane-wave reflection and transmission coefficients. Then, we used synthetic acoustic and elastic wavefield data generated by elastodynamic finite differences to verify the recovery, in the reverse time propagation, of spherical waves and illustrated the salient differences between the incident wavefields reconstructed from reflection data only and from the combination of reflection and transmission data. These examples quantitatively verify that recovering an incident plane or a spherical wave requires the reverse time propagation of all reflections and transmissions in a model with the correct velocity and density. Accurate reconstruction of an incident wave is not possible by backward propagation of only reflections. As an application, we removed downgoing internal multiple reflections generated by upgoing waves incident at reflectors shallower than a horizontal well, in which geophones are deployed. The subtraction of the downgoing reflection involves wavefield reconstruction at depths shallower than the horizontal well and separation of upgoing and downgoing wavefields. This approach assumes that the correct acoustic (or elastic) velocity and density models are available in, and shallower than, the layer where the horizontal well is located. Incident-wave reconstruction works equally well for smooth models, as for models with sharp boundaries. Uncertainties in the model used for reconstruction, and incompleteness of the data aperture are propagated into the equivalent uncertainties, and incompleteness of the reconstruction.

Utopian visions for American independent filmmaking

Abstract Today’s American independent movie scene, like the contemporary world system, is plagued by political economic contradictions and crises. Thousands of indie filmmakers across the country find their works increasingly lost in a rapidly changing marketplace where cinematic supply, bolstered by a proliferation of digital production technologies and online distribution platforms, far exceeds audience demand. And yet, a group of besieged artists working in the shadows of Hollywood are envisioning a utopian tomorrow despite the unprecedented challenges they face today. In fact, these unsung directors, producers, and writers express enthusiasm about precisely the new business models, media shifts and social relations that are threatening their very livelihoods. They are using their filmmaking imaginations to engage in ‘dialectical dreaming’, a playful process of generating ideal possibilities out of everyday limitations. Examples from an ethnographic study of more than twenty such filmmakers living in Los Angeles include the invention of cognitive storytelling software to level the resource playing field, the development of completely collaborative film projects through crowd-sourcing, and the transformation of single-family households into creative community workspaces. These dialectical dreams of socialized means of production, collectively owned intellectual property, and public/private sphere synthesis are placed within the context of recent innovations in Marxist theorizing about utopia in the hopes of inspiring not only new forms of cinematic practice, but also radical action among creative professionals across the current socio-economic structure.

Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators

We consider intertwining relations of the augmented q-Onsager algebra introduced by Ito and Terwilliger, and obtain generic (diagonal) boundary K-operators in terms of the Cartan element of Uq(sl2). These K-operators solve reflection equations. Taking appropriate limits of these K-operators in Verma modules, we derive K-operators for Baxter Q-operators and corresponding reflection equations.

Theoretical and experimental study of mirrorless fiber optics refractometer based on quasi-Gaussian approach

A mirrorless refractometer was studied and analyzed using the quasi-Gaussian beam approach. The Fresnel equation for reflectivity at the interface between two mediums with different refractive indices was used to calculate the directional reflectivity, R. Various liquid samples from 1.3325 to 1.4657 refractive indices units were used. Experimentally, a fiber bundle probe with a concentric configuration of 16 receiving fibers and a single transmitting fiber was employed to verify the developed models. The sensor performance in term of sensitivity, linear range, and resolution, were analyzed and calculated. It has been shown that the developed theoretical models are capable of providing quantitative guidance of the output of the sensor with high accuracy. The highest resolution of the sensor was 4.39 × 10−3 refractive indices units, obtained by correlating the peak voltage along the refractive index. The resolution is sufficient for determining the specific refractive index increment of most polymer solutions, certain proteins, and also in monitoring bacterial growth. The accuracy, simplicity, and effectiveness of the proposed sensor over a long period of time while having non-contact measurements reflect a good potential for commercialization.

A note on optimality of quantum circuits over metaplectic basis

Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate R = |0ih0| + |1ih1| − |2ih2|. It is arguably, a ternary basis with the simplest geometry. Recently Cui, Kliuchnikov, Wang and the Author have proposed a compilation algorithm to approximate any twolevel Householder reflection to precision ε by a metaplectic circuit of R-count at most C log3 (1/ε) + O(log log 1/ε) with C = 8. A new result in this note takes the constant down to C = 5 for non-exceptional target reflections under a certain credible numbertheoretical conjecture. The new method increases the chances of obtaining a truly optimal circuit but may not guarantee the true optimality. Efficient approximations of an important ternary quantum gate proposed by Howard, Campbell and others is also discussed.

N-Dimensional LLL Reduction Algorithm with Pivoted Reflection.

The Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm and many of its variants have been widely used by cryptography, multiple-input-multiple-output (MIMO) communication systems and carrier phase positioning in global navigation satellite system (GNSS) to solve the integer least squares (ILS) problem. In this paper, we propose an n-dimensional LLL reduction algorithm (n-LLL), expanding the Lovász condition in LLL algorithm to n-dimensional space in order to obtain a further reduced basis. We also introduce pivoted Householder reflection into the algorithm to optimize the reduction time. For an m-order positive definite matrix, analysis shows that the n-LLL reduction algorithm will converge within finite steps and always produce better results than the original LLL reduction algorithm with n > 2. The simulations clearly prove that n-LLL is better than the original LLL in reducing the condition number of an ill-conditioned input matrix with 39% improvement on average for typical cases, which can significantly reduce the searching space for solving ILS problem. The simulation results also show that the pivoted reflection has significantly declined the number of swaps in the algorithm by 57%, making n-LLL a more practical reduction algorithm.

Matrix product solutions to the G2 reflection equation

We study the $G_2$ reflection equation for the three particles in $1+1$ dimension that undergo a special scattering/reflections described by the Pappus theorem. It is a sixth order equation and serves as a natural $G_2$ analogue of the Yang-Baxter and the reflection equations corresponding to the cubic and the quartic Coxeter relations of type $A$ and $BC$, respectively. We construct matrix product solutions to the $G_2$ reflection equation by exploiting a connection to the representation theory of the quantized coordinate ring $A_q(G_2)$.

Photocathode quantum efficiency of ultrathin Cs2Te layers on Nb substrates

The quantum efficiencies (QE) of photocathodes consisting of bulk Nb substrates coated with thin films of Cs2Te are reported. Using the standard recipe for Cs2Te deposition developed for Mo substrates (220 {\AA} Te thickness), a QE ~11% - 13% at light wavelength of 248 nm is achieved for the Nb substrates, consistent with that found on Mo. Systematic reduction of the Te thickness for both Mo and Nb substrates reveals a surprisingly high residual QE ~ 6% for a Te layer as thin as 15 {\AA}. A phenomenological model based on the Spicer 3-Step model along with a solution of the Fresnel equations for reflectance, R, leads to a reasonable fit of the thickness dependence of QE and suggests that layers thinner than 15 {\AA} may still have a relatively high QE. Preliminary investigation suggests an increased operational lifetime as well. Such an ultra-thin, semiconducting Cs2Te layer may be expected to produce minimal ohmic losses for RF frequencies ~ 1 GHz. The result thus opens the door to the potential development of a Nb (or Nb3Sn) superconducting photocathode with relatively high QE and minimal RF impedance to be used in a superconducting radiofrequency (SRF) photoinjector.

On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

Let be the space of bilinear forms on with defining matrices endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action of the Poisson–Lie group on . A classification is given of all possible quadratic brackets on preserving the Poisson property of the action, thus endowing with the structure of a Poisson homogeneous space. Besides the product Poisson structure on , there are two other (mutually dual) structures, which (unlike the product Poisson structure) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples with the Poisson action , and it is shown that then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid. Bibliography: 30 titles.

Hydrodynamic conditions in front of a vertical wall with an overhanging horizontal cantilever slab

Transforming wave heights from offshore to the shoreline is the first step of any coastal engineering work. Wave breaking is analyzed to understand hydrodynamic conditions. For vertical breakwaters and sea walls, wave reflection is an important process that affects the determination of the wave height. Many of the design formulas presented in the literature depend on empirical studies based on the structures tested. In this study, the hydrodynamic conditions in front of a vertical wall with an overhanging horizontal cantilever slab with a foreshore slope of 1/20 are determined experimentally under regular wave conditions to assess the applicability of the formulas of Goda (2000) for predicting the nearshore wave height and breaker index equation (Goda, 2010). The selection of wave measurements used to determine the design wave height, the reflection coefficients, and wave breaking is also analyzed, and the reflection equations are derived from the dataset covering different breaker types. Small-scale tests show that the incident wave height is a good representative of the design wave height and that the values predicted by Goda are in good agreement with actual measurements. However, the predicted Hmax values are overestimated. In addition, the inception of the wave breaking point is postponed because of the reflection and/or turbulence left over from preceding waves, which is an effect of the vertical wall. At higher water levels, the effect of the vertical wall on the inception point becomes more significant.