Tensor Operators Research Articles

Testing lepton flavour universality with (semi)-leptonic varvec{D_{(s)}} decays

Data in B-meson decays indicate violations of lepton flavour universality, thereby raising the question about such phenomena in the charm sector. We perform a model-independent analysis of NP contributions in (semi)-leptonic decays of D_{(s)} mesons which originate from c rightarrow d bar{{ell }} nu _l and c rightarrow s bar{{ell }} nu _{ell } charged-current interactions. Starting from the most general low-energy effective Hamiltonian containing four-fermion operators and the corresponding short-distance coefficients, we explore the impact of new (pseudo)-scalar, vector and tensor operators and constrain their effects through the interplay with current data. We pay special attention to the elements |V_{cd}| and |V_{cs}| of the Cabibbo–Kobayashi–Maskawa matrix and extract them from the D_{(s)} decays in the presence of possible NP decay contributions, comparing them with determinations utilizing unitarity. We find a picture in agreement with the Standard Model within the current uncertainties. Using the results from our analysis, we make also predictions for leptonic D_{(s)}^+ rightarrow e^+ nu _e modes which could be hugely enhanced with respect to their tiny Standard Model branching ratios. It will be interesting to apply our strategy at the future high-precision frontier.

Probability distributions for space and time averaged quantum stress tensors

We extend previous work on quantum stress tensor operators which have been averaged over finite time intervals to include averaging over finite regions of space as well. The space and time averaging can be viewed as describing a measurement process for a stress tensor component, such as the energy density of a quantized field in its vacuum state. Although spatial averaging reduces the probability of large vacuum fluctuations compared to time averaging alone, we find that the probability distribution decreases more slowly than exponentially as the magnitude of the measured energy density increases. This implies that vacuum fluctuations can sometimes dominate over thermal fluctuations and potentially have observable effects.

Leading multi-stress tensors and conformal bootstrap

Near lightcone correlators are dominated by operators with the lowest twist. We consider the contributions of such leading lowest twist multi-stress tensor operators to a heavy-heavy-light-light correlator in a CFT of any even dimensionality with a large central charge. An infinite number of such operators contribute, but their sum is described by a simple ansatz. We show that the coefficients in this ansatz can be determined recursively, thereby providing an operational procedure to compute them. This is achieved by bootstrapping the corresponding near lightcone correlator: conformal data for any minimal­ twist determines that for the higher minimal-twist and so on. To illustrate this procedure in four spacetime dimensions we determine the contributions of double- and triple-stress tensors. We compute the OPE coefficients; whenever results are available in the literature, we observe complete agreement. We also compute the contributions of double-stress tensors in six spacetime dimensions and determine the corresponding OPE coefficients. In all cases the results are consistent with the exponentiation of the near lightcone correlator. This is similar to the situation in two spacetime dimensions for the Virasoro vacuum block.

Horizon constraints on holographic Green\u2019s functions

We explore a new class of general properties of thermal holographic Green’s functions that can be deduced from the near-horizon behaviour of classical perturbations in asymptotically anti-de Sitter spacetimes. We show that at negative imaginary Matsubara frequencies and appropriate complex values of the wavenumber the retarded Green’s functions of generic operators are not uniquely defined, due to the lack of a unique ingoing solution for the bulk perturbations. From a boundary perspective these ‘pole-skipping’ points correspond to locations in the complex frequency and momentum planes at which a line of poles of the retarded Green’s function intersects with a line of zeroes. As a consequence the dispersion relations of collective modes in the boundary theory at energy scales ω ∼ T are directly constrained by the bulk dynamics near the black-brane horizon. For the case of conserved U (1) current and energy-momentum tensor operators we give examples where the dispersion relations of hydrodynamic modes pass through a succession of pole- skipping points as real wavenumber is increased. We discuss implications of our results for transport, hydrodynamics and quantum chaos in holographic systems.

Emitted radiation and geometry

In conformal mathcal{N} = 2 Super Yang-Mills theory, the energy emitted by an accelerated heavy particle is computed by the one-point function of the stress tensor operator in the presence of a Wilson line. In this paper, we consider the theory on the ellipsoid and we prove a conjectured relation between the stress tensor one-point function and the first order expansion of the Wilson loop expectation value in the squashing parameter. To do this, we analyze the behavior of the Wilson loop for a small deformation of the background geometry and, at first order in the deformation, we fix the kinematics using defect CFT constraints. In the final part of the paper, we analyze the consequences of our results for the weak coupling perturbative expansion. In particular, comparing the weakly coupled matrix model with the ordinary Feynman diagram expansion, we find a natural transcendentality driven organization for the latter.

Joint Spatial-Spectral Smoothing in a Minimum-Volume Simplex for Hyperspectral Image Super-Resolution

The limitations of hyperspectral sensors usually lead to coarse spatial resolution of acquired images. A well-known fusion method called coupled non-negative matrix factorization (CNMF) often amounts to an ill-posed inverse problem with poor anti-noise performance. Moreover, from the perspective of matrix decomposition, the matrixing of remotely-sensed cubic data results in the loss of data’s structural information, which causes the performance degradation of reconstructed images. In addition to three-dimensional tensor-based fusion methods, Craig’s minimum-volume belief in hyperspectral unmixing can also be utilized to restore the data structure information for hyperspectral image super-resolution. To address the above difficulties simultaneously, this article incorporates the regularization of joint spatial-spectral smoothing in a minimum-volume simplex, and spatial sparsity—into the original CNMF, to redefine a bi-convex problem. After the convexification of the regularizers, the alternating optimization is utilized to decouple the regularized problem into two convex subproblems, which are then reformulated by separately vectorizing the variables via vector-matrix operators. The alternating direction method of multipliers is employed to split the variables and yield the closed-form solutions. In addition, in order to solve the bottleneck of high computational burden, especially when the size of the problem is large, complexity reduction is conducted to simplify the solutions with constructed matrices and tensor operators. Experimental results illustrate that the proposed algorithm outperforms state-of-the-art fusion methods, which verifies the validity of the new fusion approach in this article.

CP asymmetry in τ→KSπντ decays within the standard model and beyond

Motivated by the $2.8\sigma$ discrepancy observed between the BaBar measurement and the Standard Model prediction of the CP asymmetry in $\tau\to K_S\pi\nu_\tau$ decays, as well as the prospects of future measurements at Belle II, we revisit this observable in this paper. Firstly, we reproduce the known CP asymmetry due to $K^0 -\bar{K}^0$ mixing by means of the reciprocal basis, which is convenient when a $K_{S(L)}$ is involved in the final state. As the $K\pi$ tensor form factor plays a crucial role in generating a non-zero direct CP asymmetry that can arise only from the interference of vector and tensor operators, we then present a dispersive representation of this form factor, with its phase obtained in the context of chiral theory with resonances, which fulfills the requirements of unitarity and analyticity. Finally, the $\tau\to K_S\pi\nu_\tau$ decays are analyzed both within a model-independent low-energy effective theory framework and in a scalar leptoquark scenario. It is observed that the CP anomaly can be accommodated in the model-independent framework, even at the $1\sigma$ level, together with the constraint from the branching ratio of $\tau^-\to K_S\pi^-\nu_\tau$ decay; it can be, however, marginally reconciled only at the $2\sigma$ level, due to the specific relation between the scalar and tensor operators in the scalar leptoquark scenario. Once the combined constraints from the branching ratio and the decay spectrum of this decay are taken into account, these possibilities are however both excluded, even without exploiting further the stronger bounds from the (semi-)leptonic kaon decays under the assumption of lepton-flavour universality, as well as from the neutron electric dipole moment and $D-\bar{D}$ mixing under the assumption of $SU(2)$ invariance of the weak interactions.

Quantum Groups in Nuclear Spectra and in Metal Clusters

Quantum algebras (quantum groups), which are nonlinear generalizations of the usual Lie algebras, provide a rich variety of symmetries finding applications in the description of several physical systems [1]. Using irreducible tensor operators under sug(2) a rotationally invariant Hamiltonian which provides a good description of nuclear rotational spectra is constructed and its relation to existing nuclear models is considered. Using the same techniques a 3-dimensional ç-deformed harmonic oscillator with ug(3)Dsoq(3) symmetry is constructed, compared to the modified oscillator of Nilsson, and used for the successful description of magic numbers [2] and supershells [3] in metal clusters.

Encapsulated PGD Algebraic Toolbox Operating with High-Dimensional Data

In its original conception, proper generalized decomposition (PGD) provides explicit parametric solutions, denoted as computational vademecums or digital abacuses, to parametric boundary value problems. The PGD approach is extended here to devise a set of algebraic tools enabling to operate with multidimensional tensor data. These tools are designed to store, compress and perform basic operations (in particular divisions) with tensors in separable format. These tools are directly producing the computational vademecums for the resulting high-dimensional tensor data. Thus, the general methodology enables performing nontrivial operations (storage, compression, division, solving linear systems of equations...) for multidimensional tensor data. The idea is based on the principle of the PGD separation, that produces a separable least squares approximation of any multidimensional function. The PGD compression is a particular case, extensively used in practice to compact the separable solution without loss of accuracy. Here, this concept is applied to algebraic tensor structures that are also seen as functions in multidimensional Cartesian domains. Moreover, a straightforward extension of this concept is devised to operate with multidimensional objects stored in the separable format. That allows creating a toolbox of PGD arithmetic operators that is publicly released at https://git.lacan.upc.edu/zlotnik/algebraicPGDtools . Numerical tests demonstrate the performance and efficiency of the toolbox, both for tensor data handling and operation and also in applications pertaining to the discretized version of boundary value problems.

Intelligent-ReaxFF: Evaluating the reactive force field parameters with machine learning

Machine learning has been widely used in quantum chemistries, such as data mining in quantum mechanics calculation and representations of potential energy surface by neural networks. In this study, we report our efforts on the optimization of the ReaxFF parameters with machine learning frameworks. Although deep neural network potentials like High-Dimensional Neural Network Potentials (HDNNP) have achieved much success in applications such as materials modeling, factors like the memory usage, training time, and accuracies are still problems when the training data set is big. On the other hand, classical potentials like ReaxFF and REBO does not have these problems, and a combination of two is an ideal solution. Machine learning has generated techniques such as automatic differentiation and backpropagation, with which we can optimize deep neural networks or complexed interatomic potentials like ReaxFF. With the TensorFlow coding platform, we have constructed an Intelligent ReaxFF (I-ReaxFF) model with terms of matrix (or tensor) operations that can optimize ReaxFF parameters automatically with gradient-based optimizers like adaptive moment solver (Adam) and backpropagations. As inherited from TensorFlow, one significant feature of our code is the GPU acceleration. The training speed can be five times faster with GPU acceleration than pure CPU calculation. Another feather is that it can directly use the ab initio molecular dynamics trajectories with surrounding periodic images as training data, therefore, allowing the data set can be prepared with ease.

Derivation of Lanthanide Series Crystal Field Parameters From First Principles.

Two series of lanthanide complexes have been chosen to analyze trends in the magnetic properties and crystal field parameters (CFPs) along the two series: The highly symmetric LnZn16 (picHA)16 series (Ln=Tb, Dy, Ho, Er, Yb; picHA=picolinohydroxamic acid) and the [Ln(dpa)3 ](C3 H5 N2 )3 ⋅3H2 O series (Ln=Ce-Yb; dpa=2,6-dipicolinic acid) with approximate three-fold symmetry. The first series presents a compressed coordination sphere of eight oxygen atoms whereas in the second series, the coordination sphere consists of an elongated coordination sphere formed of six oxygen atoms. The CFPs have been deduced from ab initio calculations using two methods: The AILFT (ab initio ligand field theory) method, in which the parameters are determined at the orbital level, and the ITO (irreducible tensor operator) decomposition, in which the problems are treated at the many-electron level. It has been found that the CFPs are transferable from one derivative to another, within a given series, as a first approximation. The sign of the second-order parameter differs in the two series, reflecting the different environments. It has been found that the use of the strength parameter S allows for an easy comparison between complexes. Furthermore, in both series, the parameters have been found to decrease in magnitude along the series, and this decrease is attributed to covalent effects.

Black holes and conformal Regge bootstrap

Highly energetic particles traveling in the background of an asymptotically AdS black hole experience a Shapiro time delay and an angle deflection. These quantities are related to the Regge limit of a heavy-heavy-light-light four-point function of scalar operators in the dual CFT. The Schwarzschild radius of the black hole in AdS units is proportional to the ratio of the conformal dimension of the heavy operator and the central charge. This ratio serves as a useful expansion parameter; its power counts the number of stress tensors in the multi-stress tensor operators which contribute to the four-point function. In the cross-channel the four-point function is determined by the OPE coefficients and anomalous dimensions of the heavy-light double-trace operators. We explain how this data can be obtained and explicitly compute the first and second order terms in the expansion of the anomalous dimensions. We observe perfect agreement with known results in the lightcone limit, which were obtained by computing perturbative corrections to the energy eigenstates in AdS spacetimes.

Splitting of Initial Boundary Value Problems in Anisotropic Linear Elasticity Theory

The splitting of initial boundary value problems in the theories of elasticity is considered for some anisotropic media. In particular, the initial boundary value problems of the micropolar classical theory of elasticity are represented using the tensor-block matrix operators (or tensor operators). In the case of isotropic micropolar elastic media known also as isotropic or transversally isotropic classical media, we propose the tensor-block matrix operators of algebraic cofactors corresponding to the tensor-block matrix operators of the initial boundary value problems, which allows us to split these problems.

Nonlinear Magnetooptical Resonance in 87Rb Vapor: The Influence of Stray Magnetic Fields and Excitation Radiation Intensity on the Basic Characteristics of the Effect in Cells with Antirelaxation Coating

The nonlinear Hanle effect is generally considered in the modern literature as a particular case of coherent population trapping (CPT) at degenerate Zeeman sublevels of the ground atomic state. The shape, amplitude, and sign of observed resonances and their dependence on the excitation parameters and observation geometry were investigated theoretically and experimentally in many studies because of the wide range of application of this effect. The sensitivity of Hanle resonances to the experimental conditions increases significantly in cells with antirelaxation coating, because the atomic ensemble retains its coherence after a large number of collisions with the cell walls. In this paper, the main attention is paid to the joint influence of the excitation radiation intensity and spurious magnetic fields on CPT resonances observed in fluorescence. The theoretical description is based on the numerical solution of an algebraic system of equations for density matrix $$\hat {\rho }$$ in the formalism of irreducible tensor operators. The parameters relating different polarization moments in each equation are visualized using the modified sparse matrix of the system. It is established by numerical simulation that the shape, width, amplitude, and sign of nonlinear magnetooptical resonances change nonmonotonically with a change in the parameters of the excitation radiation and atomic system. The presented theoretical results are in good agreement with the experimental data obtained under various conditions.

Multivariate Multi-Order Markov Multi-Modal Prediction With Its Applications in Network Traffic Management

Predicting the future network traffic through big data analysis technologies has been one of the important preoccupations of network design and management. Combining Markov chains with tensors to implement predictions has received considerable attention in the era of big data. However, when dealing with multi-order Markov models, the existing approaches including the combination of states and Z-eigen decomposition still face some shortcomings. Therefore, this paper focuses on proposing a novel multivariate multi-order Markov transition to realize multi-modal accurate predictions. First, we put forward two new tensor operations including tensor join and unified product (UP). Then a general multivariate multi-order (2M) Markov model with its UP-based state transition is proposed. Afterwards, we develop a multi-step transition tensor for 2M Markov models to implement the multi-step state transition. Furthermore, an UP-based power method is proposed to calculate the stationary joint probability distribution tensor (i.e., stationary joint eigentensor, SJE) and realize SJE based multi-modal accurate predictions. Finally, a series of experiments under various Markov models on real-world network traffic datasets are conducted. Experimental results demonstrate that the proposed SJE based approach can improve the prediction accuracy for network traffic by highest up to 38.47 percentage points compared with the Z-eigen based approach.

Judd-Ofelt parametrization from emission spectra: The case study of the Eu3+ 5D1 emitting level

Traditional applications of the Judd-Ofelt (JO) theory to the analysis of the Eu3+ optical spectra make use of the emission transitions originating from the 5D0 manifold. In the present paper, we report an alternative method of evaluating the JO intensity parameters from the Eu3+ emission spectra based on the 5D1 → 7F0,1 transitions. The reduced matrix elements of the unit tensor operators are re-calculated for the 5D0,1,2 → 7F0,1,…,6 Eu3+ transitions in the intermediate coupling approximation using the average electrostatic and spin-orbit coupling parameters. The suggested method was tested by analyzing the emission spectra of the Eu3+ doped GdAlO3, LaF3, NaYF4, Y2O3, ZrO2, YNbO4, ZBLA and PIGLZ hosts. It is shown that the developed method is more accurate for the hosts with relatively high 5D1 level population, which emphasizes its high potential and applicability. In addition to the JO analysis, the CIE chromaticity coordinates are calculated for the investigated spectra.

GPGPU Linear Complexity t-SNE Optimization.

In recent years the t-distributed Stochastic Neighbor Embedding (t-SNE) algorithm has become one of the most used and insightful techniques for exploratory data analysis of high-dimensional data. It reveals clusters of high-dimensional data points at different scales while only requiring minimal tuning of its parameters. However, the computational complexity of the algorithm limits its application to relatively small datasets. To address this problem, several evolutions of t-SNE have been developed in recent years, mainly focusing on the scalability of the similarity computations between data points. However, these contributions are insufficient to achieve interactive rates when visualizing the evolution of the t-SNE embedding for large datasets. In this work, we present a novel approach to the minimization of the t-SNE objective function that heavily relies on graphics hardware and has linear computational complexity. Our technique decreases the computational cost of running t-SNE on datasets by orders of magnitude and retains or improves on the accuracy of past approximated techniques. We propose to approximate the repulsive forces between data points by splatting kernel textures for each data point. This approximation allows us to reformulate the t-SNE minimization problem as a series of tensor operations that can be efficiently executed on the graphics card. An efficient implementation of our technique is integrated and available for use in the widely used Google TensorFlow.js, and an open-source C++ library.

Grothendieck constant is norm of Strassen matrix multiplication tensor

We show that two important quantities from two disparate areas of complexity theory --- Strassen's exponent of matrix multiplication $\omega$ and Grothendieck's constant $K_G$ --- are intimately related. They are different measures of size for the same underlying object --- the matrix multiplication tensor, i.e., the $3$-tensor or bilinear operator $\mu_{l,m,n} : \mathbb{F}^{l \times m} \times \mathbb{F}^{m \times n} \to \mathbb{F}^{l \times n}$, $(A,B) \mapsto AB$ defined by matrix-matrix product over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$. It is well-known that Strassen's exponent of matrix multiplication is the greatest lower bound on (the log of) a tensor rank of $\mu_{l,m,n}$. We will show that Grothendieck's constant is the least upper bound on a tensor norm of $\mu_{l,m,n}$, taken over all $l, m, n \in \mathbb{N}$. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck's inequality as a norm inequality \[ \lVert\mu_{l,m,n}\rVert_{1,2,\infty} =\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\lVert X\rVert_{1,2}\lVert Y\rVert_{2,\infty}\lVert M\rVert_{\infty,1}}\le K_G. \] We prove that Grothendieck's inequality is unique: If we generalize the $(1,2,\infty)$-norm to arbitrary $p,q, r \in [1, \infty]$, \[ \lVert\mu_{l,m,n}\rVert_{p,q,r}=\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\|X\|_{p,q}\|Y\|_{q,r}\|M\|_{r,p}}, \] then $(p,q,r )=(1,2,\infty)$ is, up to cyclic permutations, the only choice for which $\lVert\mu_{l,m,n}\rVert_{p,q,r}$ is uniformly bounded by a constant independent of $l,m,n$.

Nonlinear finite element formulation based on invariant-free hyperelasticity for orthotropic materials

The conventional representation of isotropic hyperelastic strain energy densities as functions of scalar invariants of finite deformation tensors does not naturally extend to the field of anisotropic mechanics. Formulating an invariant-free representation of the strain energy function, fourth-order Orthotropic Lamé tensors define the constitutive law whilst naturally collapsing to the transversely isotropic and fully isotropic case where necessary simply as a by-product of known material symmetries. In this study, a simple linear isoparametric hexahedral finite element capable of describing anisotropic invariant-free hyperelasticity is presented. Careful conversion of the fourth-order tensor operations present in the strain energy function to computational arrays then applying to the principle of virtual work generates a weak formulation for finite element analyses. The finite element is then applicable to materials of any degree of anisotropy or compressibility and is particularly useful for predicting highly nonlinear responses such as the stiffening of fibrous biological tissues. A discussion of simple shear experimentation and modelling follows, as well as remarks on modelling nearly-incompressible materials.

The Ocean Tensor Package

The Ocean Tensor Package is an open-source package for matrix and tensor operations on CPU and GPU. The package aims to serve as a foundational layer for applications that require dense tensor operations on a variety of device types. All operations are available through a unified interface that is carefully designed to be powerful, extensible, and at the same time easy to use. The package has a modular implementation in C and provides a light-weight Python interface. Modularity of the package facilitates the addition of new operations as well as new device types.