Full Operator Research Articles

On LAMDA clustering method based on typicality degree and intuitionistic fuzzy sets

The learning algorithm for multivariable data analysis (LAMDA) is a learning method to group or classify quantitative and qualitative historical data. LAMDA can be applied for self-learning and directed learning. Considering self-learning, LAMDA groups historical data through the global membership degrees of a sample with respect to a class, considering the contribution of all features for each sample. Such contribution is called marginal adequacy degree (MAD). To mix all MADs, several aggregation functions such as linear and geometrical connectives are used. The aggregation function calculates the global adequacy degree (GAD) of a sample with respect to a class. This procedure avoids assigning the class number as initial parameter due to that LAMDA applies an automatic threshold called non-informative class (NIC) which finds the best assignment of a portion of data to one class. One improvement for the LAMDA algorithm is the use of a full reinforcement operator called Yager–Rybalov Triple Pi, which can reduce the number of classes. However, this kind of solution ignores the degree to which one sample belongs to one class relative to all samples. Another limitation is that the classes reduction is not controlled by a parameter or an algorithm, which removes some classes required by a human expert. Such removed classes can be useful for the human expert in the data analysis of complex processes or for intelligent systems when the searching of additional information is required in decision-making. In this paper, the typicality degree and the intuitionistic fuzzy sets (IFSs) are used to propose one improvement of the LAMDA method in order to develop a strategy that avoids the loss of significant classes. This paper shows two functions: global typicality degree (GTD) and intuitionistic global adequacy degree (IGAD). By integrating both functions, a new function called typicality and intuitionistic global adequacy degree (TIGAD) is proposed. The proposal is applied in three study cases. The results show an improvement of clustering which were validated by the human expert and the information of data. Furthermore, the impact of the research is focused on data analysis, where the expert should make the best decision.

Quantum coherence generating power, maximally abelian subalgebras, and Grassmannian geometry

We establish a direct connection between the power of a unitary map in d-dimensions (d < ∞) to generate quantum coherence and the geometry of the set Md of maximally abelian subalgebras (of the quantum system full operator algebra). This set can be seen as a topologically non-trivial subset of the Grassmannian over linear operators. The natural distance over the Grassmannian induces a metric structure on Md, which quantifies the lack of commutativity between the pairs of subalgebras. Given a maximally abelian subalgebra, one can define, on physical grounds, an associated measure of quantum coherence. We show that the average quantum coherence generated by a unitary map acting on a uniform ensemble of quantum states in the algebra (the so-called coherence generating power of the map) is proportional to the distance between a pair of maximally abelian subalgebras in Md connected by the unitary transformation itself. By embedding the Grassmannian into a projective space, one can pull-back the standard Fubini-Study metric on Md and define in this way novel geometrical measures of quantum coherence generating power. We also briefly discuss the associated differential metric structures.

A drift-kinetic analytical model for scrape-off layer plasma dynamics at arbitrary collisionality

A drift-kinetic model to describe the plasma dynamics in the scrape-off layer region of tokamak devices at arbitrary collisionality is derived. Our formulation is based on a gyroaveraged Lagrangian description of the charged particle motion, and the corresponding drift-kinetic Boltzmann equation that includes a full Coulomb collision operator. Using a Hermite–Laguerre velocity space decomposition of the gyroaveraged distribution function, a set of equations to evolve the coefficients of the expansion is presented. By evaluating explicitly the moments of the Coulomb collision operator, distribution functions arbitrarily far from equilibrium can be studied at arbitrary collisionalities. A fluid closure in the high-collisionality limit is presented, and the corresponding fluid equations are compared with previously derived fluid models.

How Does Passive Optical Network Tackle Radio Access Network Evolution?

This paper describes full service access network operator group perspectives on the feasibility of optical access networks to carry radio access network traffic for the existing and future mobile generations. The paper discusses topics related to the evolution of radio access interfaces, such as backhaul, midhaul, and fronthaul, their transmission through optical distribution networks, and passive optical network technology feasibility based on both time and wavelength division multiplexing and multiple accesses. Technology extensions are also discussed to provide guidance for future research.

A stiffly accurate integrator for elastodynamic problems

We present a new integration algorithm for the accurate and efficient solution of stiff elastodynamic problems governed by the second-order ordinary differential equations of structural mechanics. Current methods have the shortcoming that their performance is highly dependent on the numerical stiffness of the underlying system that often leads to unrealistic behavior or a significant loss of efficiency. To overcome these limitations, we present a new integration method which is based on a mathematical reformulation of the underlying differential equations, an exponential treatment of the full nonlinear forcing operator as opposed to more standard partially implicit or exponential approaches, and the utilization of the concept of stiff accuracy which ensures that the efficiency of the simulations is significantly less sensitive to increased stiffness. As a consequence, we are able to tremendously accelerate the simulation of stiff systems compared to established integrators and significantly increase the overall accuracy. The advantageous behavior of this approach is demonstrated on a broad spectrum of complex examples like deformable bodies, textiles, bristles, and human hair. Our easily parallelizable integrator enables more complex and realistic models to be explored in visual computing without compromising efficiency.

Low lying spectral gaps induced by slowly varying magnetic fields

We consider a periodic Schrödinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schrödinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure.First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere.Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.

A Fourier dimensionality reduction model for big data interferometric imaging

Data dimensionality reduction in radio interferometry can provide savings of computational resources for image reconstruction through reduced memory footprints and lighter computations per iteration, which is important for the scalability of imaging methods to the big data setting of the next-generation telescopes. This article sheds new light on dimensionality reduction from the perspective of compressed sensing theory and studies its interplay with imaging algorithms designed in the context of convex optimization. We propose a post-gridding linear data embedding to the space spanned by the left singular vectors of the measurement operator, providing a dimensionality reduction below image size. This embedding preserves the null space of the measurement operator and hence also its sampling properties as per compressed sensing theory. We show that this can be approximated by first computing the dirty image and then applying a weighted subsampled discrete Fourier transform to obtain the final reduced data vector. This Fourier dimensionality reduction model ensures a fast implementation of the full measurement operator, essential for any iterative image reconstruction method. The proposed reduction also preserves the i.i.d. Gaussian properties of the original measurement noise. For convex optimization-based imaging algorithms, this is key to justify the use of the standard L2-norm as the data fidelity term. Our simulations confirm that this dimensionality reduction approach can be leveraged by convex optimization algorithms with no loss in imaging quality relative to reconstructing the image from the complete visibility data set. Reconstruction results in simulation settings with no direction dependent effects or calibration errors show promising performance of the proposed dimensionality reduction. MATLAB code implementing the proposed reduction method is available on GitHub.

Black hole singularity, generalized (holographic) c-theorem and entanglement negativity

In this paper we revisit the question that in what sense empty AdS5 black brane geometry can be thought of as RG-flow. We do this by first constructing a holographic c-function using causal horizon in the black brane geometry. The UV value of the c-function is aUV and then it decreases monotonically to zero at the curvature singularity. Intuitively, the behavior of the c-function can be understood if we recognize that the dual CFT is in a thermal state and thermal states are effectively massive with a gap set by the temperature. In field theory, logarithmic entanglement negativity is an entanglement measure for mixed states. For example, in two dimensional CFTs at finite temperature the renormalized entanglement negativity of an interval has UV (Low-T) value cUV and IR (High-T) value zero. So this is a potential candidate for our c-function. In four dimensions we expect the same thing to hold on physical grounds. Now since the causal horizon goes behind the black brane horizon the holographic c-function is sensitive to the physics of the interior. Correspondingly the field theory c-function should also contain information about the interior. So our results suggest that high temperature (IR) expansion of the negativity (or any candidate c-function) may be a way to probe part of the physics near the singularity. Negativity at finite temperature depends on the full operator content of the theory and so perhaps this can be done in specific cases only.The existence of this c-function in the bulk is an extreme example of the paradigm that space-time is built out of entanglement. In particular the fact that the c-function reaches zero at the curvature singularity correlates the two facts: loss of quantum entanglement in the IR field theory and the end of geometry in the bulk which in this case is the formation of curvature singularity.

Theory and computation of electromagnetic transition matrix elements in the continuous spectrum of atoms

The present study examines the mathematical properties of the free-free ( f-f) matrix elements of the full electric field operator, of the multipolar Hamiltonian. Special methods are developed and applied for their computation, for the general case where the scattering wavefunctions are calculated numerically in the potential of the term-dependent (N-1) electron core, and are energy-normalized. It is found that, on the energy axis, the f-f matrix elements of the full operator have singularities of first order in the case of equal photoelectron energies. The numerical applications are for f-f transitions in Hydrogen and Neon, obeying electric dipole and quadrupole selection rules. In the limit of zero photon wave-number, the full operator reduces to the length form of the electric dipole approximation (EDA). It is found that the results for the EDA agree with those of the full operator, with the exception of a photon wave-number region about the singularity.

Nuclear matrix elements for neutrinoless double-beta decay in covariant density functional theory

We review the first fully relativistic description for the neutrinoless double-beta decay by the beyond-mean-field covariant density functional theory. The calculations of the nuclear transition matrix elements are based on a full relativistic transition operator and nuclear wave functions in which the dynamic effects of particle-number, angular-momentum, and parity conservations as well as shape fluctuations are incorporated. We find that the commonly used nonrelativistic approximation for the transition operator is justified within the mechanism of light-Majorana-neutrino exchange. We stress that reducing the discrepancies in the predicted nuclear matrix elements by different nuclear models is a major challenge in nuclear physics.

EUV two-photon-ionization cross sections of helium from the solution of the time-dependent Schrödinger equation, and comparison with measurements using free-electron lasers

Two recent experimental papers reported the first measurements of absolute two-photon-ionization cross sections $\ensuremath{\sigma}(2)$ of helium, for EUV wavelengths, using free-electron laser (FEL) pulses [Sato et al., J. Phys. B 44, 161001 (2011); Fushitani et al., Phys. Rev. A 88, 063422 (2013)]. The wavelengths correspond to transitions that are off resonance as well as on resonance with the $1s2p$ and $1s3p{\phantom{\rule{0.16em}{0ex}}}^{1}{P}^{o}$ Rydberg states. Inspection of their results reveals considerable discrepancies, while their comparison with theoretical results obtained earlier from time-independent calculations, one perturbative and two nonperturbative ones, cannot lead to secure conclusions as to the true values of $\ensuremath{\sigma}(2)$. We examined this prototypical problem by implementing a time-dependent approach, which utilizes the nonperturbative solution of the time-dependent Schr\odinger equation. This solution was obtained in terms of the state-specific expansion approach, in an upgraded version where the coupling matrix elements are computed using the full electric operator of the multipolar Hamiltonian. The $\ensuremath{\sigma}(2)$ were obtained for pulses of 300 fs, as in the 2011 FEL experiment. Their computation was achieved by fitting the time-dependent ionization survival probability to ${e}^{\ensuremath{-}\mathrm{\ensuremath{\Gamma}}t}$, where $\mathrm{\ensuremath{\Gamma}}$ is the rate of ionization. The wavelengths and intensities are those of the FEL experiments, as well as others, such as the wavelengths 52.22 and 51.56 nm, for which the $1s4p\phantom{\rule{0.28em}{0ex}}{}^{1}{P}^{o}$ and $1s5p\phantom{\rule{0.28em}{0ex}}{}^{1}{P}^{o}$ levels are on resonance with the initial $^{1}S$ state. Apart from the predictions for these wavelengths, the paper contains characteristic comparisons among all the results on these EUV $\ensuremath{\sigma}(2)$, experimental and theoretical. In general, the trends predicted by nonperturbative methods are confirmed by the FEL measurements. However, discrepancies exist among the absolute numbers. Furthermore, comparison among the results of the three nonperturbative approaches (present time-dependent and two earlier time-independent ones published in 1999 and 2005) indicates an overall consistency, although quantitative differences for individual cases are apparent.

Probabilistic Unawareness

The modeling of awareness and unawareness is a significant topic in the doxastic logic literature, where it is usually tackled in terms of full belief operators. The present paper aims at a treatment in terms of partial belief operators. It draws upon the modal probabilistic logic that was introduced by Aumann (1999) at the semantic level, and then axiomatized by Heifetz and Mongin (2001). The paper embodies in this framework those properties of unawareness that have been highlighted in the seminal paper by Modica and Rustichini (1999). Their paper deals with full belief, but we argue that the properties in question also apply to partial belief. Our main result is a (soundness and) completeness theorem that reunites the two strands—modal and probabilistic—of doxastic logic.

Geometrical and electronic structure of the molybdenum and tungsten halides MX3 and MX4 (M = Mo, W; X = F, Cl): Jahn-Teller effect and spin-orbit coupling

The ground and lower-lying excited electronic states of MX3 and MX4 (M = Mo, W; X = F, Cl) molecules were systematically studied by the complete active space self-consistent field (CASSCF) and multiconfigurational quasi-degenerate second-order perturbation (XMCQDPT2) methods. Scalar-relativistic effects and spin-orbit coupling were taken into account employing the third-order Douglas-Kroll-Hess (DKH) Hamiltonian and full Breit-Pauli operator. The ground orbital state of MoF3 and MoCl3 complexes is quadruplet 4A2′ state in the equilibrium configuration with D3h symmetry. If the spin-orbit interaction is taken into account the calculations result in two 4E1/2 and 4E3/2 spin-orbit states with 11–14 cm−1 energy differences instead of one 4A2′ state. Spin-orbit interaction quenches the Jahn-Teller effect in 2E″ ground orbital state of WF3 complex in D3h configuration, and C2v configuration with (4E1/2 + 4E3/2) spin-mixed state corresponds to the minimum on the PES. The WCl3 complex reveals strong interaction of the spin-orbit states of different multiplicity. For WCl3 complex the configuration with D3h symmetry corresponds to the minimum on the PES for four lowest spin-mixed electronic states. The MoF4, MoCl4 and WCl4 complexes possess the tetrahedral equilibrium configuration with 3A2 orbital electronic state. If the spin-orbit interaction is taken into account the ground state changes to the threefold degenerated 3T2 spin-orbit state with weak Jahn-Teller effect. In the case of WF4 complex the strong Jahn-Teller effect has been discovered in the first excited 1E singlet electronic state of the tetrahedral structure, and D2d configuration with 1A1 electronic state possesses the lowest energy. The comparison with available experimental data has been performed. The tendencies in molecular parameters of complexes studied were analyzed.

Efficient Computation of the Hartree–Fock Exchange in Real-Space with Projection Operators

We describe an efficient projection-based real-space implementation of the nonlocal single-determinant exchange operator. Through a matrix representation of the projected operator, we show that this scheme works equally well for both occupied and virtual states. Our scheme reaches a speedup of 2 orders of magnitude and has no significant loss of accuracy compared to an implementation of the full nonlocal single-determinant exchange operator. We find excellent agreement upon comparing Hartree-Fock eigenvalues, dipoles, and polarizabilities of selected molecules calculated using our method to values in the literature. To illustrate the efficiency of this scheme we perform calculations on systems with up to 240 carbon atoms.

WE‐AB‐207A‐04: Random Undersampled Cone Beam CT: Theoretical Analysis and a Novel Reconstruction Method

Purpose:Reducing x‐ray exposure and speeding up data acquisition motived studies on projection data undersampling. It is an important question that for a given undersampling ratio, what the optimal undersampling approach is. In this study, we propose a new undersampling scheme: random‐ray undersampling. We will mathematically analyze its projection matrix properties and demonstrate its advantages. We will also propose a new reconstruction method that simultaneously performs CT image reconstruction and projection domain data restoration.Methods:By representing projection operator under the basis of singular vectors of full projection operator, matrix representations for an undersampling case can be generated and numerical singular value decomposition can be performed. We compared properties of matrices among three undersampling approaches: regular‐view undersampling, regular‐ray undersampling, and the proposed random‐ray undersampling. To accomplish CT reconstruction for random undersampling, we developed a novel method that iteratively performs CT reconstruction and missing projection data restoration via regularization approaches.Results:For a given undersampling ratio, random‐ray undersampling preserved mathematical properties of full projection operator better than the other two approaches. This translates to advantages of reconstructing CT images at lower errors. Different types of image artifacts were observed depending on undersampling strategies, which were ascribed to the unique singular vectors of the sampling operators in the image domain. We tested the proposed reconstruction algorithm on a Forbid phantom with only 30% of the projection data randomly acquired. Reconstructed image error was reduced from 9.4% in a TV method to 7.6% in the proposed method.Conclusion:The proposed random‐ray undersampling is mathematically advantageous over other typical undersampling approaches. It may permit better image reconstruction at the same undersampling ratio. The novel algorithm suitable for this random‐ray undersampling was able to reconstruct high‐quality images.

Publisher's Note: “Full Coulomb collision operator in the moment expansion” [Phys. Plasmas 16, 102108 (2009)

Publisher's Note: “Full Coulomb collision operator in the moment expansion” [Phys. Plasmas <b>16</b>, 102108 (2009)

Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

Spectra of sigma models on semi-symmetric spaces

Sigma models on semi-symmetric spaces provide the central building block for string theories on AdS backgrounds. Under certain conditions on the global supersymmetry group they can be made one-loop conformal by adding an appropriate fermionic Wess–Zumino term. We determine the full one-loop dilation operator of the theory. It involves an interesting new XXZ-like interaction term. Eigenvalues of our dilation operator, i.e. the one-loop anomalous dimensions, are computed for a few examples.

Extensive ab initio study of the electronic states of BSe radical including spin–orbit coupling

The internally contracted multi-reference configuration interaction method (MRCI) with Davidson modification and the Douglas–Kroll scalar relativistic correction has been used to calculate the BSe molecule at the level of aug-cc-pV5Z basis set. The calculated electronic states, including 9 doublet and 6 quartet Λ-S states, are correlated to the dissociation limit of B(2Pu)+Se(3Pg) and B(2Pu)+Se(1Dg). The Spin-orbit coupling (SOC) interaction is taken into account via the state interaction approach with the full Breit-Pauli Hamiltonian operator, which causes the entire 15 Λ-S states to split into 32Ω states. This is the first time that the spin-orbit coupling calculation has been carried out on BSe. The potential energy curves of the Λ-S and Ω electronic states are depicted with the aid of the avoided crossing rule between electronic states of the same symmetry. The spectroscopic constants of the bound Λ-S and Ω states were determined, which are in good agreement with the experimental data. The transition dipole moments (TDMs) and the Franck-Condon factors (FCs) of the transitions from the low-lying bound Ω states A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 to the ground state X2Σ+1/2 have also been presented. Based on the previous calculations, the radiative lifetimes of the A2Π(I)3/2, B2Π(I)1/2 and C2Δ(I)3/2 were evaluated.

Probabilistic frames for non-Boolean phenomena.

Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.