General Linear Transformation Research Articles

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions.

Fuzzy Differential Subordination Associated with a General Linear Transformation

In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator BSα,λm,t:An→An in the open unit disc U. The second step is to develop fuzzy differential subordination for the operator BSα,λm,t. By considering linear transformations of the operator BSα,λm,t, we define a new fuzzy class of analytic functions in U which we denote by Tϝλ,t(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination and the operator BSα,λm,t for the function f in the class Tϝλ,t(m,α,δ). In addition, we explore a number of examples and corollaries to illustrate the implications of our key findings. Finally, we highlight several established results to demonstrate the connections between our work and existing studies.

Amplitude-phase modulated composite waves in coupled nonlocal system with self-induced PT symmetric potential.

We consider a coupled nonlocal nonlinear Schrödinger equation (nNLSE) with self-induced parity-time (PT) symmetric potential and investigate abundant amplitude-phase modulated composite waves manifesting diverse evolution patterns. It is found that the coupled nonlocal model can be decoupled into nNLSEs with self-induced PT symmetric potential under certain constraints through a general linear transformation with amplitude and phase modulation. Based on the exact solutions of the nNLSEs with self-induced PT potential, we study various composite waves superposed by bright and/or dark soliton solutions, rogue waves, bright/dark soliton and periodic soliton, and present the abundant evolution patterns under amplitude-phase modulation. The results here only demonstrate the characteristics of limited superposed composite waves. In fact, there exist infinite possible evolution patterns of composite waves due to the arbitrary amplitude-phase modulation in coupled nonlocal nonlinear system with self-induced PT symmetric potential.

Transformed singular values and spectral norms of third-order tensors

The tensor-tensor product (T-product) and tensor singular value decomposition (T-SVD) for third-order tensors, which are based upon the general revertible linear transformation, are widely used in engineering and data science. This paper focuses on studying singular values and spectral norms of third-order tensors under unitary transformation. After presenting the decomposition for the third-order Hermitian tensors and the QR factorization of third-order tensors, with the help of the involved invertible linear transformation, several basic properties of singular values for third-order tensors are studied, which include the Lipschitz continuity that characterizes the singular values perturbation. Based upon these, two kinds of spectral norms for third-order tensors are introduced, one of which is the norm induced by Frobenius norm of tensors in the sense of the T-product.

Iterative Synchrosqueezing-Based General Linear Chirplet Transform for Time-Frequency Feature Extraction

Short-time Fourier transform (STFT)-based methods are widely applied in industrial areas. However, these methods are inadequate to process non-stationary signals under variable operating conditions. An improved general linear chirplet transform method is developed by iteratively upgrading the instantaneous frequency (IF) and introducing a synchrosqueezing operator simultaneously. Initially, an iterative upgrading strategy is adopted to improve the estimation accuracy of the IF curves. Then, a synchrosqueezing operator is employed to enhance the concentration of the time-frequency representation under variable operating conditions. Finally, experiments that utilize simulated data are conducted to verify the effectiveness. Experimental results show that the enhanced time-frequency analysis (TFA) method can sharpen IF curves and enhance the time-frequency readability compared with other advanced TFA methods. Moreover, the feature extraction ability of the present method is superior to other commonly used methods.

Interference of Gaussian and/or Airy beams in coupled PT-symmetric nonlocal system

We investigate the interference of Gaussian and/or Airy beams based on coupled parity-time (PT) symmetric nonlocal nonlinear Schrödinger (NNLS) equation, which can be decoupled into two paraxial diffraction equations through a general linear transformation under the balance among nonlocal nonlinear effects. Various superposed solutions of the coupled NNLS equation can be obtained with the help of linear transformation and the solutions of linear diffraction equation. Based on these solutions, we analytically explore the abundant interference phenomena of Gaussian beam train, two Gaussian and/or Airy beams, which can be modulated by the linear transformation, the transverse displacement and relative phase difference between adjacent beams. The results presented here may be helpful to design complex interferometric patterns that can be applied to photoetching or optical printing etc.

Bayesian transformation models with partly interval‐censored data

In many scientific fields, partly interval-censored data, which consist of exactly observed and interval-censored observations on the failure time of interest, appear frequently. However, methodological developments in the analysis of partly interval-censored data are relatively limited and have mainly focused on additive or proportional hazards models. The general linear transformation model provides a highly flexible modeling framework that includes several familiar survival models as special cases. Despite such nice features, the inference procedure for this class of models has not been developed for partly interval-censored data. We propose a fully Bayesian approach coped with efficient Markov chain Monte Carlo methods to fill this gap. A four-stage data augmentation procedure is introduced to tackle the challenges presented by the complex model and data structure. The proposed method is easy to implement and computationally attractive. The empirical performance of the proposed method is evaluated through two simulation studies, and the model is then applied to a dental health study.

Location-Sensitive Embedding for Knowledge Graph Embedding

<p indent=0mm>Knowledge graph embedding maps the discrete symbolic entities and relations into continuous low-dimensional space, enabling the downstream task (e.g., inference, knowledge graph completion and etc.) to be carried on in an algebraic manner. Roughly speaking, there are two main-streams towards knowledge graph embedding, translational distance models and semantic matching models. The learning bottleneck of the former caused by special graph structures is analyzed. And it is found out that translational distance models are to be blamed for treating head and tail entities from a same viewpoint. So, a location-sensitive embedding (LSE) model is proposed. Unlike previous models, it just transforms the head entity with relational-specific mapping. And it models relation as a general linear transformation instead of a translational operator. Its representation capacity and the relationship to current models are theoretically analyzed. Also, to make it efficient, the model is simplified by assuming a diagonal matrix for transformation, forming a simplified model LSE<italic>_d</italic>. It is evaluated that the proposed model on four large-scale knowledge graph datasets for link prediction task. Experiments have shown that the proposed model achieves highest, or at least competitive, performance compared with the state-of-the-art models.

Tangent space symmetries in general relativity and teleparallelism

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonormality. We borrow techniques from the methodology of non-linear realizations to analyze this group-subgroup structure. “Parallel maps” are used to relate tangent space at different points. “Parallelisms” across a finite region of the manifold may be built up from them. These are used to define Weitzenböck connections and Levi-Civita connections. This provides a new formulation of teleparallel gravity, in which the tetrad field is viewed as a field-valued group element relating the coordinate basis to the frame basis used in defining a parallelism. This formulation separates the metric degrees of freedom from those associated with the choice of parallelism. The group element can be combined by matrix multiplication with Lorentz transformations of frame or with other Jacobian matrices. We show how this facilitates a new understanding of inertial forces and local Lorentz transformations. The analysis is also applied to translations of the coordinates. If they are constant across spacetime, this has no effect on the tangent space bases. If the translation parameters become fields, they induce general linear transformations of the coordinate basis; however, the tetrad components can only be expressed in terms of translations on a flat spacetime.

Transient-Based Internet of Things Emitter Identification Using Convolutional Neural Networks and Optimized General Linear Chirplet Transform

This letter investigates the application of the General Linear Chirplet Transform (GLCT), a time-frequency representation recently proposed in the literature, in combination with Convolutional Neural Network (CNN) for the identification of Internet of Things Internet of Things (IOT) wireless devices. The identification is performed by analyzing the radio frequency emissions from the IOT devices during transmission. We apply the novel combination of CNN and GLCT (CNN-GLCT) to the transient portions of the radio frequency emissions by proposing an optimization procedure for GLCT which is specifically designed for emitter identification. Our empirical results show that this combination provides a superior identification performance as compared to other Deep CNN approaches available in the literature, as well as to approaches based on shallow machine learning methods, like SVM and KNN, especially under low Signal Noise Ratio (SNR) and fading conditions.

English

In this paper, we define bi-linear games as a generalization of the bimatrix games. In particular, we generalize concepts like the value and equilibrium of a bimatrix game to the general linear transformations defined on a finite dimensional space. For a special type of Z-transformation we observe relationship between the values of the linear and bi-linear games. Using this relationship, we prove some known classical results in the theory of linear complementarity problems for this type of Z-transformations.

Application of hybrid GLCT-PICA de-noising method in automated EEG artifact removal

The electrical activities associated with non-cerebral biological origins are usually having high amplitude (order of 230–350 micro-volts) and effect on-going cerebral activity (order of 7–20 micro-volts) adversely. The frequent occurrence of multiple artifactual origins makes it imperative to adopt adequate artifacts removal methodologies prior to feature estimation in Brain-Computer Interface applications. The present work proposes a novel artifact removal methodology using a joint application of Fast-Power Independent Component Analysis and General Linear Chirplet Transform for automatic identification and rejection of artifactual origins. After segregating on-going Electroencephalogram activity into Independent Components, Katz-Fractal Sparsity criterion is employed to identify artifactual components. The identified artifactual components are treated by General Linear Chirplet Transform-based EEG de-noising method to recover useful cerebral information leaked with artifactual origins. Thereafter, Inverse Independent Component Analysis yields artifact corrected clean Electroencephalogram activity for further analysis. The effectiveness of the proposed methodology is validated with simulated and empirical Electroencephalogram dataset. The experimental results establish the proposed method as a potential candidate for non-cerebral artifacts correction and noise suppression from Electroencephalogram records.

Irreducible matrix resolution for symmetry classes of elasticity tensors

In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

Micro-Doppler based target classification in ground surveillance radar systems

In this paper, we investigate effect of different Time-Frequency (TF) analysis of micro-Doppler signature for robust and efficient automatic target classification in next-generation radar systems. Indeed, we provide a novel mathematical baseband model for the non-stationary reflected signal of the targets and explore how Short Time Fourier Transform (STFT) and General Linear Chirplet Transform (GLCT) are useful in categorization of different ground moving targets, including walking person, animal, light vehicles and drone. To simulate the micro-Doppler signature, we will introduce a simple, but sufficiently realistic signal model. Simulation and results which are provided for various scenarios, illustrate the importance of appropriate TF representation in target classification.

An Integrated Urban Flood Vulnerability Index for Sustainable Planning in Arid Zones of Developing Countries

Floods are among the most recurring and devastating natural hazards, impacting human lives and causing severe economic damage. Urbanization can increase the risk of flooding due to increased peak discharge and volume. Over arid urban areas of developing countries, flood disaster management is reactive, responding to prevailing disaster situations, mainly because of the lack of budget, equipment, facilities, and human resources. The approach required in a new city requires a different operative planning process, ruled by different kinds of specific indicators to be incorporated in the sustainable planning process. This study focuses on an approach to assess flood vulnerability as a planning tool using an integrated flood vulnerability index (FVI) with variables that are accessible in developing countries and arid urban areas. The research took place in the city of Mexicali, Baja, California. México. This index was determined by coupling the variables of three components: social, economic, and physical. The FVI reflects the status of an urban scale’s vulnerability. Variables were obtained from government data for the social and economic components, and a hydrological and hydraulic model approach as a physical component. The correlation of each variable to the flood was taken into account by using a general linear transformation. GIS was used as a tool for the development of spatial analysis. The results showed the spatial distribution of vulnerability at an urban district scale. It was found that 55% of the population is exposed to a vulnerability above the average value of the urban area. Integrating all the components will help decision-makers to implement strategies to improve the resilience of the area by attending the needs of the particular component that is more vulnerable.

Antenna Array Calibration for a General Linear Transformation of the Array Manifold

We consider the problem of calibrating an antenna array using received signals from either known or unknown directions. A complex matrix is used to map the assumed array manifold to the true array manifold. This matrix may be completely unstructured. Algorithms are presented for joint estimation of this matrix and the directions-of-arrival, using data collected during multiple time intervals.

An Efficient Scrambled Estimator of Population Mean of Quantitative Sensitive Variable Using General Linear Transformation of Non-sensitive Auxiliary Variable

In the present paper, we propose an efficient scrambled estimator of population mean of quantitative sensitive study variable, using general linear transformation of non-sensitive auxiliary variable. Efficiency comparisons with the existing estimators have been carried out both theoretically and numerically. It has been found that our optimal scrambled estimator is always more efficient than most of the existing scrambled estimators and also it is more efficient than few other scrambled estimators under some conditions.

Linear Transformation Recognition Using Radon Transform

Given two regular functions (images) f and g on R2 where g is formed from f by a general linear transformation, g(x) = f (Ax + b). We present a procedure to determine the transformation ‘parameters’ A and b using Radon projections of f and only two projections of g. We use these projections together with simple facts on matrix vector multiplication to recover the matrix A. The assumptions we have here are: f is nonnegative and A is nonsingular. Commonly used transformations in image processing such as rotation, scaling and others are special cases of our approach.

On the linearization of nonlinear supersymmetry based on the commutator algebra

We discuss a linearization procedure of nonlinear supersymmetry (NLSUSY) based on the closure of the commutator algebra for variations of functionals of Nambu-Goldstone fermions and their derivative terms under NLSUSY transformations in Volkov-Akulov NLSUSY theory. In the case of a set of bosonic and fermionic functionals, which leads to (massless) vector linear supermultiplets, we explicitly show that general linear SUSY transformations of basic components defined from those functionals are uniquely determined by examining the commutation relation in the NLSUSY theory.

Generalized Singular Value Decomposition with Additive Components

The singular value decomposition (SVD) technique is extended to incorporate the additive components for approximation of a rectangular matrix by the outer products of vectors. While dual vectors of the regular SVD can be expressed one via linear transformation of the other, the modified SVD corresponds to the general linear transformation with the additive part. The method obtained can be related to the family of principal component and correspondence analyses, and can be reduced to an eigenproblem of a specific transformation of a data matrix. This technique is applied to constructing dual eigenvectors for data visualizing in a two dimensional space.