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Model-based hyperspectral image (HSI) denoising methods have attracted continuous attention in the past decades, due to their effectiveness and interpretability. In this work, we aim at advancing model-based HSI denoising, through sophisticated investigation for both the fidelity and regularization terms, or correspondingly noise and prior, by virtue of several recently developed techniques. Specifically, we formulate a novel unified probabilistic model for the HSI denoising task, within which the noise is assumed as pixel-wise non-independent and identically distributed (non-i.i.d) Gaussian predicted by a pre-trained neural network, and the prior for the HSI image is designed by incorporating the deep image prior (DIP) with total variation (TV) and spatio-spectral TV. To solve the resulted maximum a posteriori (MAP) estimation problem, we design a Monte Carlo Expectation–Maximization (MCEM) algorithm, in which the stochastic gradient Langevin dynamics (SGLD) method is used for computing the E-step, and the alternative direction method of multipliers (ADMM) is adopted for solving the optimization in the M-step. Experiments on both synthetic and real noisy HSI datasets have been conducted to verify the effectiveness of the proposed method.

Time–frequency analysis is an important tool used for the processing and interpretation of non-stationary signals, such as seismic data and remote sensing data. In this paper, based on the novel short-time fractional Fourier transform (STFRFT), a new modified STFRFT is first proposed which can also generalize the properties of the modified short-time Fourier transform (STFT). Then, in the modified STFRFT domain, we derive the instantaneous frequency estimator for the chirp signal and present a new type of synchrosqueezing STFRFT (FRSST). The proposed FRSST presents many results similar to those of the synchrosqueezing STFT (FSST), and it extends the harmonic signal to a chirp signal that offers attractive new features. Furthermore, we provide a detailed analysis of the signal reconstruction, theories, and some properties of the proposed FRSST. Several experiments are conducted, and all of the results illustrate that the proposed FRSST is more effective than the FSST. Finally, based on the linear amplitude modulation and frequency modulation signal, we present a derivation for analyzing the limitations of the FRSST.

The rapid development of big data breeds data science. Understanding and mastering the internal pattern of the value generation of big data is important for improving digitization and the covergence of data science with management science, computer science, and other disciplines. In this study, we discuss the significance of data science for the development of science and technology and social progress. Based on the interpretation of the connotation of data science, we introduce the definition of data science and review its formation by summarizing the major progress of related disciplines. We also discuss the research methodologies, development patterns and trends, and technical directions of data science. Finally, we make suggestions for data science to promote the development of data-based science and technology.

In hyperspectral image (HSI) processing, a fundamental issue is to restore HSI data from various degradations such as noise corruption and information missing. However, most existing methods more or less ignore the abundant prior knowledge on HSIs and the embedded noise, leading to suboptimal performance in practice. In this article, we propose a novel HSI restoration method by fully considering the intrinsic image structures and the complex noise characteristics. For HSIs, the global correlation is captured by the Kronecker-basis-representation-based tensor low-rankness measure, which integrates the insights delivered by both CP and Tucker decompositions; the local regularity is depicted by a plug-and-play spatial-spectral convolutional neural network with strong fitting ability to complex image features. For realistic noise, its statistical characteristics are encoded by a nonidentical and nonindependent distributed mixture of Gaussians distribution with flexible fitting capability. Then, we incorporate these image and noise priors into a probabilistic model based on the maximum a posteriori principle, and develop a solving scheme by combining expectation-maximization and alternating direction method of multipliers. Extensive experimental results on both simulated and real scenarios demonstrate the effectiveness of the proposed method and its superiority over the compared state-of-the- arts.

Noise removal is a fundamental problem in remote sensing image processing. Most existing methods, however, have not yet attained sufficient robustness in practice, due to more or less neglecting the intrinsic structures of remote sensing images and/or underestimating the complexity of realistic noise. In this paper, we propose a new remote sensing image denoising method by integrating intrinsic image characterization and robust noise modeling. Specifically, we use low-Tucker-rank tensor approximation to capture the global multi-factor correlation within the underlying image, and adopt a non-identical and non-independent distributed mixture of Gaussians (non-i.i.d. MoG) assumption to encode the statistical configurations of the embedded noise. Then, we incorporate the proposed image and noise priors into a full Bayesian generative model and design an efficient variational Bayesian algorithm to infer all involved variables by closed-form equations. Moreover, adaptive strategies for the selection of hyperparameters are further developed to make our algorithm free from burdensome hyperparameter-tuning. Extensive experiments on both simulated and real multispectral/hyperspectral images demonstrate the superiority of the proposed method over the compared state-of-the-art ones.

In this paper, a new supervised classification algorithm which simultaneously considers spectral and spatial information of a hyperspectral image (HSI) is proposed. Since HSI always contains complex noise (such as mixture of Gaussian and sparse noise), the quality of the extracted feature inclines to be decreased. To tackle this issue, we utilize the low-rank property of local three-dimensional, patch and adopt complex noise strategy to model the noise embedded in each local patch. Specifically, we firstly use the mixture of Gaussian (MoG) based low-rank matrix factorization (LRMF) method to simultaneously extract the feature and remove noise from each local matrix unfolded from the local patch. Then, a classification map is obtained by applying some classifier to the extracted low-rank feature. Finally, the classification map is processed by Markov random field (MRF) in order to further utilize the smoothness property of the labels. To ease experimental comparison for different HSI classification methods, we built an open package to make the comparison fairly and efficiently. By using this package, the proposed classification method is verified to obtain better performance compared with other state-of-the-art methods.

This note discusses the recovery of signals from undersampled data in the situation that such signals are nearly block sparse in terms of an overcomplete and coherent tight frame D. By introducing the notion of block D-restricted isometry property (D-RIP), we establish several sufficient conditions for the proposed mixed l2/l1-analysis method to guarantee stable recovery of nearly block-sparse signals in terms of D. One of the main results of this note shows that if the measurement matrix satisfies the block D-RIP with constants δk

Within the conventional framework of a native space structure, a smooth kernel generates a small native space, and radial basis functions stemming from the smooth kernel are intended to approximate only functions from this small native space. In this paper, we embed the smooth radial basis functions in a larger native space generated by a less smooth kernel and use them to interpolate the samples. Our result shows that there exists a linear combination of spherical radial basis functions that can both exactly interpolate samples generated by functions in the larger native space and near best approximate the target function.

We study the problem of estimating the frequencies of several complex sinusoids with constant amplitude (CA) (also called constant modulus) from multichannel signals of their superposition. To exploit the CA property for frequency estimation in the framework of atomic norm minimization (ANM), we introduce multiple positive-semidenite block matrices composed of Hankel and Toeplitz submatrices and formulate the ANM problem as a convex structured low-rank approximation (SLRA) problem. The proposed SLRA is a semidenite programming and has substantial differences from existing such formulations without using the CA property. The proposed approach is termed as SLRA-based ANM for CA frequency estimation (SACA). We provide theoretical guarantees and extensive simulations that validate the advantages of SACA. [ABSTRACT FROM AUTHOR]

Parallel imaging (PI), relying on multicoils to sense k-space data, is an effective technique to accelerate magnetic resonance imaging by exploiting spatial sensitivity coding of multiple coils, with an integrated compressive sensing (CS) technology to achieve higher acceleration. In this paper, we propose a novel nonconvex reconstruction model and its proximal alternating linearized minimization (PALM) algorithm for PI in a blind setting that MR image and multichannel sensitivity maps are jointly estimated, regularized by image and sensitivity regularizers. Instead of hand-crafting the image and sensitivity regularizers, we propose unrolling the PALM algorithm to be a deep network for Blind Parallel MRI, dubbed as BPMRI-Net, with two learnable subnetworks to substitute the proximal operators of the image and sensitivity regularizers. We theoretically prove the linear convergence of BPMRI-Net as an iterative algorithm, which alternately updates two variables based on the learnable proximal operators. The learned BPMRI-Net can simultaneously output the MR image and sensitivity maps from undersampled multichannel k-space data even when the number of low-frequency sampling lines in the center of k-space is small. Numerical results demonstrate the effectiveness of our method with state-of-the-art reconstruction accuracy. [ABSTRACT FROM AUTHOR]

As a kind of cultural art, calligraphy and painting are not only an important part of traditional culture but also has important value of art collection and trade. The existence of forgeries has seriously affected the fair trade, protection, and inheritance of calligraphy and painting. There is an urgent need for the efficient, accurate, and intelligent technical identification method. By combining the advantages of material attribute recognition and imaging detection of hyperspectral imaging technology with the powerful feature expression ability and classification ability of the convolutional neural network, it can greatly improve the comprehension efficiency of calligraphy and painting identification; meanwhile, in order to reduce the redundancy and the amount of parameters in the method of directly using the hyperspectral image, an objective convex dimensionality reduction method should be used for compressing the original hyperspectral image before deep learning. Based on this, we propose a kind of deep learning method to classify author and authenticity based on the multichannel images obtained by minimum noise fraction (MNF) dimensionality reduction to calligraphy and painting hyperspectral data, and its core is the 2D-CNN or 3D-CNN model with the basic network of "4 convolution layers + 4 pooling layers + 2 full-link layers." The experimental results show that the identification accuracy of the 2D-CNN calligraphy and painting identification with MNF pseudocolor image mosaic as input and the 2D-CNN calligraphy and painting identification with multichannel MNF dimensionality reduced images direct as input have high accuracy, while the 3D-CNN calligraphy and painting identification with multichannel MNF dimensionality reduced images direct as input not only maintains excellent identification accuracy but also has better learning convergence (step number) and stability compared with the 2D-CNN model. Especially, the 3D-CNN identification accuracy of calligraphy and painting's author and authenticity on the test set can reach 93.2% and 95.2%, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts--the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm--we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems. Index Terms--Contractive mappings, exponential bound, exponential stability, lip constant, nonlinear discrete dynamical systems, strongly equivalent metric.

A set of sufficient and necessary conditions are presented for global exponential stability (GES) of a class of generic discrete-time recurrent neural networks. By means of the uncovered conditions, GES and convergence properties of the neural networks are analyzed quantitatively. It is shown that exact equivalences exist among the GES property of the neural networks, the contractiveness of the deduced nonlinear operators, and the global asymptotic stability (GAS) of the neural networks plus the spectral radius of Jacobian matrix of the neural networks at the unique equilibrium point less than one. When the neural networks have small state feedback coefficients, it is shown further that the infimum of exponential bounds of the trajectories of the neural networks equals exactly the spectral radius of Jacobian matrix of the neural networks at the unique equilibrium point. The obtained results are helpful in understanding essence of GES and clarifying difference between GES and GAS of the discrete-time recurrent neural networks. Index Terms--Discrete-time recurrent neural networks, exponential bound, global asymptotic stability (GAS), global exponential stability (GES), nonlinear difference equations.

The Bayesian approach, a useful tool for quantifying uncertainties, has been extensively employed to solve the inverse problems of partial differential equations (PDEs). One of the main difficulties in employing the Bayesian approach to such problems is how to extract information from the posterior probability measure. Compared with conventional sampling-type methods, the variational Bayes method (VBM) has been intensively examined in the field of machine learning attributed to its ability in extracting approximately the posterior information with lower computational cost. In this paper, we generalize the conventional finite-dimensional VBM to the infinitedimensional space to rigorously solve the inverse problems of PDEs. We further establish a general infinite-dimensional mean-field approximate theory and apply it to the linear inverse problems under the Gaussian and Laplace noise assumptions at the abstract level. The results of some numerical experiments substantiate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

One of the existing sparse clustering approaches, ℓ1-k-means, maximizes the weighted between-cluster sum of squares subject to the ℓ1 penalty. In this paper, we propose a sparse clustering method based on an ℓ1=ℓ0 penalty, which we call ℓ0-k-means. We design an efficient iterative algorithm for solving it. To compare the theoretical properties of ℓ1 and ℓ0-k-means, we show that they can be explained explicitly from a thresholding perspective based on different thresholding functions. Moreover, ℓ1 and ℓ0-k-means are proven to have a screening consistent property under Gaussian mixture models. Experiments on synthetic as well as real data justify the outperforming results of ℓ0 with respect to ℓ1-k-means. [ABSTRACT FROM AUTHOR]

Motivation: Results from genome-wide association studies (GWAS) suggest that a complex phenotype is often affected by many variants with small effects, known as 'polygenicity'. Tens of thousands of samples are often required to ensure statistical power of identifying these variants with small effects. However, it is often the case that a research group can only get approval for the access to individual-level genotype data with a limited sample size (e.g. a few hundreds or thousands). Meanwhile, summary statistics generated using single-variant-based analysis are becoming publicly available. The sample sizes associated with the summary statistics datasets are usually quite large. How to make the most efficient use of existing abundant data resources largely remains an open question. Results: In this study, we propose a statistical approach, IGESS, to increasing statistical power of identifying risk variants and improving accuracy of risk prediction by integrating individual level genotype data and summary statistics. An efficient algorithm based on variational inference is developed to handle the genome-wide analysis. Through comprehensive simulation studies, we demonstrated the advantages of IGESS over the methods which take either individual-level data or summary statistics data as input. We applied IGESS to perform integrative analysis of Crohns Disease from WTCCC and summary statistics from other studies. IGESS was able to significantly increase the statistical power of identifying risk variants and improve the risk prediction accuracy from 63.2% (60:4%) to 69.4% (60:1%) using about 240 000 variants. [ABSTRACT FROM AUTHOR]

Recently, a new framework, Fredholm learning, was proposed for semisupervised learning problems based on solving a regularized Fredholm integral equation. It allows a natural way to incorporate unlabeled data into learning algorithms to improve their prediction performance. Despite rapid progress on implementable algorithms with theoretical guarantees, the generalization ability of Fredholm kernel learning has not been studied. In this letter, we focus on investigating the generalization performance of a family of classification algorithms, referred to as Fredholm kernel regularized classifiers. We prove that the corresponding learning rate can achieve O(l-1) (l is the number of labeled samples) in a limiting case. In addition, a representer theorem is provided for the proposed regularized scheme, which underlies its applications. [ABSTRACT FROM AUTHOR]

Motivation: Imaging genetics combines brain imaging and genetic information to identify the relationships between genetic variants and brain activities. When the data samples belong to different classes (e.g. disease status), the relationships may exhibit class-specific patterns that can be used to facilitate the understanding of a disease. Conventional approaches often perform separate analysis on each class and report the differences, but ignore important shared patterns. Results: In this paper, we develop a multivariate method to analyze the differential dependency across multiple classes. We propose a joint sparse canonical correlation analysis method, which uses a generalized fused lasso penalty to jointly estimate multiple pairs of canonical vectors with both shared and class-specific patterns. Using a data fusion approach, the method is able to detect differentially correlated modules effectively and efficiently. The results from simulation studies demonstrate its higher accuracy in discovering both common and differential canonical correlations compared to conventional sparse CCA. Using a schizophrenia dataset with 92 cases and 116 controls including a single nucleotide polymorphism (SNP) array and functional magnetic resonance imaging data, the proposed method reveals a set of distinct SNP-voxel interaction modules for the schizophrenia patients, which are verified to be both statistically and biologically significant. Availability and Implementation: The Matlab code is available at https://sites.google.com/site/jian fang86/JSCCA. [ABSTRACT FROM AUTHOR]

Regularization is a well-recognized powerful strategy to improve the performance of a learning machine and lq regularization schemes with 0 < q < ∞ are central in use. It is known that different q leads to different properties of the deduced estimators, say, l² regularization leads to a smooth estimator, while l¹ regularization leads to a sparse estimator. Then how the generalization capability of lq regularization learning varies with q is worthy of investigation. In this letter, we study this problem in the framework of statistical learning theory. Our main results show that implementing lq coefficient regularization schemes in the sampledependent hypothesis space associated with a gaussian kernel can attain the same almost optimal learning rates for all 0 < q < ∞. That is, the upper and lower bounds of learning rates for l q regularization learning are asymptotically identical for all 0 < q < ∞. Our finding tentatively reveals that in some modeling contexts, the choice of q might not have a strong impact on the generalization capability. From this perspective, q can be arbitrarily specified, or specified merely by other nongeneralization criteria like smoothness, computational complexity or sparsity. [ABSTRACT FROM AUTHOR]

This letter proposes a new method for cooperative spectrum sensing by exploiting sparsity. The novel scheme uses the theory of Bayesian hierarchical prior modeling in the framework of sparse Bayesian learning. This model has sparsity-inducing penalization terms leading to sparser solutions compared with typically l1 norm based ones. Based on the factor graph that represents the signal model of the hierarchical prior models, the variational message passing (VMP) algorithm is implemented to estimate the power spectral density (PSD) map. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel spatial and spectral fusion model (SASFM) that uses sparse matrix factorization to fuse remote sensing imagery with different spatial and spectral properties. By combining the spectral information from sensors with low spatial resolution (LSaR) but high spectral resolution (HSeR) (hereafter called HSeR sensors), with the spatial information from sensors with high spatial resolution (HSaR) but low spectral resolution (LSeR) (hereafter called HSaR sensors), the SASFM can generate synthetic remote sensing data with both HSaR and HSeR. Given two reasonable assumptions, the proposed model can integrate the LSaR and HSaR data via two stages. In the first stage, the model learns from the LSaR data a spectral dictionary containing pure signatures, and in the second stage, the desired HSaR and HSeR data are predicted using the learned spectral dictionary and the known HSaR data. The SASFM is tested with both simulated data and actual Landsat 7 Enhanced Thematic Mapper Plus (ETM+) and Terra Moderate Resolution Imaging Spectroradiometer (MODIS) acquisitions, and it is also compared to other representative algorithms. The experimental results demonstrate that the SASFM outperforms other algorithms in generating fused imagery with both the well-preserved spectral properties of MODIS and the spatial properties of ETM+. Generated imagery with simultaneous HSaR and HSeR opens new avenues for applications of MODIS and ETM+. [ABSTRACT FROM PUBLISHER]

Structure of data set is of critical importance in identifying clusters, especially the density difference feature. In this paper, we present a clustering algorithm based on density consistency, which is a filtering process to identify same structure feature and classify them into same cluster. This method is not restricted by the shapes and high dimension data set, and meanwhile it is robust to noises and outliers. Extensive experiments on synthetic and real world data sets validate the proposed the new clustering algorithm. [ABSTRACT FROM AUTHOR]

Two strategies for selecting the kernel parameter (σ) and the penalty coefficient (C) of Gaussian support vector machines (SVMs) are suggested in this paper. Based on viewing the model parameter selection problem as a recognition problem in visual systems, a direct parameter setting formula for the kernel parameter is derived through finding a visual scale at which the global and local structures of the given data set can be preserved in the feature space, and the difference between the two structures can be maximized. In addition, we propose a heuristic algorithm for the selection of the penalty coefficient through identifying the classification extent of a training datum in the implementation process of the sequential minimal optimization (SMO) procedure, which is a well-developed and commonly used algorithm in SVM training. We then evaluate the suggested strategies with a series of experiments on 13 benchmark problems and three real-world data sets, as compared with the traditional 5-cross validation (5-CV) method and the recently developed radius-margin bound (RM) method. The evaluation shows that in terms of efficiency and generalization capabilities, the new strategies outperform the current methods, and the performance is uniform and stable. [ABSTRACT FROM AUTHOR]

Classical BCH codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes. But for given restricted length n, good quantum BCH codes are very sparse. In this paper, by puncturing and pasting check matrices of Hermitian dual containing BCH codes over the quaternary field, we construct many linear quantum codes with good parameters, and some of them have parameters exceeding the finite Gilbert-Varshamov bound for stabilizer quantum codes. [ABSTRACT FROM AUTHOR]

Abstract-This paper reveals two important characterizations of global exponential stability (GES) of a generic class of continuous-time recurrent neural networks. First, we show that GES of the neural networks can be fully characterized by global asymptotic stability (GAS) of the networks plus the condition that the maximum abscissa of spectral set of Jacobian matrix of the neural networks at the unique equilibrium point is less than zero. This result provides a very useful and direct way to distinguish GES from GAS for the neural networks. Second, we show that when the neural networks have small state feedback coefficients, the supremum of exponential convergence rates (ECRs) of trajectories of the neural networks is exactly equal to the absolute value of the maximum abscissa of spectral set of Jacobian matrix of the neural networks at the unique equilibrium point. Here, the supremum of ECRs indicates the potentially fastest speed of trajectory convergence. The obtained results are helpful in understanding the essence of GES and clarifying the difference between GES and GAS of the continuous-time recurrent neural networks. [ABSTRACT FROM AUTHOR]

The classification of binary [n, k, d] codes with d ⩾ s2k -1 and without zero coordinates is reduced to the classification of binary [(2k - 1)c(k,s,t) + t,k,d] code for n =(k - 1)s + t, s ⩾ 1 and 1 ⩽ t ⩽ - 2k - 2, where c(k, s, t) ⩽ min{s, t} is a function of k, s, and t. Binary [15s + t, 4] optimal self-orthogonal codes are characterized by systems of linear equations. Based on these two results, the complete classification of [15s + t,4] optimal self-orthogonal codes fort t Є {1,2,6,7,8,9,13,14} and s ⩾ 1 is obtained, and the generator matrices and weight polynomials of these 4-dimensional optimal self-orthogonal codes are also given. [ABSTRACT FROM AUTHOR]

In this correspondence, we use a symplectic geometry over the binary field to discuss the equivalence of additive codes over the quaternary field and the equivalence of additive quantum codes. We establish the existence of a standard form of the stabilizer and the normalizer matrices for additive quantum codes. Thus, we present the quantum analogue of standard forms of generator and parity check matrices for systematic linear codes in classical coding theory for additive quantum codes. [ABSTRACT FROM AUTHOR]

A new method, which is called decomposition- composition (D-C) method, is proposed for the nonlinear dimensionality reduction (NLDR) of data lying on the multicluster manifold. The main idea is first to decompose a given data set into clusters and independently calculate the low-dimensional embeddings of each cluster by the decomposition procedure. Based on the intercluster connections, the embeddings of all clusters are then composed into their proper positions and orientations by the composition procedure. Different from other NLDR methods for multicluster data, which consider associatively the intracluster and intercluster information, the D-C method capitalizes on the separate employment of the intracluster neighborhood structures and the intercluster topologies for effective dimensionality reduction. This, on one hand, isometrically preserves the rigid-body shapes of the clusters in the embedding process and, on the other hand, guarantees the proper locations and orientations of all clusters. The theoretical arguments are supported by a series of experiments performed on the synthetic and real-life data sets. In addition, the computational complexity of the proposed method is analyzed, and its efficiency is theoretically analyzed and experimentally demonstrated. Related strategies for automatic parameter selection are also examined. [ABSTRACT FROM AUTHOR]

A new algorithm for finding the inverse of a nonsingular scaled factor circulant matrix is presented by the Euclid's algorithm. Extension is made to compute the group inverse and the Moore-Penrose inverse of the singular scaled factor circulant matrix. Numerical examples are presented to demonstrate the implementation of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors, so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S is the combinatorial Brandt semigroup with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely 0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain. [ABSTRACT FROM AUTHOR]

We call a semigroup S eventually inverse (or π-inverse) if for every element a of S there exists m ∈ Z+(the set of positive integers) such that am is regular and every regular element of S possesses a unique inverse. In this paper, we investigate some basic properties of eventually inverse semigroups. In the first section, we provide some characters of idempotents of eventually inverse semigroups and some equivalent conditions of eventually inverse semigroups. In the second section, we study the full eventually inverse suhsemigroups generated by some special subsets of an eventually inverse semigroup'. Finally, we describe a π-group congruence and a minimum π-group congruence on an eventually inverse semigroup. [ABSTRACT FROM AUTHOR]

In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions, Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient. [ABSTRACT FROM AUTHOR]