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This article concentrates in analyzing optimal controls for stochastic integrodifferential equation (SIDE) in Hilbert space. Necessary parameters are imposed to demonstrate the system that follows a unique variation of parameter formula using Leray Schauder Alternative. Subsequently, the existence of optimal control is investigated for the considered Lagrange control problem. The theoretical example with the mechanical example of ethanol fuelled engine are discussed to validate the results obtained. [ABSTRACT FROM AUTHOR]

In this article, we present two inertial modifications of regularized algorithms for the split variational inclusion problem (SVIP, for short) in real Hilbert spaces (RHSs). When the circumstances are right, strong convergence theorems are demonstrated. The major findings are used to solve the split minimization, split common fixed point problem (SCFPP), split minimization problem (SMP), and split feasibility problems (SFP) in applications. The proposed algorithms are contrasted with a number of other existing algorithms in the literature in order to test their numerical performances. Finally, the computer tests demonstrate that the suggested algorithms outperform alternative strategies in terms of speed and efficiency. [ABSTRACT FROM AUTHOR]

In this work, our aim is to ensure the existence of mild solutions and study the controllability of fractional evolution inclusions of Clarke's subdifferential type of order q ∈ (1 , 2) with nonlocal conditions in the setting of Hilbert spaces. Using fixed point techniques, fractional calculus, multivalued maps, cosine and sine function operators, we discuss the existence of mild solutions for the considered system. Moreover, under suitable conditions, we investigate the controllability of the proposed system. Finally, we present an illustrated example. [ABSTRACT FROM AUTHOR]

This paper proposes an extension of the KLMS algorithm to the Vector-Valued Reproducing Kernel Hilbert Space (VV-RKHS), enabling the use of the operator-valued kernel (OV-kernel) in the online identification of multi-input multi-output (MIMO) nonlinear systems. The yielded multivariate kernel model offers more design flexibility and involves fewer parameters than other vector-valued KLMS algorithms present in the literature. Conditions ensuring the convergence of the proposed OV-KLMS algorithm are given. Experiments on a multivariate chaotic attractor as well as numerical simulations are carried out and show the effectiveness of the proposed algorithm • A Kernel LMS algorithm in Vector Valued Reproducing Kernel Hilbert Space is proposed. • The proposed algorithm makes use of Operator Valued Kernels. • Proofs of the algorithm convergence are given. • The yielded kernel model is more accurate compared to state-of-the-art algorithms. • It is a low time consumption algorithm and is suitable for online identification of nonlinear Multi Input Multi Output systems. [ABSTRACT FROM AUTHOR]

In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings. [ABSTRACT FROM AUTHOR]

For stochastic nonlinear systems (SNSs) perturbed by compound uncertainties, the conventional model-based control approaches assume that the evolution behavior of uncertain variables is known. Unfortunately, such approaches are often conservative for most practical scenarios with the slow convergence speed and unsatisfactory anti-interference performance. For this sake, an adaptive control scheme based on deep deterministic policy gradient (DDPG) and multi-dimensional Taylor network (MTN) is proposed here to address the tracking problem for a category of SNSs subject to fast time-varying uncertainties, stochastic disturbance and unknown time-varying delays. The effect of time delay is embedded in the reproducing kernel Hilbert space through the error coordinate transformation. In the framework of DDPG, the MTN-based surrogate is utilized to construct the online network and target network via the temporal-difference method, which promises more desirable real-time performance due to its concise structure than conventional NN-based surrogates. In order to enhance the robustness of the system under fast time-varying uncertainties, a novel persistent excitation (PE) mechanism is designed to ensure that the control policy is appropriately rewarded or punished. Based on the PE condition, weights of MTNs converge exponentially and animate the system to evolve towards the target persistently. The tracking error and closed-loop state signals are proved theoretically to be uniformly ultimately bounded (UUB) via Lyapunov–Krasovskii functional. A numerical simulation from the process industry verifies the effectiveness of the proposed method. • An adaptive optimal control strategy based on DDPG is developed. • A novel persistent excitation (PE) mechanism is designed to incentivize the evolution. • PE-based evaluation ensures the fast maneuver encounters time-varying uncertainties. • MTNs are utilized as surrogates for better real-time performance. [ABSTRACT FROM AUTHOR]

It is notable that, the nonlocal reaction-diffusion equation carries math and computational physics to the core of extremely dynamic multidisciplinary studies that emerge from a huge assortment of uses. In this investigation, a totally new methodology for building a locally numerical pointwise solution is given by the agent the reproducing kernel algorithm. This is done utilizing a couple of generalized Hilpert spaces and their corresponding Green functions. The proposed calculation algorithm is applied to certain scalar issues problems to figure the arrangement solutions with Dirichlet constraints. By applying the procedures of the Gram–Schmidt process, orthonormalizing the basis, and truncating the optimized series, the approximate solutions are drawn, tabulated, and sketched. Introduced mathematical outcomes not only show the hidden superiority of the algorithm but also show its accurate efficiency. Finally, focused notes and futures planning works are mentioned with the most-used references. [ABSTRACT FROM AUTHOR]

The article analyzes the existence of Caputo fractional evolution integrodifferential equations of order 1 < r < 2 in Hilbert space with delay. A new set of adequate requirements for the existence outcomes of fractional delay evolution integrodifferential equations have been developed and are shown using the fractional derivative, Krasnoselskii's fixed point theorem, and Henry-Gronwall inequalities. In addition, for the provided system, we developed continuous dependence results. Afterward, we apply our findings to the concept of nonlocal conditions. Then, to demonstrate our primary outcomes, two examples are given. [ABSTRACT FROM AUTHOR]

We consider the reproducing kernel Hilbert space method to construct numerical solutions for some basic fractional ordinary differential equations (FODEs) under fractal fractional derivative with the generalized Mittag–Leffler (M-L) kernel. Deriving the analytic and numerical solutions of this new class of differential equations are modern trends. To apply this method, we use reproducing kernel theory and two important Hilbert spaces. We provide three problems to illustrate our main results including the profiles of different representative approximate solutions. The computational results are compared with the exact solutions. The results obtained clearly show the effect of the fractal fractional derivative with the M-L kernel in the obtained outcomes. Meanwhile, the compatibility between the approximate and exact solutions confirms the applicability and superior performance of the method. [ABSTRACT FROM AUTHOR]

The widespread use of spatial location-based services not only provides considerable convenience, but also exposes the downsides of location privacy leakage. Most of the existing user-side location privacy protection techniques are limited to planar locations. However, the extensive use of aircraft, sensor equipment and acquisition devices with positioning functions promotes the urgency of protecting the privacy of 3D spatial locations. Therefore, this study suggests a local differential privacy protection approach for 3D spatial locations. A 3D spatial decomposition and Hilbert encoding method are designed to reduce the 3D location data into one-dimensional encoding. The optimized random response mechanism was utilized to perturb the dimensional-reduced location encoding, which not only achieves user-side location privacy protection but also improves the accuracy of aggregated data on the server-side. Experiments on the real spatial location datasets show that the suggested method can reduce spatial location service quality loss, maintain the availability of perturbed spatial location and improve the operation efficiency of the spatial location perturbation algorithm. [ABSTRACT FROM AUTHOR]

This manuscript concerns the approximate controllability results in Hilbert space of Hilfer fractional differential inclusions. We show that the Hilfer fractional neutral differential inclusions are approximately controllable using Bohnenblust-Karlin's fixed point theorem. Further, nonlocal conditions are extended to the existence results. Furthermore, an application is shown to demonstrate the theoretical conclusions utility. [ABSTRACT FROM AUTHOR]

A compact self-isolated Multi Input Multi Output (MIMO) antenna array is presented for 5G mobile phone devices. The proposed antenna system is operating at the 3.5 GHz band (3400-3600 MHz) and consists of eight antenna elements placed along two side edges of a mobile device, which meets the current trend requirements of full-screen smartphone devices. Each antenna element is divided into two parts, a front part and back part. The front part consists of an I-shaped feeding line and a modified Hilbert fractal monopole antenna, whereas the back part is an L-shaped element shorted to the system ground by a 0.5 mm short stub. A desirable compactness can be obtained by utilizing the Hilbert space-filling property where the antenna element's overall planar size printed on the side-edge frame is just (9.57 mm × 5.99 mm). The proposed MIMO antenna system has been simulated, analyzed, fabricated and tested. Based on the selfisolated property, good isolation (better than 15 dB) is attained without employing additional decoupling elements and/or isolation techniques, which increases system complexity and reduces the antenna efficiency. The scattering parameters, antenna efficiencies, antenna gains, and antenna radiation characteristics are investigated to assess the proposed antenna performance. For evaluating the proposed antenna array system performance, the Envelope Correlation Coefficients (ECCs), Mean Effective Gains (MEGs) and channel capacity are calculated. Desirable antenna and MIMO performances are evaluated to confirm the suitability of the proposed MIMO antenna system for 5G mobile terminals. [ABSTRACT FROM AUTHOR]

We propose a general framework for policy representation for reinforcement learning tasks. This framework involves finding a low-dimensional embedding of the policy on a reproducing kernel Hilbert space (RKHS). The usage of RKHS based methods allows us to derive strong theoretical guarantees on the expected return of the reconstructed policy. Such guarantees are typically lacking in black-box models, but are very desirable in tasks requiring stability and convergence guarantees. We conduct several experiments on classic RL domains. The results confirm that the policies can be robustly represented in a low-dimensional space while the embedded policy incurs almost no decrease in returns. [ABSTRACT FROM AUTHOR]

In this study, we will discuss numerical solutions of conformable fractional systems of second-order integrodifferential equations concerning couple types Volterra and Fredholm. These conformable fractional systems will be considered in the sense of periodic boundary conditions in two points. The reproducing kernel Hilbert space method is used with the help of Mathematica 11 to utilize different tabulated data and graphical results. Series representation and required theorems are utilized and proved in the constructed Hilbert spaces. Several applications of linear and nonlinear function types have been studied and solved to ensure the applicability and power of the method. From the gained results, one can note that the approach algorithm is accurate and provides numerical results in a faster time and with less effort. Several notes and highlights with the latest references are discussed and presented at the end of the article. [ABSTRACT FROM AUTHOR]

In this paper we deal with the problem of uniform exponential stabilization for a class of distributed bilinear parabolic systems with time delay in a Hilbert space by means of a bounded feedback control. The uniform exponential stabilization problem of such a system reduces to stabilizing only its projection on a suitable finite dimensional subspace. Furthermore, the stabilizing feedback control depends only on the state projection on the finite dimensional subspace. An explicit decay rate estimate of the stabilized state is given provided that a nonstandard weaker observability condition is satisfied. Illustrative examples for partial functional differential equations are displayed. [ABSTRACT FROM AUTHOR]

The Support Vector Regression (SVR) technique can approximate intricate systems by addressing learning and estimation challenges within a reproducing kernel Hilbert space, devoid of reliance on specific parameter assumptions. However, when dealing with correlated data like time series, the SVR method often falls short in accounting for underlying temporal structures, leading to limited enhancements in prediction efficiency. We introduce an enhanced SVR method that considers temporal correlations (TemporalSVR) to overcome this constraint. Our proposed method extends kernel functions to include additional linear kernels, facilitating learning temporal patterns. Additionally, we develope an iterative training procedure for the augmented regression model. During model training, we estimate the hyper-parameter in the corresponding loss function using a 'working' likelihood approach, enhancing the generalization capabilities of the proposed regression. To demonstrate superior forecasting performance, we conduct extensive numerical simulations on both linear and nonlinear systems and the TemporalSVR achieves improvements ranging from 8% to 114% based on the RMSE ratio from the AR-X model. Furthermore, we investigate the forecasting performance of three basic models (NARX-NN, Statistical SVR, and SVR-ARIMA) and four deep learning (DL) techniques (Transformer, Informer, Reformer, Autormer, and Autoformer) by using a WTI forecasting study. Our proposed TemporalSVR achieves the smallest RMSE at 2.22 and attains the highest success ratio of stock direction prediction at 71.30%. All these numerical results highlight the effectiveness and advantages of our TemporalSVR in handling temporal data and making accurate predictions. • An augmented support vector regression model with an autoregressive process is designed to model temporal data. • A robust iterative procedure is developed for training the proposed SVR. • The working likelihood method is used for parameter estimation. [ABSTRACT FROM AUTHOR]

Low-channel EEG devices are crucial for portable and entertainment applications. However, the low spatial resolution of EEG presents challenges in decoding low-channel motor imagery. This study introduces TSFF-Net, a novel network architecture that integrates time–space–frequency features, effectively compensating for the limitations of single-mode feature extraction networks based on time-series or time–frequency modalities. TSFF-Net comprises four main components: time–frequency representation, time–frequency feature extraction, time–space feature extraction, and feature fusion and classification. Time–frequency representation and feature extraction transform raw EEG signals into time–frequency spectrograms and extract relevant features. The time–space network processes time-series EEG trials as input and extracts temporal–spatial features. Feature fusion employs Maximum Mean Discrepancy (MMD) loss to constrain the distribution of time–frequency and time–space features in the Reproducing Kernel Hilbert Space, subsequently combining these features using a weighted fusion approach to obtain effective time–space–frequency features. Moreover, few studies have explored the decoding of three-channel motor imagery based on time–frequency spectrograms. This study proposes a shallow, lightweight decoding architecture (TSFF-img) based on time–frequency spectrograms and compares its classification performance in low-channel motor imagery with other methods using two publicly available datasets. Experimental results demonstrate that TSFF-Net not only compensates for the shortcomings of single-mode feature extraction networks in EEG decoding, but also outperforms other state-of-the-art methods. Overall, TSFF-Net offers considerable advantages in decoding low-channel motor imagery and provides valuable insights for algorithmically enhancing low-channel EEG decoding. • A feature fusion architecture for 3-channel motor imagery classification. • Compensating for the limitations of single-mode EEG feature extraction. • A shallow and efficient network for EEG spectrogram feature extraction. [ABSTRACT FROM AUTHOR]

The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives. [ABSTRACT FROM AUTHOR]

Current work defines Schmidt representation of a bilinear operator T : H 1 × H 2 → K , where H 1 , H 2 and K are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if T is compact, and its singular values are ordered, then T has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental. [ABSTRACT FROM AUTHOR]

The law of large numbers for weakly correlated random elements with values in a separable Hilbert space is proved. [ABSTRACT FROM AUTHOR]

We study infinite-dimensional stochastic port-Hamiltonian systems (SPHSs) having multiplicative and boundary input noise. Using Itô and Stratonovich integrals on Hilbert spaces, a formal mathematical description of this specific class of stochastic systems is presented, and some properties, including almost sure and weak passivity, are investigated. By considering dissipative effects, we derive a condition for SPHSs to be weakly passive. Finally, the stochastic port-Hamiltonian framework and the passivity concepts are illustrated on the example of an inhomogeneous vibrating string subject to random damping and state noises. [ABSTRACT FROM AUTHOR]

This paper proposes an advanced numerical-analytical approach for handling a class of fuzzy fractional differential equations involving Caputo-Fabrizio derivative with a non-singular kernel arsing in the medical sector. The solution methodology relies on the reproducing-kernel algorithm to generate analytical solutions in the form of a uniformly convergent series in the direct sum of the desired Hilbert spaces. The effectiveness of the method is analyzed by studying some theoretical, analytical, and stability results of the derived solutions based on the reproducing kernel theory. Numerical simulations are also provided in tables and graphs to demonstrate the reliability of this algorithm in solving fuzzy models using the new Caputo-Fabrizio fractional operator, especially for the drug pharmacokinetic model. The obtained results show the ability of the applied algorithm to solve a wide range of nonlinear fractional models emerging in pharmacology, medicine, and biochemistry. [ABSTRACT FROM AUTHOR]

Among all the techniques combining multi-carrier modulation and spread spectrum, the multi-carrier code division multiple access (MC-CDMA) system is by far the most widely studied. In this paper, we present the performance of the MC-CDMA system associated with key single-user detection techniques. We are interested in problems related to identification and equalization of mobile radio channels, using the kernel method in Hilbert space with a reproducing kernel, and a linear adaptive algorithm, for MC-CDMA systems. In this context, we tested the efficiency of these algorithms, considering practical frequency selective fading channels, called broadband radio access network (BRAN), standardized for MC-CDMA systems. As far as the equalization problem encountered after channel identification is concerned, we use the orthogonality restoration combination (ORC) and the minimum mean square error (MMSE) equalizer techniques to correct the distortion of the channel. Simulation results demonstrate that the kernel algorithm is efficient for practical channels. [ABSTRACT FROM AUTHOR]

In this paper we discuss a general way of improving the bias properties of nonparametric kernel regression estimators. The procedure involves choosing a parametric model and constructing a semiparametric estimator that consists of a parametric component and a nonparametric adjustment, where the parametric component is picked from the parametric model in such a way that the resulting estimator has the best bias performance. We study the method for response variables taking values in a general Hilbert space and for local linear smoother. We show that the procedure always improves the bias of the local linear estimator regardless of the choice of parametric model. We also illustrate the method via a real data example where the response variable is a random density. [ABSTRACT FROM AUTHOR]

Aiming at the minority samples cannot be effectively diagnosed when the samples are limited and imbalanced, a multiple classifier ensemble of the weighted and balanced distribution adaptation method (MC-W-BDA) is presented to solve the rolling bearing's fault diagnosis problem under the limited samples imbalance. We adopt random sampling to obtain enough different training sample sets whose base classifiers are trained in the Reproducing Kernel Hilbert Space. The appropriate base classifiers are integrated into strong classifiers by multiple classifier ensemble strategy to obtain the final result of classification. In addition, we propose A-distance method to automatically set the optimal parameter (balance factor) in MC-W-BDA. Experimental verification verifies the feasibility and effectiveness of proposed approach. • A multiple classifier ensemble of the weighted and balanced distribution adaptation is proposed. • A-distance method to automatically set the optimal parameter is proposed. • The appropriate base classifiers are integrated into strong classifiers. • The fitness factor of edge distribution and conditional distribution is calculated dynamically. • The proposed method can realize bearing fault diagnosis at different speeds. [ABSTRACT FROM AUTHOR]

To achieve more appropriate fault feature representation for bearing, a statistical-enhanced covariance matrix (SECM) is proposed to extract the global–local features and the interaction of them. Besides, three statistical parameters are introduced to SECM to enhance its statistical characteristics. For fully mining the Riemannian geometric information embedded in SECMs, a Riemannian maximum margin flexible convex hull (RMMFCH) classifier with Log-Euclidean metric (LEM) is designed, where a set of Riemannian kernel mapping functions map SECMs to a higher-dimensional Hilbert space. In this space, the RMMFCH can be directly solved, which reduces the extra computation cost. Hence, we design a fault diagnosis scheme of bearing with SECM and RMMFCH. Experiment results prove the promising performance of our method for bearing fault diagnosis. • A statistical-enhanced covariance matrix (SECM) is used as bearing fault feature. • A RMMFCH classifier is proposed with Riemannian manifold kernels. • A bearing fault diagnosis scheme is proposed with SECM and RMMFCH. • Experiment results verify the great effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

Many rice-growing areas are affected by high concentrations of arsenic (As). Rice varieties that prevent As uptake and/or accumulation can mitigate As threats to human health. Genomic selection is known to facilitate rapid selection of superior genotypes for complex traits. We explored the predictive ability (PA) of genomic prediction with single-environment models, accounting or not for trait-specific markers, multi-environment models, and multi-trait and multi-environment models, using the genotypic (1600K SNPs) and phenotypic (grain As content, grain yield and days to flowering) data of the Bengal and Assam Aus Panel. Under the base-line single-environment model, PA of up to 0.707 and 0.654 was obtained for grain yield and grain As content, respectively; the three prediction methods (Bayesian Lasso, genomic best linear unbiased prediction and reproducing kernel Hilbert spaces) were considered to perform similarly, and marker selection based on linkage disequilibrium allowed to reduce the number of SNP to 17K, without negative effect on PA of genomic predictions. Single-environment models giving distinct weight to trait-specific markers in the genomic relationship matrix outperformed the base-line models up to 32%. Multi-environment models, accounting for genotype × environment interactions, and multi-trait and multi-environment models outperformed the base-line models by up to 47% and 61%, respectively. Among the multi-trait and multi-environment models, the Bayesian multi-output regressor stacking function obtained the highest predictive ability (0.831 for grain As) with much higher efficiency for computing time. These findings pave the way for breeding for As-tolerance in the progenies of biparental crosses involving members of the Bengal and Assam Aus Panel. Genomic prediction can also be applied to breeding for other complex traits under multiple environments. [ABSTRACT FROM AUTHOR]

In the paper approximate controllability of second order infinite dimensional system with damping is considered. Applying linear operators in Hilbert spaces general mathematical model of second order dynamical systems with damping is presented. Next, using functional analysis methods and concepts, specially spectral methods and theory of unbounded linear operators, necessary and sufficient conditions for approximate controllability are formulated and proved. General result may be used in approximate controllability verification of second order dynamical system using known conditions for approximate controllability of first order system. As illustrative example using Green function approach approximate controllability of distributed dynamical system is also discussed. [ABSTRACT FROM AUTHOR]

The aim of this article is to offer a concise and unitary vision upon the algebraic connections between classical logic and its generalizations, such as fuzzy logic and quantum logic. The mathematical concept which governs any kind of logic is that of lattice. Therefore, the lattices are the basic tools in this presentation. The Hilbert spaces theory is important in the study of quantum logic and it has also been used in the present paper. [ABSTRACT FROM AUTHOR]

In this paper, a modified reproducing kernel algorithm is proposed to solve a class of quadratic and cubic logistic equations with Caputo-Fabrizio fractional derivative in Hilbert space. These equations are the generalization of Verhulst's model which describes population growth taking in account that individuals will compete for limited resources. A novel reproducing kernel function is constructed to create an orthogonal system and to calculate the analytical and approximate solutions in the desirable Sobolev space. The stability, convergence, and complexity of the proposed approach are discussed. Furthermore, the effects of the Caputo-Fabrizio fractional derivatives are studied in solving the population growth model comparing with those of the classical Caputo derivatives. The main motivation for using the proposed technique is high accuracy and low computational cost compared to other existing methods especially when involving fractional differentiation operators. In this orientation, the effectiveness, applicability, and feasibility of this technique are verified by numerical examples. In a numerical viewpoint, the obtained results indicate that the suggested intelligent method has many advantages in accuracy and stability using the new Caputo-Fabrizio derivative. [ABSTRACT FROM AUTHOR]

To address the instruction order effects of dynamic decision-making for landing signal officers (LSO) of carrier-based aircraft in a comparative semantic context, in this paper, we propose a prediction approach of LSO guidance decision-making based on quantum interference. By constructing the quantum decision-making model of incompatible events, the restraint of law of total probability is shaken. Comparing the decision deviation of alternatives with non-comparative and comparative contexts, we analyze the interference nature of order effects. Then, we represent the similarity of events based on projective similarity index in Hilbert space and design a q-test prediction that satisfies the order effects of quantum decision-making. The experimental results show that the modified approach is accurate in predicting for assimilation effects and contrast effects. The theoretical basis for improving the accuracy of LSO decision-making is confirmed in a less projective similarity comparative semantic context. As such, the obtained results have certain reference value for personnel allocation, work periods, and landing safety of carrier-based aircraft. [ABSTRACT FROM AUTHOR]

Breast carcinoma is the most prevalent type of malignancy among women worldwide. Breast cancer grading often termed as Nuclear Atypia Scoring (NAS) forms a significant factor in determining individualized treatment plans and in the prognosis of the disease. For addressing the problem of breast cancer grading, we attempt to model the variations in features between histopathological images of different cancer grades and thereby explore the discriminative information concealed in these variations. In this regard, we aggregate multiple correlated features from the images using the geodesic geometric mean of the region covariances, to obtain the gmRC descriptors. As these gmRC descriptors are symmetric positive definite (SPD) matrices lying on the non-Euclidean Riemannian manifold, the discriminant analysis techniques developed for the Euclidean framework may not be appropriate. Hence, we propose a kernel-based Fisher discriminant analysis on the Riemannian manifold (KFDAR), that exploits the kernel trick for embedding the non-linear Riemannian manifold M into a higher dimensional linear Hilbert space H , which are then reduced to a low-dimensional and more discriminative subspace, where the samples become linearly separable. The kernel approach formulated for the Hilbert space embedding and for the kernel discriminant analysis is based on three Riemannian distance metrics: the log-Euclidean metric and the two symmetrized Bregman divergences – Stein and Jeffrey divergences. The experimental results show that this mapping to a highly discriminative space has succeeded in well-separating the histopathological images belonging to different cancer grades and hence it qualitatively and quantitatively outperforms the existing algorithms for cancer grading. [ABSTRACT FROM AUTHOR]

Hilbert-Schmidt Independence Criterion (HSIC) measures statistical independence between two random variables. However, instead of measuring the statistical independence between two random variables directly, HSIC first transforms two random variables into two Reproducing Kernel Hilbert Spaces (RKHS) respectively and then measures the kernelled random variables by using Hilbert-Schmidt (HS) operators between the two RKHS. Since HSIC was first proposed around 2005, HSIC has found wide applications in machine learning. In this paper, a HSIC regularized Local Tangent Space Alignment algorithm (HSIC-LTSA) is proposed. LTSA is a well-known dimensionality reduction algorithm for local homeomorphism preservation. In HSIC-LTSA, behind the objective function of LTSA, HSIC between high-dimensional and dimension-reduced data is added as a regularization term. The proposed HSIC-LTSA has two contributions. First, HSIC-LTSA implements local homeomorphism preservation and global statistical correlation during dimensionality reduction. Secondly, HSIC-LTSA proposes a new way to apply HSIC: HSIC is used as a regularization term to be added to other machine learning algorithms. The experimental results presented in this paper show that HSIC-LTSA can achieve better performance than the original LTSA. [ABSTRACT FROM AUTHOR]

• A novel methodology for automated myocardial infarction detection and localization. • Modeling of multi-lead ECG through a higher-order LDS and the projection of LDS parameters to a Grassmann manifold. • VLAD encoding of multi-lead ECG signal after a dyadic DWT and multiscale higher-order SVD analysis on sub-band tensors. • Two fusion approaches for mapping the extracted feature representations in a common Hilbert space. Electrocardiogram is commonly used as a diagnostic tool for the monitoring of cardiac health and the detection of possible heart diseases. However, the procedure followed for the diagnosis of heart abnormalities is time consuming and prone to human errors. Thus, the development of computer-aided techniques for the automatic analysis of electrocardiogram signals is of vital importance for the diagnosis and prevention of heart diseases. The most serious outcome of coronary heart disease is the myocardial infarction, i.e., the rapid and irreversible damage of cardiac muscles, which, if not diagnosed and treated in time, continues to damage further the myocardial structure and function. In this paper we propose a novel approach for the automatic detection and localization of myocardial infarction from multi-lead electrocardiogram signals. The proposed method initially reshapes the multidimensional signal into a third-order tensor structure and subsequently extracts feature representations in both Euclidean and Grassmannian space. In addition, two different methods are proposed for the mapping of the two different feature representations into a common Hilbert space before the final classification of signals. The first approach is based on the mapping of both Grassmannian and Euclidean features in a Reproducing Kernel Hilbert Space (RKHS), while the second one attempts to initially apply Vector of Locally Aggregated Descriptors (VLAD) encoding directly to Grassmann manifold and then concatenate the two VLAD representations. For the evaluation of the proposed method, we have conducted extensive tests using a publicly available dataset, namely PTB Diagnostic ECG database, containing 549 multi-lead ECG data recordings from 290 subjects and from different diagnostic classes. The method provides an excellent detection rate of 100%, and localization rate, i.e., 100% with the first fusion method and 99.7% with the second one. The Experimental results presented in this paper show the superiority of the proposed methodology against a number of state-of-the-art approaches. The main advantage of the proposed approach is that it exploits better the intercorrelations between signals of different ECG leads, by extracting feature representations that lie in different geometrical spaces and contain complementary information with regard to the dynamics of signals. [ABSTRACT FROM AUTHOR]

Abstract The present paper is devoted to the study of robust control problems of parabolic stochastic partial differential equations under model uncertainty. To be more precise, the robust control problem under investigation is expressed as a stochastic differential game in a real separable infinite dimensional Hilbert space. By resorting to the theory of mild solutions, we prove that the elliptic partial differential equation associated with the problem at hand, also known as the Hamilton-Jacobi-Bellman-Isaacs equation, admits a unique solution, which is the value function of the game. Furthermore, we investigate the problem of existence of an optimal control pair that satisfies a saddle point property. Finally, as a demonstration of the proposed approach, we apply our results to the study of a certain robust control problem arising in the spatiotemporal management of natural resources. [ABSTRACT FROM AUTHOR]