Search Results (1,642)

The graph Hilbert transform (GHT) is a key tool in constructing analytic signals and extracting envelope and phase information in graph signal processing. However, its utility is limited by confinement to the graph Fourier domain, a fixed phase shift, information loss for real-valued spectral components, and the absence of tunable parameters. The graph fractional Fourier transform introduces domain flexibility through a fractional order parameter $\alpha$ but does not resolve the issues of phase rigidity and information loss. Inspired by the dual-parameter fractional Hilbert transform (FRHT) in classical signal processing, we propose the graph FRHT (GFRHT). The GFRHT incorporates a dual-parameter framework: the fractional order $\alpha$ enables analysis across arbitrary fractional domains, interpolating between vertex and spectral spaces, while the angle parameter $\beta$ provides adjustable phase shifts and a non-zero real-valued response ($cos\beta$) for real eigenvalues, thereby eliminating information loss. We formally define the GFRHT, establish its core properties, and design a method for graph analytic signal construction, enabling precise envelope extraction and demodulation. Experiments on anomaly identification, speech classification and edge detection demonstrate that GFRHT outperforms GHT, offering greater flexibility and superior performance in graph signal processing. Full article

The high penetration of renewable energy brings significant power fluctuations and operational uncertainties to distribution networks. Traditional power allocation methods for hybrid energy storage systems (HESSs) exhibit strong parameter dependency, limited frequency-domain recognition accuracy, and poor dynamic coordination capability. To overcome these limitations, this study proposes an adaptive power allocation strategy for HESSs under renewable energy integration scenarios. The proposed method employs the Grey Wolf Optimizer (GWO) to jointly optimize the mode number and penalty factor of the Variational Mode Decomposition (VMD), thereby enhancing the accuracy and stability of power signal decomposition. In conjunction with the Hilbert transform, the instantaneous frequency of each mode is extracted to achieve a natural allocation of low-frequency components to the battery and high-frequency components to the supercapacitor. Furthermore, a multi-objective power flow optimization model is formulated, using the power commands of the two storage units as optimization variables and aiming to minimize voltage deviation and network loss cost. The model is solved through the Particle Swarm Optimization (PSO) algorithm to realize coordinated optimization between storage control and system operation. Case studies on the IEEE 33-bus distribution system under both steady-state and dynamic conditions verify that the proposed strategy significantly improves power decomposition accuracy, enhances coordination between storage units, reduces voltage deviation and network loss cost, and provides excellent adaptability and robustness. Full article

This paper introduces a novel multidimensional half-discrete Hardy–Hilbert-type inequality that simultaneously addresses several key extensions in the literature. The inequality incorporates a general parameterized kernel involving a scalar term and the $\beta$-norm of a vector, and replaces the traditional discrete coefficient with a partial sum. Under suitable parameter conditions, the resulting inequality is sharper and preserves the optimal constant factor. The proof employs a systematic combination of weight-function techniques, parameter introduction, real-analysis methods, and the Euler–Maclaurin summation formula. Equivalent characterizations of the best possible constant are provided, and several meaningful corollaries are deduced, thereby unifying and generalizing a series of earlier inequalities. Full article

In quasi-Hermitian quantum mechanics (QHQM) of unitary systems, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, $H\ne H^{†}$. This makes its physical interpretation ambiguous. Without altering H, this ambiguity can be resolved either via a transformation of H into its isospectral Hermitian form via a so-called Dyson map $\mathrm{\Omega }:H\to \mathfrak{h}$, or via a (formally equivalent) specification of a nontrivial physical inner-product metric $\mathrm{\Theta }$ in Hilbert space. Here, we focus on the former strategy. Our present construction of the Hermitian isospectral twins $\mathfrak{h}$ of H is exhaustive. As a byproduct, it not only restores the conventional correspondence principle between quantum and classical physics, but it also provides a framework for a systematic classification of all of the admissible probabilistic interpretations of quantum systems using a preselected H in QHQM framework. Full article

We study how convolutional neural networks reorganize information during learning in natural image classification tasks by tracking mutual information (MI) between inputs, intermediate representations, and labels. Across VGG-16, ResNet-18, and ResNet-50, we find that label-relevant MI grows reliably with depth while input MI depends strongly on architecture and activation, indicating that “compression’’ is not a universal phenomenon. Within convolutional layers, label information becomes increasingly concentrated in a small subset of channels; inference-time knockouts, shuffles, and perturbations confirm that these high-MI channels are functionally necessary for accuracy. This behavior suggests a view of representation learning driven by selective concentration and decorrelation rather than global information reduction. Finally, we show that a simple dependence-aware regularizer based on the Hilbert–Schmidt Independence Criterion can encourage these same patterns during training, yielding small accuracy gains and consistently faster convergence. Full article

Alzheimer’s Disease (AD) is an advanced brain illness that affects millions of individuals across the world. It causes gradual damage to the brain cells, leading to memory loss and cognitive dysfunction. Although Magnetic Resonance Imaging (MRI) is widely used in AD diagnosis, the existing studies rely solely on the visual representations, leaving alternative features unexplored. The objective of this study is to explore whether MRI sonification can provide complementary diagnostic information when combined with conventional image-based methods. In this study, we propose a novel dual-stream multimodal framework that integrates 2D MRI slices with their corresponding audio representations. MRI images are transformed into audio signals using a multi-scale, multi-orientation Gabor filtering, followed by a Hilbert space-filling curve to preserve spatial locality. The image and sound modalities are processed using a lightweight CNN and YAMNet, respectively, then fused via logistic regression. The experimental results of the multimodal achieved the highest accuracy in distinguishing AD from Cognitively Normal (CN) subjects at 98.2%, 94% for AD vs. Mild Cognitive Impairment (MCI), and 93.2% for MCI vs. CN. This work provides a new perspective and highlights the potential of audio transformation of imaging data for feature extraction and classification. Full article

The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality mapping. In this paper, we propose and analyze an RFBS algorithm in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. To the best of our knowledge, this work presents the first strong (R-linear) convergence result for the RFBS method in such Banach spaces, achieved under a newly adapted notion of strong monotonicity. Our results thus establish a foundational theoretical guarantee for RFBS in Banach spaces under strengthened monotonicity conditions, while highlighting the open problem of proving weak convergence for the general monotone case. Full article

This paper introduces a rigorously defined fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives. By deriving a compatible fractional Pearson-type equation, a new weight function and Hilbert space setting are established, ensuring the operator’s self-adjointness under natural fractional boundary conditions. Within this framework, we prove the existence of a real, discrete spectrum and demonstrate that the corresponding eigenfunctions form a complete orthogonal system in $L_{{\omega }_{\alpha }}^2(a,b)$. The central theoretical result shows that the fractional eigenpairs $({\lambda }_n^{\left(\alpha \right)},u_n^{\left(\alpha \right)})$ converge continuously to their classical Heun counterparts $({\lambda }_n^{\left(1\right)},u_n^{\left(1\right)})$ as $\alpha \to 1^{-}$. This provides a rigorous analytic bridge between fractional and classical spectral theories. A numerical study based on the fractional Legendre case confirms the predicted self-adjointness and spectral convergence, illustrating the smooth deformation of the classical eigenfunctions into their fractional counterparts. The results establish the fractional Heun operator as a mathematically consistent generalization capable of generating new families of orthogonal fractional functions. Full article

Heart rate variability (HRV) comprises several components driven by various internal processes, the least understood of which is the ultra-low frequency (ULF) one. Recently published research has shown that the HRV frequency distribution in this range is bimodal. The main aims of this work were to verify this finding, to determine the basic characteristics of these two components and to analyze their potential physiological couplings. For this purpose, two components within the conventional ULF band (below 4 mHz) were extracted from HRVs of 25 patients with apnea using adaptive variational mode decomposition (AVMD) and continuous wavelet transform (CWT), and then analyzed with the Hilbert transform (HT), Savitzky–Golay filter, and empirical distributions of instantaneous amplitudes and frequencies. These studies have demonstrated the existence of both components in HRVs of all subjects and apnea groups: extremely low frequencies (ELFs) in the range of 0.01–0.4 mHz and narrowed ultra-low frequencies (nULFs) in the range of 0.1–4 mHz. The independence of both components is also shown. Concluding, heart rate variability is separately regulated by circadian rhythms (ELF bound) and ultradian fluctuations (nULF bound), which can be assessed by decomposing HRV, and the obtained components may be helpful to better understand the underlying homeostatic mechanisms, as well as in the long-term monitoring of patients. Full article

A conditional framework of Conditional Cosmological Recurrence (CCR) is introduced, as follows: if a causal patch admits a finite operational Hilbert space dimension D (as motivated by holographic and entropy bounds), then unitary quantum dynamics guarantee almost-periodic evolution, leading to recurrences. The central contribution is the explicit formulation of a micro-to-macro bridge, as follows: (i) finite regions discretize field modes; (ii) gravitational bounds cap entropy and energy; and (iii) the number of accessible states is finite, yielding CCR. The analysis differentiates global microstate recurrences (with double-exponential timescales in $S_{max}$) from operationally relevant coarse-grained returns (exponential in subsystem entropy), with conservative timescale estimates. For predictivity in eternally inflating settings, a causal-diamond measure with xerographic typicality and a single no-Boltzmann-brain constraint is employed, thereby avoiding volume-weighting pathologies. The scope is explicitly conditional: if future quantum gravity demonstrates $D=\infty$ for causal patches, CCR is falsified. Full article

Exploring the design of beneficial nonlinear restoring force structures has become a highly popular topic due to their extensive applications in energy harvesting, actuation, energy absorption, robotics, etc. However, the current literature lacks a systematic review and classification that addresses the design, modeling, and parameter identification of nonlinear restoring forces. Thus, the present paper provides a thorough examination of the latest advancements in the design of nonlinear restoring forces, as well as modeling and parameter identification in contemporary beneficial nonlinear designs. The seven design methodologies, namely magnetic coupling, oblique spring linkages, static or dynamic preloading, metamaterials, bio-inspired, MEMS (Micro-Electromechanical Systems) manufacturing, and dry friction applied approaches, are classified. The polynomial, hysteretic, and piecewise linear models are summarized for nonlinear restoring force characterization. The system parameter identification methods covering restoring force surface, Hilbert transform, time-frequency analysis, nonlinear subspace identification, unscented Kalman filter, optimization algorithms, physics-informed neural networks, and data-driven sparse regression are reviewed. Moreover, possible enhancement strategies for nonlinear system identification of nonlinear restoring forces are presented. Finally, broader implications and future directions for the design, characterization, and identification of nonlinear restoring forces are discussed. Full article

This paper presents a unified framework for controllability and minimum-energy control of linear fractional differential systems with Caputo derivative order $\gamma \in (0,1)$ and fully time-varying state and control delays. An explicit mild solution representation is derived using the fractional fundamental matrix, and a new controllability Gramian is introduced. Using analytic properties of the matrix-valued Mittag-Leffler function, we prove a fractional Kalman-type theorem showing that bounded time-varying delays do not change the algebraic controllability structure determined by $(F,G,K)$. The minimum-energy control problem is solved in closed form through Hilbert space methods. Efficient numerical strategies and several examples—including delayed viscoelastic, neural, and robotic models—demonstrate practical applicability and computational feasibility. Full article

The cosmological constant (CC, $\Lambda$) problem stands as one of the most profound puzzles in the theory of gravity, representing a remarkable discrepancy of about 120 orders of magnitude between the observed value of dark energy and its natural expectation from quantum field theory. This paper synthesizes two innovative paradigms—holographic naturalness (HN) and pre-geometric gravity (PGG)—to propose a unified and natural resolution to the problem. The HN framework posits that the stability of the CC is not a matter of radiative corrections but rather of quantum information and entropy. The large entropy $S_{\mathrm{dS}}\sim M_{\mathrm{P}}^2/\Lambda$ of the de Sitter (dS) vacuum (with $M_{\mathrm{P}}$ being the Planck mass) acts as an entropic barrier, exponentially suppressing any quantum transitions that would otherwise destabilize the vacuum. This explains why the universe remains in a state with high entropy and relatively low CC. We then embed this principle within a pre-geometric theory of gravity, where the spacetime geometry and the Einstein–Hilbert action are not fundamental, but emerge dynamically from the spontaneous symmetry breaking of a larger gauge group, SO($1,4$)→SO($1,3$), driven by a Higgs-like field ${\varphi }^A$. In this mechanism, both $M_{\mathrm{P}}$ and $\Lambda$ are generated from more fundamental parameters. Crucially, we establish a direct correspondence between the vacuum expectation value (VEV) v of the pre-geometric Higgs field and the de Sitter entropy: $S_{\mathrm{dS}}\sim v$ (or $v^3$). Thus, the field responsible for generating spacetime itself also encodes its information content. The smallness of $\Lambda$ is therefore a direct consequence of the largeness of the entropy $S_{\mathrm{dS}}$, which is itself a manifestation of a large Higgs VEV v. The CC is stable for the same reason a large-entropy state is stable: the decay of such state is exponentially suppressed. Our study shows that new semi-classical quantum gravity effects dynamically generate particles we call “hairons”, whose mass is tied to the CC. These particles interact with Standard Model matter and can form a cold condensate. The instability of the dS space, driven by the time evolution of a quantum condensate, points at a dynamical origin for dark energy. This paper provides a comprehensive framework where the emergence of geometry, the hierarchy of scales and the quantum-information structure of spacetime are inextricably linked, thereby providing a novel and compelling path toward solving the CC problem. Full article

Given Hilbert space operators $A,B$, and X, let ${▵}_{A,B}$ and ${\delta }_{A,B}$ denote, respectively, the elementary operators ${▵}_{A,B}\left(X\right)=I-AXB$ and the generalised derivation ${\delta }_{A,B}\left(X\right)=AX-XB$. This paper considers the structure of operators $D_{d_1,d_2}^m\left(I\right)=0$ and $D_{d_1,d_2}^m$ compact, where m is a positive integer, $D=▵$ or $\delta$, $d_1={▵}_{A^{*},B^{*}}$ or ${\delta }_{A^{*},B^{*}}$ and $d_2={▵}_{A,B}$ or ${\delta }_{A,B}$. This is a continuation of the work performed by C. Gu for the case where ${▵}_{{\delta }_{A^{*},B^{*}},{\delta }_{A,B}}^m\left(I\right)=0$, and the author with I.H. Kim for the cases where ${▵}_{{\delta }_{A^{*},B^{*}},{\delta }_{A,B}}^m\left(I\right)=0$ or ${▵}_{{\delta }_{A^{*},B^{*}},{\delta }_{A,B}}^m$ is compact, and ${\delta }_{{▵}_{A^{*},B^{*}},{▵}_{A,B}}^m\left(I\right)=0$ or ${\delta }_{{▵}_{A^{*},B^{*}},{\delta }_{A,B}}^m$ is compact. Operators $D_{d_1,d_2}^m\left(I\right)=0$ are examples of operators with a finite spectrum; indeed, the operators $A,B$ have at most a two-point spectrum, and if $D_{d_1,d_2}^m$ is compact, then (the non-nilpotent operators) $A,B$ are algebraic. $D_{d_1,d_2}^m\left(I\right)=0$ implies $D_{d_1,d_2}^n\left(I\right)=0$ for integers $n\ge m$: the reverse implication, however, fails. It is proved that $D_{d_1,d_2}^m\left(I\right)=0$ implies $D_{d_1,d_2}\left(I\right)=0$ if and only if of A and B (are normal and hence) satisfy a Putnam–Fuglede commutativity property. Full article

This article introduces an iterative algorithm that is created by integrating the $S^{*}$-iteration process with the inertial forward-backward algorithm. The algorithm is designed to solve optimization problems formulated as variational inclusions in a real Hilbert space. We establish the weak convergence of the algorithm under conventional assumptions. One of the applications of the algorithm is to solve the extreme learning machine, which can be transformed into the variational inclusion problem. Different algorithms, with all parameters set to be identical, are employed to solve the stroke classification problem in order to evaluate the algorithm’s performance. The results indicate that our algorithm converges faster than others and achieves a precision of 93.90%, a recall of 100%, and an F1-score of 96.58%. Full article