Spectral Dimension Research Articles

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates through the prime sequence and define an emergent arithmetic geometry. From these spectra we extract observables such as the heat trace, entropy, and eigenvalue growth, which reveal persistent spectral compression): eigenvalues grow sublinearly, entropy scales slowly, and the inferred dimension remains strictly below one. This rigidity appears across logarithmic, entropic, and fractal-type kernels, reflecting intrinsic arithmetic constraints. Analytically, we show that for the unnormalized Laplacian, the continuum limit of its squared Hamiltonian corresponds to the one-dimensional bi-Laplacian, whose heat trace follows a short-time scaling proportional to t−1/4. Under the spectral dimension convention ds=−2dlogΘ/dlogt, this result produces ds=1/2 directly from first principles, without fitting or external hypotheses. This value signifies maximal spectral compression and the absence of classical diffusion, indicating that arithmetic sparsity enforces a coherence-limited, non-Euclidean geometry linking spectral and number-theoretic structure.

Nearly minimax robust estimator of the mean vector by iterative spectral dimension reduction

Spatial-spectral multi-order gated aggregation network with bidirectional interactive fusion for hyperspectral image classification.

We investigate, using Langevin dynamics simulations, the Rouse-type dynamics of active harmonic bead-spring percolation clusters of square and triangular lattices. Two types of active stochastic forces, modeled as a random telegraph process with correlation time τ, are considered: force monopoles, acting on individual nodes in random directions, and force dipoles, where extensile or contractile forces act between pairs of nodes. For force monopoles, a dynamical steady state is reached where the network is dynamically swollen and the mean square displacement (MSD) shows sub-diffusive behavior at t > τ, MSD ∼ tν, with ν=1-ds2 where ds is the spectral dimension, in accord with a previously advanced general analytic theory. In contrast, dipolar forces require diverging times to reach steady state and lead to network shrinkage. Within a quasi-steady-state approximation, the MSD is found to saturate at the same temporal regime t > τ, which is followed by ballistic-like and/or diffusive behaviors. We further extend our study of dipolar forces to dilution regimes above the isostatic threshold, also known as "rigidity percolation". Here, weak dipolar forces effectively do not shrink the network in steady state. Instead, they induce "rotational swimming" of the network. Yet, for the triangular lattice, an incipient discontinuous collapse transition occurs above a critical force amplitude value. Conversely, we find a continuous crossover to a collapsed state for the non-diluted square lattice, resulting from its marginal stability. We suggest that disordered solids be poised above the isostatic point to be stable against active dipolar forces, provided that the dynamical persistent length remains lower than the spring rest length.

Rapid detection of quarantine diseases in apples is essential for import–export control but remains difficult because routine inspections rely on manual visual checks that limit automation at port scale. A fast, non-destructive system suitable for deployment at customs is therefore needed. In this study, three common apple quarantine pathogens were targeted using hyperspectral images acquired by a close-range hyperspectral camera and analyzed with a convolutional neural network (CNN). Symptoms of these diseases often appear similar in RGB images, making reliable differentiation difficult. Reflectance from 400 to 1000 nm was recorded to provide richer spectral detail for separating subtle disease signatures. To quantify stage-dependent differences, average reflectance curves were extracted for apples infected by each pathogen at early, middle, and late lesion stages. A CNN tailored to hyperspectral inputs, termed HSC-Resnet, was designed with an increased number of convolutional channels to accommodate the broad spectral dimension and with channel and spatial attention integrated to highlight informative bands and regions. HSC-Resnet achieved a precision of 95.51%, indicating strong potential for fast, accurate, and non-destructive detection of apple quarantine diseases in import–export management.

The high-performance identification of insulating materials is crucial for reducing resource waste, minimizing pollution, and promoting resource recycling. To achieve this, a novel method based on laser-induced breakdown spectroscopy (LIBS), named the generalized spectrum method (GSM-LIBS), was proposed in this study. Compared to traditional dimensionality reduction methods such as PCA, GSM-LIBS outperforms by integrating multiple spectral features, preserving both global and local information that may be lost in PCA-based methods. GSM-LIBS not only effectively reduces the spectral dimensions but also extracts more key features, such as peak intensity, integral intensity, intensity ratio, radiation background, and spectral shape. These features help retain important information from the spectrum, providing more precise details such as plasma state, element concentration, and spectral characteristics, thereby significantly improving analysis performance. To verify the effectiveness of GSM-LIBS, this method was applied to the classification study of seven types of insulating materials and compared with principal component analysis (PCA-LIBS). To ensure the general applicability of this study, two traditional machine learning models, k-nearest neighbor (KNN) and support vector machine (SVM), and one deep learning model, neural network (NN), were used. For the machine learning models, the accuracy of the KNN and SVM classification models on the prediction set improved from 0.935 and 0.965 to 0.979 and 0.996, respectively. For the deep learning model, the performance of the NN classification model was also significantly improved, with accuracy increasing from 0.984 to 0.994. These experimental results strongly demonstrate the feasibility and effectiveness of GSM-LIBS in effectively reducing the spectral dimensions while retaining key information.

ABSTRACT Current hyperspectral image classification methods face two major challenges: convolutional neural networks struggle to effectively capture long-range dependencies in the spectral dimension, while Transformer architectures suffer from inefficiency due to quadratic computational complexity when processing hyperspectral data with hundreds of bands. This paper proposes DualPathMamba, a state space model-based method featuring: (1) a dual-path architecture that simultaneously processes spectral sequences through SSM operations and spatial features through convolutional operations, achieving complementary feature fusion through element-wise addition; (2) linear computational complexity O(L) achieved through state space modelling with four core operations – time modulation, state transition, input mapping, and output mapping – compared to quadratic complexity O(L 2) in attention-based methods; (3) efficient processing of high-dimensional spectral data while maintaining approximately 2.0 M parameters. Experiments demonstrate that DualPathMamba achieves overall accuracies of 93.85%, 98.85%, and 94.52% on the Indian Pines, Pavia University, and Kennedy Space Center benchmark datasets, respectively, significantly outperforming existing methods while demonstrating superior computational efficiency and robustness.

Modern reconnaissance technologies, including hyperspectral and multispectral intensity imaging across optical, thermal infrared, terahertz, and microwave bands, can detect the shape, material composition, and temperature of targets. Consequently, developing a camouflage technique that seamlessly integrates both spatial and spectral dimensions across all key atmospheric windows to outsmart advanced surveillance has yet to be effectively developed and remains a significant challenge. In this study, we propose a digital camouflage strategy that covers the optical (0.4-2.5 μm) hyperspectra and thermal infrared-terahertz-microwave (thermal IR (MWIR and LWIR)/THz/MW) tri-bands, encompassing over 80% of atmospheric windows. In the optical band, the hyperspectral digital camouflage can simulate various vegetational spectra as primary colors, with deviation rate less than 0.2 (can be regarded as the same type of plant). In the tri-bands, it also produces multilevel intensity digital camouflage within each band. The average structural similarity among multiple digital camouflage patterns is approximately 0.52, which is favorable for multispectral pattern-background matching. This work introduces a new paradigm in ultra-broadband electromagnetic wave manipulation by combining hyper/multi-spectra and spatial distribution, offering deeper insights into imaging, image processing, and information encryption technologies.

The classification of hyperspectral images (HSI) is an important foundation in the field of remote sensing. Mamba architectures based on state space model (SSM) have shown great potential in the field of HSI processing due to their powerful long-range sequence modeling capabilities and the efficiency advantages of linear computing. Based on this theoretical basis, We propose a novel deep learning framework: long-sequence Mamba (EchoMamba), which combines the powerful long sequence processing capabilities of Long Short-Term Memory(LSTM) and Mamba to further explore the spectral dimension of HSI, and carry out more in-depth mining and learning of the spectral dimension of HSI. Compared with the previous HSI classification model, the experimental results show that EchoMamba can significantly reduce the training time cost of HSI and effectively improve the performance of the classification task.This study not only advances the current state of HSI classification but also provides a robust foundation for future research in spectral-spatial feature extraction and large-scale remote sensing applications.

Background: Efficient decoding of motor imagery (MI) electroencephalogram (EEG) signals is essential for the precise control and practical deployment of brain-computer interface (BCI) systems. Owing to the complex nonlinear characteristics of EEG signals across spatial, spectral, and temporal dimensions, efficiently extracting multidimensional discriminative features remains a key challenge to improving MI-EEG decoding performance. Methods: To address the challenge of capturing complex spatial, spectral, and temporal features in MI-EEG signals, this study proposes a multi-branch deep neural network, which jointly models these dimensions to enhance classification performance. The network takes as inputs both a three-dimensional power spectral density tensor and two-dimensional time-domain EEG signals and incorporates four complementary feature extraction branches to capture spatial, spectral, spatial-spectral joint, and temporal dynamic features, thereby enabling unified multidimensional modeling. The model was comprehensively evaluated on two widely used public MI-EEG datasets: EEG Motor Movement/Imagery Database (EEGMMIDB) and BCI Competition IV Dataset 2a (BCIIV2A). To further assess interpretability, gradient-weighted class activation mapping (Grad-CAM) was employed to visualize the spatial and spectral features prioritized by the model. Results: On the EEGMMIDB dataset, it achieved an average classification accuracy of 86.34% and a kappa coefficient of 0.829 in the five-class task. On the BCIIV2A dataset, it reached an accuracy of 83.43% and a kappa coefficient of 0.779 in the four-class task. Conclusions: These results demonstrate that the network outperforms existing state-of-the-art methods in classification performance. Furthermore, Grad-CAM visualizations identified the key spatial channels and frequency bands attended to by the model, supporting its neurophysiological interpretability.

Abstract We study unitary orthonormal bases in the sense of Pimsner and Popa for inclusions $(\mathcal{B}\subseteq \mathcal{A}, E),$ where $\mathcal{A}, \mathcal{B}$ are finite dimensional von Neumann algebras and $E$ is a conditional expectation map from $\mathcal{A}$ onto $\mathcal{B}$. It is shown that existence of such bases requires that the associated inclusion matrix satisfies a spectral condition forcing dimension vectors to be Perron–Frobenius eigenvectors and the conditional expectation map preserves the Markov trace. Subject to these conditions, explicit unitary orthonormal bases are constructed if either one of the algebras is abelian or simple. They generalize complex Hadamard matrices, Weyl unitary bases, and a recent work of Crann et al., which correspond to the special cases of $\mathcal{A}$ being abelian, simple, and general multi-matrix algebras respectively with $\mathcal{B}$ being the algebra of complex numbers. For the first time $\mathcal{B}$ is more general. As an application of these results it is shown that if $(\mathcal{B}\subseteq \mathcal{A}, E),$ admits a unitary orthonormal basis then the Connes–Størmer relative entropy $H(\mathcal{A}_{1}|\mathcal{A})$ equals the logarithm of the square of the norm of the inclusion matrix, where $\mathcal{A}_{1}$ denotes the Jones basic construction of the inclusion. As a further application, we prove the existence of unitary orthonormal bases for a large class of depth 2 subfactors with abelian relative commutant.

We investigate critical transport and the dynamical exponent through the spreading of an initially localized particle in quadratic Hamiltonians with short-range hopping in lattice dimension d_{l}. We consider critical dynamics that emerges when the Thouless time, i.e., the saturation time of the mean-squared displacement, approaches the typical Heisenberg time. We establish a relation, z=d_{l}/d_{s}, linking the critical dynamical exponent z to d_{l} and to the spectral fractal dimension d_{s}. This result has notable implications: it says that superdiffusive transport in d_{l}≥2 and diffusive transport in d_{l}≥3 cannot be critical in the sense defined above. Our findings clarify previous results on disordered and quasiperiodic models and, through Fibonacci potential models in two and three dimensions, provide nontrivial examples of critical dynamics in systems with d_{l}≠1 and d_{s}≠1.

Context. M stars are preferred targets for upcoming studies of terrestrial exoplanets aimed at obtaining their atmosphere spectra over the next decade. However, M dwarfs have long been known for their strong magnetic activity and the ability to frequently produce optical and broadband emission flares. Aims. We aim to characterise the flaring behaviour of young M dwarfs in the temporal, spectral, and energy dimensions, as well as examine the stellar parameters governing this behaviour. In this way, we aim to improve our understanding of the energy and frequency of the flare events capable of shaping the exoplanet atmosphere. Methods. Members of young moving groups (YMGs) provide a unique age-based perspective on stellar activity. By examining their flare behaviour in conjunction with rotation, mass, and Hα data, we can obtain a comprehensive understanding of flare-activity drivers in young stars. Results. We demonstrate that young stars sharing similar stellar parameters could also exhibit a broad range of flare frequency distributions. We also find that the flare behaviour shows indications of difference between optical and far ultraviolet (FUV). We propose that the period of rotation (and not the age of the star) can serve as a good proxy for assessing flaring activity. Furthermore, we recommend that instead of a simple power law for describing the flare frequency distribution, a piecewise power law can be used to describe mid-size and large flare distributions in young and active M dwarfs. Conclusions. Using known periods of rotation and fine-tuned power laws governing the flare frequency, we can produce a realistic sequence of flare events to study whether the atmosphere of small exoplanets orbiting M dwarf could withstand such activity until the emergence of life.

S3-Net: Learning spectral-spatio self-similarity for hyperspectral image super-resolution.

Imaging techniques have revolutionized cultural heritage analysis, particularly for objects that cannot be sampled. This study investigated the utilization of spectral imaging for the identification of pigments in artifacts from the Arxiu Valencià del Disseny, in conjunction with other portable spectroscopy techniques such as XRF, Raman, FT-NIR, and FT-MIR. Four early 1930s watercolors were examined using point-wise elemental and molecular spectroscopic data for pigment classification. Initially, the data cubes obtained with the spectral camera were processed using various methods. The spectral behavior was analyzed pixel-point, and the reflectance curves were qualitatively compared with a set of standards. Subsequently, a computational approach was applied to the data cube to produce RGB, false-color infrared (IRFC), and principal component (PC) images. Algorithms, such as the Vector Angle (VA) mapper, were also employed to map the pigment spectra. Consequently, 19th-century pigments such as Prussian blue, chrome yellow, and alizarin red were distinguished according to their composition, combining the spatial and spectral dimensions of the data. Elemental analysis and infrared spectroscopy supported these findings. In this context, the use of reflectance imaging spectroscopy (RIS), despite its technical limitations, emerged as an essential tool for the documentation and conservation of design heritage.

This paper presents a conceptual and methodological framework for using Geographical Information Systems (GIS) to “deep map” cultural heritage sites along Ireland’s Wild Atlantic Way, with a focus on the 1588 Spanish Armada wrecks in County Kerry and archaeological landscapes in County Sligo’s “Yeats Country.” Drawing on interdisciplinary dialogues from the humanities, social sciences, and geospatial sciences, it illustrates how digital spatial technologies can excavate, preserve, and sustain intangible cultural knowledge embedded within such palimpsestic landscapes. Using MAXQDA 24 software to mine and code historical, literary, folkloric, and environmental texts, the study constructed bespoke GIS attribute tables and visualizations integrated with elevation models and open-source archaeological data. The result is a richly layered cartographic method that reveals the spectral and affective dimensions of heritage landscapes through climate, memory, literature, and spatial storytelling. By engaging with “deep mapping” and theories such as “Spectral Geography,” the research offers new avenues for sustainable heritage conservation, cultural tourism, and public education that are sensitive to both ecological and cultural resilience in the West of Ireland.

In this work we study the spectral dimensionality of spacetime around a radiating Schwarzschild black hole using a recently introduced formalism of quantum gravity, where the alterations of the gravitational field produced by the radiation are represented on an extended manifold, and describe a non-commutative and nonlinear quantum algebra. The relation between classical and quantum perturbations of spacetime can be measured by the parameter z≥0. In this work we have found that when z=(1+3)/2≃1.3660, a relativistic observer approaching the Schwarzschild horizon perceives a spectral dimension N(z)=4θ(z)−1≃2.8849, which is related to quantum gravitational interference effects in the environment of the black hole. Under these conditions, all studied Schwarzschild black holes with masses ranging from the Planck mass to 1046 times the Planck mass present the same stability configuration, which suggests the existence of a universal property of these objects under those particular conditions. The difference from the spectral dimension previously obtained at cosmological scales leads to the conclusion that the spacetime dimensionality is scale-dependent. Another important result presented here is the fundamental alteration of the effective gravitational potential near the horizon due to Hawking radiation. This quantum phenomenon prevents the potential from diverging to negative infinity as the observer approaches the Schwarzschild horizon.

Hyperspectral imaging (HSI) collects detailed spectral information across hundreds of narrow bands, providing valuable datasets for applications such as medical diagnostics. However, the large size of HSI datasets, often exceeding several gigabytes, creates significant challenges in data transmission, storage, and processing. We aim to develop a wavelet-based compression method that addresses these challenges while preserving the integrity and quality of spectral information. The proposed method applies wavelet transforms to the spectral dimension of hyperspectral data in three steps: (1)wavelet transformation for dimensionality reduction, (2)spectral cropping to eliminate low-intensity bands, and (3)scale matching to maintain accurate wavelength mapping. Daubechies wavelets were used to achieve up to 32× compression while ensuring spectral fidelity and spatial feature retention. The wavelet-based method achieved up to 32× compression, corresponding to a 96.88% reduction in data size without significant loss of important data. Unlike principal component analysis and independent component analysis, the method preserved the original wavelength scale, enabling straightforward spectral interpretation. In addition, the compressed data exhibited minimal loss in spatial features, with improvements in contrast and noise reduction compared with spectral binning. We demonstrate that wavelet-based compression is an effective solution for managing large HSI datasets in medical imaging. The method preserves critical spectral and spatial information and therefore facilitates efficient data storage and processing, providing a way for the practical integration of HSI technology in clinical applications.

Perovskite‐based detectors for γ‐rays have emerged relatively recently and have attracted limited attention. Although increasing efforts are being directed towards their development, the absence of an adaptive processor, systematic theoretical framework, and various applications limit its further development. In this paper, drawing inspiration from the commercial detectors, electrode configuration is improved to reshape the bias electric field, guide the effective signal carrier, and mitigate the noise from the side section. Through theoretical analysis, simulations, and experimental comparisons, the clear improvement of energy resolution from 7% to ≈5% is demonstrated, and the highest resolution of 1.9% is observed. Furthermore, to fill up the research framework of radiation, the definition of X‐ray sensitivity is extended to quantitatively describe the γ‐ray response, and assess the performance of γ‐ray detectors, reaching a sensitivity ≈105 µC Gyair−1 cm−2. Based on these, a spectral‐enhanced imaging strategy is proposed to broaden the application of γ‐Ray detectors, where the spectral dimension of images is utilized to improve contrast and enhance imaging quality.

Abstract A two-dimensional spectrogram with oscillations along the temporal dimension and partially overlapping peaks along the spectral dimension is the typical outcome of interferometric measurements, e.g. the reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) experiment of complex systems. It is necessary to retrieve the oscillation phases of the individual components in order to extract the attosecond photoionisation time delays. One can use either the global-fit method (simulating the oscillations of each component and adding them together) or the complex-fit method (first Fourier transforming along the temporal dimension and then fitting the Fourier coefficients at the relevant frequency in the spectral dimension). Here, we prove that the two methods are mathematically equivalent in the frame of least-squares fitting, and we derive the formula for the variance of the extracted phases based on the Poisson distribution. The fitting and the uncertainty formula are not limited to specific peak shapes. For the special case of two Gaussian peaks, there is a relatively simple expression of the phase uncertainty. The method can be further extended to fitting with peaks that have known phase structures or peaks with relative phase constraints. The uncertainty formula (with multiple peaks and a background) is verified by numerical simulations, and the results show that phase retrieval is possible as long as the peaks do not fully overlap (having exactly the same shape, position and phase structure), although the uncertainty rises with the degree of overlap. We also find that the correctness of fitting relies on properly assigning all the peaks in the energy domain, which is particularly important for extracting the phase from a relatively weak peak overlapped with other peaks.