Spectral Sequence Research Articles

Dimensional Reduction of Cobordism and K-theory

It has been proposed that cobordism and K-theory groups, which can be mathematically related in certain cases, are physically associated to generalised higher-form symmetries. As a consequence, they should be broken or gauged in any consistent theory of quantum gravity, in accordance with swampland conjectures. We provide further support to this idea by showing that cobordism and K-theory groups of a general manifold X reproduce the pattern of symmetries expected from the dimensional reduction of the theory on X, as well as their breaking and gauging. To this end, we employ the Atiyah-Hirzebruch spectral sequence to compute such groups for common choices of X in string compactifications.

Some refinements of the Deligne–Illusie theorem

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the de Rham complex in $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo $p^2$). Consequently, these truncations are decomposable if $X$ admits a lifting to $W_2(k)$, in which case the first nonzero differential in the conjugate spectral sequence appears no earlier than on page $\max(p,3)$ (these corollaries have been recently strengthened by Drinfeld, Bhatt-Lurie, and Li-Mondal). Without assuming the existence of a lifting, we describe the gerbes of splittings of two-term truncations and the differentials on the second page of the conjugate spectral sequence, answering a question of Katz. The main technical result used in the case $p>2$ belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us "abstract Koszul complexes", of which the de Rham complex in characteristic $p$ is an example. In the appendix, we use the aforementioned stronger decomposition result to prove that Kodaira-Akizuki-Nakano vanishing and Hodge-de Rham degeneration both hold for $F$-split $(p+1)$-folds.

Zebra finches (Taeniopygia guttata) demonstrate cognitive flexibility in using phonology and sequence of syllables in auditory discrimination

Zebra finches rely mainly on syllable phonology rather than on syllable sequence when they discriminate between two songs. However, they can also learn to discriminate two strings containing the same set of syllables by their sequence. How learning about the phonological characteristics of syllables and their sequence relate to each other and to the composition of the stimuli is still an open question. We compared whether and how the zebra finches’ relative sensitivity for syllable phonology and syllable sequence depends on the differences between syllable strings. Two groups of zebra finches were trained in a Go-Left/Go-Right task to discriminate either between two strings in which each string contained a unique set of song syllables (‘Different-syllables group’) or two strings in which both strings contained the same set of syllables, but in a different sequential order (‘Same-syllables group’). We assessed to what extent the birds in the two experimental groups attend to the spectral characteristics and the sequence of the syllables by measuring the responses to test strings consisting of spectral modifications or sequence changes. Our results showed no difference in the number of trials needed to discriminate strings consisting of either different or identical sets of syllables. Both experimental groups attended to changes in spectral features in a similar way, but the group for which both training strings consisted of the same set of syllables responded more strongly to changes in sequence than the group for which the training strings consisted of different sets of syllables. This outcome suggests the presence of an additional learning process to learn about syllable sequence when learning about syllable phonology is not sufficient to discriminate two strings. Our study thus demonstrates that the relative importance of syllable phonology and sequence depends on how these features vary among stimuli. This indicates cognitive flexibility in the acoustic features that songbirds might use in their song recognition.

On BV supermanifolds and the super Atiyah class

We study global and local geometry of forms on odd symplectic BV supermanifolds, constructed from the total space of the bundle of 1-forms on a base supermanifold. We show that globally 1-forms are an extension of vector bundles defined on the base supermanifold. In the holomorphic category, we prove that this extension is split if and only if the super Atiyah class of the base supermanifold vanishes. This is equivalent to the existence of a holomorphic superconnection: we show how this condition is related to the characteristic non-split geometry of complex supermanifolds. From a local point of view, we prove that the deformed de Rham double complex naturally arises as a de-quantization of the de Rham/Spencer double complex of the base supermanifold. Following Ševera, we show that the associated spectral sequence yields semidensities on the BV supermanifold, together with their differential in the form of a super BV Laplacian.

A new family in the stable homotopy groups of spheres

In this paper, the nontriviality of homotopy elements β 1 ϱ s \beta _1\varrho _s in the p p -primary component of stable homotopy groups of spheres is shown by virtue of the Adams spectral sequence and the May spectral sequence, where ϱ s \varrho _s is represented by b 0 g 0 γ ~ s b_0g_{0}\widetilde {\gamma }_{s} in the Adams spectral sequence, p p is a prime number greater than 5 5 , and 3 ≤ s > p 3\leq s>p .

The Toral Rank Conjecture and variants of equivariant formality

An action of a compact Lie group is called equivariantly formal, if the Leray–Serre spectral sequence of its Borel fibration degenerates at the E2-term. This term is as prominent as it is restrictive.In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: “MOD-formal actions” and “actions of formal core”.We investigate and characterize these new terms in many different ways involving various tools from rational homotopy theory, Hirsch–Brown models, A∞-algebras, etc., and, in particular, we provide different applications ranging from actions on symplectic manifolds and rationally elliptic spaces to manifolds of non-negative sectional curvature.A major motivation for the new definitions was that an almost free action of a torus Tn↷X possessing any of the two new properties satisfies the toral rank conjecture, i.e. dim⁡H⁎(X;Q)≥2n. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces.

Semantic and spatial‐spectral feature fusion transformer network for the classification of hyperspectral image

AbstractRecently, transformer‐based networks have been introduced for the classification of hyperspectral image (HSI). Although transformer‐based methods can well capture spectral sequence information, their ability to fuse different types of information contained in HSI is still insufficient. To exploit rich spectral, spatial and semantic information in HSI, a novel semantic and spatial‐spectral feature fusion transformer (S3FFT) network is proposed in this study. In the proposed S3FFT method, spatial attention and efficient channel attention (ECA) modules are employed for the extraction of shallow spatial‐spectral features. Then, a transformer‐based module is designed to extract advanced fused features and to produce the pseudo‐label and class probability of each pixel for semantic feature extraction. Finally, the semantic, spatial and spectral features are combined by the transformer for classification. Compared with traditional deep learning methods and recently transformer‐based methods, the proposed S3FFT shows relatively better results on three HSI datasets.

HyFormer: Hybrid Transformer and CNN for Pixel-Level Multispectral Image Land Cover Classification.

To effectively solve the problems that most convolutional neural networks cannot be applied to the pixelwise input in remote sensing (RS) classification and cannot adequately represent the spectral sequence information, we propose a new multispectral RS image classification framework called HyFormer based on Transformer. First, a network framework combining a fully connected layer (FC) and convolutional neural network (CNN) is designed, and the 1D pixelwise spectral sequences obtained from the fully connected layers are reshaped into a 3D spectral feature matrix for the input of CNN, which enhances the dimensionality of the features through FC as well as increasing the feature expressiveness, and can solve the problem that 2D CNN cannot achieve pixel-level classification. Secondly, the features of the three levels of CNN are extracted and combined with the linearly transformed spectral information to enhance the information expression capability, and also used as the input of the transformer encoder to improve the features of CNN using the powerful global modelling capability of the Transformer, and finally the skip connection of the adjacent encoders to enhance the fusion between different levels of information. The pixel classification results are obtained by MLP Head. In this paper, we mainly focus on the feature distribution in the eastern part of Changxing County and the central part of Nanxun District, Zhejiang Province, and conduct experiments based on Sentinel-2 multispectral RS images. The experimental results show that the overall accuracy of HyFormer for the study area classification in Changxing County is 95.37% and that of Transformer (ViT) is 94.15%. The experimental results show that the overall accuracy of HyFormer for the study area classification in Nanxun District is 95.4% and that of Transformer (ViT) is 94.69%, and the performance of HyFormer on the Sentinel-2 dataset is better than that of the Transformer.

Annular Khovanov homology and meridional disks

We exhibit infinite families of annular links for which the maximum nonzero annular Khovanov grading grows infinitely large but the maximum nonzero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the decategorified level for some of the infinite families. Our computations provide further evidence for the wrapping conjecture of Hoste–Przytycki and its categorified analogue. Additionally, we show that certain satellite operations be used to construct counterexamples to the categorified wrapping conjecture. We also extend the Batson–Seed link splitting spectral sequence to the setting of annular Khovanov homology.

Small Sample Coherent DOA Estimation Method Based on S2S Neural Network Meta Reinforcement Learning.

Aiming at the existing Direction of Arrival (DOA) methods based on neural network, a large number of samples are required to achieve signal-scene adaptation and accurate angle estimation. In the coherent signal environment, the problems of a larger amount of training sample data are required. In this paper, the DOA of coherent signal is converted into the DOA parameter estimation of the angle interval of incident signal. The accurate estimation of coherent DOA under the condition of small samples based on meta-reinforcement learning (MRL) is realized. The meta-reinforcement learning method in this paper models the process of angle interval estimation of coherent signals as a Markov decision process. In the inner loop layer, the sequence to sequence (S2S) neural network is used to express the angular interval feature sequence of the incident signal DOA. The strategy learning of the existence of angle interval under small samples is realized through making full use of the context relevance of spatial spectral sequence through S2S neural network. Thus, according to the optimal strategy, the output sequence is sequentially determined to give the angle interval of the incident signal. Finally, DOA is obtained through one-dimensional spectral peak search according to the angle interval obtained. The experiment shows that the meta-reinforcement learning algorithm based on S2S neural network can quickly converge to the optimal state by only updating the gradient of S2S neural network parameters with a small sample set when a new signal environment appears.

FRÖLICHER SPECTRAL SEQUENCE AND HODGE STRUCTURES ON THE COHOMOLOGY OF COMPLEX PARALLELISABLE MANIFOLDS

For complex parallelisable manifolds Γ\\G, with G a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.

(Co)homology of crossed products by weak Hopf algebras

We obtain a mixed complex simpler than the canonical one that computes the cyclic type homologies of a crossed product with invertible cocycle A×ρfH, of a weak module algebra A by a weak Hopf algebra H. This complex is endowed with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in [12]. When f takes its values in a separable subalgebra of A that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.

Morse inequalities for the Koszul complex of multi-persistence

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.

HOMOLOGY THEORY OF MULTIPLICATIVE HOM-LIE ALGEBRAS

In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abe

Adaptively Dictionary Construction for Hyperspectral Target Detection

The task of hyperspectral images (HSIs) target detection is to identify whether the target spectral sequences present in the HSI. Recently, the topic of representation models has received much interest in hyperspectral target detection. The performance of representation models depends on whether the corresponding dictionary and sparse matrix can correctly recover the original spectrum. Therefore, the background dictionary of these models should contain the spectra of all classes except the target spectrum, i.e., the dictionary should be over-complete. However, most representation models cannot satisfy this condition. Moreover, due to the potentially large spectral similarity between the target and the background, representation models perform poorly in background suppression. Aiming to solve these issues, a novel adaptively dictionary constructing (ADC) strategy with background suppression sparse representation (BSSR) module is proposed in this letter, called adaptively dictionary construction for target detection (ADCTD). Specifically, the proposed ADC is adopted to segment the HSI into superpixels consisting of pixels with similar spectrums. This process can be considered as an unsupervised coarse classification process, which can construct an over-complete background dictionary. In addition, the BSSR is adopted to improve the separation of the target and background by a linear function. Experiments on three datasets demonstrate the superiority of the proposed ADCTD.

Exploiting Discriminative Advantage of Spectrum for Hyperspectral Image Classification: SpectralFormer Enhanced by Spectrum Motion Feature

As for hyperspectral images (HSIs), the discrepancy of contiguous spectral information should be the main basis for the identification of ground objects. Due to the difficulty of spectral sequence coding and the spectrum similarity between categories, successful deep-learning-based classification methods always attempt to capture the spatial information to improve the accuracy by convolutional neural networks (CNNs) or other excellent spatial feature extractors. However, extracting spatial features is generally accompanied by the distortion of ground objects distribution and categories boundary. To effectively represent spectral features, the SpectralFormer based on transformer backbone can better capture the long-term dependence of the spectrum, which improves the performance of spectral feature methods significantly. However, it is still unable to compete with advanced spectral–spatial feature methods. To exploit the discriminative advantage of the spectrum fully, this letter introduces an efficient sparse-to-dense optical flow estimation method to track the spectrum variation in the HSI. Then, we take such a variation as a spectrum motion feature to enhance the original spectrum. At last, we continue to use the SpectralFormer to encode the concatenated spectrum sequence for classification. Extensive experiments show that the SpectralFormer enhanced by the spectrum motion feature (SF-SMF) significantly improves the performance of spectral feature methods, even surpassing advanced spectral–spatial feature methods. SF-SMF can avoid interference with additional spatial information to obtain exquisite whole-domain classification maps, showing its practical value. The codes will be public at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/sssssyf/SF-SMF</uri> .

Global–Local 3-D Convolutional Transformer Network for Hyperspectral Image Classification

Benefiting from powerful feature extraction capabilities, convolutional neural networks (CNNs) have gained prominence in hyperspectral image (HSI) classification. Nevertheless, with restricted receptive fields of convolution kernels, CNN-based methods fail to learn complex characteristics of long-range sequences. Meanwhile, vision transformer allows us to learn long-range dependencies in a global view, but local region features are ignored. To overcome these limitations, we propose a novel method entitled global-local three-dimensional convolutional transformer network (GTCT), where 3-D convolution is embedded in a dual-branch transformer to simultaneously capture global-local associations in both spectral and spatial domains. In particular, the global-local spectral convolutional transformer (GECT) is designed to exploit global spectral sequence signatures and local spectral relationships between bands. Symmetrically, the global-local spatial convolutional transformer (GACT) is devised to exploit local spatial context features and global interactions among different pixels. In addition, multiscale global-local spectral-spatial information is adaptively fused with trainable weights by the weighted multiscale spectral-spatial feature interaction (WMSFI) module. It is worth noting that a spectral-spatial global attention mechanism (SSGAM) is incorporated into multi-head convolutional attention to further integrate discriminative spectral-spatial information. Extensive experiments on four HSI datasets, including GF-5 and ZY1-02D satellite hyperspectral images, demonstrate the superiority of the proposed GTCT method over other state-of-the-art algorithms with fewer parameters and lower floating-point operations (FLOPs) in practical applications.

Dual Self-Attention Swin Transformer for Hyperspectral Image Super-Resolution

Spatial resolution is a crucial indicator for measuring the quality of hyperspectral imaging (HSI) and obtaining high-resolution (HR) hyperspectral images without any auxiliary information has become increasingly challenging. One promising approach is to use deep-learning (DL) techniques to reconstruct HR hyperspectral images from low-resolution (LR) images, namely super-resolution (SR). While convolutional neural networks are commonly used for hyperspectral image SR (HSI-SR), they often lead to unavoidable performance degradation due to the lack of long-range dependence learning ability. In this article, we propose a dual self-attention Swin transformer SR (DSSTSR) network that utilizes the ability of the shifted windows (Swin) transformer in the spatial representation of both global and local features and learns spectral sequence information from adjacent bands of HSI. Additionally, DSSTSR incorporates an image denoising module using the wavelet transformation method to mitigate the impact of stripe noise on HSI-SR. Our extensive experiments using publicly close-range datasets demonstrate that DSSTSR outperforms other state-of-art HSI-SR methods in terms of three image quality metrics. Furthermore, we applied DSSTSR to the SR of satellite hyperspectral images and achieved improved classification results. Compared to its competitors, DSSTSR exhibits superior performance in enhancing spatial resolution while preserving spectral information. These results suggest that the DSSTSR network has great potential for standardization in remote-sensing image processing and practical applications.

Fiber integration of gerbes and Deligne line bundles

Let $\pi: X \to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne's line bundle $\langle{L,M}\rangle$ can be obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in a previous work by the authors via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $\pi$. Our construction provides a full account of the biadditivity properties of $\langle {L,M}\rangle$. The functorial description of the low degree maps in the Leray spectral sequence for $\pi$ we develop are of independent interest, and along the course we provide an example of their application to the Brauer group.

Spectral Token Guidance Transformer for Multisource Images Change Detection

With the development of Earth observation technology, more multisource remote sensing (RS) images are obtained from various satellite sensors and significantly enrich the data source of change detection (CD). However, the utilization of multisource bitemporal images frequently introduces challenges during featuring or representing the various physical mechanisms of the observed landscapes and makes it more difficult to develop a general model for homogeneous and heterogeneous CD adaptively. In this article, we propose an adaptive spatial-spectral transformer change detection network based on spectral token guidance, named STCD-Former. Specifically, a spectral transformer with dual-branch first encodes the diverse spectral sequence in spectral-wise to generate a corresponding spectral token. And then, the spectral token is used as guidance to interact with the patch token to learn the change rules. More significantly, to optimize the learning of difference information, we design a difference amplification module to highlight discriminative features by adaptively integrating the difference information into the feature embedding. Finally, the binary CD result is obtained by multilayer perceptron (MLP). The experimental results on three homogeneous datasets and one heterogeneous dataset have demonstrated that the proposed STCD-Former outperforms the other state-of-the-art methods qualitatively and visually.