Spectral Sequence Research Articles

The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere

AbstractWe compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields.

Breaking dimensional barriers in hyperspectral target detection: Atrous convolution with Gramian Angular field representations

Hyperspectral images contain extensive spectral bands with rich spectral information that reflects object properties. Leveraging state-of-the-art deep learning techniques has proven to be effective in hyperspectral target detection. However, compared to two-dimensional matrix data, the one-dimensional nature of spectral sequence limits the information that can be extracted, posing a challenge for deep learning-based hyperspectral target detection methodologies. To address this issue, a novel hyperspectral target detection method employing atrous convolution with gramian angular field representations is proposed in this paper. This approach breaks the barrier between one-dimensional vector and two-dimensional matrix by gramian angular field, transforming the spectral sequences from one-dimensional vectors into two-dimensional matrices, enabling the exploration of multidimensional relationships within spectral band relations through an atrous convolution-based spectral feature extraction network. The proposed model transcends the traditional one-dimensional spectral target detection limitations, offering a new perspective for spectral-based hyperspectral target detection. Experimental results on four real-world hyperspectral datasets demonstrate that the proposed method significantly outperforms existing state-of-the-art methods in detection performance, showcasing its potential for advancing hyperspectral target detection.

HTD-TS3: Weakly Supervised Hyperspectral Target Detection Based on Transformer via Spectral-Spatial Similarity.

As an advanced technique in remote sensing, hyperspectral target detection (HTD) is widely concerned in civilian and military applications. However, the limitation of prior and heterogeneous backgrounds makes HTD models sensitive to data corruption under various interference from the environment. In this article, a novel united HTD framework based on the concept of transformer is proposed to extract [HTD based on transformer via spectral-spatial similarity (HTD-TS3)] under weak supervision, which opens up more flexible ways to study HTD. For the first time, the transformer mechanism is introduced into the HTD task to extract spectral and spatial features in a unified optimization procedure. By modeling long-range dependence among spectra, it realizes spectral-spatial joint inference based on long-range context, which addresses the issues of insufficient utilization of spatial information. To provide samples for weakly supervised learning (WSL), the coarse sample selection and spectral sequence construction in an efficient way are proposed, which makes full use of limited prior information. Finally, an exponential constrained nonlinear function is adopted to acquire pixel-level prediction via combining discriminative spectral-spatial features and coarse spatial information. Experiments on real hyperspectral images (HSIs) captured by different sensors at various scenes verify the effectiveness and efficiency of HTD-TS3.

Bidirectional Mamba with Dual-Branch Feature Extraction for Hyperspectral Image Classification.

The hyperspectral image (HSI) classification task is widely used in remote sensing image analysis. The HSI classification methods based on convolutional neural networks (CNNs) have greatly improved the classification performance. However, they cannot well utilize the sequential properties of spectral features and face the challenge of increasing computational cost with the increase in network depth. To address these shortcomings, this paper proposes a novel network with a CNN-Mamba architecture, called DBMamba, which uses a bidirectional Mamba to process spectral feature sequences at a linear computational cost. In the DBMamba, principal component analysis (PCA) is first used to extract the main features of the data. Then, a dual-branch CNN structure, with the fused features from spectral-spatial features by 3D-CNN and spatial features by 2D-CNN, is used to extract shallow spectral-spatial features. Finally, a bidirectional Mamba is used to effectively capture global contextual information in features and significantly enhance the extraction of spectral features. Experimental results on the Indian Pines, Salinas, and Pavia University datasets demonstrate that the classification performance surpasses that of many cutting-edge methods, improving by 1.04%, 0.15%, and 0.09%, respectively, over the competing SSFTT method. The research in this paper enhances the existing knowledge on HSI classification and provides valuable insights for future research in this field.

The Lubin–Tate theory of configuration spaces: I

AbstractWe construct a spectral sequence converging to the Lubin–Tate theory, that is, Morava ‐theory, of unordered configuration spaces and identify its ‐page as the homology of a Chevalley–Eilenberg‐like complex for Hecke Lie algebras. Based on this, we compute the ‐theory of the weight summands of iterated loop spaces of spheres (parameterizing the weight operations on ‐algebras), as well as the ‐theory of the configuration spaces of points on a punctured surface. We read off the corresponding Morava ‐theory groups, which appear in a conjecture by Ravenel. Finally, we compute the ‐homology of the space of unordered configurations of particles on a punctured surface.

RS-Net: Hyperspectral Image Land Cover Classification Based on Spectral Imager Combined with Random Forest Algorithm

Recursive neural networks and transformers have recently become dominant in hyperspectral (HS) image classification due to their ability to capture long-range dependencies in spectral sequences. Despite the success of these sequential architectures, mainstream deep learning methods primarily handle two-dimensional structured data. However, challenges such as the curse of dimensionality, spectral variability, and confounding factors in hyperspectral remote sensing images limit their effectiveness, especially in remote sensing applications. To address this issue, this paper proposes a novel land cover classification algorithm that integrates random forests with a spectral transformer network structure (RS-Net). Firstly, this paper presents a combination of the Gramian Angular Field (GASF) and Gramian Angular Difference Field (GADF) algorithms, which effectively maps the multidimensional time series constructed for each pixel onto two-dimensional image features, enabling precise extraction and recognition in the backend network algorithms and improving the classification accuracy of land cover types. Secondly, to capture the relationships between features at different scales, this paper proposes a SpectralFormer network architecture using the Context and Structure Encoding (CASE) module to effectively learn dependencies between channels. This architecture enhances important features and suppresses unimportant ones, thereby addressing the semantic gap and improving the recognition capability of land cover features. Finally, the final prediction results are determined by a voting mechanism from the Random Forest algorithm, which synthesizes predictions from multiple decision trees to enhance classification stability and accuracy. To better compare the performance of RS-Net, this paper conducted extensive experiments on three benchmark HS datasets obtained from satellite and airborne imagers, comparing various classic neural network models. Surprisingly, the RS-Net algorithm achieves high performance and efficiency, offering a new and effective tool for land cover classification.

Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

In this paper we focus on the interplay between the behaviour of the Frölicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Frölicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Frölicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Frölicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.

The discrete flow category: structure and computation

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Cohomological and motivic inclusion–exclusion

We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

Idempotents and homology of diagram algebras

AbstractThis paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

Non-invertible symmetries and higher representation theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra \mathfrak{so}(N)𝔰𝔬(N), and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed ’t Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

Sinha’s spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad

Sinha’s spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad

Spectral prediction method based on the transformer neural network for high-fidelity color reproduction

Color distortion often occurs during transmission and reproduction processes, and existing spectral prediction methods have the disadvantage of low prediction accuracy in halftone reproduction. Addressing this issue, this paper establishes a halftone dataset composed of four-color inks (CMYK) mixtures. Based on this, the transformer network is introduced to model and characterize the spectral features of mixed inks, and a forward color formulation prediction model and a reverse spectral prediction model combining halftone reproduction with spectral sequences are proposed, namely the spectrum-color transformer (SC-Former). Color reproduction quality assessment experiments are conducted using the dataset established in this paper and the international standard Ugra/Fogra Media Wedge V3.0 test set. The experimental results show that the SC-Former model outperforms traditional physical models and data-driven prediction models in terms of color reproduction effects and spectral prediction accuracy. This research contributes to the development of high-fidelity color reproduction techniques.

SupSpec: A Python Package for Measuring the Doppler Shifts of Supernova Spectral Lines

Here we present an open-source Python package, SupSpec, which aids in finding the velocity structure of spectral features throughout supernova (SN) ejecta. This code is designed to require minimal user interaction and can ingest an arbitrarily large number of spectra. By fitting a Gaussian distribution to the spectral sequence of a single SN over time, it finds the “velocity structure”—the evolution of the velocity of a specific absorption feature over time. SupSpec is available on GitHub and on Zenodo.

A spectral sequence for spaces of maps between operads

A spectral sequence for spaces of maps between operads

The localization spectral sequence in the motivic setting

The localization spectral sequence in the motivic setting

What Bordism-Theoretic Anomaly Cancellation Can Do for U

We perform a bordism computation to show that the E7(7)(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7(7)}({\\mathbb {R}})$$\\end{document} U-duality symmetry of 4d N=8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}} = 8$$\\end{document} supergravity could have an anomaly invisible to perturbative methods; then we show that this anomaly is trivial. We compute the relevant bordism group using the Adams and Atiyah–Hirzebruch spectral sequences, and we show the anomaly vanishes by computing η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document}-invariants on the Wu manifold, which generates the bordism group.

Künneth formulas for Cotor

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the [Formula: see text]th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the [Formula: see text]-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

abstract: In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.

RO(G)-Graded Homotopy Fixed Point Spectral Sequence for Height 2 Morava E-Theory

RO(G)-Graded Homotopy Fixed Point Spectral Sequence for Height 2 Morava E-Theory