Stable Theory Research Articles

The Adams isomorphism revisited

We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more generally in the settings of G-Mackey functors, G-global homotopy theory, and equivariant Kasparov categories.

The Sure Thing Principle leads to instability

Abstract Orthodox causal decision theory is unstable. Its advice changes as you make up your mind about what you will do. Several have objected to this kind of instability and explored stable alternatives. Here, I’ll show that explorers in search of stability must part with a vestige of their homeland. There is no plausible stable decision theory which satisfies Savage’s Sure Thing Principle. So those in search of stability must learn to live without it.

Supervised machine learning for understanding and predicting the status of bistable eukaryotic plankton community in urbanized rivers

Understanding and predicting the ecological status of urbanized rivers is crucial for their restoration and management. However, the complex and nonlinear nature of ecological responses poses a challenge to the development of predictive models. Here, the study investigated and predicted the status of eukaryotic plankton communities in urbanized rivers by coupling environmental DNA metabarcoding, the alternative stable states theory, and supervised machine learning (SML) models. The results revealed two distinct states of eukaryotic plankton communities under similar environmental conditions: one state was characterized by the enrichment of a diverse phytoplankton population and the high relative abundance of protozoa, whereas the alternative state was characterized by abundant phytoplankton and fungi with an associated risk of algal blooms. Turbidity was identified as a key driver based on the SML model and Mantel test. Potential analysis demonstrated that the response pattern of eukaryotic plankton communities to turbidity was thresholds with hysteresis (Threshold1 = 17 NTU, Threshold2 = 24 NTU). A reduction in turbidity induced a regime shift in the eukaryotic plankton community toward an alternative state associated with a risk of algal blooms. In the prediction of ecological status, both SML models showed excellent performance (R2 > 0.80, RMSE < 0.1, Kappa > 0.70). Additionally, SHapley Additive exPlanations analysis identified turbidity, chlorophyll-a, chemical oxygen demand (COD), ammonia nitrogen and green algae's amplicon sequence variants as crucial features for prediction, with turbidity and COD showing a synergistic effect on ecological status. A framework was further proposed to enhance the understanding and prediction of ecological status in urbanized rivers. The obtained results of this study demonstrated the feasibility of using SML models to predict and explain the ecological status of urbanized rivers with alternative stable states. This provides valuable insights for the application of SML models in the restoration and management of urbanized rivers.

On the Auslander–Bridger–Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ⁎torsionfreeness and ⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

Adaptive fixed‐time fault‐tolerant fuzzy control of AUVs with asymmetric output constraints

AbstractThe fast and safe formation tracking problem for multiple autonomous underwater vehicle (AUV) systems (MAUVS) with asymmetric output constraints and actuator faults is studied in this article. Actuator faults are composed of loss of effectiveness and time‐varying unknown bias faults, an adaptive fixed‐time fault‐tolerant controller (AFFTC) is designed by employing fixed‐time stable theory and fuzzy logic systems. Under the designed control algorithm, the MAUVS can be practically fixed‐time stable, the tracking errors among the follower AUVs and the virtual leader AUV can converge to a small area near the origin within the fixed time, and the settling time is unaffected by the initial state of the system. To enhance the safety of MAUVS, a novel asymmetric barrier function is utilized to constrain the trajectory tracking errors within the prescribed range. Finally, the simulation results demonstrate the effectiveness of the proposed control algorithm.

Causality and stability in relativistic hydrodynamic theory - a choice to be endured

In this work, it has been indicated that the key features requisite for preserving causality and stability of the popularly existing relativistic hydrodynamic theories, can be translated into each other. It has been shown here, that a generic ‘fluid frame transformation’ including all orders of gradient corrections can recast a stable-causal hydrodynamic theory that (i) only includes fundamental fluid variables (velocity and temperature) but requires to be in a general hydrodynamic frame other than the Landau or Eckart, to a theory that is (ii) pathology free in Landau frame but needs newer degrees of freedom. Since frame choice provides the first principle field definitions and degrees of freedom indicate the number of conserved quantities, the causality and stability of a theory seem to require a consensus between the two unless the derivative correction goes to infinity. The key finding of this work is to indicate a connection between these different formalisms, that lead to a causal and stable hydrodynamic theory.

On metric approximate subgroups

Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset [Formula: see text] of [Formula: see text] such that the pairwise product set [Formula: see text] is covered by at most [Formula: see text] translates of [Formula: see text]. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of [Formula: see text] finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on [Formula: see text] replacing finiteness. In particular, if [Formula: see text] has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].

Hysteretic and asynchronous regime shifts of bacterial and micro-eukaryotic communities driven by nutrient loading

Nutrient pollution is pervasive in many urban rivers, while restoration measures that reduce nutrient loading but fail to improve biological communities often lack effectiveness due to the indispensable role of biota, especially multi-taxa, in enhancing ecosystem stability and function. The investigation of the response patterns of multi-taxa to the nutrient loading in urban rivers is important for the recovery of biota structure and thus ecosystem function. However, little is known about the response patterns of multi-taxa and their impact on ecosystem structure and function in urban rivers. Here, the study, from the perspective of alternative stable states theory, showed the hysteretic response of both bacterial and micro-eukaryotic communities to nutrient loading based on the field investigation and environmental DNA metabarcoding. Bistability was shown to exist in both bacterial and micro-eukaryotic communities, demonstrating that the response of microbiota to nutrient loading was a regime shifts with hysteresis. Potential analysis then indicated that the increased nutrient loading drove regime shifts in the bacterial community and the micro-eukaryotic community towards a state dominated by anaerobic bacteria and benthic Bacillariophyta, respectively. High nutrient loading was found to reduce the relative abundance of metazoan, but increase that of eukaryotic algae, which made the trophic pyramid top-lighter and bottom-heavier, probably exacerbating the degradation of ecosystem function. It should be noted that, in response to the reduced nutrient loading, the recovery threshold of micro-eukaryotic communities (nutrient loading = ∼0.5) was lower than that of bacterial communities (nutrient loading = ∼1.2), demonstrating longer hysteresis of micro-eukaryotic communities. In addition, the markedly positive correlation between the status of microbial communities and N-related enzyme activities suggested the recovery of microbial communities probably will benefit the improvement of N-cycling functionality. The obtained results provide a deep insight into the collapse and recovery trajectories of multi-trophic microbiota to the nutrient loading gradient and their impact on the N transformation potential, therefore benefiting the restoration and management of urban rivers.

The interplay of facilitation and competition drives the emergence of multistability in dryland plant communities.

Within communities, species are wrapped in a set of feedbacks with each other and with their environment. When such feedbacks are strong enough they can generate alternative stable states. So far, research on alternative stable states has mostly focused on systems with a small number of species and a limited diversity of interaction types. Here, we analyze a spatial model of plant community dynamics in stressed ecosystems such as drylands, where each species is characterized by a strategy, and the different species interact through facilitation and competition for space and resources, such as water. We identify three different types of multistability emerging from the interplay of competition and facilitation. Under low-stress levels, plant communities organize in small groups of coexisting species, maintained by space, competition and facilitation ("cliques"). Under higher stress levels, positive feedback from facilitation lead to the dominance of a single facilitating species ("mutual exclusion states"). At the highest stress levels, the single facilitating species left in the system coexists with the desert state. By linking community ecology and alternative stable states theory using a spatial plant community model for stressed ecosystems, our study contributes to highlight the importance of positive feedback loops for the stability of ecological communities.

The Tarski–Lindenbaum algebra of the class of strongly constructivizable models with $$\omega $$-stable theories

The Tarski–Lindenbaum algebra of the class of strongly constructivizable models with $$\omega $$-stable theories

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 $H&lt;G$ 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.

Causal third-order viscous hydrodynamics within relaxation-time approximation

In the present work, we derive a linearly stable and causal theory of relativistic third-order viscous hydrodynamics from the Boltzmann equation with relaxation-time approximation. We employ viscous correction to the distribution function obtained using a Chapman-Enskog like iterative solution of the Boltzmann equation. Our derivation highlights the necessity of incorporating a new dynamical degree of freedom, specifically an irreducible tensor of rank three, within this framework. This differs from the recent formulation of causal third-order theory from the method of moments which requires two dynamical degrees of freedom: an irreducible third-rank and a fourth-rank tensor. We verify the linear stability and causality of the proposed formulation by examining perturbations around a global equilibrium state. Published by the American Physical Society 2024

Stratified Noncommutative Geometry

We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable ∞ \infty -categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as E n \mathbb {E}_n -monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group G G , we give a symmetric monoidal stratification of genuine G G -spectra. In the case that G G is finite, this expresses genuine G G -spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory.

Some semilattices of definable sets in continuous logic

In continuous first-order logic, the union of definable sets is definable but generally the intersection is not. This means that in any continuous theory, the collection of $\varnothing$-definable sets in one variable forms a join-semilattice under inclusion that may fail to be a lattice. We investigate the question of which semilattices arise as the collection of definable sets in a continuous theory. We show that for any non-trivial finite semilattice $L$ (or, equivalently, any finite lattice $L$), there is a superstable theory $T$ whose semilattice of definable sets is $L$. We then extend this construction to some infinite semilattices. In particular, we show that the following semilattices arise in continuous theories: $\alpha+1$ and $(\alpha+1)^\ast$ for any ordinal $\alpha$, a semilattice containing an exact pair above $\omega$, and the lattice of filters in $L$ for any countable meet-semilattice $L$. By previous work of the author, this establishes that these semilattices arise in stable theories. The first two are done in languages of cardinality $\aleph_0 + |\alpha|$, and the latter two in countable languages.

A novel multi-criteria conflict evidence combination method and its application to pattern recognition

In recent years, the Dempster–Shafer Theory (DST) has been widely applied in areas such as target classification and multi-modal fusion due to its advantages in uncertain reasoning. However, in DST, when there exists highly conflicts between Sources of Evidence (SoEs), it often leads to counterintuitive fusion results, thereby affecting the performance of the final fused decision-making. To eliminate the potential impact of highly conflicts during the fusion process, this paper primarily focuses on modifying the SoEs themselves to achieve the discounting fusion. Currently, existing literature has proposed numerous discounting fusion methods to address highly conflict fusion. However, in these discounting strategies, the determination of discounting factors is typically based on a single criterion. Considering that a single indicator cannot comprehensively and accurately assess the reliability of each evidence source, this paper introduces a novel conflict evidence combination method based on a multi-criteria evaluation strategy. In this method, the Best and Worst Method (BWM) is initially used to prioritize the best SoE, determining the solution points for each criterion and subsequently selecting the ideal solution. Then, combining the selected criteria: distance, relative entropy, and divergence metrics, reliability calculation is performed based on the Stable Preference Order Theory for Ideal Solution (SPOTIS), which achieves a stable preference order strategy towards the ideal solutions. Finally, leveraging the discounting fusion method in DST, modification and fusion of the original highly conflict SoEs are achieved. Through extensive experiments, the effectiveness and practicality of the proposed method in this paper have been validated.

Theories of Relativistic Dissipative Fluid Dynamics.

Relativistic dissipative fluid dynamics finds widespread applications in high-energy nuclear physics and astrophysics. However, formulating a causal and stable theory of relativistic dissipative fluid dynamics is far from trivial; efforts to accomplish this reach back more than 50 years. In this review, we give an overview of the field and attempt a comparative assessment of (at least most of) the theories for relativistic dissipative fluid dynamics proposed until today and used in applications.

Schwinger-Keldysh effective field theory for stable and causal relativistic hydrodynamics

We construct stable and causal effective field theories (EFTs) for describing statistical fluctuations in relativistic diffusion and relativistic hydrodynamics. These EFTs are fully non-linear, including couplings to background sources, and enable us to compute n-point time-ordered correlation functions including the effects of statistical fluctuations. The EFTs we construct are inspired by the Maxwell-Cattaneo model of relativistic diffusion and Müller-Israel-Stewart model of relativistic hydrodynamics respectively, and have been derived using both the Martin-Siggia-Rose and Schwinger-Keldysh formalisms. The EFTs non-linearly realise the dynamical Kubo-Martin-Schwinger (KMS) symmetry, which ensures that n-point correlation functions and interactions in the theory satisfy the appropriate fluctuation-dissipation theorems. Since these EFTs typically admit ultraviolet sectors that are not fixed by the low-energy infrared symmetries, we find that they simultaneously admit multiple realisations of the dynamical KMS symmetry. We also comment on certain obstructions to including statistical fluctuations in the recently-proposed stable and causal Bemfica-Disconzi-Noronha-Kovtun model of relativistic hydrodynamics.

Comparative analysis on heat and moisture transfer of composite insulation walls

Abstract Firstly, the influence of heat and moisture transfer on composite insulation walls was initially considered. Relying on the established theory of heat and moisture coupled transfer in walls and taking into account the specific climate conditions of regions with hot summers and cold winters, a non-steady-state heat and moisture coupled model was developed to simulate the processes of heat and moisture transfer. Subsequently, based on the one-dimensional stable heat transfer theory, the heat transfer situation for winter insulation was calculated. Then, practical methods for on-site temperature and humidity measurements were explored and implemented. Through analysis, comparison, and mutual verification of the calculated values and the measured values, it was observed that both sets of data showed consistent trends, thereby verifying their accuracy and correlation. The findings of this study provide valuable reference data for understanding the heat and moisture migration within building walls and offer solutions for optimizing thermal insulation strategies.

Cohomology of the Morava stabilizer group through the duality resolution at 𝑛=𝑝=2

We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava E E -theory, H ∗ ( G 2 , E t ) H^*(\mathbb {G}_2, \mathbf {E}_t) , at p = 2 p=2 , for 0 ≤ t &gt; 12 0\leq t &gt; 12 , using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the d 3 d_3 -differentials in the homotopy fixed point spectral sequence for the K ( 2 ) K(2) -local sphere spectrum. These cohomology groups and differentials play a central role in K ( 2 ) K(2) -local stable homotopy theory.

Generic Stability Independence and Treeless Theories

Abstract We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP $_{3}$ theory is simple.