Stable Theory Research Articles

Enumerating partitions arising in homotopy theory

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory.

The springback penalty for robust signal recovery

We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex $\ell_1$ penalty and many of its non-convex surrogates that have been well studied in the literature. We establish the exact and stable recovery theory for the recovery model using the springback penalty for both sparse and nearly sparse signals, respectively, and derive an easily implementable difference-of-convex algorithm. In particular, we show its theoretical superiority to some existing models with a sharper recovery bound for some scenarios where the level of measurement noise is large or the amount of measurements is limited. We also demonstrate its numerical robustness regardless of the varying coherence of the sensing matrix. The springback penalty is particularly favorable for the scenario where the incomplete and inaccurate measurements are collected by coherence-hidden or -static sensing hardware due to its theoretical guarantee of recovery with severe measurements, computational tractability, and numerical robustness for ill-conditioned sensing matrices.

Exact saturation in pseudo-elementary classes for simple and stable theories

We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.

Suboptimal attitude tracking control law and eigenvalue analysis for a near-space hypersonic vehicle based on Koopman operator and stable manifold method

This paper proposes a novel strategy to design a suboptimal attitude tracking control law for a near-space hypersonic vehicle (NSHV) based on the Koopman operator and stable manifold theory. The nonlinear vector field of the NSHV attitude model is locally Lipschitz continuous and can be approximated by a high-dimensional linear system over a compact set. Linear and nonlinear parts of the attitude dynamics are determined based on this system. Subsequently, the stable manifold theory is applied to determine the unconstrained approximated optimal control law that is used to further consider the control input constraints of the NSHV attitude model. The suboptimality of the control law is analyzed, and the local exponential stability of the closed-loop system with input constraints is proven. Furthermore, the eigenvalues for the closed-loop nonlinear attitude error dynamics are analyzed. After the control input saturation, the nonlinear closed-loop error dynamics of the NSHV can be approximated by a high-dimensional linear system with a minimal dimension. The eigenvalues of this linear system indicate the stability and time response characteristics of the attitude error dynamics of the NSHV. The numerical simulation results demonstrate the effectiveness and suboptimality of the proposed attitude tracking control law and workflow of the eigenvalue analysis.

Motivic stable homotopy theory is strictly commutative at the characteristic

We show that mapping spaces in the p-local motivic stable category over an Fp-scheme are strictly commutative monoids (whence HZ-modules) in a canonical way.

Symmetric-hyperbolic quasihydrodynamics

We set up a general framework for systematically building and classifying, in the linear regime, causal and stable dissipative hydrodynamic theories that, alongside with the usual hydrodynamic modes, also allow for an arbitrary number of nonhydrodynamic modes with complex dispersion relation (such theories are often referred to as ``quasihydrodynamic''). To increase the number of nonhydrodynamic modes, one needs to add more effective fields to the model. The system of equations governing this class of quasihydrodynamic theories is symmetric hyperbolic, thermodynamically consistent (i.e., the entropy is a Lyapunov function) and can be derived from an action principle. As a first application of the formalism, we prove that, in the linear regime, the Israel-Stewart theory in the Eckart frame and the Israel-Stewart theory in the Landau frame are exactly the same theory. In addition, with an Onsager-Casimir analysis, we show that in strongly coupled plasmas the nonequilibrium degrees of freedom typically appear in pairs, whose members acquire opposite phase under time reversal. We use this insight to modify Cattaneo's model for diffusion, in a way to make its initial transient consistent with the transient dynamics of holographic plasmas.

Transient stable region of power systems incorporating stochasticity of variable renewable energies and system contingencies

The stochasticity of grid-connected renewable energies, such as wind farms, is challenging the deterministic engineering practices of power system operation. In this paper, a stochastic transient energy function (STEF) model based on the Lyapunov theory is proposed to evaluate the stability of the power system with doubly-fed-induction generator (DFIG). Based on the STEF, the methodology for construction of the stable region is proposed. And the stability degree of each bus under the different fault types and locations are evaluated by the bus stable nominal value (BSNV). To this end, the STEF is derived first from the operation mechanism of synchronous generators and DFIGs. Sequentially, the stable region of the power system with DFIG is constructed by using the stable region theory; the operation locations of every buses are marked on the stable region by the stochastic matrix theory. Case study based on the New England 9-Bus System with DFIG demonstrates the robustness of the proposed STEF to determine the stable region and the results show its potential applications for transient stability incorporating stochasticity of the wind farms based power grid.

Double nested Hilbert schemes and the local stable pairs theory of curves

We propose a variation of the classical Hilbert scheme of points, thedouble nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss$K$-theoretic refinements à la Nekrasov–Okounkov.

Adaptive Fuzzy Fast Finite-Time Formation Control for Second-Order MASs Based on Capability Boundaries of Agents

This article addresses a new adaptive fuzzy fast finite-time state-constraint protocol for leader-follower formation control. Each agent in uncertain nonlinear dynamic multiagent systems is represented by second-order integrator, which synchronously governs its position and velocity. The fuzzy logic systems are employed to compensate and approximate uncertain functions. On the premise of maintaining formation structure and coupling communication topology, time-varying transformation equations containing exponential signals are introduced to ensure that state capability boundaries for different physical quantities of agents are not violated. It not only guarantees own state performance and collision avoidance among agents, but also realizes the specified transient and steady formation performance. Furthermore, focusing on convergence rate, the adaptive fuzzy fast finite-time strategy is designed that can guarantee all agents will follow the desired formation configuration in fast finite-time. Through the abovementioned approaches provide a good way to improve the convergence and ensure the security for decentralized formation control. Finally, the validity of the theoretical method is proved by fast finite-time stable theory and Lyapunov stability theory. The effectiveness of the protocol is verified by digital simulation and simulation comparison.

DISTALITY RANK

AbstractBuilding on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rank m such that $1\leq m \leq \omega $ . For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called m-determinacy and show that theories of distality rank m require certain products to be m-determined. Furthermore, for NIP theories, this behavior characterizes m-distality. If we narrow the scope to stable theories, we observe that m-distality can be characterized by the maximum cycle size found in the forking “geometry,” so it coincides with $(m-1)$ -triviality. On a broader scale, we see that m-distality is a strengthening of Saharon Shelah’s notion of m-dependence.

Event-Triggered Security Consensus for Multi-Agent Systems with Markov Switching Topologies under DoS Attacks

This paper studies secure consensus control for multi-agent systems subject to denial-of-service (DoS) attacks. The DoS attacks cause changes in topologies, which will destroy the channels of communication and result in network paralysis. Unlike the existing publications with Markov switching, this paper mainly studies the topological structure changes of the subsystem models after DoS attacks. To ensure the consensus of systems, this paper designs an event triggered to reduce the use of the controller and decrease the influence of channel breaks off caused by DoS attacks. On this basis, different Lyapunov functions are established in each period of attack. Then, stochastic and Lyapunov stable theory is used to form the consensus criteria. Moreover, Zeno behavior is excluded by theoretical analysis. Finally, the simulation experiment proves the effectiveness of the proposed protocol.

Climatic legacy effects on the drought response of the Amazon rainforest.

Extreme precipitation and drought events are predicted to become more intense and more frequent over the Amazon rainforest. Because changes in forest dynamics could prompt strong feedback loops to the global climate, it is of crucial importance to gain insight into the response of tropical forests to these recurring extreme climatic events. Here, we evaluated the Amazon forest stability (resistance and resilience) to drought in the context of past dry and wet climatic events using MODIS EVI satellite imagery and cumulative water deficit anomalies. We observed large spatial differences in the occurrence of extreme climatic events from 1980 to 2019, with an increase in drought frequency in the central and northern Amazon and drought intensity in the southern Amazon basin. An increasing trend in the occurrence of wet events was found in the western, southern, and eastern Amazon. Furthermore, we found significant legacy effects of previous climatic events on the forest drought response. An extreme drought closely preceding another drought decreased forest resilience, whereas the occurrence of a recent drier-than-usual event also decreased the forest resistance to later droughts. Both wetter-than-usual and extreme wet events preceding an extreme drought increased the resistance of the forest, and with similar effects sizes as dry events, indicating that wet and dry events have similarly sized legacy effects on the drought response of tropical forests. Our results indicate that the predicted increase in drought frequency and intensity can have negative consequences for the functioning of the Amazon forest. However, more frequent wet periods in combination with these droughts could counteract their negative impact. Finally, we also found that more stable forests according to the alternative stable states theory are also more resistant and resilient to individual droughts, showing a positive relationship between the equilibrium and non-equilibrium stability dynamics.

Is first-order relativistic hydrodynamics in a general frame stable and causal for arbitrary interactions?

We derive a first-order, stable and causal, relativistic hydrodynamic theory from the microscopic kinetic equation using the gradient expansion technique in a general frame. The general frame is introduced from the arbitrary matching conditions for hydrodynamic fields. The interaction is introduced in the relativistic Boltzmann equation through the momentum-dependent relaxation time approximation (MDRTA) with the proposed collision operator that preserves the conservation laws. We demonstrate here for the first time that not only the general frame choice, but also the momentum dependence of microscopic interaction rate, captured through MDRTA, is imperative for producing the essential field corrections that give rise to a causal and stable first-order relativistic theory.

Adaptive formation control for autonomous surface vessels with prescribed-time convergence

This article is dedicated to solving the problem of predefined-time cooperative control for autonomous surface vessels encountering model uncertainties and external perturbations. By virtue of the prescribed-time stable theory, a robust formation controller is constructed, with which the settling time of the cooperative system can be prescribed in advance. The controller is developed under the backstepping framework, where the dynamic surface control is applied to generate the real-time command. Considering the unmodeled autonomous surface vessel dynamics, the neural network-based nonlinear approximator is incorporated with minimum-learning-parameter technique. Under this scenario, the real-time control can be pursed with one parameter being estimated. Finally, comparative simulation examples are provided to exhibit the effectiveness and advantages of designed control strategies.

Kim-independence in positive logic

An important dividing line in the class of unstable theories is being NSOP$_1$, which is more general than being simple. In NSOP$_1$ theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP$_1$ theories to positive logic -- a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP$_1$ and various other properties. We prove that, in a thick positive NSOP$_1$ theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in a first-order NSOP$_1$ theory. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP$_1$ by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable. In first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in first-order theories. In particular, we only work with trees of finite height.

Matching with Externalities

AbstractWe incorporate externalities into the stable matching theory of two-sided markets. Extending the classical substitutes condition to markets with externalities, we establish that stable matchings exist when agent choices satisfy substitutability. We show that substitutability is a necessary condition for the existence of a stable matching in a maximal-domain sense and provide a characterization of substitutable choice functions. In addition, we extend the standard insights of matching theory, like the existence of side-optimal stable matchings and the deferred acceptance algorithm, to settings with externalities even though the standard fixed-point techniques do not apply.

A Simplified Method for Leakage Estimation of Clay Core Dams with Different Groundwater Levels

Clay core dams are widely applied in reservoir construction, regulating water resource and provide electric power. Leakage is a common problem in reservoir construction, and the leakage amount, which not only affects the economic benefits of the project, but also relates to the safety of the dam body, is difficult to estimate. According to Darcy’s law and stable seepage theory, an analytical method can be proposed to calculate the leakage of the clay core dam to gain the seepage flux in a short time. By making some reasonable assumptions, we propose formulae for seepage calculation in different conditions of the position of the groundwater levels, below or above the reservoir bottom. Both sets of formulae contain two parts of leakage calculation, i.e., leakage from the reservoir bottom and leakage from the dam body. By using the proposed analytical method, the leakage of clay core dams can be estimated considering the influence of the groundwater level. To prove the rationality of the analytical method, a simple numerical model can be established using Geo-studio 2020 to calculate the seepage flux of the clay core dam, where relative errors between numerical solutions and analytical solutions are less than 10%. To verify the feasibility in engineering applications, the proposed method was applied to calculate the seepage of a clay core dam in Sichuan, China, which was also calculated using numerical methods by establishing a three-dimensional model. The results show the rationality of the analytical method, which can strike a balance between precision and efficiency.

η$\eta$‐Periodic motivic stable homotopy theory over Dedekind domains

We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer–Witt K $K$ -theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence, we lift the fundamental fiber sequence of η $\eta$ -periodic motivic stable homotopy theory established in Bachmann and Hopkins (2020) from fields to arbitrary base schemes, and use this to determine (among other things) the η $\eta$ -periodized algebraic symplectic and SL ${\rm SL}$ -cobordism groups of mixed characteristic Dedekind schemes containing 1 / 2 $1/2$ .

Heat transport in Rayleigh-Bénard convection with linear marginality.

Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh-Bénard convection (RBC) exhibits the classical [Formula: see text] scaling as the Rayleigh number [Formula: see text] with Prandtl number unity, consistent with Malkus-Howard's marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin-Doering-Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same [Formula: see text] scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the [Formula: see text] scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Modeling nosocomial infection of COVID-19 transmission dynamics

COVID-19 epidemic has posed an unprecedented threat to global public health. The disease has alarmed the healthcare system with the harm of nosocomial infection. Nosocomial spread of COVID-19 has been discovered and reported globally in different healthcare facilities. Asymptomatic patients and super-spreaders are sough to be among of the source of these infections. Thus, this study contributes to the subject by formulating a SEIHR mathematical model to gain the insight into nosocomial infection for COVID-19 transmission dynamics. The role of personal protective equipment θ is studied in the proposed model. Benefiting the next generation matrix method, R0 was computed. Routh–Hurwitz criterion and stable Metzler matrix theory revealed that COVID-19-free equilibrium point is locally and globally asymptotically stable whenever R0<1. Lyapunov function depicted that the endemic equilibrium point is globally asymptotically stable when R0>1. Further, the dynamics behavior of R0 was explored when varying θ. In the absence of θ, the value of R0 was 8.4584 which implies the expansion of the disease. When θ is introduced in the model, R0 was 0.4229, indicating the decrease of the disease in the community. Numerical solutions were simulated by using Runge–Kutta fourth-order method. Global sensitivity analysis is performed to present the most significant parameter. The numerical results illustrated mathematically that personal protective equipment can minimizes nosocomial infections of COVID-19.