The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$-free graph $G$ is close to the extremal graphs for $H$, then $G$ has a well-defined structure but may be far away from the extremal graphs for $H$. As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.

This paper examines the aeroelastic stability of uniform flexible wings imperfectly supported at one end and free at the other. Real-world aircraft wings inevitably exhibit imperfections, including non-ideal end supports. This work is motivated by the critical need to fundamentally understand how end-support imperfections influence the aeroelastic behaviour of fixed wings. The equations of motion are obtained via the extended Hamilton’s principle. The bending-torsional dynamics of the wing is approximated using the Euler-Bernoulli beam theory. The aerodynamic lift and pitching moment are modelled using the unsteady aerodynamics for the arbitrary motion of thin aerofoils in the time domain, extended by the strip flow theory. The imperfect support is modelled via rotational springs (with linear stiffness) for both bending and torsional degrees of freedom. The Galerkin method is used for the spatial discretisation. The stability analysis is performed by solving the resulting eigenvalue problem, and the numerical results are presented in Argand diagrams. The numerical results presented in this study are novel and offer great insights. It is demonstrated that support imperfections can substantially influence the critical flow velocity for both flutter and divergence, as well as alter the sequence of instabilities and the unstable mode. The extent of these effects directly depends on the magnitude of the imperfections. Interestingly–and counterintuitively–in certain cases, a reduction in the flutter speed is observed as the imperfections decrease.

Food insecurity affects the health of college-aged individuals, but its impact on the gut microbiome (GM) over time is poorly understood. This study explored the association between food insecurity and the GM in eighty-five college students, identifying microbial taxa, metabolites and pathways linked to food security status and examining GM stability and microbe–metabolite interactions. Longitudinal GM and metabolomic data were collected from first-year students over an academic year, encompassing periods of variable food security status. Participants were categorised into three groups: food insecure (FI, n 13), food secure (FS, n 44) and variable (VAR, n 28) status. GM composition varied significantly between FS classifications (Bray–Curtis dissimilarity, P ≤ 0·005). Stability analysis revealed correlations between stability scores and microbial features, pathways and metabolites. Specific microbes (e.g. Bifidobacterium species, Faecalibacterium prausnitizii D and Lachnospiraceae), pathways (energy and microbial turnover) and metabolites (cadaverine, N-acetylcadaverine, putrescine, testosterone sulfate and creatine) associated with FI status were identified. Multi-omic integration revealed metabolic pathways influenced by differentially abundant microbial species and co-occurring fecal metabolites in FI participants related to the microbial production of polyamines, detoxification and energy metabolism. The transition from FS to FI showed no significant differences at specific taxonomic, functional or metabolite levels. This study uncovers complex interactions between food security, GM composition and metabolism. Significant differences were found in microbial community variability and metabolic pathways associated with food security status, but the transition from food security to insecurity disrupted the GM without clear taxonomic or functional distinctions, emphasising the need for further research into these mechanisms.

Cable-driven parallel robots (CDPRs) have been widely used as motion executers for their large workspace and lower inertia. However, there are few studies on structural optimization design considering its stability. This paper proposes a stability optimization method based on force-position workspace for a reconfigurable cable-driven parallel robot (RCDPR). First, the structural optimization analysis of RCDPR is carried out. Then, the forces distribution algorithm based on the feasibility of real-time control is determined, and the boundary contour algorithm (BCA) of the RCDPR force feasible workspace (FFW) on the central plane is proposed. Second, the stiffness and cables driving force space (CFS) models of RCDPR are established. Subsequently, the stability evaluation function is established to optimize the structure of RCDPR, which uses FFW and main task feasible workspace (MFW) as carriers and stiffness and CFS as weights. Finally, an experimental prototype of the developed robot is constructed, and motion performance and workspace verification experiments are conducted. The results demonstrate that the developed RCDPR has good motion accuracy and stable workspace, and the results also verify the feasibility of the stability evaluation function and BCA.

A real variety whose real locus achieves the Smith–Thom equality is called maximal. This paper introduces new constructions of maximal real varieties, by using moduli spaces of geometric objects. We establish the maximality of the following real varieties:

• – moduli spaces of stable vector bundles of coprime rank and degree over a maximal real curve (recovering Brugallé–Schaffhauser’s theorem with a short new proof), which extends to moduli spaces of parabolic vector bundles;

• – moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal real curve, providing maximal hyper-Kähler manifolds in every even dimension;

• – if a real variety has maximal Hilbert square, then the variety and its Hilbert cube are maximal, which happens for all maximal real cubic 3-folds, but never for maximal real cubic 4-folds;

• – punctual Hilbert schemes on a maximal real surface with vanishing first $\mathbb {F}_2$-Betti number and connected real locus, such as $\mathbb {R}$-rational maximal real surfaces and some generalized Dolgachev surfaces;

• – moduli spaces of stable sheaves on an $\mathbb {R}$-rational maximal Poisson surface (e.g. the real projective plane).

Measurements of the bunch arrival times at the European X-ray free-electron laser show noise contributions in the spectral range between 0.05 and 0.5 Hz with peak-to-peak jitter of up to 25 fs. Correlation with distributed acoustic sensing measurements confirms the seismic origin. The seismic noise in this frequency band is known to be ocean-generated microseism. Both primary and secondary ocean-generated microseisms were identified using seismometers and a numerical ocean wave model. Whereas secondary microseism has a strong impact on the bunch arrival time, primary microseism has no notable effect. Rayleigh waves cause the effect, while Love waves have minimal impact. In the presented cases, the noise originates from the North Atlantic and/or the North Sea. The amplitude of the noise depends on the local weather conditions and is much stronger in winter. Ocean-generated microseism is a significant bottleneck that must be addressed to achieve femtosecond bunch arrival time stability in the sub-Hz regime.

This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

In a series of laboratory experiments, we explore the impact of different market features (the level of information, search costs, and the level of commitment) on agents’ behavior and on the outcome of decentralized matching markets. In our experiments, subjects on each side of the market actively search for a partner, make proposals, and are free to accept or reject any proposal received at any time throughout the game. Our results suggest that a low information level does not affect the stability or the efficiency of the final outcome, although it boosts market activity, unless coupled with search costs. Search costs have a significant negative impact on stability and on market activity. Finally, commitment harms stability slightly but acts as a disciplinary device to market activity and is associated with higher efficiency levels of the final outcome.

In this paper, we use experimental data to study players’ stability in normal-form games where subjects have to report beliefs and choose actions. Subjects saw each of 12 games four times in a regular or isomorphic form spread over two days without feedback. We document a high degree of stability within the same (strategically equivalent) game, although time and changes in the presentation of the game do lead to less stability. To look at stability across different games, we adopt the level-k theory, and show that stability of both beliefs and actions is significantly lower. Finally, we estimate a structural model in which players either apply a consistent level of reasoning across strategically different games, or reasoning levels change from game to game. Our results show that approximately 23% of subjects apply a consistent level of reasoning across the 12 games, but that they assign a low level of sophistication to their opponent. The remaining 77% apply different levels of reasoning to different games. We show that this may be due to subjects being attracted to the action with the highest possible payoff.

For a nondegenerate r-graph F, large n, and t in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on F, we present a general approach for determining the maximum number of edges in an n-vertex r-graph that does not contain $t+1$ vertex-disjoint copies of F. In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions.

Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, the expanded triangles, and hypergraph books. Our results extend old results of Erdős [13], Simonovits [76], and Moon [58] on complete graphs, and can be viewed as a step toward a general density version of the classical Corrádi–Hajnal [10] and Hajnal–Szemerédi [32] theorems.

Our method relies on a novel understanding of the general properties of nondegenerate Turán problems, which we refer to as smoothness and boundedness. These properties are satisfied by a broad class of nondegenerate hypergraphs and appear to be worthy of future exploration.

We experimentally investigate in the laboratory prominent mechanisms that are employed in school choice programs to assign students to public schools and study how individual behavior is influenced by preference intensities and risk aversion. Our main results show that (a) the Gale–Shapley mechanism is more robust to changes in cardinal preferences than the Boston mechanism independently of whether individuals can submit a complete or only a restricted ranking of the schools and (b) subjects with a higher degree of risk aversion are more likely to play “safer” strategies under the Gale–Shapley but not under the Boston mechanism. Both results have important implications for enrollment planning and the possible protection risk averse agents seek.