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.

This study investigates the stability and instability of the language control network in bilinguals using longitudinal resting-state functional magnetic resonance imaging (rs-fMRI) data. We compared the language control network of Chinese university students majoring in English with those not, using three other functional networks as controls. Results indicate that the English major group exhibits reduced stability and increased instability in the language control network compared with the non-English major group. This suggests that second language (L2) learning experience may induce adaptive neural changes. Moreover, the coexistence of stability and instability in the language control network appears less modular in the English major group, implying a more integrated response to language experience. Notably, these results were not observed in the control networks. Overall, these findings enhance the understanding of bilingual language control and the impact of L2 learning on neural plasticity.

Additive main effects and multiplicative interactive effect stability model (AMMI) was used in the present study to understand the impact of season × genotype interaction (SGI) on pod yield and its attributing traits. A total of 86 determinate growth habit type French bean germplasm were evaluated in randomized block design with two replications in three different seasons. Significant variability was observed for genotypes, seasons and SGI. The component ‘seasons’ contributed more than 50% of variability to pod yield, pod number per plant and days to flowering (DFL), and ‘genotypes’ accounted more than 50% of phenotypic variation for pod length and pod width. The SGI signals were observed for pod yield per plant, number of pods per plant, pod weight and DFL, and SGI accounted for more than 20% phenotypic variability for all traits. We identified IIHR-155 and IIHR-11 as the promising genotypes across three seasons based on their position on AMMI biplots, stability indices combined with high trait mean, estimates of best linear unbiased prediction and minimal crossover interaction. The results from the present study are highly useful for utilization in crop improvement programmes to evolve the season-specific varieties and varieties with wide adaptability in French bean.

Recently it has been shown that the unique local perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens, hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem.

Bootstrap and jackknife techniques are used to estimate ellipsoidal confidence regions of group stimulus points derived from INDSCAL. The validity of these estimates is assessed through Monte Carlo analysis. Asymptotic estimates of confidence regions based on a MULTISCALE solution are also evaluated. Our findings suggest that the bootstrap and jackknife techniques may be used to provide statements regarding the accuracy of the relative locations of points in space. Our findings also suggest that MULTISCALE asymptotic estimates of confidence regions based on small samples provide an optimistic view of the actual statistical reliability of the solution.

Proteins are vital biological macromolecules that execute biological functions and form the core of synthetic biological systems. The history of de novo protein has evolved from initial successes in subordinate structural design to more intricate protein creation, challenging the complexities of natural proteins. Recent strides in protein design have leveraged computational methods to craft proteins for functions beyond their natural capabilities. Molecular dynamics (MD) simulations have emerged as a crucial tool for comprehending the structural and dynamic properties of de novo-designed proteins. In this study, we examined the pivotal role of MD simulations in elucidating the sampling methods, force field, water models, stability, and dynamics of de novo-designed proteins, highlighting their potential applications in diverse fields. The synergy between computational modeling and experimental validation continued to play a crucial role in the creation of novel proteins tailored for specific functions and applications.

Three-dimensional short-crested water waves are known to host harmonic resonances (HRs). Their existence depends on their sporadicity versus their persistency. Previous studies, using a unique yet hybrid solution, suggested that HRs exhibit sporadic instability, with the domain of instability exhibiting a bubble-like structure which experiences a loss of stability followed by a re-stabilization. Through the calculation of their complete multiple solution structures and normal forms, we discuss the particular harmonic resonance (2,6). The (2,6) resonance was chosen, not only because it is of lower order, and thus more likely to be significant, but also because it is representative of a fully developed three-dimensional water wave field. Its appearance, growth rate and persistency are discussed. On our converged solutions, we show that, at an incidence angle for which HR (2,6) occurs, the associated superharmonic instability is no longer sporadic. It was also found that the multiple solution operates a subcritical pitchfork bifurcation, so regardless of the value of the control parameter, the wave steepness, a stable branch of the solution always exists. As a result, the analysis reveals two competing processes that either provoke and enhance HRs, or inhibit their appearance and development.

Traditionally, the role of general topology in model theory has been mainly limited to the study of compacta that arise in first-order logic. In this context, the topology tends to be so trivial that it turns into combinatorics, motivating a widespread approach that focuses on the combinatorial component while usually hiding the topological one. This popular combinatorial approach to model theory has proved to be so useful that it has become rare to see more advanced topology in model-theoretic articles. Prof. Franklin D. Tall has led the re-introduction of general topology as a valuable tool to push the boundaries of model theory. Most of this thesis is directly influenced by and builds on this idea.

The first part of the thesis will answer a problem of T. Gowers on the undefinability of pathological Banach spaces such as Tsirelson space. The topological content of this chapter is centred around Grothendieck spaces.

In a similar spirit, the second part will show a new connection between the notion of metastability introduced by T. Tao and the topological concept of pseudocompactness. We shall make use of this connection to show a result of X. Caicedo, E. Dueñez, J. Iovino in a much simplified manner.

The third part of the thesis will carry a higher set-theoretic content as we shall use forcing and descriptive set theory to show that the well-known theorem of M. Morley on the trichotomy concerning the number of models of a first-order countable theory is undecidable if one considers second-order countable theories instead.

The only part that did not originate from model-theoretic questions will be the fourth one. We show that $\operatorname {ZF} + \operatorname {DC} +$“all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations. This result provides evidence in favour of a long-standing conjecture asking whether Turing determinacy implies the axiom of determinacy.

Abstract prepared by Clovis Hamel

E-mail: [email protected]

We analyze the limit of stable solutions to the Ginzburg-Landau (GL) equations when ${\varepsilon }$, the inverse of the GL parameter, goes to zero and in a regime where the applied magnetic field is of order $|\log {\varepsilon } |$ whereas the total energy is of order $|\log {\varepsilon }|^2$. In order to do that, we pass to the limit in the second inner variation of the GL energy. The main difficulty is to understand the convergence of quadratic terms involving derivatives of functions converging only weakly in $H^1$. We use an assumption of convergence of energies, the limiting criticality conditions obtained by Sandier-Serfaty by passing to the limit in the first inner variation, and properties of limiting vorticities to find the limit of all the desired quadratic terms. At last, we investigate the limiting stability condition we have obtained. In the case with magnetic field, we study an example of an admissible limiting vorticity supported on a line in a square ${{\Omega }}=(-L,L)^2$ and show that if L is small enough, this vorticiy satisfies the limiting stability condition, whereas when L is large enough, it stops verifying that condition. In the case without magnetic field, we use a result of Iwaniec-Onninen to prove that every measure in $H^{-1}({{\Omega }})$ satisfying the first-order limiting criticality condition also verifies the second-order limiting stability condition.

The fixed points of the generalized Ricci flow are the Bismut Ricci flat (BRF) metrics, i.e., a generalized metric (g, H) on a manifold M, where g is a Riemannian metric and H a closed 3-form, such that H is g-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each Gi is a compact simple Lie group and K is a closed subgroup of them holding some extra assumption, we consider $M=G_1\times G_2/\Delta K$. Recently, Lauret and Will proved the existence of a BRF metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on M among a subset of G-invariant metrics and, if $G_1=G_2$, then it is globally stable.