For an arbitrary negative Schwarzian unimodal map with a non-flat critical point, we establish the level-2 large deviation principle for empirical distributions. We also give an example of a bimodal map for which the level-2 large deviation principle does not hold.

Let $G(\mathbb {R})$ be a real reductive group. Suppose $\pi $ is an irreducible representation of $G(\mathbb {R})$ having a Whittaker model, and consider three invariants of $\pi $ related to nilpotent elements of the Lie algebra of G (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi $ has a Whittaker model. If $\pi $ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related matters. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant–Sekiguchi correspondence.

In this paper, we describe étale Boolean right restriction monoids in terms of Boolean inverse monoids. We show that the Thompson groups arise naturally in this context.

We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal, we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order. Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley–Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices.

The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues $\lambda_1$ and $\lambda_2$ being given. Applying the method of critical equations in $L^p[0,1]$ for $p \gt 1$ and the inverse spectral theory of Sturm–Liouville problems with measure coefficients, we find that the optimal weight can be uniquely determined if and only if $\lambda_2 \ge 2\lambda_1$ provided that the weight is non-negative and symmetrical. As an application, we provide an estimation of the extremum for partial trace of the first two eigenvalues on a sphere in $L^1[0,1]$.

Biochemical reaction networks (RNs) are widely applied across scientific disciplines to model complex dynamic systems. We investigate the diffusion approximation of RNs with mass-action kinetics, focusing on the identifiability of the stochastic differential equations associated to the reaction network. We derive conditions under which the law of the diffusion approximation is identifiable and provide theorems for verifying identifiability in practice. Notably, our results show that some RNs have non-identifiable reaction rates, even when the law of the corresponding stochastic process is completely known. Moreover, we show that RNs with distinct graphical structures can generate the same diffusion law under specific choices of reaction rates. Finally, we compare our framework with identifiability results in the deterministic ordinary differential equation setting and the discrete continuous-time Markov chain models for RNs.

We study the dynamics of a delayed predator–prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is the demonstration that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller’s space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ordinary differential equation model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics.

For a $G$-equivariant fibration $p \colon E\to B$, we introduce and study the invariant analogue of Cohen, Farber, and Weinberger’s parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. When $G$ is a compact Lie group acting freely on $E$, we show that the invariant parametrized topological complexity of the $G$-fibration $p \colon E\to B$ coincides with the parametrized topological complexity of the induced fibration $\overline{p} \colon \overline{E} \to \overline{B}$ between the orbit spaces. Furthermore, we compute the invariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations, which measures the complexity of motion planning in the presence of obstacles with unknown positions, where the order of their placement is irrelevant. In addition, we study the equivariant sectional category and the equivariant parametrized topological complexity, which serve as essential tools for obtaining several results in this paper.

In this paper, we investigate dynamical properties of monoid actions on symbolic systems. First, we establish mixing properties and an entropy formula for such actions. To describe chaotic behavior, we introduce and distinguish several types of Li–Yorke chaos. Our main result shows that positive entropy is equivalent to locally Li–Yorke chaos, a notion that strengthens the classical definition of Li–Yorke chaos.

We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

In this paper, we show the existence, uniqueness and stability of nontrivial solutions to the following Minkowski-curvature problems on unbounded domains:

$$ \begin{align*} \bigg(\frac{x'}{\sqrt{1-x^{\prime2}}}\bigg)'=f(t,\ x),\quad t\geq t_0, \quad \lim_{t\to\infty}x(t)=\psi_0,\quad \lim_{t\to\infty}x'(t)e^{t}=0, \end{align*} $$

$f:\ [t_0, \infty )\times \mathbb {R}\rightarrow \mathbb {R}$

$t_0>0$

$\psi _0\in \mathbb {R}$

Interval consensus is an important generalization of conventional consensus problems by allowing each agent to individually nominate an acceptable interval for their consensus value. However, as other consensus problems, agents in the network exchange information explicitly among neighbours and disclose their values without any protection for sensitive information causing serious privacy concerns in many applications in distributed multiagent systems. We propose a privacy-preserving approach consisting of decomposition and weighting mechanisms. Based on this approach, we show that the agents in the network can achieve interval consensus with the final consensus value within the intersection of all proposed intervals if the intersection is nonempty and the network is connected. Moreover, the privacy of the initial states of the agents is guaranteed against internal and external adversaries. The proposed consensus protocol is simple and efficient, and it can be implemented in a distributed manner over the network.

This paper proves that the integrality of algebraic Witt vectors over imaginary quadratic fields is decidable; based on this, some related problems are also discussed.

We consider a subshift of finite type endowed with a Markov measure that is given by a stochastic matrix. We introduce a Markov hole determined by a finite collection of allowed words in the subshift. We first present a simple yet precise formula to compute the escape rate into the hole as the spectral radius of a perturbed stochastic matrix, where the rule of perturbation is governed by the hole. The combinatorial nature of the subshift comes to our aid in obtaining another formulation of the escape rate as the logarithm of the smallest real pole of a certain rational function, by way of recurrence relations. This proves crucial in comparing the escape rates into cylinders based at words of fixed length. Merits of both the formulas are illustrated through examples.

We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and $\mu $-limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.

In this paper, we define and study noncommutative affine pencils of conics, and give a complete classification result. We also fully classify four-dimensional Frobenius algebras. It turns out that the classification of noncommutative affine pencils of conics is the same as the classification of four-dimensional Frobenius algebras.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.