The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^{*}$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupoids. We construct here countably many nonisomorphic finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids. At this time, there is not a clear $C^{*}$-algebraic technique to prove these inverse semigroups have stably finite complex algebras.

We prove the existence of $\mathrm {GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$, arising from forms of ${\mathrm {GSO}}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.

In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems, a class that includes systems in which all orbit closures are minimal, have isomorphic K-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the associated crossed product $C^*$-algebras as well. We additionally explore the K-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.

We study equilibrium states for a class of non-uniformly expanding skew products, and show how a family of fiberwise transfer operators can be used to define the conditional measures along fibers of the product. We prove that the pushforward of the equilibrium state onto the base of the product is itself an equilibrium state for a Hölder potential defined via these fiberwise transfer operators.

We consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$. Astérisque 287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

We compute $ku^*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ and $ku_*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$, the connective $KU$-cohomology and connective $KU$-homology groups of the mod-$p$ Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,2\right)$, using the Adams spectral sequence. We obtain a striking interaction between $h_0$-extensions and exotic extensions. The mod-$p$ connective $KU$-cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups.

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.

In this work, we carry out an analytical and numerical investigation of travelling waves representing arced vegetation patterns on sloped terrains. These patterns are reported to appear also in ecosystems which are not water deprived; therefore, we study the hypothesis that their appearance is due to plant–soil negative feedback, namely due to biomass-(auto)toxicity interactions.

To this aim, we introduce a reaction-diffusion-advection model describing the dynamics of vegetation biomass and toxicity which includes the effect of sloped terrains on the spatial distribution of these variables. Our analytical investigation shows the absence of Turing patterns, whereas travelling waves (moving uphill in the slope direction) emerge. Investigating the corresponding dispersion relation, we provide an analytic expression for the asymptotic speed of the wave. Numerical simulations not only just confirm this analytical quantity but also reveal the impact of toxicity on the structure of the emerging travelling pattern.

Our analysis represents a further step in understanding the mechanisms behind relevant plants‘ spatial distributions observed in real life. In particular, since vegetation patterns (both stationary and transient) are known to play a crucial role in determining the underlying ecosystems’ resilience, the framework presented here allows us to better understand the emergence of such structures to a larger variety of ecological scenarios and hence improve the relative strategies to ensure the ecosystems’ resilience.

In this article, we prove a generalized Rodrigues formula for a wide class of holonomic Laurent series, which yields a new linear independence criterion concerning their values at algebraic points. This generalization yields a new construction of Padé approximations including those for Gauss hypergeometric functions. In particular, we obtain a linear independence criterion over a number field concerning values of Gauss hypergeometric functions, allowing the parameters of Gauss hypergeometric functions to vary.

This paper is devoted to the study of the propagation dynamics of a mutualistic model of mistletoes and birds with nonlocal dispersal. By applying the theory of asymptotic speeds of spread and travelling waves for monotone semiflows, we establish the existence of the asymptotic spreading speed $c^*$, the existence of travelling wavefronts with the wave speed $c\ge c^*$ and the nonexistence of travelling wavefronts with $c\lt c^*$. It turns out that the spreading speed coincides with the minimal wave speed of travelling wavefronts. Moreover, some lower and upper bound estimates of the spreading speed $c^*$ are provided.

We study cofinal systems of finite subsets of $\omega _1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

We study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under $C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$ conjugacy.

Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions $\omega (q)$, $\nu (q)$, and $\psi ^{(3)}(q)$, introduced by Ramanujan and Watson.

Let $n$ be an integer congruent to $0$ or $3$ modulo $4$. Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$. The same result is obtained unconditionally in special cases.

In this paper, we present a sufficient condition for almost Yamabe solitons to have constant scalar curvature. Additionally, under some geometric scenarios, we provide some triviality and rigidity results for these structures.

This paper introduces a generalization of the $dd^c$-condition for complex manifolds. Like the $dd^c$-condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such $dd^c$-type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.

We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma $. For $p\in \left [1,2\right )\cup \left (2,\infty \right )$, we show that the isometry classes of $L_p[0,1]$ and $\ell _p$ are $G_\delta $-complete sets and $F_{\sigma \delta }$-complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma \delta }$-complete set.

Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal {L}_{p,\lambda +}$-spaces, for $p,\lambda \geq 1$, is shown to be a $G_\delta $-set, the class of superreflexive spaces is shown to be an $F_{\sigma \delta }$-set, and the class of spaces with local $\Pi $-basis structure is shown to be a $\boldsymbol {\Sigma }^0_6$-set. The paper is concluded with many open problems and suggestions for a future research.