The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the stability analysis for traveling wave solutions to evolutionary PDEs.

In contrast, we use Evans function as a flexible tool yielding general sufficient condition for local bifurcations of specific bounded entire solutions to (Carathéodory) differential equations. These bifurcations are intrinsically nonautonomous in the sense that the assumptions implying them cannot be fulfilled for autonomous or periodic temporal forcings. In addition, we demonstrate that Evans functions are strictly related to the dichotomy spectrum and hyperbolicity, which play a crucial role in studying the existence of bounded solutions on the whole real line and therefore the recent field of nonautonomous bifurcation theory. Finally, by means of non-trivial examples we illustrate the applicability of our methods.

We consider the Perron–Frobenius operator defined on the space of functions of bounded variation for the beta-map $\tau _\beta (x)=\beta x$ (mod $1$) for $\beta \in (1,\infty )$, and investigate its isolated eigenvalues except $1$, called non-leading eigenvalues in this paper. We show that the set of $\beta $ such that the corresponding Perron–Frobenius operator has at least one non-leading eigenvalue is open and dense in $(1,\infty )$. Furthermore, we establish the Hölder continuity of each non-leading eigenvalue as a function of $\beta $ and show in particular that it is continuous but non-differentiable, whose analogue was conjectured by Flatto, Lagarias and Poonen in [The zeta function of the beta transformation. Ergod. Th. & Dynam. Sys. 14 (1994), 237–266]. In addition, for an eigenfunctional of the Perron–Frobenius operator corresponding to an isolated eigenvalue, we give an explicit formula for the value of the functional applied to the indicator function of every interval. As its application, we provide three results related to non-leading eigenvalues, one of which states that an eigenfunctional corresponding to a non-leading eigenvalue cannot be expressed by any complex measure on the interval, which is in contrast to the case of the leading eigenvalue $1$.

We strengthen known results on Diophantine approximation with restricted denominators by presenting a new quantitative Schmidt-type theorem that applies to denominators growing much more slowly than in previous works. In particular, we can handle sequences of denominators with polynomial growth and Rajchmann measures exhibiting arbitrary slow decay, allowing several applications. For instance, our results yield non-trivial lower bounds on the Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers (each with restricted denominators) and enable the construction of Salem subsets of well-approximable numbers of arbitrary Hausdorff dimension.

We prove that in the space of $C^r$ maps $(r=2,\ldots ,\infty ,\omega )$ of a smooth manifold of dimension at least 4, there exist open regions where maps with infinitely many corank-2 homoclinic tangencies of all orders are dense. The result is applied to show the existence of maps with universal two-dimensional dynamics, that is, maps whose iterations approximate the dynamics of every map of a two-dimensional disk with an arbitrarily good accuracy. We show that maps with universal two-dimensional dynamics are $C^r$-generic in the regions under consideration.

It was proved in [11, J. Funct. Anal., 2020] that the Cauchy problem for some Oldroyd-B model is well-posed in $\dot{B}^{d/p-1}_{p,1}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,1}(\mathbb{R}^d)$ with $1\leq p \lt 2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\dot{B}^{d/p-1}_{p,r}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,r}(\mathbb{R}^d)$ with $1\leq p\leq \infty$ and $1 \lt r\leq\infty$ due to the lack of continuous dependence of the solution.

We study the K-stability of blow-ups of the weighted projective plane ${\mathbb P}(1,1,n)$, where n is a positive integer.

The influence of certain arithmetic conditions on the sizes of conjugacy classes of a finite group on the group structure has been extensively studied in recent years. In this paper, we explore analogous properties for fusion categories. In particular, we establish an Ito-Michler-type result for modular fusion categories.

Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $\Phi$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(\Phi )$ of objects admitting a composition series-like filtration with factors in $\Phi$ has the Jordan-Hölder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series for an extriangulated category, in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise Jordan-Hölder, length, weakly idempotent complete extriangulated categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system $\Phi$ in an extriangulated category is part of a minimal projective one $(\Phi ,Q)$. We prove that $\mathcal{F}(\Phi )$ is a length, Jordan-Hölder extriangulated category when $(\Phi ,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto–Saito in the negative.

Using an original method, we find the algebra of generalised symmetries of a remarkable (1+2)-dimensional ultraparabolic Fokker–Planck equation, which is also called the Kolmogorov equation and is singled out within the entire class of ultraparabolic linear second-order partial differential equations with three independent variables by its wonderful symmetry properties. It turns out that the essential subalgebra of this algebra, which consists of linear generalised symmetries, is generated by the recursion operators associated with the nilradical of the essential Lie invariance algebra of the Kolmogorov equation, and the Casimir operator of the Levi factor of the latter algebra unexpectedly arises in the consideration. We also establish an isomorphism between this algebra and the Lie algebra associated with the second Weyl algebra, which provides a dual perspective for studying their properties. After developing the theoretical background of finding exact solutions of homogeneous linear systems of differential equations using their linear generalised symmetries, we efficiently apply it to the Kolmogorov equation.

We prove that the proximal unit normal bundle of the subgraph of a $W^{2,n} $-function carries a natural structure of Legendrian cycle. This result is used to obtain an Alexandrov-type sphere theorem for hypersurfaces in $ \mathbf{R}^{n+1} $, which are locally graphs of arbitrary $W^{2,n} $-functions. We also extend the classical umbilicality theorem to $ W^{2,1} $-graphs, under the Lusin (N) condition for the graph map.

Electricity supply operators offer financial incentives to encourage large energy users to reduce their power demand during declared periods of increased demand from energy users such as residential homes. This demand flexibility enables electricity system operators to ensure adequate power supply and avoid the construction of peaking power plants.

Railway operators can sometimes reduce their power demand during specified peak demand periods without disrupting the train schedules. For trains with infrequent stops, such as intercity trains, it is possible to speed up trains prior to the peak demand period, slow down during the peak demand period, then speed up again after the peak demand period. We use simple train models to develop an optimal strategy that minimizes energy use for a fleet of trains subject to energy-use constraints during specified peak demand intervals. The strategy uses two sets of interacting parameters to find an optimal solution—a Lagrange multiplier for each energy-constrained time interval to control the speed of trains during each interval, and a Lagrange multiplier for each train to control the relative train speeds and ensure each train completes its journey on time.

For every , we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures, and thus, the ergodic measure space is not connected, while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti [11, Conjecture 2] in $C^r$-topology for Lorenz attractors.

We are concerned with the following coupled Schrödinger system with a Hardy potential in the critical case

\begin{align*}\begin{cases}-\Delta u_{i}-\frac{\lambda_{i}}{|x|^2}u_{i}=|u_i|^{2^*-2}u_i+\sum_{j\neq i}^{3}\beta_{ij}|u_{j}|^{\frac{2^*}{2}}|u_i|^{\frac{2^*}{2}-2}u_i, ~x\in \mathbb{R}^N,\\u_i\in D^{1,2}(\mathbb{R}^N),\,\, N\geq 3,\,\, i=1,2,3,\end{cases}\end{align*}

$2^*=\frac{2N}{N-2}$

$\lambda_i\in (0,\Lambda_N), \Lambda_N:= \frac{(N-2)^2}{4}$

$\beta_{ij}=\beta_{ji}$

$\beta_{ij} \gt 0$

$\beta_{i_{1}j_{1}} \gt 0$

$\beta_{i_{2}j_{2}} \lt 0$

$(i_{1}, j_{1})$

$(i_{2}, j_{2})$

$\beta_{ij}\ge0$

$N\ge5$

$\beta_{ij} \gt 0$

$N=3,4$

$N=3, 4$

$N\geq 5$

$N\geq 5$

$N\ge5$

In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right least common multiple (LCM) monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear if and only if the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens if and only if the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over $ax+b$-monoids of integral domains and over Baumslag–Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide for m and n relatively prime.

We investigate a Leslie-type prey–predator system with an Allee effect to understand the dynamics of populations under stress. First, we determine stability conditions and conduct a Hopf bifurcation analysis using the Allee constant as a bifurcation parameter. At low densities, we observe that a weak Allee effect induces a supercritical Hopf bifurcation, while a strong effect leads to a subcritical one. Notably, a stability switch occurs, and the system exhibits multiple Hopf bifurcations as the Allee effect varies. Subsequently, we perform a sensitivity analysis to assess the robustness of the model to parameter variations. Additionally, together with the numerical examples, the FAST (Fourier amplitude sensitivity test) approach is employed to examine the sensitivity of the prey–predator system to all parameter values. This approach identifies the most influential factors among the input parameters on the output variable and evaluates the impact of single-parameter changes on the dynamics of the system. The combination of detailed bifurcation and sensitivity analysis bridges the gap between theoretical ecology and practical applications. Furthermore, the results underscore the importance of the Allee effect in maintaining the delicate balance between prey and predator populations and emphasize the necessity of considering complex ecological interactions to accurately model and understand these systems.

In this paper, we initiate the study of higher rank Baumslag–Solitar (BS) semigroups and their related C*-algebras. We focus on two rather interesting classes—one is related to products of odometers and the other is related to Furstenberg’s $\times p, \times q$ conjecture. For the former class, whose C*-algebras are studied in [32], we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting that encompasses (single-vertex) generalized BS semigroups. One of our main tools in this paper is from self-similar higher rank graphs and their C*-algebras.

For a connected Lie group G and an automorphism T of G, we consider the action of T on Sub$_G$, the compact space of closed subgroups of G endowed with the Chabauty topology. We study the action of T on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of G. We relate the distality of the T-action on Sub$^p_G$ with that of the T-action on G and characterise the same in terms of compactness of the closed subgroup generated by T in Aut$(G)$ when T acts distally on the maximal central torus and G is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$ and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group G acts distally on Sub$^p_G$ by conjugation if and only if G is either compact or is isomorphic to a direct product of a compact group and a vector group. Some of our results generalise those of Shah and Yadav.

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space contains two disjoint subsets: one is a dense $G_\delta $ set for which all maximizing measures have ‘relatively small’ entropy; the other is the set of functions having uncountably many, fully supported ergodic maximizing measures with ‘relatively large’ entropy. This result generalizes and unifies the results of Morris [Discrete Contin. Dyn. Syst. 27 (2010), 383–388] and Shinoda [Nonlinearity 31 (2018), 2192–2200] on symbolic dynamics, and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without the Bowen specification property, including any transitive piecewise monotonic interval map, some coded shifts, and multidimensional $\beta $-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.