This paper is concerned with the $ L^2 $-critical minimization problem of the coupled planar Gross-Pitaevskii system with a harmonic potential and logarithmic convolutions. We first establish the existence and nonexistence of constraint minimizers for the system, which is connected with the unique positive solution of $ -\Delta u+u-u^3 = 0 $ in $ \mathbb{R}^2 $. The refined limiting behavior and exponential decay of nonnegative constraint minimizers are also analyzed. We further obtain the uniqueness of nonnegative constraint minimizers by overcoming the non-invariance under translations of the potential and the sign-changing property of the logarithmic convolutions.

This paper investigates the long-term dynamics of solutions to a parabolic-elliptic chemotaxis system with weakly singular sensitivity and a logistic source under the Neumann boundary conditions in a smooth bounded domain $ \Omega \subset \mathbb{R}^N $ with $ N \geq 2 $$ \begin{equation*} \begin{cases} u_t = \Delta u-\chi \nabla\cdot \Big(\frac{u}{v^{\lambda}} \nabla v \Big) +ru- \mu u^2, \quad &x\in \Omega, \\ 0 = \Delta v- \alpha v +\beta u, \quad &x\in \Omega, \end{cases} \end{equation*} $where the parameters $ \chi, \, r, \, \mu, \, \alpha, \, \beta $ are positive constants and $ \lambda \in (0, 1). $ The present study improves previously established results on the global existence and boundedness of classical solutions; and constitutes the first investigation of the large-time dynamics of globally bounded solutions. In particular, it has provide a detailed analysis of their uniform boundedness, pointwise persistence, and asymptotic stability.For all suitably smooth initial data $ u_0\in C^0(\bar\Omega) $ with $ u_0 \not \equiv 0, $ the following results have been established. First, there exists $ \mu > \mu_1^*(N, \lambda, \chi, \beta) $ such that all classical solutions exist globally and remain bounded. Next, there exists $ \mu > \mu_2^*(N, \lambda, \chi, \beta) $ such that every global positive solution is uniformly bounded from above and below by positive constants independent of its initial function $ u_0. $ Last, there exists $ \mu > \mu_3^*(N, \lambda, \chi, \alpha, \beta, r, \Omega) $ such that any globally bounded classical solution exponentially converges to the constant steady state $ (\frac{r}{\mu}, \frac{\beta}{\alpha}\frac{r}{\mu}). $

For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fréchet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.

Faure and Tsujii recently proposed a new quantization theory for symplectic Anosov diffeomorphisms. It combines prequantization with the study of the Pollicott–Ruelle resonances of an associated transfer operator. We apply this framework to the hyperbolic symplectic automorphisms of the $ 2n $-dimensional torus, the so-called cat maps. Our main result gives an explicit relation between the resonances of the prequantum transfer operator and the eigenvalues of the standard quantum cat maps, generalizing the case $ n = 1 $ previously treated by Faure.

We consider the problem $ -\Delta u+\lambda u = |u|^{p-2}u $, where $ u\in H^1_0(\Omega) $ satisfies $ |u|_{2} = m>0 $, $ \lambda\in \mathbb{R} $ and $ \Omega $ is a smooth exterior domain. We prove the existence of a positive solution with a constrained Morse index less or equal than $ N+1 $ and $ \lambda\ge 0 $. We treat both the cases $ m $ fixed and $ \mathbb{R}^N\setminus\Omega $ small and $ \Omega $ fixed and $ m $ large.

Appeals to randomness in various number-theoretic constructions appear regularly in modern scientific publications. Such famous names as V.I. Arnold, M. Katz, Ya.G. Sinai, and T. Tao are just a few examples. Unfortunately, all of these approaches rely on various, although often very non-trivial and elegant, heuristics. A new analytical approach is proposed to address the issue of randomness/complexity of an individual deterministic sequence. This approach demonstrates the expected high complexity of quadratic residues and the unexpectedly low complexity in the case of prime numbers. Technically, our approach is based on a new construction of the dynamical entropy of a single trajectory, which measures its complexity, in contrast to classical Kolmogorov-Sinai and topological entropies, which measure the complexity of the entire dynamical system.

This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second-order elliptic operator. We discuss sufficient and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal mean curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

This paper investigates semilinear partial differential equations under Dirichlet-Neumann boundary conditions in a planar kite. By imposing specific geometric conditions on the kite, we establish that every positive solution exhibits symmetry with respect to the symmetry axis and maintains monotonicity along this axis. Our approach relies on the moving plane method, coupled with a strategic application of the maximum principle for mixed boundary conditions in a narrow domain.

We prove an effective slope gap distribution result for translation surfaces whose Veech group is a lattice. As a corollary, we obtain a dynamical proof for an effective gap distribution result for the Farey fractions. As an intermediate step, we prove an effective equidistribution result for the intersection points of long horocycles with a particular transversal of the horocycle flow in $ {\rm SL}_2 (\mathbb{R})/\Gamma $ where $ \Gamma $ is a lattice.