Acta Applicandae Mathematicae最新文献

When taking a regular planar polygon of (M) sides and length (2pi ) as the initial datum of the vortex filament equation, (mathbf{X}_{t}= mathbf{X}_{s}wedge mathbf{X}_{ss}), the solution becomes polygonal at times of the form (t_{pq} = (p/q)(2pi /M^{2})), with (gcd (p,q)=1), and the corresponding polygon has (Mq) sides, if (q) is odd, and (Mq/2) sides, if (q) is even. Moreover, that polygon is skew (except when (q = 1) or (q = 2), where the initial shape is recovered), and the angle (rho ) between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time (t_{pq}), (cos ^{q}(rho /2) = cos (pi /M)), if (q) is odd, and (cos ^{q}(rho /2) = cos ^{2}(pi /M)), if (q) is even. Since the transition of one side of the polygon to the next one is given by a rotation in (mathbb{R}^{3}) determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle (2pi /M), which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.

In this paper, we consider Cauchy problem for the 2D incompressible Keller-Segel-Navier-Stokes equations with the fractional diffusion

where (wedge :=(-Delta )^{frac{1}{2}}) and (alpha in [frac{1}{2},1]). We get the global well-posedness for the above system with the rough initial data by a new priori estimate of the solutions.

Total (absolute) curvature is defined for any curve in a metric space. Its properties, finiteness, local boundedness, Lipschitz continuity, depending whether there are satisfied or not, permit a classification of curves alternative to the classical regularity classes. In this paper, we are mainly interested in the total curvature estimation. Under the sole assumption of curve simpleness, we prove the convergence, as (epsilon to 0), of the naive turn estimators which are families of polygonal lines whose vertices are at distance at most (epsilon ) from the curve and whose edges are in (Omega (epsilon ^{alpha })cap text{O}(epsilon ^{beta })) with (0<beta le alpha <frac{1}{2}). Besides, we give lower bounds of the speed of convergence under an additional assumption that can be summarized as being “convex-or-Lipschitz”.

The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method (Carrillo et al. in J. Comput. Phys. 7:100066, 2020) has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies Bobylev-Krook-Wu (BKW) solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.

Recently, lots of studies demonstrated that the signals are not only sparse in some system (e.g. shearlets) but also reveal a certain structure such as sparsity in levels. Therefore, sampling strategy is designed as a variable subsampling strategy in order to use this extra structure, for example magnetic resonance imaging (MRI) and etc. In this paper, we investigate the uniform recovery guarantees on the signals which possess sparsity in levels with respect to a general dual frame. First, we prove that the stable and robust recovery is possible when the weighted (l^{2} )-robust null space property in levels is satisfied. Second, we establish sufficient conditions under which subsampled isometry satisfies the weighted (l^{2} )-robust null space property in levels.

This paper studies the Liouville type theorems for stationary the tropical climate model on the whole space (mathbb{R}^{3}). It shows that if ((u,v,theta )) satisfies certain anisotropic integrability conditions on the components of the (u) or ((u,v)), also (theta ) satisfies certain isotropic integrability conditions, there is only a trivial solution to the stationary tropical climate model. The results are a further extension of the recent work by Chae (Appl. Math. Lett. 142:108655, 2023).

In this article we address the problem of locating point forces within a time-dependent singular Brinkman flow. The context of the study is framed as an approximation of cerebrospinal fluid (CSF) around the central nervous system, with the point forces representing a model for the blood-brain barrier. We approach the problem by reformulating the identification task as an optimization problem, employing a tracking shape functional. A notable challenge in this study arises from the irregularity in the solution of the partial differential equation (PDE), which complicates the exploration of sensitivity analysis. To overcome this issue, we employ a relaxation method and compute the topological derivative of the cost function. The topological derivative, commonly used in shape optimization problems, offers insights into how the cost function responds to small perturbations in the domain. To determine the optimal position of the point forces, we employ a one-shot algorithm based on the derived topological gradient. Finally, we present numerical results that showcase the efficiency of our method in addressing the identified problem.

This paper investigates the fast chemical diffusion limit from a parabolic-parabolic Keller-Segel system to the corresponding parabolic-elliptic Keller-Segel system by constructing approximate solutions with an appropriate order via an asymptotic expansion. Nonlinear stability of the precise initial layer is characterized with an exact convergence rate by using basic energy method.

We investigate the following nonlinear bosonic equation on Euclidean space arising in string theory and cosmology:

where (nge 3), (m>0), (c>0) and (frac{f(x,u)}{u}) tends to a positive function (h(x)) independent of (u) as (urightarrow +infty ), (e^{-cDelta }) is given by a power series with (Delta ) is the Euclidean Laplace operator. Here, the nonlinear term (f(x,u)) does not satisfy the usual condition:

for (theta >0) and (|u|) is large, which is important in using the mountain pass theorem, see Alves et al. (J. Differ. Equ. 323:229-252, 2022) and Corrêa et al. (J. Differ. Equ. 363:491-517, 2023). This paper is devoted to discuss how to use the mountain pass theorem to obtain the existence of nontrivial solution to problem (P) without the (AR) condition.

This paper investigates the significance of employing random subgraphs and analyzing the expected values of Zagreb ndices within chemical graphs. By examining smaller, representative subsets, we uncover valuable insights into the properties and characteristics of complex networks. The expected values of Zagreb indices serve as critical mathematical measures for quantifying the structural complexity of chemical graphs, providing essential information about connectivity and branching patterns within molecules. Our primary contribution includes deriving theoretical expressions for these indices and validating them through extensive computational experiments on fullerene graphs (C_{20}) and (C_{60}). The results demonstrate that our theoretical predictions closely align with experimental findings, affirming the robustness of Zagreb indices in characterizing molecular structures. Additionally, our analysis of specific cases, such as complete graphs and complete bipartite graphs, is consistent with previous studies, further reinforcing our methodology. This research emphasizes the relevance of random subgraphs and expected values of Zagreb indices in advancing our understanding of molecular behavior and stability, with important implications for materials science and drug design.