<p>We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $ \delta $-dimensional Hausdorff content for any $ \delta > 0 $. In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.</p>

We introduce and analyse a variant of the two-dimensional $ XY $-model energy which is suited to detect both topological defects and geometric defects in form of fractional vortices and domain walls, respectively. In contrast to previously introduced variants, the energies we consider here are defined without using an angular lifting of the $ \mathbb{S}^1 $-valued spin variables. Moreover, they combine in an explicit way the features of the $ XY $-model energy on the one hand and weak-membrane energies on the other hand. This leads to simplified proofs of compactness and lower bound in the $ \Gamma $-convergence analysis.

We establish a Harnack inequality for weak solutions of nonlocal nonlinear equations in a disconnected region. The inequality compares the value of a solution on one connected component with that on another, capturing a purely nonlocal phenomenon with no local analogue. We provide three different approaches based on a new weak Harnack inequality, the Poisson formula and the localized maximum principle.

A rigorous unified perspective of cohesive zone models is presented, including and comparing potential-based and non potential-based formulations, and encompassing many of the known examples studied in literature. The main novelty of the work consists in the natural inclusion of loading and unloading effects in a general mixed-mode framework, incorporated through an intrinsic construction of energy densities or tensions. The proposed mathematical investigation identifies and proves the limitations of variational models with respect to non-variational ones, the latter yielding a feasible description of real instances in all relevant situations and regimes. This validates existing empirical and numerical observations. An application to a mechanical process of two elastic laminates sliding one on each other along their cohesive interface is finally analyzed, and existence results in both potential-based and non potential-based versions are obtained, extending previous contributions.

The connection between curvature and topology is a very well-studied theme in the subject of differential geometry. By suitably defining curvature on networks, the study of this theme has been extended into the domain of network analysis as well. In particular, this has led to curvature-based community detection algorithms. In this paper, we reveal the relation between community structure of a network and the curvature of its edges. In particular, we give apriori bounds on the curvature of intercommunity edges of a graph.

<abstract><p>We construct time almost-periodic solutions (global in time) with finite regularity to the incompressible Euler equations on the torus $ \mathbb{T}^d $, with $ d = 3 $ and $ d\in\mathbb{N} $ even.</p></abstract>

<abstract><p>We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.</p></abstract>

<p>The paper presents an augmented curvilinear virtual element method to determine homogenized in-plane shear material moduli of long-fibre reinforced composites in the framework of asymptotic homogenization method. The new virtual element combine an exact representation of the curvilinear computational geometry for complex fibre cross section shapes through an innovative two-dimensional cubature suite for NURBS-like polygonal domains. A selection of representative numerical tests supports the accuracy and efficiency of the proposed approach for both doubly periodic and random fibre arrangement with matrix domain.</p>

<abstract><p>This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.</p></abstract>

<p>Distributed real time system has developed into an outstanding computing platform for parallel, high-efficiency applications. A real time system is a kind of planning where tasks must be completed with accurate results within a predetermined amount of time. It is well known that obtaining an optimal assignment of tasks for more than three processors is an NP-hard problem. This article examines the issue of assigning tasks to processors in heterogeneous distributed systems with a view to reduce cost and response time of the system while maximizing system reliability. The proposed method is carried out in two phases, Phase Ⅰ provides a hybrid HPSOGAK, that is an integration of particle swarm optimization (PSO), genetic algorithm (GA), and <italic>k</italic>-means technique while Phase Ⅱ is based on GA. By updating cluster centroids with PSO and GA and then using them like initial centroids for the <italic>k</italic>-means algorithm to generate the task-clusters, HPSOGAK produces 'm' clusters of 'r' tasks, and then their assignment onto the appropriate processor is done by using GA. The performance of GA has been improved in this article by introducing new crossover and mutation operators, and the functionality of traditional PSO has been enhanced by combining it with GA. Numerous examples from various research articles are employed to evaluate the efficiency of the proposed technique, and the numerical results are contrasted with well-known existing models. The proposed method enhances PIR values by 22.64%, efficiency by 6.93%, and response times by 23.8 on average. The experimental results demonstrate that the suggested method outperforms all comparable approaches, leading to the achievement of superior results. The developed mechanism is acceptable for an erratic number of tasks and processors with both types of fuzzy and crisp time.</p>