In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of interest, given that the numerical analysis usually takes place on large input domains having more than one dimension. After introducing the general reaction-diffusion model, we discuss a numerical approximation scheme for the diffusion term, based on a finite difference method. In the next sections we propose two algorithms to solve the numerical approximation scheme, focusing on finding a way to improve the time performance. While the first algorithm (sequential) is used as a baseline for performance measurement, the second algorithm (parallel) is implemented using two different memory-sharing parallelization technologies: Open Multi-Processing (OpenMP) and CUDA. All the results were obtained by using the model in image processing applications such as image restoration and segmentation.

<p style='text-indent:20px;'>In our current paper we are introducing a new method to enhance semantic image segmentation accuracy of a U-Net neural network model by integrating it with a mathematical model based on reaction-diffusion equations.</p><p style='text-indent:20px;'>The methods currently used for semantic image segmentation, including U-Net neural networks, are processing images as blocks of pixels in which the boundaries, the colors and patterns are all mixed together as inputs to the transformations that take place inside the layers of the convolutional neural networks. In our method we are modifying the architecture of a U-Net network and introduce a new data input feed in parallel to the image feed that needs to be segmented. The new input feed is mathematically extracted from the input image and contains the edges (shape) information of the image to be processed. The new input feed it's used during the U-Net decoding phase in order to help shape more precisely the up-scaled output edges, thus leading to improved accuracy performance of the network.</p><p style='text-indent:20px;'>Introducing the parallel feed shows an improvement of accuracy metrics up to 4% (if compared to the U-Net model) and has a limited impact on computational resources consumed at training, because we are only adding a small number of new parameters to be calculated.</p>

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The C-Nail® system is a novel intramedullary fixation method for displaced intra-articular calcaneal fractures. The aim of this study was to evaluate the biomechanical performance of the C-Nail® system and compare it with conventional plate fixation for the treatment of displaced intra-articular calcaneal fractures using finite element analysis. The geometry of a Sanders type-IIB fracture was constructed using the computer-aided design software Ansys SpaceClaim. The C-Nail® system (Medin, Nové Mesto n. Morave, Czech Republic) and the calcaneal locking plate (Auxein Inc., 35 Doral, Florida) and screws were designed according to the manufacturer specifications. Vertical loading of 350 N and 700 N were applied to the subtalar joint surfaces to simulate partial weight bearing and full weight bearing. Construct stiffness, total deformation, and von Mises stress were assessed. The maximum stress on the C-Nail® system was lower compared with the plate (110 MPa vs. 360 MPa). At the bone level the stress was found to have higher values in the case of the plate compared to the C-Nail® system. The study suggests that the C-Nail® system can provide sufficient stability, making it a viable option for the treatment of displaced intra-articular calcaneal fractures.