By protein structure prediction (PSP) we refer to the prediction of the 3-dimensional (3D) folding of a protein, known as tertiary structure, starting from its amino acid sequence, known as primary structure. The state-of-the-art in PSP is currently achieved by complex deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids in space positively impacts the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of Geometric Algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA-based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as an additional input features to a Graph Transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle-based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative, and can be (i) closely associated to protein secondary structures and (ii) more readily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way.

This article focuses on studying mild solutions of an original frac- tional partial differential equation disturbed by multiplicative white noise. We employ techniques of semi group theory, Hausdorff measure, and Darbo xed point theorem.

In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum $\Lambda=\left\{0, r/N\right\}+2\mathbb Z$, where $N\ge 1$ is an integer and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

In this paper, we investigate the controllability under positivity constraints from the birth of a size-structured population system with delayed birth process. Firstly, we establish the well-posedness and positivity property of the model. Secondly, and more importantly, we derive a sufficient condition for boundary approximate controllability under state and control positivity constraints. The method relies on the feedback theory of infinite-dimensional positive systems.

The developed model of diffusion-limited and diffusionless solidification of eutectic alloy describes the relation “undercooling (ΔT)–velocity(V)–interlamellar spacing (λ)” for two cases. Namely, if the lamellar velocity V is smaller than the solute diffusion speed in bulk liquid VD, VVD, the solidification is mainly controlled by kinetic and thermal undercoolings. We show an influence of model parameters on the growth kinetics of eutectic solidification. Comparison of the model predictions with experimental data obtained on Fe-B samples processed in melt-glass fluxing is given.

The accuracy and repeatability of the color reproduction in print is determined by the fine tuning of the tone reproduction curves of the basic printing colorants (most often this is CMYK). However, the diversity of manufacturers of printing equipment and dyes introduces an element of significant uncertainty about color uniformity. In addition, the traditional approach does not take into account the effect of hue change when applying the original dyes, as well as, the non-linearity of the hue rise in high and low density areas. Determining the color of base colorants that produces the most uniform tone change is an important engineering challenge. Previously, there was no scientific basis for such calculations. We recently proposed an alternative color correction model based on gradation trajectories as an analogue of gradation curves in the CIE Lab space. We have also described the extension of the approach to double color overlay (gradation surfaces) and its analytical and discrete implications. The trajectories are the geodetic lines on gradation surfaces. In this paper, we propose using the gradation trajectories to determine “ideal” or “true” initial printing dyes for electrophotography. To simplify calculations, natural color discretization in digital printing is used.

In this paper, we address the global asymptotic synchronization (GAS) problem of the Master-Slave fractional order multi-linked memristive neural networks (FOMMNNs). Firstly, we propose the FOMMNNs which incorporate the fractional calculus into multi-linked memristive neural networks (MMNNs) for the first time. Then, by utilizing the fractional differential inclusions and set-valued mapping theories, the addressed FOMMNNs with discontinuous state switching at the right-hand side and time-varying delays are converted into the continuous FOMMNNs. Under the frameworks of fractional Caputo derivative and fractional Fillipov solution, by the way of building up appropriate Lyapunov functionals and utilizing some synchronous analysis technology, several sufficient criteria ensuring that the Master-Slave FOMMNNs can realize global asymptotic synchronization (GAS) under two different state-feedback controllers are obtained.

In this paper, a fourth-order accurate conservative finite difference scheme for solving the coupled nonlinear Schr\”odinger (CNLS) equations is proposed. Conservation of the discrete energy and masses, priori estimates, existence and uniqueness of numerical solutions, convergence with second-order in time and fourth-order in space as well as stability of the present scheme are proved by discrete energy method. A convergent iterative method for the present scheme is developed. Numerical experiments are given to support the theoretical analysis.

The source term identification of the time-fractional diffusion equation under Robin boundary condition is studied. This problem is ill-posed. Therefore, we apply Landweber iterative regularization method, Fractional Landweber iterative regularization method, TSVD method, combining TSVD method and Fractional Landweber iterative regularization method, respectively. The comparisons of these four methods are given, which can help us select the most effective method. The error estimates between the regularized approximate solutions and the exact solution are given under the a priori and a posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the methods.

In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

Pursuit-evasion games are the next logical stage in the exploring of powerful, intelligent, adaptive performance. In fact the optimal strategy is known for games in an infinitely sized playing field. The quality of the machine learning methods can thus be compared to the optimal performance possible. Therefore, we consider in this study a pursuit-evasion differential game in Hilbert space l2 with a hybrid system of dynamics. The game consists of a non-inertial pursuer and an inertial evader where controls of the pursuer and the evader are satisfied to the integral constraints. The duration of the game, φ, is fixed. The position of the evader at time φ satisfies to the phase constraint. We obtain attainability domains of the players and then we make a winning strategy for the pursuer which guarantees capturing the evader. We show that our constructed strategy is admissible as well.

Studying spread of epidemics and diseases are world wide problems especially during the current time where the whole world is suffering from COVID-19 pandemic. Antimicrobial resistance (AMR) and waning vaccination are classified as world wide problems. Both depend on the exposure time to antibiotic and vaccination. Here, a simple model for competition between drug resistant and drug sensitive bacteria is given. Conditions for local stability are investigated which agree with observation. Existence of positive solution in the AMR complex networks is proved. Dynamics of the identical AMR models are explored with different topologies of complex networks such as global, star, line and unidirectional line networks coupled through their susceptible states. Chaotic attractors are shown to be existed as the AMR models are located on all the indicted topologies of complex networks. Thus, it is found that the dynamics of the AMR model become more complicated as it is located on either integer-order or fractional-order complex networks. Furthermore, a discretized version of the fractional AMR model is presented. Complex dynamics such as existence of Neimark–Sacker, flip bifurcations, coexistence of multi attractors, homoclinic connections and multi closed invariant curves are investigated. Basin sets of attraction are also computed. Finally, the discretized system is located on complex networks with different topologies which also show rich variety of complex dynamics. Also, 0-1 test is used to verify the existence of unpredictable dynamics. So, studying the dynamics of AMR models on complex networks is very helpful to understand the mechanism of spread of diseases.

We establish nonexistence of nontrivial solutions (including sign-changing ones) for some partial differential inequalities of elliptic and parabolic type containing nonlinear terms that depend on the positive and negative part of the sought function in different ways. Systems of elliptic inequalities with similar structure are also considered. The proofs are based on the test function method.

An SIS epidemic reaction-diffusion model with saturated incidence rate and spontaneous infection is considered. We establish the existence of endemic equilibrium by using a fixed point theorem. We mainly investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium. Our analysis shows that the spontaneous infection can enhance persistence of infectious disease.

This paper attempts to focus on the split common fixed point problem for demicontractive mappings. We give an accelerated hybrid projection algorithm which combines the hybrid projection method and the inertial technique. The strong convergence theorems of this algorithm are obtained under mild conditions by a self-adaptive step-size sequence, which does not need prior knowledge of operator norms. Some numerical experiments in infinite Hilbert space are provided to illustrate the reliability and robustness of the algorithm and also to compare it with existin

In this paper, we introduced a new fractional derivative operator based on Lonezo Hartely function, which is called G-function. With the help of the operator, we solved a fractional diffusion equations. Some applications related to the operator is also discussed as form of corollaries.

Assuming non-Fourier thermal effects, Tzou's dual-phase-lag model has been applied to introduce the governing heat conduction equation in the presented mathematical model. Moreover, in order to design a well-posed stable dual-phase-lag model, the governing time fractional dual-phase-lag heat equation has been established by introducing conductive temperature and thermodynamical temperature, satisfying the two-temperature theory. Due to the application of phase-lags the heat conduction equation became hyperbolic. The corresponding governing equations of motion and stresses have been considered in two-dimensional bounded spherical domain. The spherical boundaries are assumed to be traction free. The Laplace and the Legendre integral transforms have been applied to obtain the analytical solutions of conductive and thermodynamical temperatures, displacement components and thermal stresses. The Gaver-Stehfest algorithm has been employed to achieve the time domain inversions of Laplace transforms numerically, satisfying the Kuznetsov convergence criteria. Classical, fractional and generalized thermoelasticity theories has been recovered theoretically and numerically as well for various fractional orders and phase-lags values.

By the Lagrangian multiplier and constraint variational derivative, a relationship between conserved quantities and multi-Hamiltonian structures is built. Making using the relation a method is founded to prove the infinite-dimensional Liouville integrability of evolution equations with continuous variables. As the application, the conservation laws of the Kundu equation are firstly obtained. Its conserved quantities are deduced for comparing by Fokas' method different from the method used in the existed literature. The integrability of the equation is proved through taking the conservation laws as a starting point.