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A one-dimensional space-time fractional reaction-diffusion problem is considered. We present a complete theory for the solution of the time-space fractional reaction-diffusion model, including existence and uniqueness in the case of using the spectral representation of the fractional Laplacian operator. An optimal error estimate is presented for the Galerkin spectral approximation of the problem under consideration. [ABSTRACT FROM AUTHOR]

The purpose of this research is to establish the generalised fractional Bloch-Torrey equation for better simulating anomalous diffusion in heterogeneous biological tissues. The introduction of the distributed-order time fractional derivative allows for an improved interpretation of the complex diffusion behaviours with multi-scale effects. The use of variable coefficients in the model increases its applicability for describing the spatial heterogeneity evident in the cellular structures. The proposed distributed-order time and space fractional Bloch-Torrey equation is discretised in time and space by the L 2-1 σ formula and the finite element method, respectively. The stability and convergence analyses of these numerical methods are provided. To further improve the computational efficiency, a reduced-order extrapolation scheme is developed. We verify the effectiveness of the proposed methods by numerical examples. Moreover, the coupled fractional dynamic system solution behaviour is explored on a human brain-like domain divided into the white matter and grey matter regions. Compared with the model having constant coefficients, solution behaviours suggest that variable diffusion coefficients offer an effective way to differentiate the distinct diffusion phenomena evolving in different tissue micro-environments. Furthermore, to evaluate the impacts of the weight function in the distributed-order operator, we choose three types of beta distributions with the same mean but different values of the variance. The results indicate that the larger value of the variance leads to a more remarkable fluctuation and a slower decay of the transverse magnetisation. This generalised distributed-order fractional model may provide further insights into capturing anomalous diffusion in heterogeneous media. • A distributed-order time-space FBTE with variable coefficients is developed. • The fractional dynamic system is solved by the L 2 − 1 σ formula and FEM. • The stability and convergence of numerical methods are provided. • An improved fast algorithm reduces the computational cost. • This research provides more insights into capturing the diffusion complexity. [ABSTRACT FROM AUTHOR]

Consider an arbitrary n -dimensional lattice Λ such that Z n ⊂ Λ ⊂ Q n . Such lattices are called rational and can always be obtained by adding m ≤ n rational vectors to Z n. The defect d (E , Λ) of the standard basis E of Z n (n unit vectors going in the directions of the coordinate axes) is defined as the smallest integer d such that certain (n − d) vectors from E together with some d vectors from the lattice Λ form a basis of Λ. Let ‖ ⋅ ‖ be L 1 -norm on Q n. Suppose that for each non-integer x ∈ Λ inequality ‖ x ‖ > 1 holds. Then the unit octahedron O E n = x ∈ R n : ‖ x ‖ ⩽ 1 is called admissible with respect to Λ and d (E , Λ) is also called the defect of the octahedron O E n with respect to E and is denoted by d (O E n , Λ). Let d n = max Λ ∈ A n d (O E n , Λ) , where A n is the set of all rational lattices Λ such that O E n is admissible w.r.t. Λ. In this article we show that n − d n = Θ (log n). Let d n m = max Λ ∈ A m d (O E n , Λ) , where A m is the set of all rational lattices Λ such that 1) O E n is admissible w.r.t. Λ and 2) Λ can be obtained by adding m rational vectors to Z n : Λ = Z n , a 1 , ... , a m Z for some a 1 , ... , a m ∈ Q n. In this article we show that for any 0 < ϵ < 1 and large enough n we have m < n ϵ 8 (1 ϵ + 1) ⟹ d n m < n − n 1 − ϵ. Finally we show that for any B > 0 there exists a positive constant C > 0 such that m < C n log log n log 2 n ⟹ d n m < n − B log n. [ABSTRACT FROM AUTHOR]

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme. [ABSTRACT FROM AUTHOR]

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L -stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A -stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented. [ABSTRACT FROM AUTHOR]

In this manuscript, a numerical method with high accuracy and efficiency based on the difference-Legendre spectral method is proposed for obtaining the numerical solutions of the two-dimensional space-time distributed-order fractional diffusion-wave equations with the Riesz space fractional derivative. The difference method by an approximate formula respect to the time variable is used to discrete the distribution-order integral part consisting of Caputo fractional derivative. Also, the Gauss quadrature formula respect to the space variable is used to estimate the distribution-order integral part consisting of Riesz space derivative. Further, the stability and convergence analyses are studied for the numerical estimation. Some numerical examples are displayed to show the effectiveness of proposed methods. [ABSTRACT FROM AUTHOR]

In this article, the black box dynamic model is presented for forecasting the performance of the PEM (Proton-exchange membrane) fuel cell (FC). An optimized deep artificial neural network has been used to build the experimental nonlinear model of the polymer membrane FC series that functions with hydrogen and oxygen. This research investigates predictability for a gate recurrent unit (GRU) optimized by a modified Prairie Dog Optimizer in PEMFCs. The results obtained have been validated by applying a case study and then a comparison is conducted among the outcomes of the offered technique and 2 other published methods: modified relevance vector machine and Lattice Gated Recurrent Unit (LGRU). The voltage clearly changes significantly, as demonstrated by simulations, even though the FC is handled with a low starting temperature and current. Also, the voltage point distribution has become more concentrated when the current and temperature are high. In both the training and prediction phases, the MAPE is reduced to approximately 0.0043 and 0.0047, respectively, showing that the proposed GRU technique produces superior prediction results when the operational settings approach the optimum operating conditions. According to simulations, the proposed IPDO/GRU with a 0.004 root mean square has the least error, followed by the mRVM and GRU with 0.009 and 0.010 root mean square values. The outcomes show that using the offered procedure does provide the finest verification of the empirical data. • A method for forecasting PEMFC performance. • Optimized CNN to design an experimental nonlinear model of the PEMFC. • Using an optimized gate recurrent unit (GRU) for this purpose. • Optimizing GRU by a modified Prairie Dog Optimizer in PEMFCs. [ABSTRACT FROM AUTHOR]

In this paper, we consider fast numerical methods for linear systems arising from implicit Runge-Kutta temporal discretization methods (based on the fourth-order, 2-stage Gauss method) for one- and two-dimensional Riesz space fractional advection-diffusion equations (RSFADEs). An implicit Runge-Kutta-standard/shifted Grünwald difference scheme for RSFADEs is introduced, and its stability and convergence are also studied. In the one-dimensional case, the coefficient matrix of the discretized linear system is the sum of an identity matrix, a Toeplitz matrix and a square of Toeplitz matrix. We construct a class of Toeplitz times Toeplitz splitting (TTTS) iteration methods to solve the corresponding linear systems. We prove that it converges uniformly to the exact solution without imposing any additional condition, and the optimal parameters for the TTTS iteration method are given. Meanwhile, we design an induced sine transform based preconditioner for two-dimensional problems to accelerate the convergence rate of the conjugate gradient method. Theoretically, we prove that the spectra of the preconditioned matrices of the proposed methods are clustering around 1. Numerical results are presented to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the analysis and the numerical solution of the time-space fractional Schnakenberg reaction-diffusion model with a fixed time delay. This model is a natural system of autocatalysis, which often occurs in a variety of biological systems. The numerical solutions are obtained by constructing an efficient numerical algorithm to approximate Riesz-space and Caputo-time fractional derivatives. More precisely, the L 1 approximation is applied to discretize the temporal Caputo fractional derivative, while the Legendre-Galerkin spectral method is used to approximate the spatial fractional operator. The described method is shown to be unconditionally stable, with a 2 − β convergent order in time and an exponential rate of convergence in space in case of the smoothness of the solution. The error estimates for the obtained solution are derived by applying a proper discrete fractional Grönwall inequality. Moreover, we offer numerical simulations that demonstrate a close match with the theoretical study to evaluate the efficacy of our methodology. [ABSTRACT FROM AUTHOR]

Nano-electrochemistry, i.e. , the research of the properties of nano-(structured) electrodes and their influence on electrochemical processes when immersed inside an electrolyte, represents a hot topic in view of applications in nano-electronics, electro-catalysis and energy storage devices. The role of physical confinement in the electrochemical fabrication and performances of the respective systems have been recently addressed in the context of metal-organic networks on surfaces, but rarely of nano-structured bare metal surfaces, for instance, regularly stepped (vicinal) surfaces. In this work we investigate the interplay between physical confinement and adsorbate induced restructuring by the electrochemical adsorption of sulfate anions on the flat and two distinctly different vicinal Cu(111) surfaces. Sulfate adsorption on the flat Cu(111) surface is known to create a long-range ordered Moiré-superstructure with lattice parameters in the 2–4 nm range due to an expansion of the topmost layer of copper atoms with respect to the underlying crystal planes. This restructuring is also observed on a vicinal Cu(111) surface whose original terrace width is considerably smaller than the lattice vectors of the sulfate induced Moiré-structure. The results clearly indicate not only that the Moiré formation lifts the physical confinement imposed by the initial terrace width, but also shine more light on the Moiré formation process itself. Such adsorbate induced restructuring, of course, depends on the respective adsorbate – electrode combination, but must, in principle, always be taken into account in order to understand electrochemical processes at nano-structured (and nano-sized) electrode surfaces. [ABSTRACT FROM AUTHOR]

• A semi-implicit energy-preserving scheme is constructed. • Generalized potential and fractional spatial diffusion are considered. • A dissipative generalization of Higgs' boson equation is studied. • The properties of stability and convergence of the numerical model are proved rigorously • The reliability of the simulations is confirmed by the preservation of the numerically modified Hamiltonian of the equations. Naturally preserving dissipative or conservative structures of a given continuous system in discrete analogs is of high demand when designing numerical schemes for dissipative or Hamiltonian partial differential equations. Armed with this fact, the proposed computational study focuses on the issue of considering energy-preserving discrete numerical schemes for the Riesz space-fractional Klein-Gordon equation with a generalized scalar potential. For the sake of more clearness, a combined numerical scheme that owes energy-preserving properties is a target to be achieved here. This is done by adapting Galerkin spectral approximation based on Legendre polynomials for Riesz space Laplacian operator side by side to invoke a Cranck–Nickolson scheme in time direction after making order reduction of the original system and a special second-order approximation of the scalar potential derivative. The numerical properties (stability, uniqueness, and convergence) of the scheme are well investigated. Numerical simulations also show that the proposed scheme can inherit the physical properties as the original problem and the numerical solution is stable and convergent to the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a τ matrix from the coefficient matrix, and use their sum to construct a class of τ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed τ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods. [ABSTRACT FROM AUTHOR]

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω = (2 − α) / α shall give an optimal temporal convergence O (N − (2 − α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme. • Efficient spatial semi-discretization using matrix transfer technique for time-space fractional diffusion problem is established. • Stability and error bounds of full discrete scheme based on fast time-stepping L1 method on graded time meshes are discussed. • Fast and accurate solver for system of equations is designed by discrete sine transform at per time level. • Commendably concise and implementation-friendly double fast algorithm is proposed. [ABSTRACT FROM AUTHOR]

We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation's versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements. [ABSTRACT FROM AUTHOR]

[Display omitted] The simplest spin-orbital model can host a nematic spin-orbital liquid state on the triangular lattice. We provide clear evidence that the ground state of the SU(4) Kugel-Khomskii model on the triangular lattice can be well described by a "single" Gutzwiller projected wave function with an emergent parton Fermi surface, despite it exhibits strong finite-size effect in quasi-one-dimensional cylinders. The finite-size effect can be resolved by the fact that the parton Fermi surface consists of open orbits in the reciprocal space. Thereby, a stripy liquid state is expected in the two-dimensional limit, which preserves the SU(4) symmetry while breaks the translational symmetry by doubling the unit cell along one of the lattice vector directions. It is indicative that these stripes are critical and the central charge is c = 3 , in agreement with the SU(4) 1 Wess-Zumino-Witten conformal field theory. All these results are consistent with the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem. [ABSTRACT FROM AUTHOR]

In this paper, time fractional Riesz space telegraph partial differential equation is proposed. By applying the variable separation condition, the main telegraph equation, which consists of two variables, is reduced to an ordinary differential equation of single variable. This simplifies the problem computationally. Then the well-known Haar wavelets method is developed to derive the approximate solution of the reduced equation which includes low cost and fast calculations. The error bounds for function approximation are established. To illustrate the reliability and capability of the method, some examples are provided. The results show that the proposed algorithm is very simple and effective. [ABSTRACT FROM AUTHOR]

Time-space Jacobi pseudospectral method is constructed to approximate the numerical solutions of the fractional Volterra integro-differential and parabolic Volterra integro-differential equations. We define fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi-Gauss-Lobatto points. The fractional derivative is defined in the modified Atangana-Baleanu derivative defined in the Caputo sense formula at JGL points. Further, we transform the domain of fractional Volterra integro-differential and parabolic Volterra integro-differential equations to the standard interval [ − 1 , 1 ] using variable transformation and function transformation. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The theory of error estimates and convergence analysis for the proposed method is also derived. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse τ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, fast numerical methods are established to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by a weighted and shifted Grünwald formula in time and a fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing a preconditioned Krylov subspace method is developed to solve the Toeplitz linear system derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the resulting Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around 1, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other testing methods. [ABSTRACT FROM AUTHOR]

The AFLOW Library of Crystallographic Prototypes has been updated to include an additional 683 entries, which now reaches 1,783 prototypes. We have also made some changes to the presentation of the entries, including a more consistent definition of the AFLOW-prototype label and a better explanation of our choice of space group when the experimental data is ambiguous. A method is presented for users to submit new prototypes for the Encyclopedia. We also include a complete index linking to all the prototypes currently in the Library. [Display omitted] [ABSTRACT FROM AUTHOR]

Signal processing is often affected by various sources of noise that can distort or modify the signals. Removing these noises from the original signal is a crucial step in signal processing, and researchers have proposed several approaches to address this issue. However, achieving an optimized solution remains a challenge. In this study, we introduce a novel approach called the Hybrid Ebola-based Reptile Search (HERS) model based on Time Fractional Diffusion Equation (TFDE). The TFDE is a conventional diffusion equation used for preserving the peak smoothness of spectra signals. In our proposed technique, we consider the processing spectrum of the signal as the reference signal, which serves as the design for the diffusion equation. By applying the diffusion function, we achieve signal peak preservation and smoothing, referred to as the filtering of diffusion. One potential challenge with the time fractional order diffusion equation is its susceptibility to variations in the time step size. To address this, we employ the HERS algorithm to select an optimal time step size that enables efficient signal smoothing. To validate the effectiveness of the proposed technique, we conduct simulations and compare the results with conventional techniques such as the wavelet model, Savitzky-Golay, and regularization techniques. The performance evaluation confirms the superiority of our proposed HERS-TFDE approach in noise removal and signal smoothing. This research aims to contribute to the development of an optimized solution for noise removal in signal processing, leveraging the Hybrid Ebola Reptile Search algorithm and TFDE. The findings have the potential to enhance various signal-processing applications where noise reduction is critical. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Sine-Gordon Equations with the Riesz space fractional derivative. Firstly, a fourth-order conservative difference scheme is proposed for the one-dimensional problem. The unique solvability, conservation and boundedness of the difference scheme are rigorously demonstrated. It is proved that the scheme is convergent at the order of O (τ 2 + h 4) in the l ∞ norm, where τ , h are the time and space step, respectively. Subsequently, the proposed difference scheme is extended to solve the two-dimensional problem. Combining the Revised Newton method, conjugate gradient method and fast Fourier transform, a fast method is proposed for the implementation of the proposed numerical schemes. Finally, several numerical examples are provided to verify the correctness of the theoretical results and the efficiency of the proposed fast algorithm. [ABSTRACT FROM AUTHOR]

Doping transparent conducting oxides in the delafossite form with Ni has been presented in the literature as an exciting candidate to improving the conductivity while maintaining the transparency of these materials. Here, the effects of 6.25% Ni doping on the electronic, structural, and hole effective masses in the 2H phase of XCrO 2 (X = Cu, Ag) is studied using spin polarized ab initio calculations. 6.25% Ni doping is found to produce an asymmetry across spin in both materials, as well as decrease the bandgaps, potentially harming transparent character. Hole effective masses are calculated to be heavier in every lattice vector direction for CuCrO 2 , but in AgCrO 2 hole effective masses are lighter overall. Considering that 6.25% Ni doping introduces an increase in hole concentration, AgCr 0.94 Ni 0.06 O 2 should have a higher conductivity as result of 6.25% Ni doping, while the same doping may reduce the conductivity of CrCr 0.94 Ni 0.06 O 2. [ABSTRACT FROM AUTHOR]

To discretize the distributed-order term of two-dimensional nonlinear Riesz space fractional diffusion equation, we consider the high accuracy Gauss-Legendre quadrature formula. By combining an s -stage implicit Runge-Kutta method in temporal direction with a spectral Galerkin method in spatial direction, we construct a numerical method with high global accuracy. If the nonlinear function satisfies the local Lipschitz condition, the s -stage implicit Runge-Kutta method with order p (p ≥ s + 1) is coercive and algebraically stable, then we can prove that the proposed method is stable and convergent of order s + 1 in time. In addition, we also derive the optimal error estimate for the discretization of distributed-order term and spatial term. Finally, numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Suppose that A is a nonnegative n × m real matrix. The NMF problem is the determination of two nonnegative real matrices F , V so that A = F V with intermediate dimension p , smaller than m i n { n , m }. In this article we present a general method for the determination of two nonnegative real factors F , V of A , based on the theory of lattice-subspaces and positive bases. [ABSTRACT FROM AUTHOR]

The security of most lattice based cryptography relies on the hardness of computing a shortest nonzero vector of lattices. We say that a lattice basis is SV-reduced if it contains a shortest nonzero vector of the lattice. In this paper, we prove that, π ∕ 6 orthogonality between the shortest vector of the basis and the vector space spanned by other vectors of the basis is enough to be SV-reduced under the assumption that a plausible condition C n holds. By using the π ∕ 6 orthogonality under C 2 , we prove a new complexity bound log 3 ∥ x ∥ 2 3 det L + 1 for Gauss–Lagrange algorithm which clarifies why the currently known complexity is so far fall short to expose the efficiency of the algorithm we experience in practice. Our experiments suggest that our complexity bound of Gauss–Lagrange algorithm is somewhat close to actual efficiency of the algorithm. We also show that LLL(δ) algorithm outputs a S V -reduced basis if δ ≥ 1 ∕ 3 for two dimensional lattice. We present an efficient three dimensional SV-reduction algorithm by using the condition C 3 and π ∕ 6 orthogonality and how to generalize the algorithm for higher dimension. [ABSTRACT FROM AUTHOR]

Ternary Al–Mg–Si alloys have been modelled based on a multi-scale approach that spans across atomistic and mesoscale models and uses theoretically determined parameters. First, a cluster expansion model for total energy has been trained for atomistic configurations (FCC lattice) based on the data from density functional simulations of electronic structure. Free energy curves as a function of solute (Mg, Si) concentrations and disorder have been obtained by using this parameterisation together with meta-dynamics Monte Carlo sampling. In addition, free energy data, surface tensions as well as strain energy using the linear elasticity theory have been collected to be combined for a mesoscale phase-field model. The application of this approach shows that the formation of a layered MgSi phase, with (100) planes, is a particularly stable solute aggregation motif within the Al host matrix. Moreover, the phase-field model demonstrates that the preferred shape of the MgSi precipitates is needle-like (in FCC), and they can act as precursors for the important and well-known β ″-type precipitates which are formed by translating one Mg column by a 1/2 lattice vector. The results provide theoretical evidence that the solute aggregation into needle-like MgSi domains (precipitates) is an inherent property of Al-Mg-Si alloys, and that it takes place even without the presence of vacancies which is a precondition for the eventual formation β ″ precipitates. Image, graphical abstract [ABSTRACT FROM AUTHOR]

A lattice (d , k) -polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let δ (d , k) be the largest diameter over all lattice (d , k) -polytopes. We develop a computational framework to determine δ (d , k) for small instances. We show that δ (3 , 4) = 7 and δ (3 , 5) = 9 ; that is, we verify for (d , k) = (3 , 4) and (3 , 5) the conjecture whereby δ (d , k) is at most ⌊ (k + 1) d ∕ 2 ⌋ and is achieved, up to translation, by a Minkowski sum of lattice vectors. [ABSTRACT FROM AUTHOR]

In the theory of aggregation, there is a trend towards the relaxation of the axiom of monotonicity and also towards the extension of the definition to other domains besides real numbers. In this work, we join both approaches by defining the concept of directional monotonicity for functions that take values in Riesz spaces. Additionally, we adapt this notion in order to work in certain convex sublattices of a Riesz space, which makes it possible to define the concept of directional monotonicity for functions whose purpose is to fuse uncertain data coming from type-2 fuzzy sets, fuzzy multisets, n -dimensional fuzzy sets, Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets, among others. Focusing on the latter, we characterize directional monotonicity of interval-valued representable functions in terms of standard directional monotonicity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical study for the multi-dimensional fractional-in-space Allen-Cahn equation with homogeneous Dirichlet boundary condition. By utilizing Strang's second-order splitting method, at each time step, the numerical scheme can be split into three sub-steps. The first and third sub-steps give the same ordinary differential equation, where the solutions can be obtained explicitly. While a multi-dimensional linear fractional diffusion equation needs to be solved in the second sub-step, and this is computed by the Crank-Nicolson scheme together with alternating directional implicit (ADI) method. Thus, instead of solving a multidimensional nonlinear problem directly, only a series of one-dimensional linear problems need to be solved, which greatly reduces the computational cost. A fourth-order quasi-compact difference scheme is adopted for the discretization of the space Riesz fractional derivative of α (1 < α ≤ 2). The proposed method is shown to be unconditionally stable in L 2 -norm, and satisfying the discrete maximum principle under some reasonable time step constraint. Finally, numerical experiments are given to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

In ICITS 2015, Walter studied the worst case computational cost to enumerate short lattice vectors on two well-known block reduced bases, i.e., BKZ reduced bases and slide reduced bases. Until then, existing works analyzed only extreme preprocessed bases, e.g., LLL reduced bases that are the weakest ones and quasi-HKZ reduced bases that are the strongest ones; hence, Walter tried to interpolate these results. For this purpose, Walter tried to calculate enumeration cost on block reduced bases of arbitrary blocksizes. The topic should be theoretically interesting since hardness of the lattice problem relates to the security of lattice-based cryptography. In this paper, we revisit the problem with refined analyses. For both BKZ and slide reduced bases, we show that the worst case enumeration costs are smaller than Walter's analyses. In particular, we show that our results are the best possible ones when we follow Walter's approach. Furthermore, we extend Walter's result for slide reduced bases. Since Walter only studied the original slide reduced bases proposed by Gama and Nguyen, he did not analyze arbitrary blocksizes. To extend the analyses to arbitrary blocksizes, we use the generalized slide reduction that was defined by Li and Wei. As a side contribution, we also show that the worst case behaviors of the generalized slide reduced bases are better than Li and Wei's analyses. We obtain all these results by exploiting better geometric properties of block reduced bases. [ABSTRACT FROM AUTHOR]

Consider an arbitrary n -dimensional lattice Λ such that Z n ⊂ Λ ⊂ Q n . Such lattices are called rational and can always be obtained by adding m ⩽ n rational vectors to Z n. The defect d (E , Λ) of the standard basis E of Z n (n unit vectors going in the directions of the coordinate axes) is defined as the smallest integer d such that certain (n − d) vectors from E together with some d vectors from the lattice Λ form a basis of Λ. Let | | ⋅ | | be L 1 -norm on Q n. Suppose that for each non-integer x ∈ Λ inequality | | x | | > 1 holds. Then the unit octahedron O n = x ∈ R n : | | x | | ⩽ 1 is called admissible with respect to Λ and d (E , Λ) is also called the defect of the octahedron O n with respect to E and is denoted by d (O E n , Λ). Let d n m = max Λ ∈ A m d (O E n , Λ) , where A m is the set of all rational lattices Λ such that (1) O n is admissible w.r.t. Λ and (2) Λ can be obtained by adding m rational vectors to Z n : Λ = Z n , a 1 , ... , a m Z for some a 1 , ... , a m ∈ Q n. In this article we show that there exists an absolute positive constant C such that for any m < n d n m ⩽ C n ln (m + 1) ln n m ln ln n m m 2 This bound was also claimed in [2] and [1] , however the proof was incorrect. In this article along with giving correct proof we highlight substantial inaccuracies in those articles. [ABSTRACT FROM AUTHOR]

We report on low twinning stress of 0.2 MPa for type 1 twin boundaries with rational (101) twinning plane and 0.1 MPa for type 2 twin boundaries with rational [ 1 ¯ 01] twinning direction, in Ni 2 MnGa 0.8 Fe 0.2 seven-layer modulated martensite exhibiting about 12% stress-induced strain. The studied samples were free of compound twins with (100) and (110) twinning planes. The martensite lattice is monoclinic and shows a long-period commensurate modulation with wave vector q = (2/7) g 110 , where g 110 is the reciprocal lattice vector. The investigated alloy has high potential for practical applications due to the low twinning stress and the elevated Curie point T C =400 K. Image, graphical abstract [ABSTRACT FROM AUTHOR]

During the last decade, automatic data analysis methods concerning different aspects of crystal analysis have been developed, e.g., unsupervised primitive unit cell extraction and automated crystal distortion and defects detection. However, an automatic, unsupervised motif extraction method is still not widely available yet. Here, we propose and demonstrate a novel method for the automatic motif extraction in real space from crystalline images based on a variational approach involving the unit cell projection operator. Due to the non-convex nature of the resulting minimization problem, a multi-stage algorithm is used. First, we determine the primitive unit cell in form of two lattice vectors. Second, a motif image is estimated using the unit cell information. Finally, the motif is determined in terms of atom positions inside the unit cell. The method was tested on various synthetic and experimental HAADF STEM images. The results are a representation of the motif in form of an image, atomic positions, primitive unit cell vectors, and a denoised and a modeled reconstruction of the input image. The method was applied to extract the primitive cells of complex μ -phase structures Nb 6.4 Co 6.6 and Nb 7 Co 6 , where subtle differences between their interplanar spacings were determined. • Novel method for unsupervised motif and unit cell extraction from crystalline images. • The performance is demonstrated on several HAADF STEM images and simulated images. • The motif is extracted in terms of a motif image and as atomic positions. • The motif image can also be extracted from non-crystalline periodic images. • Determines subtle differences in interplanar spacings of complex mu-phase structures. [ABSTRACT FROM AUTHOR]

The radial basis functions collocation method is developed to solve Riesz space fractional advection-dispersion equation (RSFADE). To do this, we first provide the Riemann-Liouville fractional derivatives of the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. Then a method of lines, implemented as a meshless method based on spatial trial spaces spanned by the RBFs is developed for the numerical solution of the RSFADE. Numerical experiments are presented to validate the newly developed method and to investigate accuracy and efficiency. The numerical rate of convergence in space is computed. The stability of the linear systems arising from discretizing the Riesz fractional derivative with RBFs is also analysed. [ABSTRACT FROM AUTHOR]

This paper studies the structural, electronic and optical properties of a new group IV-V two-dimensional (2D) material GeP under the influence of different strains by using the first-principles calculations. The strains discussed in this paper include both the uniaxial strain along the different lattice vector directions, as well as the biaxial strain. The results show that monolayer GeP is a 2D dynamically stable semiconductor with low structural symmetry. In terms of the electronic property, regardless of the compressive strain or tensile strain applied in any direction, the band-gap of the monolayer GeP exhibits a decreasing tendency, and it remains semiconductivity in the range of −10% compressive strain to 10% tensile strain. As for the optical property, it is anisotropic. When GeP is subjected to a strain, regardless of the directions, the optical property of monolayer GeP is significantly affected more by tensile strain than by compressive strain. When the monolayer GeP is subjected to a tensile strain, its optical absorption and reflectivity in the visible light region to some extent increase, while its transmittance reduces. When it is subjected to a compressive strain, the main change in optical properties such as absorption and reflectivity occurs in the deep ultraviolet region. In particular, the monolayer GeP is most susceptible to deformation by the strain in a direction, and only compressive strain in a direction will cause GeP to undergo a transition from an indirect band-gap semiconductor to a direct band-gap semiconductor, which results in the improvements of absorption efficiency and reflectivity efficiency in the low energy region. Our calculated results indicate that the electronic and optical properties of monolayer GeP can be efficiently and regularly regulated by strain. In the future, it can be a suitable 2D materials used in optically polarized devices, blue light-emitting diode devices, and strain-tunable photoelectric nanodevices. • Monolayer GeP is a 2D semiconductor with highly dynamically stable as well as low structural symmetry. • The structural, electronic and optical properties could be efficiently and regularly regulated by in-layer strain. • The optical property of monolayer GeP is anisotropic. • When GeP is subjected to strain, regardless of the direction of the strain, the optical property of monolayer GeP is significantly affected more by tensile strain than by compressive strain. • The monolayer GeP is most susceptible to deformation by strain in a direction, and only compressive strain in a direction will cause GeP to undergo a transition from an indirect band-gap semiconductor to a direct band-gap semiconductor, which results in the improvements of absorption efficiency and reflectivity efficiency in the low energy regions. [ABSTRACT FROM AUTHOR]

• A multi-term time fractional and Riesz space distributed-order wave equation is discussed. • Firstly the equation is transformed into a multi-term time-space fractional wave equation. • The equation is solved by discretising using C–N scheme and the FEM with an unstructured mesh. • Stability and convergence are investigated. • Some examples are provided to show the effectiveness and correctness of the proposed numerical method. In this paper, the numerical analysis for a multi-term time fracstional and Riesz space distributed-order wave equation is discussed on an irregular convex domain. Firstly, the equation is transformed into a multi-term time-space fractional wave equation using the mid-point quadrature rule to approximate the distributed-order Riesz space derivative. Next, the equation is solved by discretising in time using a Crank–Nicolson scheme and in space using the finite element method (FEM) with an unstructured mesh, respectively. Furthermore, stability and convergence are investigated by introducing some important lemmas on irregular convex domain. Finally, some examples are provided to show the effectiveness and correctness of the proposed numerical method. [ABSTRACT FROM AUTHOR]

A new class of fractional differential equations with the Riesz–Caputo derivative is proposed and the physical meaning is introduced in this paper. The boundary value problem is investigated under some conditions. Leray–Schauder and Krasnoselskii's fixed point theorems in a cone are adopted. Existence of positive solutions is provided. Finally, two examples with numerical solutions are given to support theoretical results. [ABSTRACT FROM AUTHOR]

The notion of d -ideal has been abstracted by Martínez and Zenk from Riesz spaces to algebraic frames. They introduced the d -nucleus and d -elements. In this paper we extend several characterizations of d -elements that parallel similar characterizations of d -ideals in rings. For instance, calling a coherent map between algebraic frames "weakly skeletal" if it maps compact elements with equal pseudocomplements to images with equal pseudocomplements, we show that an a ∈ L is a d -element if and only if a = h ⁎ (0) for some weakly skeletal map h : L → M , where h ⁎ denotes the right adjoint of h. The sublocale of L induced by the d -nucleus is denoted by dL. We characterize when d (L ⊕ M) ≅ d L ⊕ d M. We weaken the notion of d -element by defining eL to be the set of elements that are joins of double pseudocomplements of compact elements. We show that if L = d L and M = d M , then L ⊕ M = e (L ⊕ M). Clearly, d L ⊆ e L. We give an example to show that the containment can be proper. Finally, we show that dL is always a sublocale of eL. [ABSTRACT FROM AUTHOR]

An important step in the efficient computation of multi-dimensional theta functions is the construction of appropriate symplectic transformations for a given Riemann matrix assuring a rapid convergence of the theta series. An algorithm is presented to approximately map the Riemann matrix to the Siegel fundamental domain. The shortest vector of the lattice generated by the Riemann matrix is identified exactly, and the algorithm ensures that its length is larger than 3 ∕ 2. The approach is based on a previous algorithm by Deconinck et al.. using the LLL algorithm for lattice reductions. Here, the LLL algorithm is replaced by exact Minkowski reductions for small genus and an exact identification of the shortest lattice vector for larger values of the genus. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new explicit fourth-order scheme for solving Riesz space fractional nonlinear wave equations is developed. The scheme is designed by using a novel Riesz space fractional difference operator for spatial discretization and a multidimensional extended Runge–Kutta–Nyström method for time integration. The conservation law of the semi-discrete energy, stability and convergence of the semi-discrete system are investigated. Numerical experiments show the efficiency and energy conservation of the present scheme. [ABSTRACT FROM AUTHOR]

Abstract In this work we propose an algorithmic process that finds the minimum-cost insured portfolio in the case where the space of marketed securities is a subspace of C [ a, b ]. This process uses, effectively, the theory of positive bases in Riesz spaces and does not require the presence of linear programming methods. The key for finding the minimum-cost insured portfolio is the existence of a positive basis. Until know, we could check, under a rather complicated procedure, the existence of a positive basis in a prescribed interval [ a, b ]. In this paper we propose a heuristic method for computing appropriate intervals [ a, b ], so that the existence of a positive basis is guaranteed. All the proposed algorithmic processes are followed by appropriate Matlab code. [ABSTRACT FROM AUTHOR]