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Line‐scan mode enables rapid and high‐throughput imaging through the development of an appropriate optical transverse needle focus. Diffraction gratings allow the generation of a line focus, but they face the challenge of low light power utilization due to multiple high‐order diffractions. In addition, designing the focus requires the selection of functional materials. Atomically thin transition metal dichalcogenides with high dielectric constants provide significant phase shifts to incident light by utilizing phase singularity at zero reflection. However, at zero reflection, no light power is available for utilization, necessitating a balance between phase and amplitude modulation. In this work, the aforementioned issues are addressed by designing a monolayer MoS2 based Fresnel strip metalens. An optical needle primary focus with a transverse length of 40 µm (≈80 λ, the longest value reported to date), a sub‐diffraction‐limited lateral spot, and a broad range of working wavelengths are achieved using the metalens. The metalens not only concentrate light power in the primary diffraction order by overcoming the constraint of momentum conservation but also maintains a constant phase modulation between distinct strips. The novel method for optical manipulation present here holds great promise for applications in biology, oncology, nanofabrication, energy harvesting, and optical information processing. [ABSTRACT FROM AUTHOR]

The Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods. [ABSTRACT FROM AUTHOR]

Low‐energy electron diffraction patterns contain precise information about the structure of the surface studied. However, retrieving the real space lattice periodicity from complex diffraction patterns is challenging, especially when the modeled patterns originate from superlattices with large unit cells composed of several symmetry‐equivalent domains without a simple relation to the substrate. This work presents ProLEED Studio software, built to provide simple, intuitive and precise modeling of low‐energy electron diffraction patterns. The interactive graphical user interface allows real‐time modeling of experimental diffraction patterns, change of depicted diffraction spot intensities, visualization of different diffraction domains, and manipulation of any lattice points or diffraction spots. The visualization of unit cells, lattice vectors, grids and scale bars as well as the possibility of exporting ready‐to‐publish models in bitmap and vector formats significantly simplifies the modeling process and publishing of results. [ABSTRACT FROM AUTHOR]

We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation T=Ta+Tc$T = T_a + T_c$, where Ta$T_a$ is the sum of an absolutely order summable family of disjointness preserving operators and Tc$T_c$ is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators. [ABSTRACT FROM AUTHOR]

Diffraction elastic constants (DECs) describe the elastic response of a subset of orientation‐correlated grains which share a common lattice vector. DECs reflect the elastic behaviour of the single‐crystal constituents through their dependence on grain orientation. DECs furthermore depend on the behaviour of the polycrystal aggregate both through the dependence on preferred orientation and through the average elastic interaction of the grains in the subset with their surroundings. The latter is also known as grain–matrix interaction which is grain‐shape dependent. Both dependencies can make the DECs uniquely sensitive to the elastic effects of the grain shape, texture and phase composition. Several micro‐mechanical models are explored for use in calculating both DECs and overall elastic constants. Furthermore, it is shown how discrete data from electron backscatter diffraction on grain shape, grain orientations and neighbouring grains can be used for DEC calculations. Lastly, the inverse problem of calculating single‐crystal elastic constants from DECs is discussed in detail. All calculations discussed in this work can be verified using the freely available computer program IsoDEC. [ABSTRACT FROM AUTHOR]

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification EC$E_{\mathbb {C}}$ of a real vector lattice E and introduce the lateral order on EC$E_{\mathbb {C}}$. Our first main result asserts that the set of all fragments Fv$\mathfrak {F}_v$ of an element v∈EC$v\in E_{\mathbb {C}}$ of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification EC$E_{\mathbb {C}}$ of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator T:EC→X$\mathcal {T}: E_{\mathbb {C}} \rightarrow X$ from the complexification EC$E_{\mathbb {C}}$ of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator T$\mathcal {T}$ from EC$E_{\mathbb {C}}$ to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from EC$E_{\mathbb {C}}$ to a complex Banach lattice (ℓp(D)C$(\ell _p(\mathcal {D})_{\mathbb {C}}$ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators. [ABSTRACT FROM AUTHOR]

In this article, we study inverse source problems for time‐fractional diffusion equations from a posteriori boundary measurement. Using the memory effect of these class of equations, we solve these inverse problems for several class of space‐ or time‐dependent source terms. We prove also the unique determination of a general class of space–time‐dependent separated variables source terms from such measurement. Our approach is based on the study of singularities of the Laplace transform in time of boundary traces of solutions of time‐fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order (β+1)∈(1,2)${(\beta +1)}\in (1,2)$ whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order β∈(0,1)$\beta \in (0,1)$. A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models. [ABSTRACT FROM AUTHOR]

Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli‐reduced cells (based on the three shortest non‐coplanar lattice vectors) or by Delaunay‐reduced cells (based on four non‐coplanar vectors summing to zero and all meeting at obtuse or right angles) is commonly performed. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner–Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point as they are to any other lattice point. The three non‐coplanar lattice vectors chosen are here called the Niggli‐reduced cell edges. Starting from a Niggli‐reduced cell, the Dirichlet cell is characterized by the planes determined by 13 lattice half‐edges: the midpoints of the three Niggli cell edges, the six Niggli cell face‐diagonals and the four body‐diagonals, but seven of the lengths are sufficient: three edge lengths, the three shorter of each pair of face‐diagonal lengths, and the shortest body‐diagonal length. These seven are sufficient to recover the Niggli‐reduced cell. [ABSTRACT FROM AUTHOR]

The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper proposes the shifted Legendre Gauss–Lobatto collocation (SL‐GLC) scheme to solve two‐dimensional space‐fractional coupled reaction–diffusion equations (SFCRDEs). The proposed method is implemented by expressing the function and its spatial fractional derivatives as a finite expansion of shifted Legendre polynomials. Then the expansion coefficients are determined by reducing the SFCRDEs with their initial and boundary conditions to a system of ordinary differential equations for these coefficients. Subsequently, we applied the proposed method to discretize the temporal and spatial variables to convert the two‐dimensional spacetime fractional coupled reaction–diffusion equations (STFCRDEs) to a system of algebraic equations. Some results regarding the error estimation are obtained. Several examples are discussed to validate the capability and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this article, a novel fourth‐order accurate difference method is derived for the distributed‐order Riesz space fractional diffusion equation in one‐dimensional (1D) and two‐dimensional (2D) cases, respectively. First, the distributed integral terms are discretized by using the Simpson quadrature rule into the multi‐term Riesz space fractional diffusion equations. Then, a fourth‐order accurate difference scheme is presented to approximate the multi‐term Riesz fractional diffusion equations. Moreover, the proposed difference schemes are proved to be unconditionally stable and convergent in L2$$ {L}_2 $$ norm for both 1D and 2D cases. Finally, numerical experiments are given to verify the efficiency of the schemes. [ABSTRACT FROM AUTHOR]

The accurate and efficient computation of the electromagnetic response of objects made from artificial materials is crucial for designing photonic functionalities and interpreting experiments. Advanced fabrication techniques can nowadays produce new materials as 3D lattices of scattering unit cells. Computing the response of objects of arbitrary shape made from such materials is typically computationally prohibitive unless an effective homogeneous medium approximates the discrete material. In here, a homogenization method based on the effective transition (T‐)matrix, Teff${T_{eff}}$ is introduced. Such a matrix captures the exact response of the discrete material, is determined by the T‐matrix of the isolated unit cell and the material lattice vectors, and is free of spatial dispersion. The truncation of Teff${T_{eff}}$ to dipolar order determines the common bi‐anisotropic constitutive relations. When combined with quantum‐chemical and Maxwell solvers, the method allows one to compute the response of arbitrarily‐shaped volumetric patchworks of structured molecular materials and metamaterials. [ABSTRACT FROM AUTHOR]

The aggregation of molecules is usually associated with a specific type of interaction, which can be altered by thermodynamic conditions. Under normal conditions, the crystal structure of selenourea, SeC(NH2)2, phase α is trigonal, space group P31, Z = 27. Its large number of independent molecules (Zα′ = 9) can be associated with the formation of an NH...N hydrogen bond substituting one of 36 independent NH...Se hydrogen bonds, which prevail among intermolecular interactions. Phase α approximates the trigonal structure with a threefold smaller unit cell (Z = 9), which in turn approximates another still threefold smaller unit cell (Z = 3). The temperature‐induced transformations of selenourea have been characterized by calorimetry and by performing 21 single‐crystal X‐ray diffraction structural determinations as a function of temperature. At 381.0 K, phase α undergoes a first‐order displacive transition to phase γ, with space group P3121 and Z reduced to 9, when the NH...N bond is broken and an NH...Se bond is formed in its place. Previously, an analogous competition was observed between NH...N and NH...O hydrogen bonds in high‐pressure phase III of urea. The lattice vectors along the (001) plane in low‐ and high‐temperature phases of selenourea are related by a similarity rule, while the lattice dimensions along direction c are not affected. This similarity rule also applies to the structures of phase γ and hypothetical phase δ (Z = 3). The thermally controlled transition between enantiomorphic phases of selenourea contrasts with its high‐pressure transition at 0.21 GPa to a centrosymmetric phase β, where both the NH...Se and NH...N bonds are present. The compression and heating reduce the number of independent molecules from Z′ = 9 in phase α, to Z′ = 2 in phase β and to Z′ = 1.5 in phase γ. [ABSTRACT FROM AUTHOR]

The security of lattice‐based cryptosystems is generally based on the hardness of the Shortest Vector Problem (SVP). The original enumeration (ENUM) algorithm solving SVP runs in exponential time due to the exhaustive search, which is used as a subroutine for the block Korkin–Zolotarev (BKZ) algorithm. It is a critical issue to reduce the computational complexity of ENUM. In this paper, first, we improve the reordering method proposed by Wang et al. in ACISP 2018. We call our proposed method DPR, which permutates the projected dual lattice vectors by decreasing norms. Preliminary experimental results show that the proposed reordering methods can reduce the ENUM complexity compared to the predecessor; for instance, DPR reduces around 32.8% on average in 45‐dimensional lattices. Moreover, the authors' simulation shows that the higher the lattice dimension, the more DPR can reduce the ENUM complexity. In addition, we study a condition for deciding when the reordering method shall be executed or not. Finally, we improve the BKZ algorithm with DPR methods and the proposed condition. [ABSTRACT FROM AUTHOR]

A noniterative approach to generation of binary phase holograms is applied for synthesizing complex optical vortex arrays. This approach does not use inverse Fourier analysis and allows one to obtain arbitrary optical vortex arrays with specified topological charges, which cannot be obtained using conventional fork‐shaped gratings. A topological charge of individual optical vortices generated using binary phase holograms can be specified at a given spatial frequency, so that the equidistant array of optical vortices can be generated via illuminating a hologram by a beam with the zero topological charge. The results of the calculation are consistent with the experimental data, including those of the interferometric measurements. The approach also makes it possible to synthesize superimposed optical vortices, where an optical field represents well‐ordered circular arrays of optical singularities. An unusual behavior of phase dislocations in superimposed structures is found. For two optical vortices of same reciprocal lattice vectors, but different topological charges, spatial distribution of singularities are reconfigured in a such manner that a couple of concentric circular arrays of singularities appear. The proposed binary phase holograms offer new opportunities for synthesizing the complex optical vortex light fields, which can find light–matter interaction‐based applications. [ABSTRACT FROM AUTHOR]

Bragg coherent X-ray diffractive imaging is a cutting-edge method for recovering three-dimensional crystal structure with nanoscale resolution. Phase retrieval provides an atomic displacement parallel to the Bragg peak reciprocal lattice vector. The derivative of the displacement along the same vector provides the normal strain field, which typically serves as a proxy for any structural changes. In this communication it is found that the other component of the displacement gradient, perpendicular to the reciprocal lattice vector, provides additional information from the experimental data collected from nanocrystals with mobile dislocations. Demonstration on published experimental data show how the perpendicular component of the displacement gradient adds to existing analysis, enabling an estimate for the external stresses, pinpointing the location of surface dislocations, and predicting the dislocation motion in in situ experiments. [ABSTRACT FROM AUTHOR]

The epitaxial growth of molecular crystals at single‐crystalline surfaces is often strongly related to the first monolayer at the substrate surface. The present work presents a theoretical approach to compare three‐dimensional lattices of epitaxially grown crystals with two‐dimensional lattices of the molecules formed within the first monolayer. Real‐space and reciprocal‐space representations are considered. Depending on the crystallographic orientation relative to the substrate surface, proper linear combinations of the lattice vectors of the three‐dimensional unit cell result in a rhomboid in the xy plane, representing a two‐dimensional projection. Mathematical expressions are derived which provide a relationship between the six lattice parameters of the three‐dimensional case and the three parameters obtained for the two‐dimensional surface unit cell. It is found that rotational symmetries of the monolayers are reflected by the epitaxial order. Positive and negative orientations of the crystallographic contact planes are correlated with the mirror symmetry of the surface unit cells, and the corresponding mathematical expressions are derived. The method is exemplarily applied to data obtained in previous grazing‐incidence X‐ray diffraction (GIXD) measurements with sample rotation on thin films of the conjugated molecules 3,4;9,10‐perylenetetracarboxylic dianhydride (PTCDA), 6,13‐pentacenequinone (P2O), 1,2;8,9‐dibenzopentacene (trans‐DBPen) and dicyanovinyl‐quaterthiophene (DCV4T‐Et2) grown by physical vapor deposition on Ag(111) and Cu(111) single crystals. This work introduces the possibility to study three‐dimensional crystal growth nucleated by an ordered monolayer by combining two different experimental techniques, GIXD and low‐energy electron diffraction, which has been implemented in the second part of this work. [ABSTRACT FROM AUTHOR]

Nonlinear Raman–Nath diffraction is an effective method for frequency conversion in the presence of phase mismatch. In a structure featuring periodically modulated quadratic nonlinearity, the available frequency‐doubled patterns generated from nonlinear Raman–Nath diffraction constitute uniformly spaced points due to the one parametric decided reciprocal lattice vectors. Herein, nonlinear Raman–Nath second‐harmonic generation from the optical Fibonacci superlattice is theoretically investigated, and a propagation equation that describes the properties of frequency‐doubled waves is presented. Abundant diffraction peaks that are substantially more varied than the traditional Raman–Nath regime are observed. The dependence of the generated second‐harmonic signals on the structural parameters and fundamental laser performance is also comprehensively discussed. [ABSTRACT FROM AUTHOR]

The discovery of ferromagnetic 2D van der Waals (vdW) materials has opened up opportunities for exploring and harnessing magnetism in low‐dimensional limit, and developing innovative paper‐like spintronic devices. Stress engineering demonstrates unique advantages in regulating material properties, especially for 2D materials. With the aid of first‐principles calculations, the effectiveness of strain engineering for modulating the electronic and magnetic properties of MSi2N4 (M = V, Cr) monolayers from a new family of layer‐structured 2D vdW materials is explored. Their magnetic configurations are effectively tuned via the uniaxial and biaxial tensile strain of in‐plane lattice vectors. VSi2N4 exhibits half‐metallic ferromagnetic character, whereas CrSi2N4 is prone to be antiferromagnetic (AFM), which is rationalized from the competition between the direct metal–metal AFM interaction and the 900 ferromagnetic superexchange. More intriguingly, VSi2N4 demonstrates a high magnetic ordering temperature of 285 K, which is close to room temperature. The predicted appealing properties of VSi2N4 under strain await experimental confirmation. [ABSTRACT FROM AUTHOR]

This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction–diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domain x ∈ [0, L] by replacing the second‐order spatial derivatives with the fractional Laplacian operator of order 1 < α ≤ 2, which is classified as superdiffusion process. We examine the resulting coupled reaction–diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral‐like structures, and seemingly complex spatiotemporal distributions. [ABSTRACT FROM AUTHOR]

Recently, the notion of difference operators based on fractional‐order is being extensively used in linear algebra, approximation theory, the theory of fractional calculus (FC), and many others. In this paper, an attempt has been taken for studying the convergence of difference sequence and hence analyzing the consistency and validity of certain related formulas. Investigations on basic results involving convergence, linearity, exponent rule, topological properties, Leibniz, and chain rules for fractional derivatives have been incorporated. In this context, some well‐known results have been demonstrated and verified with the help of some illustrative examples. [ABSTRACT FROM AUTHOR]

The task of determining the orientations of crystals is usually performed by indexing reflections detected on diffraction patterns. The well known underlying principle of indexing methods is universal: they are based on matching experimental scattering vectors to some vectors of the reciprocal lattice. Despite this, the standard attitude has been to devise algorithms applicable to patterns of a particular type. This paper provides a broader perspective. A general approach to indexing of diffraction patterns of various types is presented. References are made to formally similar problems in other research fields, e.g. in computational geometry, computer science, computer vision or star identification. Besides a general description of available methods, concrete algorithms are presented in detail and their applicability to patterns of various types is demonstrated; a program based on these algorithms is shown to index Kikuchi patterns, Kossel patterns and Laue patterns, among others. [ABSTRACT FROM AUTHOR]

Simple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi‐dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi‐dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically. [ABSTRACT FROM AUTHOR]

We derived a new expression for the electrostatic interaction of three‐dimensional charge‐neutral systems with two‐dimensional periodic boundary conditions (slab geometry) using a fast multipole method (FMM). Contributions from all the image cells are expressed as a sum of real and reciprocal space terms, and a self‐interaction term. The reciprocal space contribution consists of two parts: zero and nonzero terms of the absolute value of the reciprocal lattice vector. To test the new expressions, electrostatic interactions were calculated for a randomly placed charge distribution in a cubic box and liquid water produced by molecular dynamics calculation. The accuracy could be controlled by the degree of expansion of the FMM. In the present expression, the computational complexity of the electrostatic interaction of N‐particle systems is order N, which is superior to that of the conventional two‐dimensional periodic Ewald method for a slab geometry and the particle mesh Ewald method with a large empty space at an interface of the unit cell. © 2020 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three‐by‐three matrices that transform three‐space lattice vectors. Using those three‐by‐three matrices when working in the six‐dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three‐space representation, apply the three‐by‐three matrices and then back‐transform to the six‐space representation, but it is much simpler to have the equivalent six‐by‐six matrices and apply them directly. The general form of the transformation from the three‐space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.) [ABSTRACT FROM AUTHOR]

The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three‐by‐three matrices that transform three‐space lattice vectors. Using those three‐by‐three matrices when working in the six‐dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three‐space representation, apply the three‐by‐three matrices and then back‐transform to the six‐space representation, but it is much simpler to have the equivalent six‐by‐six matrices and apply them directly. The general form of the transformation from the three‐space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.) [ABSTRACT FROM AUTHOR]

A full implementation of the analytical stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian‐type orbitals as basis functions. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518–2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange‐correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097–3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn–Sham matrix formation. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Atomic structures and migration mechanisms of interphase boundaries have been of scientific interest for many years owing to their significance in the field of phase transformations. Though the interphase boundary structures can be deduced from crystallographic investigations, the detailed atomic structures and migration mechanisms of interphase boundaries during phase transformations are still poorly understood. In this study, a systematic study on atomic structures and migration mechanisms of interphase boundaries in a body‐centered cubic (b.c.c.) to face‐centered cubic (f.c.c.) massive transformation was carried out using the phase‐field crystal model. Simulation results show that the f.c.c./b.c.c. interphase boundaries can be classified into faceted interphase boundaries and side surfaces. The faceted interphase boundaries are semi‐coherent with a group of dislocations, leading to a ledge migration mechanism, while the side surfaces are incoherent and thus migrate in a continuous way. After a careful analysis of the simulated migration process of interphase boundaries at atomic scales, a detailed description of the ledge mechanism based on the motion and nucleation of interphase boundary dislocations is presented. The ledge‐forming process is accompanied by the nucleation of new heterogeneous dislocations and motions of original dislocations, and thus the barrier of ledge formation comes from the hindrance of these two dislocation behaviors. Once the ledge is formed, the original dislocations continue to advance until the ledge height reaches 1/|Δg|, where Δg represents the difference in reciprocal lattice vectors between two phases. The new heterogeneous dislocation moves along the radial direction of the interphase boundary, resulting in ledge extension. The interface dislocation behaviors greatly affect the migration of the interphase boundary, leading to different migration kinetics of faceted interphase boundaries under the Kurdjumov–Sachs and the Nishiyama–Wasserman orientation relationships. This study revealed the mechanisms and kinetics of complex structure transition during a b.c.c.–f.c.c. massive phase transformation and can shed some light on the process of solid phase transformations. [ABSTRACT FROM AUTHOR]

The lattice parameters and unit‐cell orientation of an SrLaAlO4 crystal have been determined by means of energy‐dispersive X‐ray Laue diffraction (EDLD) using a pnCCD detector coupled to a columnar structure CsI(Tl) scintillator in the energy range between 40 and 130 keV. By exploiting the high quantum efficiency (QE) achieved by this combined detection system for hard X‐rays, a large number of Bragg reflections could be recorded by the relatively small detector area, allowing accurate and fast determination of the lattice parameters and the moduli of the structure factors. The experiment was performed on the energy‐dispersive diffraction (EDDI) beamline at the BESSY II synchrotron using a pnCCD detector with 128 × 128 pixels. Since the energies and positions of the Laue peaks can be recorded simultaneously by the pnCCD system, the tetragonal structure of the investigated specimen was determined without any prior information. The unit‐cell parameters and the angles between the lattice vectors were evaluated with an accuracy of better than 0.7%, while the structure‐factor moduli of the reflections were determined with a mean deviation of 2.5% relative to the theoretical values. [ABSTRACT FROM AUTHOR]

In this paper, we derive an inversion of the weighted Radon transform by Fourier transform, Riesz potential, and integral transform. We extend results of Rigaud and Lakhal to the n‐dimensional Euclidean space. Furthermore, we obtain some properties of the weighted Radon transform. Finally, we develop some estimate results of the weighted Radon transform under Sobolev space. [ABSTRACT FROM AUTHOR]

We show that balls, circles and 2‐spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution. [ABSTRACT FROM AUTHOR]

Using the Lie group analysis approach, we study the symmetry properties of 2‐dimensional space‐fractional filtration equation with the Riesz potential. This equation is derived by a fractional generalization of Darcy law and permits to describe the pressure evolution during 1‐phase fluid flow through naturally fractured porous medium. We construct the prolongation of the point transformation group on the Riesz potential and obtain a generalization of the Leibniz rule for the Riesz potential, which are necessary for investigating symmetry properties using the invariance principle. The Lie group of linearly autonomous point transformations is constructed for the considered equation. In a limiting case of zero‐order potential, the obtained symmetry group coincides with the group of point transformations admitted by the classical integer‐order 2‐dimensional linear heat equation. Also, the asymptotic behavior of a group invariant solution is investigated. [ABSTRACT FROM AUTHOR]

The space‐time spectral collocation method was initially presented for the 1‐dimensional sine‐Gordon equation. In this article, we introduce a space‐time spectral collocation method for solving the 2‐dimensional nonlinear Riesz space fractional diffusion equations. The method is based on a Legendre‐Gauss‐Lobatto spectral collocation method for discretizing spatial and the spectral collocation method for the time nonlinear first‐order system of ordinary differential equation. Optimal priori error estimates in L2 norms for the semidiscrete formulation and the uniqueness of the approximate solution are derived. The method has spectral accuracy in both space and time, and the numerical results confirm the statement. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we are concerned with Sobolev's inequality for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new approach to the problem of finding a common zero for a system of m‐accretive mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We propose an implicit iteration method and two explicit ones, based on compositions of resolvents with the steepest‐descent method. We show that our results contain some iterative methods in literature as special cases. An extension of the Xu's regularization method for the proximal point algorithm from Hilbert spaces onto Banach ones under simple conditions of convergence and a new variant for the method of alternating resolvents are obtained. Numerical experiments are given to affirm efficiency of the methods. [ABSTRACT FROM AUTHOR]

Abstract: Given a closed submanifold, or a compact regular domain, in Euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under Möbius transformations, thus giving a generalization to higher dimensions of the Möbius energy of knots. [ABSTRACT FROM AUTHOR]

Abstract: We study minimal energy problems for strongly singular Riesz kernels | x − y | α − n, where n ≥ 2 and α ∈ ( − 1 , 1 ), considered for compact ( n − 1 )‐dimensional C ∞‐manifolds Γ immersed into R n. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such minimization problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order β = 1 − α on Γ. The measures with finite energy are shown to be elements from the Sobolev space H β / 2 ( Γ ), 0 < β < 2, and the corresponding minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D. P. Hardin and E. B. Saff. [ABSTRACT FROM AUTHOR]

In this paper we consider a periodic 2-dimensional quasi-geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution [ABSTRACT FROM AUTHOR]

In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even-order numerical differential formulas (named the fractional-compact formulas) are established for the Riesz derivatives with order [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

In this paper, we consider a Riesz-Feller space-fractional backward diffusion problem with a time-dependent coefficient [ABSTRACT FROM AUTHOR]

We show that a semibounded Wiener-Hopf quadratic form is closable in the space [ABSTRACT FROM AUTHOR]

In this paper, numerical solutions of fractional Fokker-Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker-Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank-Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker-Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This article discusses the analyticity and the long-time asymptotic behavior of solutions to space-time fractional diffusion-reaction equations in [ABSTRACT FROM AUTHOR]

In [ABSTRACT FROM AUTHOR]

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine-Gordon equation (SGE). Two-dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property. Let [ABSTRACT FROM AUTHOR]

It is shown in the limit-circle case that system of root functions of the non-self-adjoint maximal dissipative (accumulative) Bessel operator and its perturbation Sturm-Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the maximal dissipative (accumulative) Bessel operators is investigated, and it is proved that system of root functions form a basis (Riesz and Bari bases) in the same Hilbert space. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We find conditions on the variable parameters [ABSTRACT FROM AUTHOR]