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Vacancy defects are universally regarded to be the main defect that limits the photoelectric conversion efficiency of perovskite solar cells. In perovskite, iodine vacancy dominates the defect proportion due to its low formation energy. However, the defect property of iodine vacancy (VI) is still in dispute. Ideally, the VI defect is considered to be a shallow level defect near conduction band minimum, meaning that it does not act as a Shockley–Read–Hall (SRH) nonradiative recombination center. Herein, we find a direct correlation between compressive strain and VI defect behavior. The compressive strain along the lattice vector b/c direction will drive the VI defect from shallow level to deep level defect, which is related to the formation of Pb-dimer. In addition, the influence of extra electrons is also considered during the structural evolution of VI, which is often observed in the experiments. Therefore, we find that the elimination of compressive strain and extra electrons can be of great significance for improving the photoelectric performance of perovskite solar cells. Our work reveals the defect properties of VI from shallow level one to the SRH recombination center and the inherent physics mechanism of defect evolution under external factors, which provides a strategy to control device defects and eliminate recombination losses. [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]

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by T, producing an asymptotic estimate as T \to \infty. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the 2-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in \mathbb R^n. Our main result on counting basis extensions also generalizes to arbitrary lattices in \mathbb R^n. Finally, we establish some basic properties of sparse representations of integers by multilinear forms. [ABSTRACT FROM AUTHOR]

We prove that for any convex polytope Ω ⊂ ℝd which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space L²(Ω). The result is new in all dimensions d greater than one. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Choquard equation with logarithmic nonlinearity ... where where Ω ∈ R3 is a smooth bounded domain with boundary ∂Ω, k > 0 is real parameter, α ∈ (0,3), Iα is the Riesz potential. Under some mild assumptions on V and H, we obtain a ground state solution by variational method and logarithmic inequality. In addition, we investigate the limit profiles of Choquard equations as α→0 or α→N. [ABSTRACT FROM AUTHOR]

We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multidimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of r independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these r estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as r increases. Numerical experiments illustrate and support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this work we review some common fixed point theorems for four non-continuous mappings in ε-metric spaces, where the metric is Riesz space valued. These results cover famous comparable results in the existing literature by considering fewer conditions. [ABSTRACT FROM AUTHOR]

Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of α-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed. [ABSTRACT FROM AUTHOR]

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima—the norms of the shortest independent vectors of a lattice—and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of sign-changing multi-bump solutions for the following nonlinear Choquard equation − (0.1) Δu + (λV(x) + 1)u = (Iα∗|u|p)|u|p−2u in RN, where Iα is the Riesz potential, λ∈R+, (N−4)+ < α < N, 2 ≤ p < (N+α)/(N−2), and V(x) is a nonnegative continuous function with a potential well Ω := int(V−1(0)) which possesses k disjoint bounded components Ω1, …, Ωk . We prove the existence of sign-changing multi-bump solutions for (0.1) if λ is large enough. [ABSTRACT FROM AUTHOR]

In this study, one explicit and one implicit finite difference scheme is introduced for the numerical solution of time-fractional Riesz space diffusion equation. The time derivative is approximated by the standard Grünwald Letnikov formula of order one, while the Riesz space derivative is discretized by Fourier transform-based algorithm of order four. The stability and convergence of the proposed methods are studied. It is proved that the implicit scheme is unconditionally stable, while the explicit scheme is stable conditionally. Some examples are solved to illustrate the efficiency and accuracy of the proposed methods. Numerical results confirm that the accuracy of present schemes is of order one. [ABSTRACT FROM AUTHOR]

Our first result asserts that, for linear regular operators acting from a Riesz space with the principal projection property to a Banach lattice with an order continuous norm, the C-compactness is equivalent to the AM-compactness. Next we prove that, under mild assumptions, every linear section of a C-compact orthogonally additive operator is AM-compact, and every linear section of a narrow orthogonally additive operator is narrow. [ABSTRACT FROM AUTHOR]

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone E+ of a Riesz space E taking values in an Archimedean Riesz space F, to the entire space E. We prove that, if E has the principal projection property and F is Dedekind σ-complete then for every e∈E+ every positive finitely additive F-valued measure defined on the Boolean algebra Fe of fragments of e has a unique positive linear extension to the ideal Ee of E generated by e. If, moreover, the measure is τ-continuous then the linear extension is order continuous. [ABSTRACT FROM AUTHOR]

We prove that continuous sentences preserved by the ultramean construction (a generalization of the ultraproduct construction) are exactly those sentences which are approximated by linear sentences. Continuous sentences preserved by linear elementary equivalence are exactly those sentences which are approximated in the Riesz space generated by linear sentences. Also, characterizations for linear Δn-sentences and positive linear theories will be given. [ABSTRACT FROM AUTHOR]

In this paper, the best proximity pair problem is considered with a cone metric. the conditions for the existence and uniqueness of the best proximity pair problem is discussed by using interesting relationships in Riesz spaces. This problem is studied for T-absolutely direct sets. Also, given the conditions considered for this problem, it is shown for the cone cyclic contraction maps, the best proximity pair problem is uniquely solvable. [ABSTRACT FROM AUTHOR]

In this paper, we will study on some topologies induced by order convergence in a Riesz space. We will investigate the relationships of them. [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]

Let E be a sublattice of a vector lattice F. (xa) Ç E is said to be order convergent to a vector x (in symbols xa --~C x), whenever there exists another net (ya) in F with the same index set satisfying ya 4 0 in F and |xa -- x| < ya for all indexes a. If F = E~~, this convergence is called b-order convergence and we write xa --C x. In this manuscript, first we study some properties of order convergence nets and we extend same results to the general case. In the second part, we introduce b-order continuous operators and we investigate some properties of this new concept. An operator T between two vector lattices E and F is said to be b-order continuous, if xa --+ 0 in E implies Txa --G 0 in F. [ABSTRACT FROM AUTHOR]

Some kinds of statistical unbounded convergence were studied and investigated with respect to the solid topology and order convergence. In this paper, we study the concept of statistical unbounded order convergence in Riesz spaces by a topology-free technique with the order convergence on Riesz spaces. Moreover, we give some relations with other kinds of statistical convergences. [ABSTRACT FROM AUTHOR]

Let be a unital -module over an -algebra . With the help of Arens extension theory, a module structure on can be defined. The paper deals mainly with properties of the Arens homomorphism , which is defined by the module structure on . Necessary and sufficient conditions for an submodule of to be an order ideal are obtained. [ABSTRACT FROM AUTHOR]

The Korkine–Zolotareff (KZ) reduction and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ constant and Schnorr's constant were defined by Schnorr in 1987. The KZ constant can be used to quantify some useful properties of KZ reduced matrices. Schnorr's constant can be used to characterize the output quality of his block 2k-reduction and is used to define his semi block 2k-reduction, which was also developed in 1987. Hermite's constant, which is a fundamental lattice constant, has many applications, such as bounding the length of the shortest nonzero lattice vector and the orthogonality defect of lattices. Rankin's constant was introduced by Rankin in 1953 as a generalization of Hermite's constant. It plays an important role in characterizing the output quality of block-Rankin reduction, proposed by Gama et al. in 2006. In this paper, we first develop a linear upper bound on Hermite's constant and then use it to develop an upper bound on the KZ constant. These upper bounds are sharper than those obtained recently by the authors, and the ratio of the new linear upper bound to the nonlinear upper bound, developed by Blichfeldt in 1929, on Hermite's constant is asymptotically 1.0047. Furthermore, we develop lower and upper bounds on Schnorr's constant. The improvement to the lower bound over the sharpest existing one developed by Gama et al. is around 1.7 times asymptotically, and the improvement to the upper bound over the sharpest existing one which was also developed by Gama et al. is around 4 times asymptotically. Finally, we develop lower and upper bounds on Rankin's constant. The improvements of the bounds over the sharpest existing ones, also developed by Gama et al., are exponential in the parameter defining the constant. [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]

Applications of metamaterials in the realization of efficient devices in the terahertz band have recently been considered to achieve wave deflection, focusing, amplitude manipulation and dynamical modulation. Terahertz metamaterials offer practical advantages since their structures have typical sizes of hundreds microns and are within the reach of current three-dimensional (3D) printing technologies. Here, we propose terahertz photonic structures composed of dielectric rods layers made of acrylonitrile styrene acrylate realized by low-cost, rapid, and versatile fused deposition modeling 3D-printing. Terahertz time-domain spectroscopy is employed for the experimental study of their spectral and dynamic response. Measured spectra are interpreted by using simulations performed by an analytical exact solution of the Maxwell equations for a general incidence geometry, by a field expansion as a sum over reciprocal lattice vectors. Results show that the structures possess specific spectral forbidden bands of the incident THz radiation depending on their optical and geometrical parameters. We also find evidence of disorder in the 3D printed structure resulting in the closure of the forbidden bands at frequencies above 0.3 THz. The size disorder of the structures is quantified by studying the dynamics diffusion of THz pulses as a function of the numbers of layers of dielectric rods. Comparison with simulations of light diffusion in photonic crystals with increasing disorder allows estimating the size distributions of elements. By using a Mean Squared Displacement model, from the broadening of the pulses' widths it is also possible to estimate the diffusion coefficient of the terahertz radiation in the photonic structures. [ABSTRACT FROM AUTHOR]

In this paper, we study the degree of convergence of the functions of Fourier series and conjugate Fourier series in Sobolev spaces using Riesz means. We also study some applications of our main results and observe that our results are much better than earlier results. [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]

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]

Given a |$d$| -dimensional Euclidean lattice we consider the random set obtained by adding an independent Gaussian vector to each of the lattice points. In this note we provide a simple procedure that recovers the lattice from a single realization of the random set. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of Pohožaev-type ground state solutions for the Choquard equation with a singular potential and a critical exponent. By virtue of a generalized version of the Lions-type theorem and the Pohožaev manifold, we obtain the existence of a Pohožaev-type ground state solution for the above problem. Some recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

We study the topological properties of max-plus convex sets in an Archimedean Riesz space E with respect to the topology and the max-plus structure associated to a given order unit u; the definition of max-plus convex sets is algebraic and we do not assume that E has an a priori given topological structure. To a given unit u one can associate . two equivalent norms on E one of which, denoted ||·||u, is classical, the other ||·||hu is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance Dhu on E associated to ||·||hu is a geodesic distance, called the Hilbert affine distance associated to u, for which max-plus convex sets in E are precisely the geodesically closed sets. Under suitable assumptions, we establish max-plus versions of some fixed points and con- tinuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact max-plus convex sets are Absolute Retracts. We formulate a max-plus version of the Knaster-Kuratowski-Mazurkiewicz Lemma from which, following A. Granas and J. Dugundji, all of the consequences of the classical KKM Lemma can be derived in a max-plus version. P. de la Harpe showed that the interior of the standard simplex Δn equipped with the classical Hilbert metric-defined by the cross- ration of four appropriate points is isometric to a finite dimensional normed space. We give an explicit proof of that result: the norm space in question is Rn with the Hilbert affine norm ||·||hu with respect to u = (1,..., 1). [ABSTRACT FROM AUTHOR]

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The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It's of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [-1, 1] x [-1, 1]. Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, we intend to make new approach to introduce and study some fractional difference sequence spaces by Riesz mean associated with inĄnite matrix and a sequence of modulus functions over n -normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored and some inclusion relations concerning these spaces are also establish. Finally, we make an effort to study the statistical convergence through fractional difference operator. [ABSTRACT FROM AUTHOR]

Type IV twins are defined and shown to exist in triclinic crystal systems, as well as in some monoclinic and trigonal systems. Here, we focus on Pericline twins in triclinic plagioclase as an example. Type IV twins are associated with the irrationality of one of the twinning elements that is rational for a type II twin. The formation of type IV twins is accomplished through the shear on a K2 plane produced by the motion of twinning disconnections on a K plane, followed by rotational partitioning. The same systems where type IV twins are present also have type III twins instead of type I. Without using the correct type IV analysis, one would deduce the wrong magnitude and direction of shear associated with the twinning process, the magnitude of which would increase with greater triclinicity. Types I and II twins form if and only if there are rational lattice translation vectors lying in the plane of distortion/shear. Otherwise, the twins are types III and IV. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the best approximation theory in Riesz space, which is not constructed by the norm, but only rely on the order structure. Based on the order structure, we propose the concept of the order best approximation in Riesz space and discuss some problems related to the order best approximation, including some sufficient and necessary conditions for satisfying the order best approximation set. Finally, we consider the order best approximation projection and its related properties. [ABSTRACT FROM AUTHOR]

We show that a constant factor approximation of the shortest and closest lattice vector problem in any ℓ p -norm can be computed in time 2 (0.802 + ε) n. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem in the ℓ 2 norm. To obtain our result, we combine the latter algorithm for ℓ 2 with geometric insights related to coverings. [ABSTRACT FROM AUTHOR]

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal ℓ 0 -norm of solutions to systems A x = b , where A ∈ Z m × n , b ∈ Z m and x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over R , to other subdomains such as Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables. [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 phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the socalled 'triangle phase') is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of coexistence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions. [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]

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice compared with conventional VCs. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity. [ABSTRACT FROM AUTHOR]

Minkowski proved that any $n$ -dimensional lattice of unit determinant has a nonzero vector of Euclidean norm at most $\sqrt {n}$ ; in fact, there are $2^{\Omega (n)}$ such lattice vectors. Lattices whose minimum distances come close to Minkowski’s bound provide excellent sphere packings and error-correcting codes in ${\mathbb {R}}^{n}$. The focus of this work is a certain family of efficiently constructible $n$ -dimensional lattices due to Barnes and Sloane, whose minimum distances are within an $O(\sqrt {\log n})$ factor of Minkowski’s bound. Our primary contribution is a polynomial-time algorithm that list decodes this family to distances approaching $1/\sqrt {2}$ of the minimum distance. The main technique is to decode Reed-Solomon codes under error measured in the Euclidean norm, using the Koetter-Vardy “soft decision” variant of the Guruswami-Sudan list-decoding algorithm. [ABSTRACT FROM AUTHOR]

In this article we present the construction of variable power (or variable concavification) of quasi-Banach lattices with a bounded variable exponent from ideal center. This construction generalizes the similar constructions with constant exponent. We also introduce the concepts of variable convexity and concavity of quasi-Banach lattices under the which the variable power of a quasi-Banach lattice is a Banach lattice. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method for Riesz space fractional telegraph equation with time delay. The Riesz fractional telegraph equation is approx- imated with the interpolating polynomial P2. First a system of fractional differential equations are obtained from the telegraph equation with respect to the time variable. Then our numerical algorithm is proposed. The convergence order and stability of the fractional order algorithms are proved. Finally, some numerical examples are con- structed to describe the usefulness and profitability of the numerical method. Numerical results show that the accuracy of order O(αt). [ABSTRACT FROM AUTHOR]