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Let L be a truncated Archimedean vector lattice whose truncation is denoted by *. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C∞(X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C∞(X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C∞(X) whose truncation is provided, again, by meet with some characteristic function on X. [ABSTRACT FROM AUTHOR]

Optical lattice clocks combine the accuracy and stability required for next-generation frequency standards. At the heart of these clocks are carefully engineered optical lattices tuned to a wavelength where the differential AC Stark shift between ground and excited states vanishes—the so called 'magic' wavelength. To date, only alkaline-earth-like atoms utilizing clock transitions with total electronic angular momentum J = 0 have successfully realized these magic wavelength optical lattices at the level necessary for state-of-the-art clock operation. In this article, we discuss two additional types of clock transitions utilizing states with J ≠ 0 , leveraging hyperfine structure to satisfy the necessary requirements for controlling lattice-induced light shifts. We propose realizing (i) clock transitions between same-parity clock states with total angular momentum F = 0 and (ii) M1/E2 clock transitions between a state with F = 0 and a second state with J = 1 / 2 , m F = 0 . We present atomic species which fulfill these requirements before giving a detailed discussion of both manganese and copper, demonstrating how these transitions provide the necessary suppression of fine structure-induced vector and tensor lattice light shifts for clock operations. Such realization of alternative optical lattice clocks promises to provide a rich variety of new atomic species for neutral atom clock operation, with applications from many-body physics to searches for new physics. [ABSTRACT FROM AUTHOR]

Post-quantum cryptography is considered to provide security against both traditional and quantum computer attacks. Dilithium is a digital signature algorithm that derives its security from the challenge of finding short vectors in lattices. It has been selected as one of the standardizations in the NIST post-quantum cryptography project. Hardware-software co-design is a commonly adopted implementation strategy to address various implementation challenges, including limited resources, high performance, and flexibility requirements. In this study, we investigate using compact instruction set extensions (ISEs) for Dilithium, aiming to improve software efficiency with low hardware overheads. To begin with, we propose tightly coupled accelerators that are deeply integrated into the RISC-V processor. These accelerators target the most computationally demanding components in resource-constrained processors, such as polynomial generation, Number Theoretic Transform (NTT), and modular arithmetic. Next, we design a set of custom instructions that seamlessly integrate with the RISC-V base instruction formats, completing the accelerators in a compact manner. Subsequently, we implement our ISEs in a chip design for the Hummingbird E203 core and conduct performance benchmarks for Dilithium utilizing these ISEs. Additionally, we evaluate the resource consumption of the ISEs on FPGA and ASIC technologies. Compared to the reference software implementation on the RISC-V core, our co-design demonstrates a remarkable speedup factor ranging from 6.95 to 9.96. This significant improvement in performance is achieved by incorporating additional hardware resources, specifically, a 35% increase in LUTs, a 14% increase in FFs, 7 additional DSPs, and no additional RAM. Furthermore, compared to the state-of-the-art approach, our work achieves faster speed performance with a reduced circuit cost. Specifically, the usage of additional LUTs, FFs, and RAMs is reduced by 47.53%, 50.43%, and 100%, respectively. On ASIC technology, our approach demonstrates 12, 412 cell counts. Our co-design provides a better tradeoff implementation on speed performance and circuit overheads. [ABSTRACT FROM AUTHOR]

Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method. [ABSTRACT FROM AUTHOR]

Practical implementations of the Ewald method used to compute Coulomb interactions in molecular dynamics simulations are hampered by the requirement to truncate its reciprocal space series. It is shown that this can be mitigated by representing the contributions from the neglected reciprocal lattice vector terms as a simple modification of the real space expression in which the real and reciprocal space series have slightly different charge spreading parameters. This procedure, called the α′ method, enables significantly fewer reciprocal lattice vectors to be taken than is currently typical for Ewald, with negligible additional computational cost, which is validated on model systems representing different classes of charged system, a CsI crystal and melt, water, and a room temperature ionic liquid. A procedure for computing accurate energies and forces based on a periodic sampling of an additional number of reciprocal lattice vectors is also proposed and validated by the simulations. The convergence characteristics of expressions for the pressure based on the forces and the potential energy are compared, which is a useful assessment of the accuracy of the simulations in reproducing the Coulomb interaction. The techniques developed in this work can reduce significantly the total computer simulation times for medium sized charged systems, by factors of up to ∼5 for those in the classes studied here. [ABSTRACT FROM AUTHOR]

We report the thermal conductivity of two dynamically stable bilayer phosphorene stackings, i.e., a twisted phase with a twist angle of ∼70.5° (or 2O-tαP phase) and a shifting phase with half of the lattice constants (or AB phase). This was achieved by using the first-principles-driven lattice dynamics calculations and a fully iterative solver of the Boltzmann transport equation, the latter including an anharmonic phonon–phonon scattering effect. At room temperature, the thermal conductivity of the 2O-tαP phase is 146 and 108 W/mK along its two orthogonal lattice basis vectors, respectively, larger than that along the armchair direction (69 W/mK) of the AB phase, while smaller than that along the AB zigzag direction (164 W/mK); with an increasing temperature, the conductivity decreases along the basis vectors of the 2O-tαP and AB stackings, and the anisotropy lessens for both stackings. The thermal transport anisotropies for the two kinds of bilayer stacking can be attributed to the different proportions of their acoustic branches along different directions. In particular, the phonon mean free path showed that the in-plane transverse acoustic branch is the main contribution of the thermal conductivity along the short lattice constant direction of the 2O-tαP phase due to the twist angle extending the propagation path of transverse acoustic waves in the direction. Finally, the thermal conductivity accumulation was revealed to increase in the form of a hyperbolic tangent with mean free path of the phonons, which can be used to evaluate the size effect of the stacking materials in practice. [ABSTRACT FROM AUTHOR]

Denote by I d c G the lattice of all principal ℓ -ideals of an Abelian ℓ -group G. Our main result is the following. Theorem For every countable Abelian ℓ-group G, every countable completely normal distributive 0 -lattice L, and every closed 0 -lattice homomorphism φ : I d c G → L , there are a countable Abelian ℓ-group H, an ℓ-homomorphism f : G → H , and a lattice isomorphism ι : I d c H → L such that φ = ι ∘ I d c f. We record the following consequences of that result: (1) A 0-lattice homomorphism φ : K → L , between countable completely normal distributive 0-lattices, can be represented, with respect to the functor I d c , by an ℓ -homomorphism of Abelian ℓ -groups iff it is closed. (2) A distributive 0-lattice D of cardinality at most ℵ 1 is isomorphic to some I d c G iff D is completely normal and for all a , b ∈ D the set { x ∈ D | a ≤ b ∨ x } has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ 1. The bound ℵ 1 is sharp. Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height. All our results are stated in terms of vector lattices over any countable totally ordered division ring. [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]

In this paper, high-order finite difference methods are proposed to solve the initial-boundary value problem for one- and two-dimensional Riesz space variable-order fractional diffusion equations. We first introduce fractional centered difference (FCD) and weighted and shifted fractional centered difference (WSFCD) schemes for Riesz space variable-order fractional derivatives. Then the Crank-Nicolson (CN) scheme and the linearly implicit conservative (LIC) difference scheme are applied to discretize the time derivative in linear and nonlinear problems, respectively. Thus, we get CN-FCD and CN-WSFCD schemes, and LIC-FCD and LIC-WSFCD schemes, respectively. Theoretical results about the stability and convergence for the above-mentioned schemes are presented and proved. Banded preconditioners are introduced to speed up GMRES methods for solving the discretization linear systems. The spectral property of the preconditioned matrix is analyzed. Numerical results show that the proposed schemes and preconditioners are very efficient. [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]

Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms. [ABSTRACT FROM AUTHOR]

Disjoint sequence methods from the theory of Riesz spaces are used to study measures of weak non-compactness in L_{1}(\mu)-spaces. A principal new result of the present paper is the following: Let E be an abstract M-space. Then \begin{align*} \omega (B)&=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {disjoint} \}\\ &=\inf \{\varepsilon >0:\exists x^{*}\in E^{*}_{+} \operatorname {so}\operatorname {that} B\subseteq [-x^{*},x^{*}]+\varepsilon B_{E^{*}}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {weakly}\operatorname {null} \}\\ &=\sup \{\operatorname {ca}_{\rho _{B}}((x_{n})_{n}):(x_{n})_{n}\subseteq (B_{E})_{+} \operatorname {increasing} \}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\|x^{*}_{n}\|:(x^{*}_{n})_{n}\subseteq \operatorname {Sol}(B)\operatorname {disjoint}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\sup \limits _{x^{*}\in B}|\langle x^{*},x_{n}\rangle |:(x_{n})_{n}\subseteq B_{E}\operatorname {disjoint} \}\\ \end{align*} for every norm bounded subset B of E^{*}. [ABSTRACT FROM AUTHOR]

The contraction condition in the Banach contraction principle forces a function to be continuous. Many authors overcome this obligation and weaken the hypotheses via metric spaces endowed with a partial order. In this paper, we present some coupled fixed point theorems for the functions having mixed monotone properties on ordered vector metric spaces, which are more general spaces than partially ordered metric spaces. We also define the double monotone property and investigate the previous results with this property. In the last section, we prove the uniqueness of a coupled fixed point for non-monotone functions. In addition, we present some illustrative examples to emphasize that our results are more general than the ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1 , in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form (1 + | v | 2) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v , with a fractional Laplacian κ (− Δ) β − d + 2 6 and a positive diffusion coefficient κ. [ABSTRACT FROM AUTHOR]

We study the space H (SO) of all homomorphisms of the vector lattice of all slowly oscillating functions on the half-line H = [0, ∞). In contrast to the case of homomorphisms of uniformly continuous functions, it is shown that a homomorphism in H(SO) which maps the unit to zero must be the zero-homomorphism. Consequently, we show that the space H(SO) without the zero-homomorphism is homeomorphic to H × (0, ∞). By describing a neighborhood base of the zero-homomorphism, we show that H(SO) is homeomorphic to the space H × (0, ∞) with one point added. [ABSTRACT FROM AUTHOR]

In this paper, we prove the boundedness of the Riesz potential Iα in local variable Morrey-Lorentz spaces. Also we apply our results to particular operators such as fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups in these spaces. [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]

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 lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the problem of finding a set of vectors in a given lattice such that the collection of all integer linear combinations of this subset is still the entire original lattice and so that the Euclidean norms of the subset are reduced. The present paper proposes simple, efficient iterations for lattice reduction which are guaranteed to reduce the Euclidean norms of the basis vectors (the vectors in the subset) monotonically during every iteration. Each iteration selects the basis vector for which projecting off (with integer coefficients) the components of the other basis vectors along the selected vector minimizes the Euclidean norms of the reduced basis vectors. Each iteration projects off the components along the selected basis vector and efficiently updates all information required for the next iteration to select its best basis vector and perform the associated projections. [ABSTRACT FROM AUTHOR]

In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method. [ABSTRACT FROM AUTHOR]

We investigate a fast algorithm for the time-fractional diffusion equation with a space-time-dependent variable order, which models, e.g., the subdiffusion with varying memory effects. In addition to the traditional L1 discretization of the time-fractional derivative, we perform a further approximation for the L1 coefficients, analyze the structures of the resulting all-at-once system, and apply the divide and conquer method to obtain a fast numerical algorithm. Due to the spatial dependence of the variable order and the further approximation to the L1 coefficients, the temporal discretization coefficients are coupled with the inner product of the finite element method and lack the monotonicity, which are rarely encountered in previous works and thus motivate novel analysis methods and computational techniques. Compared with the standard time-stepping methods with L1 discretization, the proposed algorithm reduces the complexity of solving the all-at-once system from O (M N 2) to O (M N ln 3 N) , where M stands for the spatial degree of freedom and N refers to the number of time steps. Numerical experiments are provided to substantiate the theoretical findings. [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 note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean -algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators. [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 paper, we introduce the class of demi-order norm continuous operator on a normed Riesz space. We study the relationship between order-to-norm continuous operator and demiorder norm continuous operator. We also investigate some properties of the class of demiorder norm continuous operator, and it is given a characterization of a normed Riesz space with order continuous norm by the term of the demi-order norm continuous operator. [ABSTRACT FROM AUTHOR]

By using the fixed point theorems, we give suficient conditions for the existence and uniqueness of solutions for the nonlocal fractional boundary value problem of nonlinear Riesz-Caputo differential equation. The boundedness assumption on the nonlinear term is replaced by growth conditions or by a continuous function. Finally, some examples are presented to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

Forming a hetero-interface is a materials-design strategy that can access an astronomically large phase space. However, the immense phase space necessitates a high-throughput approach for an optimal interface design. Here we introduce a high-throughput computational framework, InterMatch, for efficiently predicting charge transfer, strain, and superlattice structure of an interface by leveraging the databases of individual bulk materials. Specifically, the algorithm reads in the lattice vectors, density of states, and the stiffness tensors for each material in their isolated form from the Materials Project. From these bulk properties, InterMatch estimates the interfacial properties. We benchmark InterMatch predictions for the charge transfer against experimental measurements and supercell density-functional theory calculations. We then use InterMatch to predict promising interface candidates for doping transition metal dichalcogenide MoSe2. Finally, we explain experimental observation of factor of 10 variation in the supercell periodicity within a few microns in graphene/α-RuCl3 by exploring low energy superlattice structures as a function of twist angle using InterMatch. We anticipate our open-source InterMatch algorithm accelerating and guiding ever-growing interfacial design efforts. Moreover, the interface database resulting from the InterMatch searches presented in this paper can be readily accessed online. Predicting properties at the interface of materials is crucial for advanced materials design. Here, the authors introduce a high-throughput computational framework, InterMatch, for predicting several properties of an interface by using the databases of individual bulk materials. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the system of left translates {L(2k,l,m)g: k, l,m ∈ Z}, g ∈ L²(H), to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function gλ. Here, H denotes the Heisenberg group and gλ the inverse Fourier transform of g with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates {L(2k,l,m)g: k, l, m ∈ Z} on H. This result is also illustrated with an example. [ABSTRACT FROM AUTHOR]

In this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the Lp space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators. [ABSTRACT FROM AUTHOR]

The space fractional PDEs (SFPDEs) have attracted a lot of attention. Developing high-order and stable numerical algorithms for them is the main aim of most researchers. This research work presents a fractional spectral collocation method to solve the fractional models with space fractional derivative which is defined based upon the Riesz derivative. First, a second-order difference formulation is used to approximate the time derivative. The stability property and convergence order of the semi-discrete scheme are analyzed. Then, the fractional spectral collocation method based on the fractional Jacobi polynomials is employed to discrete the spatial variable. In the numerical results, the effect of fractional order is studied. [ABSTRACT FROM AUTHOR]

We propose a machine-learning interatomic potential for multi-component magnetic materials. In this potential we consider magnetic moments as degrees of freedom (features) along with atomic positions, atomic types, and lattice vectors. We create a training set with constrained DFT (cDFT) that allows us to calculate energies of configurations with non-equilibrium (excited) magnetic moments and, thus, it is possible to construct the training set in a wide configuration space with great variety of non-equilibrium atomic positions, magnetic moments, and lattice vectors. Such a training set makes possible to fit reliable potentials that will allow us to predict properties of configurations in the excited states (including the ones with non-equilibrium magnetic moments). We verify the trained potentials on the system of bcc Fe–Al with different concentrations of Al and Fe and different ways Al and Fe atoms occupy the supercell sites. Here, we show that the formation energies, the equilibrium lattice parameters, and the total magnetic moments of the unit cell for different Fe–Al structures calculated with machine-learning potentials are in good correspondence with the ones obtained with DFT. We also demonstrate that the theoretical calculations conducted in this study qualitatively reproduce the experimentally-observed anomalous volume-composition dependence in the Fe–Al system. [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]

It is well known that the lattice Id c G of all principal ℓ -ideals of any Abelian ℓ -group G is a completely normal distributive 0-lattice; yet not every completely normal distributive 0-lattice is a homomorphic image of some Id c G , via a counterexample of cardinality ℵ 2 . We prove that every completely normal distributive 0-lattice with at most ℵ 1 elements is a homomorphic image of some Id c G . By Stone duality, this means that every completely normal generalized spectral space with at most ℵ 1 compact open sets is homeomorphic to a spectral subspace of the ℓ -spectrum of some Abelian ℓ -group. [ABSTRACT FROM AUTHOR]

A multipole expansion of the fundamental solution of the fractional power of the Laplace operator is constructed in terms of the Gegenbauer polynomials. Based on the decomposition constructed and the idea of the fast multipole method, we propose a numerical algorithm for solving the fractional differential generalization of the Poisson equation in the two-dimensional and three-dimensional spaces. [ABSTRACT FROM AUTHOR]

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system's historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system. [ABSTRACT FROM AUTHOR]

In this manuscript, the density-dependent space-time fractional reaction–diffusion equation in the sense of conformable and M-truncated derivatives (CMD) is presented. Through fractional transformation, these nonlinear fractional equations can be converted into nonlinear ordinary differential equations (NLPDEs). Besides, with the help of the Riccati–Bernoulli sub-ODE method (RBM), new exact solutions for these nonlinear fractional equations are produced. In order to construct the comparative analysis between different type fractional derivatives, graphical representations are demonstrated for chosen values of unknown parameters. [ABSTRACT FROM AUTHOR]

We propose in this work a Sieve algorithm that we called OrthogonalInteger sieve algorithm for some orthogonal integer lattices and particularly the case of integer lattices Λ ⊂ ℤ n , root lattices of type A n (n ≥ 1) and of type D n (n ≥ 2). In these cases, we use the famous L L L algorithm to find the shortest vector of these lattices. Indeed, in general, a sieve algorithm builds a list of short random vectors which are not necessarily in the lattice, and try to produce short lattice vectors by taking linear combinations of the vectors in the list. But in our case, we built a list of short vectors in the lattice. From the first column of the L L L -reduced basis of the considered basis, we have the list of at least n and at most 2 n short vectors for the general case (where n is the dimension of the lattice) of orthogonal integer lattices Λ ⊂ ℤ n . For the lattices ℤ n , A n (n ≥ 1) and D n (n ≥ 2), we have, respectively, 2 n , n (n + 1) and 2 n (n − 1) short vectors. The proposed sieve algorithm for integer lattice ℤ n runs in space O (2 n) and the OrthogonalInteger sieve algorithm performs O (n 2 n) arithmetic operations and is polynomial in space. [ABSTRACT FROM AUTHOR]

In this paper, we extend to the setting of positive orthogonally bi-additive operators several results from Aliprantis and Burkinshaw (Math Z 185: 245-257, 1983), de Pagter (Math Anal Appl 472: 238-245, 2019), Pliev and Popov (Siberian Math J 57: 552-557, 2016), Pliev and Ramdane (Mediter J Math 15(2): 55, 2018). First, we show that a positive order bounded orthogonally bi-additive map T : I → W defined on a lateral ideal of a Cartesian product of vector lattices E and F and taking values in a Dedekind complete vector lattice W can be extended to the whole space E × F . Then we prove that a positive orthogonally bi-additive operator T : E × F → W is laterally-to-order continuous if and only if the kernel of each S with 0 ≤ S ≤ T is laterally closed. Finally, we calculate the laterally-to-order continuous part of a positive orthogonally bi-additive operator T : E × F → W . [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]

The statistically unbounded p-convergence is an abstraction of the statistical order, unbounded order, and p-convergences. We investigate the concept of the statistically unbounded convergence on lattice-normed Riesz spaces with respect to statistical p-decreasing sequences. Also, we get some relations between this concept and the other kinds of statistical convergences on Riesz spaces. [ABSTRACT FROM AUTHOR]

We construct and analyze the variable-step L1 scheme to efficiently solve the space-time fractional Cahn-Hilliard equation in two dimensions. The associated variational energy dissipation law in continuous form is developed for the space-time fractional model in continuous form, which is compatible with that of the classical Cahn-Hilliard model. Detailed energy stability and convergence analysis of the proposed fully discrete scheme are rigorously established. We further show that the discrete energy dissipation law of the space-time fractional Cahn-Hilliard model is asymptotically compatible with that of the backward Euler scheme for the classical Cahn-Hilliard model when the order of the time fractional derivative and that of the space fractional Laplacian both approach 1- . The proposed scheme is shown to achieve the optimal temporal convergence rate with the variable time steps numerically. More importantly, numerical results reveal that the energy dissipation rate depends on both the order of the time fractional derivative and that of the space fractional one. Specifically, the exponent of the power law for the energy dissipation scales linearly with the order of the space fractional Laplacian. [ABSTRACT FROM AUTHOR]

Let Δ 1 (x ; φ) denote the error term in the classical Rankin-Selberg problem. In this paper, our main results are getting the k -th (3 ≤ k ≤ 5) power moments of Δ 1 (x ; φ) in short intervals and its asymptotic formula by using large value arguments. [ABSTRACT FROM AUTHOR]

In this study, we treated the space–time-fractional diffusion equation in a semi-infinite medium using a recently developed fractional derivative introduced by Caputo and Fabrizio. Our main focus was on simulating the diffusion profiles during the creation of a p-n junction according to the obtained solution. We made an interesting observation regarding the influence of the fractional-order derivatives on the depth estimation of the p-n junction. Increasing the order of the time-fractional derivative, denoted as α , resulted in faster diffusion and deeper p-n junctions. On the other hand, increasing the order of the space fractional derivative, denoted as β , led to slower diffusion and shallower p-n junctions. These findings demonstrate the significant impact of the fractional derivative orders on the diffusion behavior and depth characteristics of the p-n junction in the studied system. [ABSTRACT FROM AUTHOR]

We establish the existence of a simultaneous extension from a majorizing sublattice in the classes of regular multilinear operators and regular homogeneous polynomials on vector lattices. By simultaneous extension from a sublattice we mean a right inverse of the restriction operator to this sublattice which is an order continuous lattice homomorphism. The main theorems generalize some earlier results for orthogonally additive polynomials and bilinear operators. The proofs base on linearization by Fremlin's tensor product and the existence of a right inverse of an order continuous operator with Levy and Maharam property. [ABSTRACT FROM AUTHOR]

In this paper, we study the continuity in law of the solutions of two linear multiplicative SPDEs (the parabolic Anderson model and the hyperbolic Anderson model) with respect to the spatial parameter of the noise. The solution is interpreted in the Skorohod sense, using Malliavin calculus. We consider two cases: (i) the regular noise, whose spatial covariance is given by the Riesz kernel of order α ∈ (0 , d) , in spatial dimension d ≥ 1 ; (ii) the rough noise, which is fractional in space with Hurst index H < 1 / 2 , in spatial dimension d = 1. We assume that the noise is colored in time. [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]

In this work, we will generalize the moment generating function to Riesz spaces. We will derive some of its properties and use it to prove Chernoff's inequality inn Riesz spaces. [ABSTRACT FROM AUTHOR]

Let E be a sublattice of a vector lattice F. A continuous operator T from E into a normed vector space X is said to be õrder-norm continuous if ... implies ... for every (xα)α∈A ⊆ E. This paper aims to investigate the properties of this new class of operators and explore their relationships with existing classifications of operators. We introduce a new class of operators called õrder weakly compact operators. A continuous operator T : E → X is considered õrder weakly compact if T(A) in X is a relatively weakly compact set for every Fo-bounded A ⊆ E. In this manuscript, we examine various properties of this class of operators and explore their connections with õrder-norm continuous operators. [ABSTRACT FROM AUTHOR]