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The given text is a technical document discussing mathematical equations and algorithms related to remote sensing and linear equation systems. It provides a precise definition of the rhombus convergence property and discusses the enhancement of Algorithm 2 for calculating vectors. The document demonstrates the application of algorithms to solve diagonal QUBO problems and includes examples using exact rational expressions. While the text may not be directly relevant to specific research topics, it may be useful for researchers studying remote sensing and mathematical algorithms. [Extracted from the article]

We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces. [ABSTRACT FROM AUTHOR]

Hidden Markov models are a class of probabilistic graphical models used to describe the evolution of a sequence of unknown variables from a set of observed variables. They are statistical models introduced by Baum and Petrie in Baum (JMA 101:789–810) and belong to the class of latent variable models. Initially developed and applied in the context of speech recognition, they have attracted much attention in many fields of application. The central objective of this research work is upon an extension of these models. More accurately, we define multiparameter hidden Markov models, using multiple observation processes and the Riesz distribution on the space of symmetric matrices as a natural extension of the gamma one. Some basic related properties are discussed and marginal and posterior distributions are derived. We conduct the Forward-Backward dynamic programming algorithm and the classical Expectation Maximization algorithm to estimate the global set of parameters. Using simulated data, the performance of these estimators is conveniently achieved by the Matlab program. This allows us to assess the quality of the proposed estimators by means of the mean square errors between the true and the estimated values. [ABSTRACT FROM AUTHOR]

A Riesz basis for a separable Hilbert space H is the image of an orthonormal basis under a bounded, linear and bijective operator acting on H. Equivalently, it is an exact frame that shares the properties of a basis for H. Let G be a σ -compact and metrizable locally compact abelian group, and s and r be positive integers. It is illustrated that the image of a matrix-valued orthonormal basis under a bijective, bounded and linear operator acting on the matrix-valued signal space L 2 (G , ℂ s × r) may not be a frame, hence not a basis of the space L 2 (G , ℂ s × r). We introduce a notion of matrix-valued Riesz basis in the space L 2 (G , ℂ s × r) , where the adjointability of a bounded linear operator in the definition of Riesz basis with respect to the matrix-valued inner product plays a crucial role. We establish the existence of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). Extending results for standard Riesz bases of separable Hilbert spaces, we give necessary and sufficient conditions, and a characterization of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). [ABSTRACT FROM AUTHOR]

In this paper, we propose numerical methods for the Riesz space distributed-order advection–diffusion equation with delay in 2D. We utilize the fractional backward differential formula method of second order (FBDF2), and weighted and shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and develop the finite difference method for the RFADED. It has been shown that the obtained schemes are unconditionally stable and convergent with the accuracy of O (h 2 + k 2 + κ 2 + σ 2 + ρ 2) , where h, k and κ are space step for x, y and time step, respectively. Also, numerical examples are constructed to demonstrate the effectiveness of the numerical methods, and the results are found to be in excellent agreement with analytic exact solution. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the existence of a marginal pricing economic equilibrium in presence of increasing returns and externalities in a commodity space general enough as to encompass the vast majority of economic situations. This extends the existing literature on competitive equilibria in vector lattices by incorporating market failures, and it also generalises several non-competitive existence results to a larger class of commodity spaces. The key features are a suitable definition for the marginal pricing rule and an adaptation of the properness condition. [ABSTRACT FROM AUTHOR]

The L^{p}, 1\le p\le \infty, spaces have been generalized to the setting of Riesz spaces as {L}^{p}(T) spaces, on which there are R(T)-valued norms. The strong sequential completeness of the space {L}^{1}(T) and the strong completeness of {L}^{\infty }(T) with resepct to their respective R(T)-valued norms were established by Kuo, Rodda, and Watson. In the current work, the T-strong completeness of {L}^{2}(T) is established via the Riesz–Fischer type theorem given by Kalauch, Kuo, and Watson. It is also shown that the conditional expectation operator T is a weak order unit for the T-strong dual. [ABSTRACT FROM AUTHOR]

Orthogonally biadditive operators preserving disjointness are studied. It is proved that, that for a Dedekind complete vector lattice and order ideals and in , the set of all orthogonally biadditive operators commuting with projections is a band in the Dedekind complete vector lattice of all regular orthogonally biadditive operators from the Cartesian product of and to . A general form of the order projection onto this band is obtained, and an operator version of the Radon–Nikodym theorem for disjointness-preserving positive orthogonally biadditive operators is proved. [ABSTRACT FROM AUTHOR]

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method. [ABSTRACT FROM AUTHOR]

Given a separation oracle SO for a convex function f defined on Rn that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most • O(n(n log log(n)/ log(n) + log(R))) calls to SO and poly(n, log(R)) arithmetic operations, or • O(n log(nR)) calls to SO and exp(O(n)) · poly(log(R)) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n² (n + log(R))) for polynomial time algorithms and O(n² log(nR)) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n³ log log(n)/ log(n)) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n2 log(n)) calls to an evaluation oracle. These improve upon the previously best O(n³ log² (n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n³ log(n)) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory. [ABSTRACT FROM AUTHOR]

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

Lattice-based Cryptography is known as one of the key technologies in modern cryptography. This encryption scheme has the basis vectors from the lattice as the public key and a short-length vector in the lattice consisting of an integer combination of the basis vectors as the secret key. To break this encryption, we need to solve the Shortest Vector Problem (SVP), known as NP-hard. Therefore, instead of finding the shortest vector, LLL algorithm is often used to find a vector of sufficiently short length to break the encryption. The LLL algorithm is a well-known method for breaking this encryption, but there is still no clear answer to the question of how many times the LLL algorithm needs to be used to obtain the desired level of secret key, the average number of the (δ, η)-LLL bases in dimension n is a tool to measure the probability that the LLL algorithm solves the SVP. We can expect that this number indicates how many times the appropriate algorithm should run. There is a formula for this, but it contains some functions that take a long time to compute. We apply linear regression to the formula of the average number of the (δ, η)-LLL bases in dimension n, and therefore we obtain some formulas to approximate the average number of the (δ, η)-LLL is based on dimension n, which contains simple functions. When the dimensions are high, our model is much better regarding the computation time. [ABSTRACT FROM AUTHOR]

We investigate a semilinear stochastic time-space fractional subdiffusion equation driven by fractionally integrated multiplicative noise. The equation involves the ψ -Caputo derivative of order α ∈ (0 , 1) and the spectral fractional Laplacian of order β ∈ (1 2 , 1 ] . The existence and uniqueness of the mild solution are proved in a suitable Banach space by using the Banach contraction mapping principle. The spatial and temporal regularities of the mild solution are established in terms of the smoothing properties of the solution operators. [ABSTRACT FROM AUTHOR]

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of demi-strongly order bounded operators on a Riesz space generalization of strongly order bounded operators. Let M be a Riesz space, an operator H from M into M is said to be a demi-strongly order bounded operator if for every net {uα} in M+ whenever 0 ≤ uα ↑ ≤ u ′′, u ′′ in M∼∼ and {uα −H(uα)} is order bounded in M, then {uα} is order bounded in M. We obtain a characterization of the b-property by the term of demi-strongly order bounded operators. In addition, we study the relationship between strongly order bounded operators and demi-strongly order bounded operators. Finally, we also investigate some properties of the class of demi-strongly order bounded operators. [ABSTRACT FROM AUTHOR]

Using the non-equilibrium molecular dynamics (NEMD) simulations and the time-dependent Ginzburg–Landau (TDGL) theory for solidification kinetics, we study the crystal-melt interface (CMI) kinetic coefficients for both the soft-sphere (SS) BCC-melt and the FCC-melt interfaces, modeled with the inverse-power repulsive potential ( n = 8 ). The collective dynamics of the interfacial liquids at four equilibrium CMIs are calculated and employed to eliminate the discrepancy between the predictions of the kinetic coefficient using the NEMD simulations and the TDGL solidification theory. The speedup of the two modes of the interfacial liquid collective dynamics (at wavenumbers equal to the principal and the secondary reciprocal lattice vector of the grown crystal) at the equilibrium FCC CMI is observed. The calculated local collective dynamics of the SS BCC CMIs are compared with the previously reported data for the BCC Fe CMIs, validating a hypothesis proposed recently that the density relaxation times of the interfacial liquids at the CMIs are anisotropic and material dependent. With the insights provided by the improved application of the TDGL solidification theory, an attempt has been made to interpret the variation physics of the crystal-structure dependence of the solidification kinetic coefficient. [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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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

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

We use a recent lattice determination of the vector and axial Ds → γ form factors at high squared momentum transfer q2 to infer their Bs → γ counterparts. To this end, we introduce a phenomenological approach summarized as follows. First, we describe the lattice data with different fit templates motivated by vector-meson dominance, that is expected to hold in the high-q2 region considered. We identify reference fit ansaetze with one or two physical poles, that we validate against alternative templates. Then, the pole residues can be unambiguously related to the appropriate couplings involving the pseudoscalar, the vector mesons concerned, and the photon — or tri-couplings — and the latter can be expressed as sums over quark magnetic moments, weighed by their e.m. charges. This description obeys a well-defined heavy-quark scaling, that allows to parametrically scale up the form factors to the Bs → γ case. We discuss a number of cross-checks of the whole approach, whose validation rests ultimately in a first-principle determination, e.g. in lattice QCD. Finally, we use our obtained form factors to reassess the SM prediction of ℬ(Bs → μ+μ−γ) in the range q 2 ∈ [4.2, 5.0] GeV, where an experimental measurement is awaited. [ABSTRACT FROM AUTHOR]

Photonic crystals (PhCs) are a kind of artificial structures that can mold the flow of light at will. Polaritonic crystals (PoCs) made from polaritonic media offer a promising route to controlling nano-light at the subwavelength scale. Conventional bulk PhCs and recent van der Waals PoCs mainly show highly symmetric excitation of Bloch modes that closely rely on lattice orders. Here, we experimentally demonstrate a type of hyperbolic PoCs with configurable and low-symmetry deep-subwavelength Bloch modes that are robust against lattice rearrangement in certain directions. This is achieved by periodically perforating a natural crystal α-MoO3 that hosts in-plane hyperbolic phonon polaritons. The mode excitation and symmetry are controlled by the momentum matching between reciprocal lattice vectors and hyperbolic dispersions. We show that the Bloch modes and Bragg resonances of hyperbolic PoCs can be tuned through lattice scales and orientations while exhibiting robust properties immune to lattice rearrangement in the hyperbolic forbidden directions. Our findings provide insights into the physics of hyperbolic PoCs and expand the categories of PhCs, with potential applications in waveguiding, energy transfer, biosensing and quantum nano-optics. Photonic crystals (PhCs) are artificial periodic materials that can be used to manipulate the flow of light. Here, the authors report the realization of asymmetric PhCs based on in-plane hyperbolic phonon polaritons in perforated α-MoO3, showing low-symmetry deep-subwavelength Bloch modes that are robust against lattice rearrangement in specific directions. [ABSTRACT FROM AUTHOR]

We use a recent lattice determination of the vector and axial Ds → γ form factors at high squared momentum transfer q2 to infer their Bs → γ counterparts. To this end, we introduce a phenomenological approach summarized as follows. First, we describe the lattice data with different fit templates motivated by vector-meson dominance, that is expected to hold in the high-q2 region considered. We identify reference fit ansaetze with one or two physical poles, that we validate against alternative templates. Then, the pole residues can be unambiguously related to the appropriate couplings involving the pseudoscalar, the vector mesons concerned, and the photon — or tri-couplings — and the latter can be expressed as sums over quark magnetic moments, weighed by their e.m. charges. This description obeys a well-defined heavy-quark scaling, that allows to parametrically scale up the form factors to the Bs → γ case. We discuss a number of cross-checks of the whole approach, whose validation rests ultimately in a first-principle determination, e.g. in lattice QCD. Finally, we use our obtained form factors to reassess the SM prediction of ℬ(Bs → μ+μ−γ) in the range q 2 ∈ [4.2, 5.0] GeV, where an experimental measurement is awaited. [ABSTRACT FROM AUTHOR]

In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps δ t the recommended method converges at a rate of O (δ t 2 + h 2) . Based on graphic results and numerical examples, the application of the model is also examined. [ABSTRACT FROM AUTHOR]

In this article we introduce the Riesz difference sequence space rpq(ΔBα) of fractional order α, defined by the composition of fractional backward difference operator ΔBα given by (ΔBαv)k = ... and the Riesz matrix Rq. We give some topological properties, obtain the Schauder basis and determine the α-, β- and γ- duals and investigate certain geometric properties of the space rpq(ΔBα). Finally, we characterize certain classes of compact operators on the space rpq(ΔBα) using Hausdorff measure of non-compactness. [ABSTRACT FROM AUTHOR]

The contact vectors of a lattice are vectors which are minimal in the -norm in their parity class. It is shown that, in the space of all symmetric matrices, the set of all contact vectors of the lattice defines the subspace containing the Gram matrix of the lattice . The notion of extremal set of contact vectors is introduced as a set for which the space is one-dimensional. In this case, the lattice is rigid. Each dual cell of the lattice is associated with a set of contact vectors contained in it. A dual cell is extremal if its set of contact vectors is extremal. As an illustration, we prove the rigidity of the root lattice for and the lattice dual to the root lattice . [ABSTRACT FROM AUTHOR]

This paper considers coherent states for the representation of Weyl commutation relations over a field of p-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proven that the bases of coherent states corresponding to different lattices are mutually unbiased, and that the operators defining the quantization of symplectic dynamics are Hadamard operators. [ABSTRACT FROM AUTHOR]

Metasurfaces are of great interest as they exhibit unique electromagnetic properties. Currently, metasurface design focuses on generating new meta-atoms and their combinations. Here a topological database, reticular chemistry structure resource (RCSR), is introduced to bring a new dimension and more possibilities for metasurface design. RCSR has over 200 two-dimensional crystal nets, among which 72 are identified as suitable for metasurface design. Using a simple metallic cross as the metaatom, 72 metasurfaces are constructed from the atom positions and lattice vectors of the crystal nets templates. The transmission curves of all the metasurfaces are calculated using the finite-difference time-domain method. The calculated transmission curves have good diversity, showing that the crystal nets approach is a new engineering dimension for metasurface design. Three clusters are found for the calculated curves using the K-means algorithm and principal component analysis. The structure–property relationship between metasurface topology and transmission curve is investigated, but no simple descriptor has been found, indicating that further work is still needed. The crystal net design approach developed in this work can be extended to three-dimensional design and other types of metamaterials like mechanical materials. [ABSTRACT FROM AUTHOR]

Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in L^1(\Omega,{\mathcal A},\mu), where \mu is a probability measure, new conditional versions of the above theorems are obtained. [ABSTRACT FROM AUTHOR]

Let M and N be Archimedean vector lattices. We introduce orthogonally additive band operators and orthogonally additive inverse band operators from M to N and examine their properties. We investigate the relationship between orthogonally additive band operators and orthogonally additive disjointness preserving operators and show that under some assumptions on vector lattices M or N, these two classes are the same. By using this relation, we show that if μ is a bijective orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator) from M into N then μ-1:N →M is an orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator). [ABSTRACT FROM AUTHOR]

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O (σ 2 + ρ 2 + τ + h x + h y) . After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus. [ABSTRACT FROM AUTHOR]

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) [ Measure Theory , Torres Fremlin, Colchester, 2004] concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions. [ABSTRACT FROM AUTHOR]

Let E be a sublattice of a vector lattice F. A net {xα}α∈A ⊆ E is said to be F-order convergent to a vector x ∈ E (in symbols xα -F-→o x in E), whenever there exists a net {yβ}β∈B in F satisfying yβ ↓ 0 in F and for each β, there exists α0 such that |xα - x| ≤ yβ whenever α ≥ α0. In this manuscript, first we study some properties of F-order convergence nets and we extend some results to the general cases. Let E and G be sublattices of vector lattices F and H, respectively. We introduce FH-order continuous operators, that is, an operator T between two vector lattices E and G is said to be FH-order continuous, if xα -F-→o 0 in E implies Txα -H-→o 0 in G. We will study some properties of this new classification of operators and its relationships with order continuous operators. [ABSTRACT FROM AUTHOR]

An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from the positive real line and the Laplace convolution to the m-dimensional Euclidean space and the Fourier convolution. To formulate the general fractional calculus in the Riesz form, the Luchko approach to construction of the GFC, which was suggested by Yuri Luchko in 2021, is used. The general fractional integrals and derivatives are defined as convolution-type operators. In these definitions the Fourier convolution on m-dimensional Euclidean space is used instead of the Laplace convolution on positive semi-axis. Some properties of these general fractional operators are described. The general fractional analogs of first and second fundamental theorems of fractional calculus are proved. The fractional calculus of the Riesz potential and the fractional Laplacian of the Riesz form are special cases of proposed general fractional calculus of the Riesz form. [ABSTRACT FROM AUTHOR]

In the present article, we have established a result on φ - N¯, Pn; δ μ｜k summability of general summability factor of Fourier series, generalizing a result on ｜ N¯, pn ｜k summability. [ABSTRACT FROM AUTHOR]