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We employ classical density-functional theory to investigate the phase diagram of an assembly of mutually penetrable, parallel ellipsoids interacting via the generalized exponential model of index four (GEM-4) pair potential. We show that the crystal phases of the system are obtained from those of the spherically symmetric GEM-4 model by rescaling the lattice vectors. Performing this rescaling in combination with an arbitrary rotation of the lattice leads to infinitely many different structures with the same free energy, thereby implying their infinite degeneracy. These findings generalize to non-zero temperature the results formerly obtained by us [Pini et al., J. Chem. Phys. 153, 164901 (2020)] for the ground state of a similar system of ellipsoids interacting via a Gaussian potential. According to the mean-field free-energy functional used here, our conclusions apply to soft-core potentials both when they form cluster crystals as the GEM-4 and when they form single-occupancy crystals as the Gaussian itself. [ABSTRACT FROM AUTHOR]

By employing non-equilibrium molecular dynamics (NEMD) simulations and time-dependent Ginzburg–Landau (TDGL) theory for solidification kinetics [Cryst. Growth Des. 20, 7862 (2020)], we predict the kinetic coefficients of FCC(100) crystal–melt interface (CMI) of soft-spheres modeled with an inverse-sixth-power repulsive potential. The collective dynamics of the local interfacial liquid phase at the equilibrium FCC(100) CMIs are calculated based on a recently proposed algorithm [J. Chem. Phys. 157, 084 709 (2022)] and are employed as the resulting parameter that eliminates the discrepancy between the predictions of the kinetic coefficient using the NEMD simulations and the TDGL solidification theory. A 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) is observed. With the insights provided by the quantitative predictive theory, the variation of the solidification kinetic coefficient along the crystal–melt coexistence boundary is discussed. The combined methodology (simulation and theory) presented in this study could be further applied to investigate the role of the inter-atomic potential (e.g., softness parameter s = 1/n of the inverse-power repulsive potential) in the kinetic coefficient. [ABSTRACT FROM AUTHOR]

Three new Ewald series are derived using a new strategy that does not start with a proposed charge spreading function. Of these, the Ewald series produced using shifted potential interactions for the Ewald real space series converges relatively slowly, while the corresponding expression using a shifted force (SF) interaction does not converge. A comparison is made between several approximations of the Ewald method and the SF route to include Coulomb interactions in molecular dynamics (MD) computer simulations. MD simulations of a model bulk molten salt and water were carried out. The recently derived α′ variant of Ewald, by K. D. Hammonds and D. M. Heyes [J. Chem. Phys. 157, 074108 (2022)], has been developed analytically and found to be more accurate and computationally efficient than SF in part due to the smaller real space truncation distance that can be used. In addition, with α′, the number of reciprocal lattice vectors required is reduced considerably compared with the standard Ewald implementations to give the same accuracy. The invention of the α′ method shifts the computational balance back toward using an Ewald construction. The SF method shows greater errors in the Coulomb pressure and time dependent fluctuation properties compared to α′. It does not conserve the shadow Hamiltonian in a microcanonical MD simulation, whereas the α′ method does, which facilitates long time stability and insignificant drift of properties over time. The speed of the Ewald computer code is improved by using a new lookup table method. [ABSTRACT FROM AUTHOR]

A real-space method using generating integers is used to determine possible commensurate lattice Moiré patterns for a bilayer of two equal hexagonal lattices, which can in principle be extended to lattice mismatched bilayers. These Moiré patterns can be classified by a pair of relatively prime integers (n , m) , wherein a rotation θ (n , m) of the top hexagonal lattice maps its lattice vector (n , m) to (m , n) of the bottom lattice. Within this formulation, the area of the commensurate supercell is proportional to (n 2 + m 2 + n m) and the number of coincident lattice sites per supercell is given by (n − m) 2. Taking bilayer boron nitride (BN) as an example, we present how to systematically generate Moiré patterns and explore the differences in local chemistry in the interstitial region by impurity intercalation. Systematic calculations of the properties of intercalated 3d transition metals were performed in an h-BN (4 , 3) bilayer, corresponding to a rotation of 9.43 degrees. These calculations reveal that local symmetry in the intercalated region significantly affect the energetics and magnetization of the intercalated species. These results highlight that Moiré pattern physics is not limited to optoelectronic/electronic phenomena, such as interfacial exciton formation or magic angle superconductivity, but it also produces chemical and magnetic atomic site selectivity, which may play important roles in adsorption, catalysis, or quantum information. [ABSTRACT FROM AUTHOR]

We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix G ∈ Q n × n , we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix B ∈ Q 4 n × n such that B ⊺ B = G . [ABSTRACT FROM AUTHOR]

In the recent surge of papers on ergodic theory within Riesz spaces, this article contributes by introducing enhanced characterizations of ergodicity. Our work extends and strengthens prior results of both the authors and Homann, Kuo, and Watson. Specifically, we show that in a conditional expectation preserving system (E, T, S, e), S can be extended to L1(T) and operates as an isometry on Lp (T) spaces. [ABSTRACT FROM AUTHOR]

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

This note is devoted to developing Morrey regularity theory for the following system of Rivière \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f belongs to some Morrey space. Our results extend the L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24]. [ABSTRACT FROM AUTHOR]

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]

Scanning probe techniques are popular, nondestructive ways to visualize the real space structure of Van der Waals moirés. The high lateral spatial resolution provided by these techniques enables extracting the moiré lattice vectors from a scanning probe image. We have found that the extracted values, while precise, are not necessarily accurate. Scan-to-scan variations in the behavior of the piezos that drive the scanning probe and thermally driven slow relative drift between probe and sample produce systematic errors in the extraction of lattice vectors. In this Letter, we identify the errors and provide a protocol to correct for them. Applying this protocol to an ensemble of ten successive scans of near-magic-angle twisted bilayer graphene, we are able to reduce our errors in extracting lattice vectors to less than 1%. This translates to extracting twist angles with a statistical uncertainty less than 0.001° and uniaxial heterostrain with uncertainty on the order of 0.002%. [ABSTRACT FROM AUTHOR]

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]

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most $ 2n $ 2 n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of Abramovich Y.A., Lipecki Z. [On ideals and sublattices in linear lattices and F-lattices. Math Proc Cambridge Philos Soc. 1990;108(1):79–87.; On lattices and algebras of simple functions. Comment Math Univ Carolin. 1990;31(4):627–635.], as well as Kakutani's characterization of closed sublattices of $ C(K) $ C (K) spaces. [ABSTRACT FROM AUTHOR]

Let X be an irreducible 2 n-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular if its Azumaya algebra End (F) deforms with X to every Kähler deformation of X. We show that if F is a slope-stable reflexive sheaf of positive rank and the obstruction map HH² (X) → Ext 2 (F, F) has rank 1, then F is very modular. We associate to such a sheaf a vector in the Looijenga-Lunts-Verbitsky lattice of rank b 2 (X) + 2. Three sources of examples of such modular sheaves emerge. The first source consists of slope-stable reflexive sheaves F of positive rank that are isomorphic to the image Φ (O X) of the structure sheaf via an equivalence Φ: Db (X) → Db (Y) of the derived categories of two irreducible holomorphic symplectic manifolds. The second source consists of such F, which are isomorphic to the image of a skyscraper sheaf via a derived equivalence. The third source consists of images Φ (L) of torsion sheaves L supported as line bundles on holomorphic lagrangian submanifolds Z such that Z deforms with X in codimension 1 in moduli, and L is a rational power of the canonical line bundle of Z. An example of the first source is constructed using a stable and rigid vector bundle G on a K 3 surface X to get the very modular vector bundle F on the Hilbert scheme X [n] associated to the equivariant vector bundle G ⊠ ⋯ ⊠ G on X n via the Bridgeland-King-Reid (BKR) correspondence. This builds upon and partially generalizes results of O'Grady for n = 2. A construction of the second source associates to a set {Gi}ni = 1 of n distinct stable vector bundles in the same two-dimensional moduli space of vector bundles on a K3 surface X the very modular vector bundle F on X [n] corresponding to the equivariant bundle ⊕ σ ∈ S n [Gσ(1)...Gσ(n)] on Xn. [ABSTRACT FROM AUTHOR]

In a viscoelastic environment, the diffusion of a particle becomes non-Markovian due to the memory effect. An open question concerns quantitatively explaining how self-propulsion particles with directional memory diffuse in such a medium. Based on simulations and analytic theory, we address this issue with active viscoelastic systems where an active particle is connected with multiple semiflexible filaments. Our Langevin dynamics simulations show that the active cross-linker displays superdiffusive and subdiffusive athermal motion with a time-dependent anomalous exponent α. In such viscoelastic feedback, the active particle always exhibits superdiffusion with α = 3/2 at times shorter than the self-propulsion time (τA). At times greater than τA, the subdiffusive motion emerges with α bounded between 1/2 and 3/4. Remarkably, active subdiffusion is reinforced as the active propulsion (Pe) is more vigorous. In the high Pe limit, athermal fluctuation in the stiff filament eventually leads to α = 1/2, which can be misinterpreted with the thermal Rouse motion in a flexible chain. We demonstrate that the motion of active particles cross-linking a network of semiflexible filaments can be governed by a fractional Langevin equation combined with fractional Gaussian noise and an Ornstein–Uhlenbeck noise. We analytically derive the velocity autocorrelation function and mean-squared displacement of the model, explaining their scaling relations as well as the prefactors. We find that there exist the threshold Pe (Pe∗) and crossover times (τ∗ and τ†) above which active viscoelastic dynamics emerge on timescales of τ∗≲ t ≲ τ†. Our study may provide theoretical insight into various nonequilibrium active dynamics in intracellular viscoelastic environments. [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]

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic. [ABSTRACT FROM AUTHOR]

It was showed by Donner in [ Extension of positive operators and Korovkin theorems , Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X may be embedded into a cone X^s, called the sup-completion of X. We show that if one represents the universal completion of X as C^\infty (K), then X^s is the set of all continuous functions from K to [-\infty,\infty ] that dominate some element of X. This provides a functional representation of X^s, as well as an easy alternative proof of its existence. [ABSTRACT FROM AUTHOR]

In this paper, new purely topological approaches are furnished in order to characterize self-majorizing elements in an Archimedean vector lattice A. More precisely, it is shown that an element 0 < f ∈ A is a self-majorizing element if and only if every f-maximal order ideal of A is relatively uniformly closed. In addition, it is proved that self-majorizing elements are characterized via the hull–kernel topology on both the set of all proper prime order ideals P and on the set of all g-maximal order ideals Q of A, for all g ∈ A +. In fact, the set of all prime order ideals of A not containing f (respectively, the set of all g-maximal order ideals of A not containing f, for all g ∈ A +) is a closed with respect to the hull–kernel topology on P (respectively, on Q) if and only if f is a self-majorizing element in A. [ABSTRACT FROM AUTHOR]

Let X be an Archimedean vector lattice and X + denote the positive cone of X. A unary operation ϖ on X + is called a truncation on X if x ∧ ϖ y = ϖ x ∧ y for all x , y ∈ X +. Let X u denote the universal completion of X with a distinguished weak element e > 0. It is shown that a unary operation ϖ on X + is a truncation on X if and only if there exists an element u ∈ X u and a component p of e such that p ∧ u = 0 and ϖ x = p x + u ∧ x for all x ∈ X +. Here, px is the product of p and x with respect to the unique lattice-ordered multiplication in X u having e as identity. As an example of illustration, if ϖ is a truncation on some L p μ -space then there exists a measurable set A and a function u ∈ L 0 μ vanishing on A such that ϖ x = 1 A x + u ∧ x for all x ∈ L p μ. [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]

In this paper, we introduce a large class of (so-called) conditional indicators, on a complete probability space with respect to a sub σ-algebra. A conditional indicator is a positive mapping, which is not necessary linear, but may share common features with the conditional expectation, such as the tower property or the projection property. Several characterizations are formulated. Beyond the definitions, we provide some non trivial examples that are used in finance and may inspire new developments in the theory of operators on Riesz spaces. [ABSTRACT FROM AUTHOR]

Given a finite subset A of a distributive lattice, its total orderization to(A) is a natural transformation of A into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, a novel of convergence named the fuzzy u-uniform convergence is defined on the basis of the fuzzy order convergence in fuzzy Riesz spaces. The relation between the fuzzy u-uniform convergence and the fuzzy order convergence in fuzzy Archimedean Riesz spaces is studied. We also discuss the existence of the fuzzy u-uniform limit for the fuzzy u-uniform Cauchy sequence in fuzzy Archimedean Riesz Spaces. We give the characterization of fuzzy order completeness in fuzzy Riesz spaces. [ABSTRACT FROM AUTHOR]

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

The 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]

This paper considers the problem of measuring technical efficiency in some class of normed vector spaces. Specifically, the paper focuses on preordered and partially ordered vector spaces by proposing a suitable encompassing netput formulation of the production possibility set. Duality theorems extending some earlier results are established in the context of infinite dimensional spaces. The paper considers directional and normed distance functions and analyzes their relationships. Among other things, overall efficiency can be derived from technical efficiency under a suitable preordered vector space structure. More importantly, it is shown that the existence of core points in partially ordered vector spaces guarantees the comparison of production vectors using the directional distance function. Although the interior of the positive cone may be empty in infinite dimensional vector spaces, it is shown that normed distance functions can also be used to measure efficiency in such spaces by providing them with a suitable preorder structure. [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]

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

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]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [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]

The present paper continues investigation of narrow operators on complex vector lattices started in a recent paper by Dzhusoeva, Huang, Pliev and Sukochev. We answer their questions and strengthen one of their main results by proving that the extension T_{\Bbb {C}}\colon E_{\Bbb {C}}\to F_{\Bbb {C}} of a linear operator T \colon E \to F acting from a Dedekind complete real Riesz space E to a real Banach lattice F to their complexifications is narrow if and only if T is. [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]

We show that a normalized capacity \nu : \mathcal {P}(\mathbf {N})\to \mathbf {R} is invariant with respect to an ideal \mathcal {I} on \mathbf {N} if and only if it can be represented as a Choquet average of \{0,1\}-valued finitely additive probability measures corresponding to the ultrafilters containing the dual filter of \mathcal {I}. This is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces. [ABSTRACT FROM AUTHOR]

In this work, we consider the stochastic Cahn-Hilliard/Allen-Cahn equation with fractional noise, which is fractional in time and white in space. We obtain the existence, uniqueness, and Hölder regularity of the solution. Also, using a weak convergence approach, we prove that the law of solution associated with the above equation with a small perturbation satisfies the large deviation principle in the Hölder norm. [ABSTRACT FROM AUTHOR]

We give an example of an Archimedean Riesz space on which every automorphism is a strictly positive multiple of the identity. [ABSTRACT FROM AUTHOR]

We introduce and study the concept of generating function for natural elements in a Dedekind complete Riesz space equipped with a conditional expectation operator. This allows us to study discrete processes in a free-measure setting. In particular we improve a result obtained by Kuo, Vardy and Watson concerning Poisson approximation. [ABSTRACT FROM AUTHOR]

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [B.A. Watson, An Andô-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543–558] and a Riesz-Frechet representation theorem for the T-strong dual, where T is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required. [ABSTRACT FROM AUTHOR]

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavž, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C0 -semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C0 -semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C0 -semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence. [ABSTRACT FROM AUTHOR]

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators. [ABSTRACT FROM AUTHOR]

It is well known that the bidual of C(X) for a compact space X, supplied with the Arens product, is isometrically isomorphic as a Banach algebra to for some compact space. The space is unique up to homeomorphism. We establish a similar result for realcompact spaces: The order bidual of C(X) for a realcompact space X, when supplied with the Arens product, is isomorphic as an f-algebra to for some realcompact space. The space is unique up to homeomorphism. [ABSTRACT FROM AUTHOR]

It is known that if L is a Dedekind complete Riesz space and (Ω, Σ) is a measurable space, then the partially ordered linear space of all L-valued, finitely additive and order bounded vector measures m on Σ is also a Dedekind complete Riesz space (for the natural operations). In particular, the modulus |m|o of m exists in this space of measures and |m|o is given by a well known formula. Some 20 years ago L. Drewnowski and W. Wnuk asked the question (for L not Dedekind complete) if there is an m for which |m|o exists but, |m|o is not given by the usual formula? We show that such a measure m does indeed exist. [ABSTRACT FROM AUTHOR]