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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]

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]

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]

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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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 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 characterize when, for any infinite cardinal α , the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind α -complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself. [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]

Using the linearization of positive multilinear operators by means of the Fremlin tensor product of vector lattices makes it possible to show that a multilinear operator from the Cartesian product of majorizing subspaces of vector lattices to Dedekind complete vector lattice admits extension to a positive multilinear operator on the Cartesian product of the ambient vector lattices. We establish that this is valid if the multilinear operator is defined on the Cartesian product of majorizing subspaces of separable Banach lattices and takes values in a topological vector lattice with the -interpolation property, provided that the Banach lattices have the property of subadditivity. The latter ensures that the algebraic tensor product of the majorizing subspaces is majorizing in the Fremlin tensor product of the Banach lattices. The open question is: whether or not the result is valid if the subadditivity property is omitted or weakened. The possibility of weakening the order completeness of the target lattice by some additional requirements on the initial vector lattices was firstly observed by Abramovich and Wickstead in proving a version of the Hahn–Banach–Kantorovich theorem. [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]

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]

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]

A linear operator in a locally solid vector lattice is a Lebesgue operator, if for every net in satisfying ; a -operator, if for every -bounded increasing net in there exists with ; a quasi -operator, if takes -bounded increasing nets in to -Cauchy nets; a Levi operator, if for every -bounded increasing net in there exists such that ; and a quasi Levi operator, if takes -bounded increasing nets in to -Cauchy ones. We address the domination problem for the quasi -operators and quasi Levi operators in locally solid vector lattices. Moreover, under study are some properties of Lebesgue, Levi, and -operators. In particular, we prove that the vector space of regularly Lebesgue operators is a subalgebra of the algebra of all regular operators. [ABSTRACT FROM AUTHOR]

Conditions under which the relatively uniform convergence in vector lattices is complete, topological, and agrees with the order convergence are presented. r-Completion of an Archimedean vector lattice is investigated. Basic properties of the Archimedization of a vector lattice are established. [ABSTRACT FROM AUTHOR]

We characterize finite elements, an order-theoretic concept in Archimedean vector lattices, in the setting of ordered Banach spaces with positive unconditional basis as vectors having finite support with respect to their basis representations. Using algebraic vector space bases, we further describe a class of infinite dimensional vector lattices in which each element is finite and even self-majorizing. [ABSTRACT FROM AUTHOR]

A nonempty set S of real-valued functions is said to be truncated if it contains with any function f its meet with the constant function 1. Very recently, the first named author and Hajji obtained a Stone-Weierstrass-type theorem for a truncated vector sublattice L of C 0 X , where X is a locally compact Hausdorff space. Relying heavily on the multiplicative version of the classical Stone-Weierstrass theorem, they proved that if L separates the points of X and vanishes nowhere on X, then L is uniformly dense in C 0 X . In this paper, we shall provide a new, intrinsic, and more natural proof of this result. We shall then apply the result to prove that any truncated vector lattice of bounded real-valued functions is essentially a C 0 X -type space for some locally compact Hausdorff space X. This yields, in particular, that for any arbitrary topological space Y, a locally compact Hausdorff space X can be found such that C 0 Y and C 0 X are essentially the same, eliminating any reason for considering Banach lattices of continuous functions vanishing at infinity on other than locally compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]

This paper presents a multiplierless image-cipher, with extendable 2048-bit key-space, based on a 4-dimensional (4D) quantized piece-wise linear cat map (PWLCM). The quantized PWLCM exhibits limit-cycles of 4-bit encoded integers with periods greater than 107. The synthesis of the PWLCM in a finite state space allows to eliminate the undesirable finite precision effect due to the hardware realization. The proposed image-cipher combines chaos, modular arithmetic, and lattice-based cryptography to encrypt a color image by performing pixel permutation and diffusion in a single operation. Further, an image-dependent confusion operation based on an 8-bit 2D-PWLCM is performed on the whole image to enhance security. In order to increase the key-space without key duplication, 16 × 16 sub-images are modified using sub-keys of different lattice length vectors generated from the external key. Both simulations and security analyses confirm that the proposed algorithm can resist common cipher attacks, in addition to its advantages such as simplicity, ease of implementation on low-end processors and extensibility of key-space that allows it to easily adapt even for future post-quantum computing attacks. [ABSTRACT FROM AUTHOR]

The asymptotically autonomous dynamics in H s (R n) for any s ∈ (0 , 1) and n ∈ N are discussed for a class of highly nonlinear nonclassical diffusion equations perturbed by colored noise. The main feature of this model is that the time-dependent drift and diffusion terms have subcritical and superlinear polynomial growth of orders p and q, respectively, satisfying 2 ≤ 2 q < p < ∞ if n = 1 and s ∈ [ 1 2 , 1) ; otherwise, 2 ≤ 2 q < p < 2 n n - 2 s. The number 2 n n - 2 s is called the fractional critical Sobolev embedding exponent. Under this setting, we prove the existence, time-dependent uniform compactness and asymptotically autonomous convergence of pathwise random attractors of the equations in H s (R n) when the time-dependent nonlinearities satisfy some new conditions. The time-dependent uniform pullback asymptotical compactness of the solution operators in H s (R n) is proved by virtue of a cut-off technique [38], a spectral decomposition approach and uniform estimates in a time-uniformly tempered attracting universe in order to overcome several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains, the weak dissipativeness of the systems and the unknown measurability of attractors. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of locally dominated orthogonally additive operators on lattice-normed spaces (LNS). In the first part of the paper, sufficient conditions for the existence of a local exact majorant of a locally dominated operator and formulas for its calculation are given. The second part shows that the -compactness of a dominated orthogonally additive operator acting from a decomposable lattice-normed space to a Banach space with mixed norm implies the -compactness of its exact majorant. [ABSTRACT FROM AUTHOR]

Birkhoff's representation theorem (Birkhoff, Duke Math J 3(3):443–454, 1937) defines a bijection between elements of a distributive lattice and the family of upper sets of an associated poset. Although not used explicitly, this result is at the backbone of the combinatorial algorithm by Irving et al. (J ACM 34(3):532-543, 1987) for maximizing a linear function over the set of stable matchings in Gale and Shapley's stable marriage model (Gale and Shapley, Am Math Monthly 69(1):9–15 1962). In this paper, we introduce a property of distributive lattices, which we term as affine representability, and show its role in efficiently solving linear optimization problems over the elements of a distributive lattice, as well as describing the convex hull of the characteristic vectors of the lattice elements. We apply this concept to the stable matching model with path-independent quota-filling choice functions, thus giving efficient algorithms and a compact polyhedral description for this model. To the best of our knowledge, this model generalizes all those for which similar results were known, and our paper is the first that proposes efficient algorithms for stable matchings with choice functions, beyond classical extensions of the Deferred Acceptance algorithm. [ABSTRACT FROM AUTHOR]

A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion. [ABSTRACT FROM AUTHOR]

The main idea behind lattice sieving algorithms is to reduce a sufficiently large number of lattice vectors with each other so that a set of short enough vectors is obtained. It is therefore natural to study vectors which cannot be reduced. In this work we give a concrete definition of an irreducible vector and study the properties of the set of all such vectors. We show that the set of irreducible vectors is a subset of the set of Voronoi relevant vectors and study its properties. For extremal lattices this set may contain as many as 2 n vectors, which leads us to define the notion of a complete system of irreducible vectors, whose size can be upper-bounded by the kissing number. One of our main results shows that modified heuristic sieving algorithms heuristically approximate such a set (modulo sign). We provide experiments in low dimensions which support this theory. Finally we give some applications of this set in the study of lattice problems such as SVP, SIVP and CVPP. The introduced notions, as well as various results derived along the way, may provide further insights into lattice algorithms and motivate new research into understanding these algorithms better. [ABSTRACT FROM AUTHOR]

Fractional Black-Scholes (FBS) equation has been widely applied in options pricing problems. However, most of the literatures focus on pricing the single-asset option using the FBS model, and research on time-space FBS with multiple assets has remained vacant. Option pricings are always governed by multiple assets. Under the assumption that fluctuation of asset price is regarded as a fractal transmission system and follows two independent geometric Lévy processes, 2D time-space fractional Black-Scholes equation (TDTSFBSE) is proposed to describe the instantaneous price in the financial market. In this work, we discrete the TDTSFBSE with implicit finite difference and present a fast parallel all-at-once iterative method for the resulting linear system. The proposed method can greatly reduce the storage requirements and computational cost. Theoretically, we discuss the convergence property of the fast iterative method. Numerical examples are given to illustrate the effectiveness and efficiency of our method. [ABSTRACT FROM AUTHOR]

We overview basic results in the field of Hausdorff operators obtained during the last decade. [ABSTRACT FROM AUTHOR]

The aim in this note is to show that the variable Riesz potential operator Iα(·)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$I_{\alpha(\cdot)}$$\end{document} embeds variable exponent grand Morrey spaces Lp(·)-0,ν(·),θ(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{p(\cdot)-0,\nu(\cdot),\theta}(G)$$\end{document} into Campanato–Morrey spaces. [ABSTRACT FROM AUTHOR]

We examine the problem of projecting subsets of a commutative, positively ordered monoid into an o-ideal. We prove that to this end one may restrict to a sufficient subset, for whose cardinality we provide an explicit upper bound. Several applications to set functions, vector lattices and other more explicit structures are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of the Riesz space fractional advection–dispersion equation. First, the Riesz fractional derivative is approximated with respect to the space variable using the spline interpolation. Furthermore, we use the Euler and the Crank–Nicolson schemes to approximate the time ordinary derivative and get two difference schemes. Second, using the matrix analysis method, we prove that the two difference schemes are unconditionally stable. Finally, some numerical results are given, which demonstrate the effectiveness of the two difference schemes. [ABSTRACT FROM AUTHOR]

A lattice homomorphism between quasi-Banach lattices is known to be compact if and only if it is a sum of a series of rank one lattice homomorphisms converging in the operator norm with pairwise disjoint images. We obtain an analogous description for the dual class of -compact and compact linear operators that almost preserve intervals and act in quasi-Banach lattices. As a corollary, we get a characterization of a pair of quasi-Banach lattices having no nonzero -compact (compact) operators that almost preserve intervals. Also, we prove some theorems of the Radon–Nikodym type for almost interval preserving -compact (compact) operators. [ABSTRACT FROM AUTHOR]

A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices. [ABSTRACT FROM AUTHOR]

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

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

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