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We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics. [ABSTRACT FROM AUTHOR]

The authors present the results of the research combining the methods of approximation theory and optimal decision theory. Namely, a solution to the optimization problem for the biharmonic Poisson integral in the upper half-plane is considered one of the most optimal solutions to the biharmonic equation in Cartesian coordinates. The approximate properties of the biharmonic Poisson operator in the upper half-plane on the classes of quasi-smooth functions are obtained in the form of an exact equality for the deviation of quasi-smooth functions from the positive operator under consideration. [ABSTRACT FROM AUTHOR]

It is known that translation-invariant Gibbs measures of a model with an uncountable set of spin values can be described by positive fixed points of a nonlinear integral operator of Hammerstein type. Significant results have been obtained on positive fixed points of a Hammerstein-type operator with a degenerate kernel, but the existence of Gibbs measures corresponding to the fixed points have not been proved for constructed kernels. We construct new degenerate kernels of the Hammerstein operator in the context of the theory of Gibbs measures, and show that each positive fixed point of the operator gives a translation-invariant Gibbs measure. [ABSTRACT FROM AUTHOR]

In the paper we give new asymptotic evaluations for sequences of linear positive operators V n : C 2 π R → C 2 π R . Our method of the proof is entirely different than those known in this area. As applications we complete and extend known asymptotic evaluations in this topic. [ABSTRACT FROM AUTHOR]

By using a constant step-size, the convergence analysis of the gradient projection method on the sphere is presented for a closed spherically convex set. This algorithm is applied to discuss copositivity of operators with respect to cones. This approach can also be used to analyse solvability of nonlinear cone-complementarity problems. To our best knowledge this is the first numerical method related to the copositivity of operators with respect to the positive semidefinite cone. Numerical results concerning the copositivity of operators are also provided. [ABSTRACT FROM AUTHOR]

This is a discussion on the second order of accuracy difference schemes for approximate solution of the integral type time-nonlocal parabolic problems. Theorems on the stability of r-modified Crank–Nicolson difference schemes and second order of accuracy implicit difference scheme for approximate solution of the integral type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second order of accuracy in t difference schemes for the one- and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given. [ABSTRACT FROM AUTHOR]

The article discusses the existence of weak solutions of the Kelvin–Voigt model in the context of the motion of water with added polymers. The model considers the delay in establishing equilibrium states due to relaxation time and internal reorganization processes. The study aims to prove the weak solvability of the initial–boundary value problem by introducing regular Lagrangian flows and defining weak solutions based on a priori estimates and the theory of the topological degree of contracting vector fields. The research was funded by the Russian Science Foundation and the author declares no conflicts of interest. [Extracted from the article]

We consider the Schrödinger operator with regular short range complex-valued potential in dimension d ≥ 1 . We show that, for d ≥ 2 , the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for d = 1 , we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for d = 3 . Some directions for further research are formulated. [ABSTRACT FROM AUTHOR]

In the space of square integrable functions on the positive semi-axis, two positive self-adjoint operators generated by a one-dimensional free Hamiltonian are constructed. These operators are employed to construct a pair of spectrally absolutely continuous bounded self-adjoint operators whose difference is an operator of rank 1. The perturbation determinant is used to find an explicit form of the M. G. Krein spectral shift function for this pair. It is shown that despite the Asmoothness of the perturbation in the sense of Hölder, the point λ = 1, where the spectral shift function has a discontinuity of the first kind, is not an eigenvalue of the perturbed operator. [ABSTRACT FROM AUTHOR]

As a leading economic center in China and an international metropolis, Shenzhen has great significance in promoting sustainable urban development. To predict its amount of domestic waste clearance, a new multivariable grey prediction model with combinatorial optimization of parameters is established in this paper. Firstly, the new model expands the value range of the order r of a grey accumulation generation operator from positive real numbers (R +) to all real numbers (R), which enlarges the optimization space of parameter and has positive significance for improving model performance. Secondly, the dynamic background-value coefficient λ is introduced into the new model to improve the smoothing effect of the nearest neighbor generated sequences. Thirdly, with the objective function of minimizing the mean absolute percentage error (MAPE), the particle swarm optimization (PSO) is employed to optimize parameters r and λ to improve the overall performance of the new model. The new model is used to simulate and predict the amount of domestic waste clearance in Shenzhen, and the MAPE of the new model is only 0.27%, which is far superior to several other similar models. Lastly, the new model is applied to predict the amount of domestic waste clearance in Shenzhen. The results indicate the amount of domestic waste clearance in 2028 could be 9.96 million tons, an increase of 20.58% compared to 2021.This highlights the significant challenge that Shenzhen faces in terms of urban domestic waste treatment. Therefore, some targeted countermeasures and suggestions have been proposed to ensure the sustainable development of Shenzhen's economy and society. [ABSTRACT FROM AUTHOR]

We obtain existence results for the Q-curvature equation of order 2k on a closed Riemannian manifold of dimension n ≥ 2 k + 1 , where k ≥ 1 is an integer. We obtain these results under the assumptions that the Yamabe invariant of order 2k is positive and the Green's function of the corresponding operator is positive, which are satisfied in particular when the manifold is Einstein with positive scalar curvature. In the case where 2 k + 1 ≤ n ≤ 2 k + 3 or the manifold is locally conformally flat, we assume moreover that the operator has positive mass. In the case where n ≥ 2 k + 4 and the manifold is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on n and k. [ABSTRACT FROM AUTHOR]

This paper explores the concept of approximate Birkhoff–James orthogonality in the context of operators on semi-Hilbert spaces. These spaces are generated by positive semi-definite sesquilinear forms. We delve into the fundamental properties of this concept and provide several characterizations of it. Using innovative arguments, we extend a widely known result initially proposed by Magajna [17]. Additionally, we improve a recent result by Sen and Paul [24] regarding a characterization of approximate numerical radius orthogonality of two semi-Hilbert space operators, such that one of them is A -positive. Here, A is assumed to be a positive semi-definite operator. [ABSTRACT FROM AUTHOR]

In this paper, a modification of general linear positive operators introduced by Ibragimov and Gadjiev in 1970 is constructed. It is shown that this modification preserves exponential mappings and also contains modified Bernstein-, Szász- and Baskakov-type operators as special cases. The convergence properties of corresponding operators on [ 0 , ∞) and in exponentially weighted spaces are investigated. Finally, the quantitative Voronovskaja theorem in terms of modulus of continuity for functions having exponential growth is examined. [ABSTRACT FROM AUTHOR]

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity. [ABSTRACT FROM AUTHOR]

Metasurface enables the generation and manipulation of multiphoton entanglement with flat optics, providing a more efficient platform for large-scale photonic quantum information processing. Here, we show that a single metasurface optical device would allow more efficient characterizations of multiphoton entangled states, such as shadow tomography, which generally requires fast and complicated control of optical setups to perform information-complete measurements, a demanding task using conventional optics. The compact and stable device here allows implementations of general positive operator valued measures with a reduced sample complexity and significantly alleviates the experimental complexity to implement shadow tomography. Integrating self-learning and calibration algorithms, we observe notable advantages in the reconstruction of multiphoton entanglement, including using fewer measurements, having higher accuracy, and being robust against experimental imperfections. Our work unveils the feasibility of metasurface as a favorable integrated optical device for efficient characterization of multiphoton entanglement, and sheds light on scalable photonic quantum technologies with ultra-thin optical devices. Shadow tomography is efficient for quantum state characterization. Here, the authors implement shadow tomography on photonic states with a single metasurface, which alleviates the complexity in measurement [ABSTRACT FROM AUTHOR]

In this paper, we investigate Banach lattices on which each positive semi-compact operator T : E → F is null almost L-weakly compact (rep. Null almost M-weakly compact). Additionally, we present certain sufficient and necessary conditions for a positive Null almost L-weakly compact operator to be semi-compact. [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]

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form (D t ρ) 2 u (t) + 2 α D t ρ u (t) + A u (t) = p (t) q + f (t) , where 0 < t ≤ T , 0 < ρ < 1 and D t ρ is the Caputo derivative. The equation contains a self-adjoint positive operator A and a time-varying multiplier p(t) in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition B [ u (t) ] = ψ (t) is imposed, where B is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator A and the functional B are discussed. [ABSTRACT FROM AUTHOR]

The AdS/CFT correspondence states that certain conformal field theories are equivalent to string theories in a higher-dimensional anti-de Sitter space. One aspect of the correspondence is an equivalence of density matrices or, if one ignores normalizations, of positive operators. On the CFT side of the correspondence, any two positive operators A, B will satisfy the trace inequality Tr(AB) ≤ Tr(A)Tr(B). This relation holds on any Hilbert space H and is deeply associated with the fact that the algebra B(H ) of bounded operators on H is a type I von Neumann factor. Holographic bulk theories must thus satisfy a corresponding condition, which we investigate below. In particular, we argue that the Euclidean gravitational path integral respects this inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions. We prove this conjecture for Jackiw-Teitelboim gravity and we also motivate it for more general theories. [ABSTRACT FROM AUTHOR]

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals. [ABSTRACT FROM AUTHOR]

According to Berezin and Faddeev, a Schrödinger operator with point interactions –Δ + is any self-adjoint extension of the restriction of the Laplace operator to the subset of the Sobolev space . The present paper studies the extensions (realizations) invariant under the symmetry group of the vertex set of a regular m-gon. Such realizations HB are parametrized by special circulant matrices . We describe all such realizations with non-trivial kernels. А Grinevich–Novikov conjecture on simplicity of the zero eigenvalue of the realization HB with a scalar matrix and an even m is proved. It is shown that for an odd m non-trivial kernels of all realizations HB with scalar are two-dimensional. Besides, for arbitrary realizations the estimate is proved, and all invariant realizations of the maximal dimension are described. One of them is the Krein realization, which is the minimal positive extension of the operator . [ABSTRACT FROM AUTHOR]

Runge's phenomenon reveals that interpolation methods lack uniform convergence of the sequence of the thus constructed polynomials to the function. The linear positive operators, however, can be relied on to ensure convergence across the whole domain. Further, linear positive operators can be modified suitably to approximate integrable functions. In this paper, we prove theoretical results on the estimates for the rate of approximation of a new sequence of operators. In addition, we find an error estimate for a larger class of functions, the functions of bounded variation. Finally, the theory is supported by suitable numerical examples, and the convergence estimates of the proposed operator are compared with the classical modified Bernstein–Durrmeyer operator. [ABSTRACT FROM AUTHOR]

The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, the case where the polyatomicity is modelled by a discrete internal energy variable was considered lately. Here we consider the corresponding case for a continuous internal energy variable. Compactness results, stating that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a decomposition, such that the terms are, or at least are uniform limits of, Hilbert–Schmidt integral operators and therefore are compact operators. Moreover, bounds on—including coercivity of—the collision frequency are obtained for hard sphere like, as well as hard potentials with cutoff like, models, from which Fredholmness of the linearized collision operator follows, as well as its domain. [ABSTRACT FROM AUTHOR]

We introduce and study new multivariate and univariate Aldaz–Kounchev–Render type operators including their convergence and the asymptotic expansions. We solve also in the positive a conjecture from the very recent paper of Acu A. M., De Marchi S., Rasa I. (Result Math 78(1):21, 2023). [ABSTRACT FROM AUTHOR]

This paper contains a power series summability method based on Korovkin type approximation theorem for a sequence of fuzzy positive linear operators in two variables. Besides that, a theorem related to fuzzy rate of convergence has been derived with the help of fuzzy modulus of continuity, which gives clarity to understanding the analogous observations between fuzzy set theory and classical theory. Additionally, an example is illustrated which gives a better idea to understand that summability in two variable-based Korovkin type theorem has less drawback over fuzzy set theory. [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 differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with a convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic stability of Riemann–Liouville fractional resolvent families (R–L resolvent families) on Banach spaces and ordered Banach spaces. If the generator A satisfies some natural requirement, we show that an α -times R–L resolvent family { R α (t) } with generator A is uniformly stable iff 0 ∈ ρ (A) ; Next, we prove the subordination principle of R–L resolvent family. For a positive t 0 -bounded α -times R–L resolvent family on an ordered Banach space, we show that it can not be uniformly stable if α ∈ (1 , 2) ; in the case of α ∈ (0 , 1) , we show that A generates a positive R–L resolvent family iff - A is a sectorial operator with ω (- A) ≤ π 2 . Several results on orbit stability are also given by using contour integrals, Tauberian theorems and subordination principles. [ABSTRACT FROM AUTHOR]

In a recent paper, Bachir et al. (Rendiconti del Circolo Matematico di Palermo, 2022. https://doi.org/10.1007/s12215-022-00823-x) introduced generalized powers of linear operators. In other words, operators are not raised to numbers but to other operators. They gave several properties as regards this notion. In this paper, we further extend their results and also answer the question regarding the monotonicity of the map T ↦ A T . We also introduce a notion of generalized logarithms. More precisely, for two positive and invertible operators A and B such that 1 ∉ σ (A) , we define the logarithm of B to base A, denoted by log A B , and investigate some of its properties. [ABSTRACT FROM AUTHOR]

In the present study, a new implicit absolute stable difference scheme (DS) for an approximate solution of the time-dependent source identification problem (SIP) for the telegraph equation (TE) is presented. The stability of difference problem is established. In applications of abstract results in a Hilbert space with a self-adjoint positive definite operator (SAPDO), theorems on stability estimates for the solution of DSs for approximate solutions of the multidimensional time-dependent SIPs for telegraph equations are obtained. Finally, these DSs are tested on stability in both two- and three-dimensional examples with different boundary conditions and some computational results are illustrated. [ABSTRACT FROM AUTHOR]

Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace–Beltrami operator has no L 2 -eigenvalues ≥ 1 / 4 . In this article we prove a generalization of this result for the joint L 2 -eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces Γ \ G / K of higher rank. We derive dynamical assumptions on the Γ -action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups. [ABSTRACT FROM AUTHOR]

In various contexts in mathematical physics, such as out-of-equilibrium physics and the asymptotic information theory of many-body quantum systems, one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, which we denote by ∆, can be related by a perturbative series to another operator ∆0, whose logarithm is known. We set up a perturbation theory for the logarithm log ∆. It turns out that the terms in the series possess a remarkable algebraic structure, which enables us to write them in the form of nested commutators plus some "contact terms". [ABSTRACT FROM AUTHOR]

We give some characterizations of disjointly weakly compact sets in Banach lattices, namely, those sets in whose solid hulls every disjoint sequence converges weakly to zero. As an application, we prove that a bounded linear operator from a Banach space to a Banach lattice is an almost (L) limited operator if and only if it is a disjointly weakly compact operator, indeed, an operator which carries bounded sets to disjointly weakly compact ones. Some results on weak precompactness and (L -, M-)weak compactness of disjointly weakly compact operators are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present an inexact primal–dual method with correction step for a saddle point problem by introducing the notations of inexact extended proximal operators with symmetric positive definite matrix D. Relaxing requirement on primal–dual step sizes, we prove the convergence of the proposed method. We also establish the O(1/N) convergence rate of our method in the ergodic sense. Moreover, we apply our method to solve TV- L 1 image deblurring problems. Numerical simulation results illustrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

In this paper, the stability of several time discrete schemes for Allen–Cahn and Cahn–Hilliard equations and an error estimate for Cahn–Hilliard equation are analyzed. In order to discuss the Allen–Cahn and Cahn–Hilliard equations, a skew symmetric positive operator is defined, where in Allen–Cahn equation and in Cahn–Hilliard equation. We analyze stabilities of some schemes for the Allen–Cahn and the Cahn–Hilliard equation. The error estimates of Cahn–Hilliard equation are based on fully discrete scheme, its main idea is to use the finite element method to discretize in space, and then use two approximate results of the elliptic projection operator to analyze. Finally, we numerically verify convergence rates of this scheme. [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]

This work deals with the derivation of a receiver structure for the quantum detection of information-carrying coherent states used in photonic communications systems, that employ the degrees of freedom of both the complex amplitude and the state of polarization in the quantum state, producing a high dimensional modulation in the form of a constellation of quantum composite coherent states to be transmitted in the optical channel. Our receiver is based on the extension of the positive operator value measurements (POVM) in the constellation Hilbert space, towards projective measurements in a larger Hilbert space. Starting from the measurement vectors obtained from an optimum detection/discrimination strategy-the square root method (SRM) in our case-, and basing our analysis on the Naimark extension, we present a procedure for building those projectors for a received modulation format of composite coherent states. We analytically derive the orthogonal and idempotent projectors, arriving at closer form expressions for the considered modulation format, their decomposition in unitary rotations, and suggest a receiver configuration for its physical realization. Finally, an error probability analysis of our modulation formats is presented. [ABSTRACT FROM AUTHOR]

In our study we construct a boundary value problem in elasticity of porous piezoelectric bodies with a dipolar structure To construct an eigenvalue problem in this context, we consider two operators defined on adequate Hilbert spaces. We prove that the two operators are positive and self adjoint, which allowed us to show that any eigenvalue is a real number and two eigenfunctions which correspond to two distinct eigenvalues are orthogonal. With the help of a Rayleigh quotient type functional, a variational formulation for the eigenvalue problem is given. Finally, we consider a disturbation analysis in a particular case. It must be emphasized that the porous piezoelectric bodies with dipolar structure addressed in this study are considered in their general form, i.e.,inhomogeneous and anisotropic. [ABSTRACT FROM AUTHOR]

In modeling the population dynamics of biological species and the transmission dynamics of infectious diseases, age-structure and nonlocal diffusion are two important components since individuals need to be mature enough to move and they disperse and interact each other nonlocally. In this paper we study the principal spectral theory of age-structured models with nonlocal diffusion within a population of multigroups. First, we provide a criterion on the existence of the principal eigenvalue by using the theory of positive resolvent operators with their perturbations. Then we define the generalized principal eigenvalue and use it to investigate the influence of diffusion rate on the principal eigenvalue. Next we establish the strong maximum principle for age-structured nonlocal diffusion operators. Finally, as an example we apply our established theory to an age-structured cooperative system with nonlocal diffusion. [ABSTRACT FROM AUTHOR]

In recent decades, the frame theory and related topics have been attracted by many researchers. Biframe, a concept related to frames, was introduced as a pair of sequences, which dual frames are a particular case of them. In this article, more topics related to biframes are discussed. The effect of operators on biframes in order to preserve the biframe property is investigated more specifically. Then, by the importance of the relationship between a frame and its duals in the frame theory, this relationship is examined in the forming a biframe. In particular, the dual frames of two frames that form a biframe are studied, and interesting results are obtained. [ABSTRACT FROM AUTHOR]

Entanglement witnesses (EWs) are a versatile tool in the verification of entangled states. The framework of mirrored EW doubles the power of a given EW by introducing its twin—a mirrored EW—whereby two EWs related by mirroring can bound the set of separable states more efficiently. In this work, we investigate the relation between the EWs and its mirrored ones, and present a conjecture which claims that the mirrored operator obtained from an optimal EW is either a positive operator or a decomposable EW, which implies that positive-partial-transpose entangled states, also known as the bound entangled states, cannot be detected. This conjecture is reached by studying numerous known examples of optimal EWs. However, the mirrored EWs obtained from the non-optimal ones can be non-decomposable as well. We also show that mirrored operators obtained from the extremal decomposable witnesses are positive semi-definite. Interestingly, the witnesses that violate the well known conjecture of Structural Physical Approximation, do satisfy our conjecture. The intricate relation between these two conjectures is discussed and it reveals a novel structure of the separability problem. [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]

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations. [ABSTRACT FROM AUTHOR]

In this note we consider the order Lozanovsky spectrum of a positive operator on a complex Banach lattice and show that it generally contains the order Weyl spectrum, settling a weaker form of a question raised in Alekhno (Positivity 13(1):3–20, 2009). [ABSTRACT FROM AUTHOR]

The spread and control of infectious diseases are inevitably influenced by the age structure of the population and media effect. In this paper, we propose a susceptible-exposure-infection-recovery type age-structured COVID-19 model with media effect. First, the existence and uniqueness of the solution are obtained using semigroup theory and the positive operator method. The basic regeneration number R 0 is computed next and the globally asymptotical stability of the disease-free steady state, as well as the locally asymptotical stability of endemic steady state is studied without any extra conditions. The influence of media effect and age structure of the population on disease transmission are also verified by numerical simulations. Our result show that additional intensity of media broadcasts not only reduces the peak of disease outbreak but also shortens the duration of the epidemic. Further more, the proportion of infected adolescents is lower, and adults should pay more attention to self-protection. [ABSTRACT FROM AUTHOR]

In this study, we aim to extend approximation results for Bernstein type rational functions from real function space to fuzzy function space by using properties of interval analysis. For this purpose, we introduce fuzzy Bernstein type rational functions and investigate their Korovkin type approximation properties in fuzzy function space. We estimate rates of fuzzy convergence by means of fuzzy modulus of continuity and Lipschitz type fuzzy functions. Moreover, we present a Voronovskaja type result for the asymptotic expansion of fuzzy Bernstein type rational functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the splitting operators into the product of two positive operators. For given operators A and with A ≥ 0 and , if T can be splitted as T = AX for some positive operator X, then the existence of X and the properties of T are studied. The related research are the solutions of operator equations T = XAX and TX = XAX, which have some particular properties and broad applications in many fields. The conditions for the existence of solutions or idempotent solutions of these kinds of equations are studied and new representations of the general solutions are given. [ABSTRACT FROM AUTHOR]

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter α > 1 . The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (α , z) -divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space. [ABSTRACT FROM AUTHOR]

The authors consider the optimization problem related to the integral representation of the deviation of positive linear operators on the classes of (ψ, β)-differentiable functions in the integral metric. The Poisson integral, which is the solution of the Laplace equation in polar coordinates with the corresponding initial conditions given on the boundary of the unit disk, is taken as a positive linear operator. The Poisson integral is an operator with a delta-like kernel; therefore, it is the best apparatus for solving many problems of applied mathematics, namely: methods of optimization and variational calculus, mathematical control theory, theory of dynamical systems and game problems of dynamics, applied nonlinear analysis and moving objects search. The classes of (ψ, β)-differentiable functions on which the asymptotic properties of the solutions to Laplace equations in a unit disk are analyzed are generalizations of the Sobolev, Weyl–Nagy, etc. classes well-known in optimization problems. The problem solved in the article will make it possible to generate high-quality mathematical models of many natural and social processes. [ABSTRACT FROM AUTHOR]

In this paper, we study partial integral operators in Kaplansky-Hilbert module. A modular and cyclic modular spectrum of such operators is described, and a corresponding example is given. In addition, a variant of the Mercer theorem for partial integral operators with a symmetric and positive kernel has been proved. [ABSTRACT FROM AUTHOR]