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We consider Kantorovich modifications of linking operators and Stancu modifications of the classical Bernstein operators. For the modified operators we determine the limits of iterates and the invariant measures. In order to find the limits we have to solve systems of linear equations and to this end we use a suitable iterative algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new sequence of Riemann-Liouville type fractional Bernstein-Kantorovich operators $ K_{n}^{\alpha}(f;x) $ depending on a parameter $ \alpha $. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first and second order modulus of continuity of these operators. Moreover, we introduce a new operator that preserves affine functions from Riemann-Liouville type fractional Bernstein-Kantorovich operators. Further, we define the bivariate case of Riemann-Liouville type fractional Bernstein-Kantorovich operators and investigate the order of convergence. Some numerical results are given to illustrate the convergence of these operators and its comparison with the classical case of these operators. [ABSTRACT FROM AUTHOR]

First, we give conditions on a Banach lattice E$E$ so that an operator T$T$ from E$E$ to any Banach space is disjoint p$p$‐convergent if and only if T$T$ is almost Dunford–Pettis. Then, we study when adjoints of positive operators between Banach lattices are disjoint p$p$‐convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices E$E$ and F$F$: (i) a positive operator T:E→F$T: E \rightarrow F$ is almost weak p$p$‐convergent if and only if T∗$T^*$ is disjoint p$p$‐convergent; (ii) E∗$E^*$ has order continuous norm or F∗$F^*$ has the positive Schur property of order p$p$. Very recent results are improved, examples are given and applications of the main results are provided. [ABSTRACT FROM AUTHOR]

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high‐low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity‐weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time‐weighted dissipation estimates via the time‐weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics. [ABSTRACT FROM AUTHOR]

The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided. [ABSTRACT FROM AUTHOR]

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]

In this work, we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a general radial weight. We give necessary conditions when the symbol is of the form g1+g2‾, where g1 and g2 are analytic on the annulus z∈ℂ,1/2<|z|<1. [ABSTRACT FROM AUTHOR]

In this work, we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form ψ+φ‾, where ψ and φ are analytic on the annulus z∈ℂ,1/2<|z|<1. [ABSTRACT FROM AUTHOR]

Let A be a positive bounded operator on a Hilbert space H and let ‖ T ‖ A , w A (T) , and m A (T) denote the A -operator seminorm, the A -numerical radius, and the A -minimum modulus of an operator T in the semi-Hilbertian space (H , ‖ ⋅ ‖ A) , respectively. In this paper, we present new improvements of certain A -Cauchy–Schwarz type inequalities and as applications of our results, we provide refinements of some A -numerical radius inequalities for semi-Hilbertian space operators. It is shown, among other inequalities, that w A (T) ≤ (1 − 1 2 inf λ ∈ C ⁡ m A 2 (I − λ T)) ‖ T ‖ A , where I is the identity operator on H. A refinement of the triangle inequality for semi-Hilbertian space operators is also given. [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]

By leveraging the Variational Quantum Eigensolver (VQE), the "quantum equation of motion" (qEOM) method established itself as a promising tool for quantum chemistry on near-term quantum computers and has been used extensively to estimate molecular excited states. Here, we explore a novel application of this method, employing it to compute thermal averages of quantum systems, specifically molecules like ethylene and butadiene. A drawback of qEOM is that it requires measuring the expectation values of a large number of observables on the ground state of the system, and the number of necessary measurements can become a bottleneck of the method. In this work, we focus on measurements through informationally complete positive operator-valued measures (IC-POVMs) to achieve a reduction in the measurement overheads by estimating different observables of interest through the measurement of a single set of POVMs. We show with numerical simulations that the qEOM combined with IC-POVM measurements ensures satisfactory accuracy in the reconstruction of the thermal state with a reasonable number of shots. [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]

The paper presents research results combining the methods of approximation theory and optimal decision theory. Namely, the optimization problem for the biharmonic Poisson integral in the upper half-plane is considered as 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]

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]

In this paper, we estimate the difference between the Hadamard product and the Karcher mean of n positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of n = 2. Moreover, we give ratio inequalities of the operator power means by the Hadamard product. [ABSTRACT FROM AUTHOR]

In this paper we explore the relation between the A -numerical range and the A -spectrum of A -bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of A -normal operator and prove that closure of the A -numerical range of an A -normal operator is the convex hull of the A -spectrum. We further prove Anderson's theorem for the sum of A -normal and A -compact operators which improves and generalizes the existing result on Anderson's theorem for A -compact operators. Finally we introduce strongly A -numerically closed class of operators and along with other results prove that the class of A -normal operators is strongly A -numerically closed. [ABSTRACT FROM AUTHOR]

The extraordinary correlation seen in entangled states in space-like separated regions is one of the most intriguing aspects of quantum states. Practical utility of entanglement as a resource for quantum key distribution, dense coding, or teleportation generally requires maximally entangled states. In practice, entanglement quality degrades substantially due to channel noise. The problem may be mitigated by entanglement distillation. The simplest distillation protocol is enforced by local filtering operations and classical communication. The local filtering operations are merely generalized positive operator valued measures utilizing additional degrees of freedom (DoFs) as ancillary qubits. In this work, we experimentally show that filtering on a single channel (unilateral) is equally effective as filtering on both channels (bilateral) for distillation of pure non-maximally entangled bipartite states. This result holds for a non-maximally entangled multi-qubit Greenberger-Horne-Zeilinger (GHZ) like states as well, as they show a straightforward extension of the Bell state structure. Further, we provide a theoretical comparison of the efficacy of unilateral and bilateral filtering for the case of mixed states resulting from local depolarizing noise introduced either in one or both of the non-maximally entangled pairs. Surprisingly, when noise is introduced in either one of the channels, we find that unilateral filtering on the noise-free channel outperforms the filtering on the noisy channel and bilateral filtering on both channels. We also find that bilateral filtering is more effective when both channels are noisy. A reduction in the number of local operations, in general, has the advantage of reducing the complexity of the experimental apparatus and the reduction of measurement related errors. [ABSTRACT FROM AUTHOR]

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

We presented some improvements of the inequalities involving the numerical radius powers for products and sums of the operators investigated in the Hilbert space. We generalized and improved numerical radius inequalities with a generalization of the mixed Schwarz inequality. Among other things, with the help of a fraction and its power, as well as the introduction of $ \xi $, we provided a very good improvement for the $ \omega^r(E) $, for $ E\in \mathcal{B}(\mathcal{H}_s) $.

Given any separable complex Hilbert space, any trace-class operator B which does not have purely imaginary trace, and any generator L of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator K and a unique completely positive map Φ such that (i) L = K(⋅) + (⋅)K∗ + Φ, (ii) the superoperator Φ(B∗(⋅)B) is trace class and has vanishing trace, and (iii) tr(B∗K) is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of determining the time‐dependent source term in the general setting of Hilbert spaces and for general additional data. We prove the well‐posedness of this inverse problem by reducing the problem to an operator equation for the source function. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [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]

The aim of this paper is to study the asymptotic convergence of the sequences. Concretely, we will give the asymptotically statistical convergence of sequences. The asymptotically statistical convergence is studied in many papers including papers [3], [5], [23], [26], [29], [39], [41]. In this paper, we introduce and investigate the concepts of asymptotically deferred statistical equivalent functions of order $ \alpha $ and strong asymptotically deferred statistical equivalent functions of order $ \alpha $ defined on discrete countable amenable semigroups, which generalizes some of the results known in the literature. This is achieved by introducing the Folner sequences and sequences of non-negative integers. Based on this concept we apply it to the approximation theory, and we have proved the Korovkin type theorem and study the rate of convergence for positive linear operators which are deferred statistically convergent of order $ \alpha $. [ABSTRACT FROM AUTHOR]

Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi's characterization of the extreme quantum channels, Parthasarathy's characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff's theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs. [ABSTRACT FROM AUTHOR]

In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free. This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operators in the Heisenberg description. The result is a quantum generalization of the Lévy–Khintchine formula; Gaussian and jump contributions are included. As a byproduct, the most general quasi-free quantum-dynamical semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov–Fokker–Planck equation are included. As a classical subsystem can be observed, in principle, without perturbing it, information can be extracted from the quantum system, even in continuous time; indeed, the whole construction is related to the theory of quantum measurements in continuous time. While the dynamics is formulated to give the hybrid state at a generic time t , we show how to extract multi-time probabilities and how to connect them to the quantum notions of positive operator-valued measure and instrument. The structure of the generator of the dynamical semigroup is analyzed, in order to understand how to go on to non-quasi-free cases and to understand the possible classical-quantum interactions; in particular, all the interaction terms which allow to extract information from the quantum system necessarily vanish if no dissipation is present in the dynamics of the quantum component. A concrete example is given, showing how a classical component can input noise into a quantum one and how the classical system can extract information on the behavior of the quantum one. [ABSTRACT FROM AUTHOR]

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]

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, 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]

For k a field of positive characteristic and X a smooth variety over k , we compute the Hochschild cohomology of Grothendieck's differential operators on X. The answer involves the derived inverse limit of the Frobenius acting on the cohomology of the structure sheaf of X. [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]

This paper introduces a new idea to obtain a better order of approximation for the Baskakov operator. We conclude two new operators from orders one and two of the Baskakov type. Also, we prove some directed results concerning the rate of convergence of these operators. We apply the Maple software to demonstrate how the operators converge to a specific function. [ABSTRACT FROM AUTHOR]

Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for the existence of general solutions, self-adjoint solutions, and positive solutions to the system of A i X B i + C i Y D i + E i Z F i = G i (i = 1 , 2) under additional conditions. In addition, we derive the representation of general solutions to the system of A i X B i + C i Y D i + E i Z F i = G i (i = 1 , 2) , and provide the matrix representation of the self-adjoint solutions and the positive solutions in the sense of the star order. [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]

This paper deals with the approximations of the functions by generalized Durrmeyer operators of Szász–Mirakjan, which are linear positive operators. Several approximation results are presented well, and we estimate the approximation properties along with the order of approximation and the convergence theorem of the proposed operators. For an explicit explanation of the operators, we determine the properties using the weight function. A quantitative approach is discussed for the operators; quantitative Voronovskaya type and Grüss type theorems are established, showing the operators' more efficient work. We investigate the A$$ A $$‐statistical convergence properties for the said operators, including the rate of approximation in a statistical sense. An important property for the rate of convergence of the operators is obtained in terms of the function with a derivative of the bounded variation. At last, the graphical representations and numerical analysis are discussed and shown to support our theoretical findings. [ABSTRACT FROM AUTHOR]

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $ T^*T $ T ∗ T of a densely defined linear transformation T and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint extensions are also revised. [ABSTRACT FROM AUTHOR]

We introduce and study the disjoint weak p-convergent operators in Banach lattices, and we give a characterization of it in terms of sequences in the positive cones. As an application, we derive the domination and the duality properties of the class of positive disjoint weak p-convergent operators. Next, we examine the relationship between disjoint weak p-convergent operators and disjoint p-convergent operators. Finally, we characterize order bounded disjoint weak p-convergent operators in terms of sequences in Banach lattices. [ABSTRACT FROM AUTHOR]

This work presents a generalized singular value and norm inequalities associated with 2 ´ 2 positive semidefinite block matrices. [ABSTRACT FROM AUTHOR]

We presented some improvements of the inequalities involving the numerical radius powers for products and sums of the operators investigated in the Hilbert space. We generalized and improved numerical radius inequalities with a generalization of the mixed Schwarz inequality. Among other things, with the help of a fraction and its power, as well as the introduction of ξ, we provided a very good improvement for the ωr(E), for E ∈ B(Hs). [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]

The purpose of the present paper is to study a sequence of linear and positive operators which was introduced by A. Lupas. First, we obtain estimate of moments of the operators and then prove a basic convergence theorem for simultaneous approximation. Further, we find error in approximation in terms of modulus of continuity of function. Finally, we establish a Voronovskaja asymptotic formula for linear combination of the above operators. [ABSTRACT FROM AUTHOR]

In this paper, we provide several techniques to explicitly calculate fractional powers of 2×2$2\times 2$ operator matrices Λ=Λ11Λ12Λ21Λ22,$$\begin{equation*}\hspace*{10pc} \Lambda = \def\eqcellsep{&}\begin{bmatrix} \Lambda _{11} & \Lambda _{12} \\ \Lambda _{21} & \Lambda _{22} \end{bmatrix}, \end{equation*}$$focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction–diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a 4×4$4\times 4$ operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp-statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder's fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. [ABSTRACT FROM AUTHOR]

In this article, we establish a version of Banach’s contraction principle in ordered Banach spaces. This version is adapted to prove existence and uniqueness results for an integral equation or a boundary value problem depending on the derivative. [ABSTRACT FROM AUTHOR]

In this work, we provide some further refinements for the following concavity inequality A σ B + C σ D ≤ (A + C) σ (B + D) for positive invertible bounded linear operators A , B , C , D on a complex Hilbert space and for an operator mean σ. To this end, we use the operator majorization intended for comparing two n + 1 -tuples of pairs of operators, and prove a Hardy-Littlewood-Pólya-Karamata (HLPK) type theorem for the triangle map induced by σ. Next, we apply a specification of the HLPK Theorem for n = 2 in order to obtain the required refinements. [ABSTRACT FROM AUTHOR]

We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi α -relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory. [ABSTRACT FROM AUTHOR]