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The Golden–Thompson trace inequality, which states that Tr eH+K ≤ Tr eHeK, has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or H = − − Δ + m and K = potential, Tr eH+(1−u)KeuK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr XsYtX1−sY1−t ≤ Tr XY for positive operators X, Y and for 1 2 ≤ s , t ≤ 1 , and s + t ≤ 3 2 . The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the 3 2 , 1 range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. [ABSTRACT FROM AUTHOR]

An analytic semigroup of operators on a Banach space is approximated by a sequence of positive integer powers of a linear-fractional operator function. It is proved that the order of the approximation error in the domain of the generating operator equals . For a self-adjoint positive definite operator decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup () by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals . [ABSTRACT FROM AUTHOR]

We constructed a summation–integral type operator based on the latest research in the linear positive operators area. We establish some approximation properties for this new operator. We highlight the qualitative part of the presented operator; we studied uniform convergence, a Voronovskaja-type theorem, and a Grüss–Voronovskaja type result. Our subsequent study focuses on a direct approximation theorem using the Ditzian–Totik modulus of smoothness and the order of approximation for functions belonging to the Lipschitz-type space. For a complete image on the quantitative estimations, we included the convergence rate for differential functions, whose derivatives were of bounded variations. In the last section of the article, we present two graphs illustrating the operator convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation on differences of two different positive linear operators (generalized Păltănea type operators and M. Heilmann type operators) with same basis functions. We estimates a quantitative difference of these operators in terms of modulus of continuity and Peetre's K-functional. We represent the rate of convergence, using modulus of continuity and Peetre's K-functional. Also, we represent Heilmann-type operators in terms of hypergeometric series. [ABSTRACT FROM AUTHOR]

Quantum incompatibility is a fundamental property in quantum physics and is considered as a resource in quantum information processing tasks. Here, we construct a framework based on the incompatibility witness originated from quantum state discrimination. In this framework, we discuss the geometrical properties of the witnesses with noisy mutually unbiased bases and construct a quantifier of quantum incompatibility associated with geometrical information. Furthermore, we explore the incompatibility of a pair of positive operator valued measurements, which only depends on the information of measurements, and discuss the incompatibility of measurements which can be discriminated by mutually unbiased bases. Finally, if we take a resource-theory perspective, the new-defined quantifier can characterize the resource of incompatibility. This geometrical framework gives the evidence that vectors can be utilized to describe resourceful incompatibility and make a step to explore geometrical features of quantum resources. [ABSTRACT FROM AUTHOR]

In the field of domain decomposition, the optimized Schwarz method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between subdomains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two subdomains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross-points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the proposed method. [ABSTRACT FROM AUTHOR]

We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.

Several numerical techniques have been developed to approximate Riemann–Liouville (R - L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R - L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and Hölder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models. [ABSTRACT FROM AUTHOR]

We study the convergence of Bernstein type operators leading to two results. The first: The kernel K n of the Bernstein–Durrmeyer operator at each point x ∈ (0 , 1) — that is K n (x , t) d t — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions f of bounded variation. [ABSTRACT FROM AUTHOR]

We relate non integer powers L s , s > 0 of a given (unbounded) positive self-adjoint operator L in a real separable Hilbert space H with a certain differential operator of order 2 ⌈ s ⌉ , acting on even curves R → H. This extends the results by Caffarelli–Silvestre and Stinga–Torrea regarding the characterization of fractional powers of differential operators via an extension problem. [ABSTRACT FROM AUTHOR]

By combining tree representation of sets with the method introduced in the previous three papers I–III [39,35,37] in the series, we give a new Π 2 1 -preserving interpretation of KP ω r + (Π n + 2 - Found) + θ (Kripke–Platek set theory with the foundation schema restricted to Π n + 2 , and augmented by θ) in Σ 1 1 - AC 0 + (Π n + 2 1 - TI) + θ for any Π 2 1 sentence θ , where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard interpretation. Thus the addition of any Π 2 1 theorem of BI ≡ Σ 1 1 - AC 0 + (Π ∞ 1 - TI) does not increase the consistency strength of KP ω. Among such Π 2 1 theorems are several fixed point principles for positive arithmetical operators and ω -model reflection (the cofinal existence of coded ω -models) for theorems of BI. The reader's familiarity to the previous works I–III in the series might help, but is not necessary. [ABSTRACT FROM AUTHOR]

Tomita-Takesaki theory associates a positive operator called the "modular operator" with a von Neumann algebra and a cyclic-separating vector. Tomita's theorem says that the unitary flow generated by the modular operator leaves the algebra invariant. I give a new, short proof of this theorem which only uses the analytic structure of unitary flows, and which avoids operator-valued Fourier transforms (as in van Daele's proof) and operator-valued Mellin transforms (as in Zsidó's and Woronowicz's proofs). The proof is similar to one given by Bratteli and Robinson in the special case that the modular operator is bounded. [ABSTRACT FROM AUTHOR]

• The presented novel upwind moving least squares approximation can satisfy positive operator condition. • Suggested upwind method can successfully overcome the problem of nonphysical oscillation encountering convection-dominated partial differential equations. • The proposed upwind method is more accurate than the existing version of mixed discrete least squares meshfree method for solving convection-dominated equations. • The suggested method can be utilized in any meshfree method. Moving Least Squares (MLS), as a series representation type of approximation, is broadly used in a wide array of meshfree methods. However, using the existing standard form of the MLS causes an unphysical oscillation for the meshfree methods encountering the convection-dominated partial differential equations (PDEs). In this study, several approaches are investigated to enhance the MLS approximation for solving convection-dominated problems. A novel upwind version of MLS approximation called shifted upward MLS (SU-MLS), is presented, which is based on strengthening the effect of upwind nodal points in approximation by wisely adjusting the weight function. Regarding the presented theoretical/numerical investigation, the proposed SU-MLS approximation can yield monotone solutions encountering the convection-dominated PDEs, unlike the existing standard form of MLS. The suggested SU-MLS approximation is, then, utilized in the mixed discrete least squares meshfree (MDLSM) method, rather than the standard MLS which is conventionally used in existing MDLSM. The novel method, which uses SU-MLS, is named upwind MDLSM (UMDLSM). Several numerical examples are investigated, and the results are compared to existing MDLSM. The obtained results indicate that the suggested UMDLSM is remarkably more accurate than the existing MDLSM in convection-dominated PDEs. Furthermore, while the existing MDLSM dramatically suffers from spurious oscillations (wiggling) when the Peclet number is high, the presented UMDLSM can yield monotone and accurate solutions. [ABSTRACT FROM AUTHOR]

We find conditions under which a generalized potential-type operator acts continuously from a Lebesgue space with a general weight to its dual space and possesses the positivity property. Based on these conditions, we prove the global existence and uniqueness theorems for various classes of nonlinear integral equations of convolution type in real weighted Lebesgue spaces using the method of monotonic (in the Browder–Minty sense) operators. Also we obtain estimates of the norms of solutions, which imply that the corresponding homogeneous equations have only a trivial solution. [ABSTRACT FROM AUTHOR]

In this paper, using the contractive maps, we present some new interpolations between the Heinz and Heron operator means for unitarily invariant norms. Let A , X , B ∈ B (H) and A , B be two positive definite operators. and let 1 2 ≤ β ≤ 1 , α ∈ [ 1 2 , ∞). If 1 4 ≤ ν ≤ μ ≤ 1 2 and μ + ν 2 ≤ r ≤ 1 − μ + ν 2 , or if 1 2 ≤ μ ≤ ν ≤ 3 4 and 1 − μ + ν 2 ≤ r ≤ μ + ν 2 , then ⫴ H r (A , X , B) ⫴ ≤ ⫴ (1 − β) H μ (A , X , B) + β H ν (A , X , B) ⫴ ≤ ⫴ ℱ α (A , X , B) ⫴. We also consider some Heinz-type inequalities that involve differences and sums of operators. We give new forms and reverse inequalities of the presented inequalities by Singh et al. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth. [ABSTRACT FROM AUTHOR]

We introduce a tensor-product kind bivariate operator of a new generalization of Bernstein-type rational functions and its GBS (generalized Boolean sum) operator, and we investigate their approximation properties by obtaining their rates of convergence. Moreover, we present some graphical comparisons visualizing the convergence of tensor-product kind bivariate operator and its GBS operator. [ABSTRACT FROM AUTHOR]

In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ( Δ 0 1 ) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ( Δ 0 0 ) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to constructive investigations of variants of determinacy statements for those games in which the players make choices from only finitely many alternatives. As a result, three formulations of open determinacy, that are all classically equivalent with each other, are equivalent to three different semi-classical principles, namely Markov's Principle, Lesser Limited Principle of Omniscience and Limited Principle of Omniscience, over a suitable constructive base theory that proves clopen determinacy. Open and clopen determinacies for these games are thus separated. Some basic results on variants of inductive definitions for Δ 0 0 positive operators will also be given. [ABSTRACT FROM AUTHOR]

Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states. [ABSTRACT FROM AUTHOR]

The distribution of ticks is essentially determined by the presence of climatic conditions and ecological contexts suitable for their survival and development. We build a model that explicitly takes into account each physiological state through a system of infinite differential equations where tick population density are structured on an infinite discrete set. We suppose that intrastage development process is temperature dependent (Arrhenius temperatures function) and that larvae hatching and adult mortality are temperature and water vapor deficit dependent. We analysed mathematically the model and have explicit the R 0 of the tick population. [ABSTRACT FROM AUTHOR]

In this paper, following the concept of AI-statistical convergence for real sequences introduced by Savas et al. [22], we deal with Korovkin type approximation theory for a sequence of positive convolution operators deĄned on C[a, b], the space of all real valued continuous functions on [a, b], in the line of Duman [6]. In the Section 3, we study the rate of AI-statistical convergence. [ABSTRACT FROM AUTHOR]

In this paper, we obtain asymptotic formulae on nilmanifolds Γ ﹨ G , where G is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup Γ. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group G (and more generally any positive Rockland operators when G is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions. [ABSTRACT FROM AUTHOR]

In the current paper a new form of Durrmeyer-type operator involving the beta function is introduced and its approximation properties are studied. We obtain the rate of convergence in different terms. Order of approximation for functions of some special class is also obtained. We establish the Voronovskaja type asymptotic result for this operator and the direct estimate. [ABSTRACT FROM AUTHOR]

The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators. [ABSTRACT FROM AUTHOR]

For any I l i , 1 I l i I n i , we denote by HT ht the function I g SB abl sb i if I b SB l sb i ( I m i + 1) I p i - 1 and the constant function I b SB l sb i if I b SB l sb i ( I m i + 1) I p i . Now by substituting function (010) into function (100) we get function (101), and by substituting function (112) into function (101) we get function (001). The function v z has the vector of values HT ht Since d i /= d j, then for each z D j the tuple of functions (v z (x), f 1 (x)) is correct. Keywords: enumeration closure operator; functions of three-valued logic EN enumeration closure operator functions of three-valued logic 105 114 10 04/15/22 20220401 NES 220401 Originally published in Diskretnaya Matematika (2021) 33, No2, 86-99 (in Russian). [Extracted from the article]

We give a general theory on well-posedness and time asymptotics for growth fragmentation equations in L1 spaces. We prove first generation of C0-semigroups governing them for unbounded total fragmentation rate and fragmentation kernel b (. ,.) such that ∫0y xb(x, y)dx = y − η(y)y (0 ≤ η (y) ≤ 1 expresses the mass loss) and continuous growth rate r (.) such that ∫0∞ 1/r(τ) dτ = + ∞. This is done in the spaces of finite mass or finite mass and number of agregates. Generation relies on unbounded perturbation theory peculiar to positive semigroups in L1 spaces. Secondly, we show that the semigroup has a spectral gap and asynchronous exponential growth. The analysis relies on weak compactness tools and Frobenius theory of positive operators. A systematic functional analytic construction is provided. [ABSTRACT FROM AUTHOR]

In this paper we consider some rational approximations to the fractional powers of self-adjoint positive operators, arising from the Gauss–Laguerre rules. We derive practical error estimates that can be used to select a priori the number of Laguerre points necessary to achieve a given accuracy. We also present some numerical experiments to show the effectiveness of our approaches and the reliability of the estimates. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let H be a Hilbert space and let A be a positive bounded operator on H. The semi-inner product hh | kiA := hAh | ki, h, k H, induces a semi-norm k·kA. This makes H into a semi-Hilbertian space. An operator T BA(H) is said to be (n,m)-A-normal if [T n, (T A)m] := T n(T A)m - (T A)mT n = 0 for some positive integers n and m. [ABSTRACT FROM AUTHOR]

Many non-local processes are modeled using mathematical models that include fractional powers of elliptic operators. The approximate solution of stationary problems with fractional powers of operators is most often based on rational approximations introduced in various versions for a fractional power of the self-adjoint positive operator. The purpose of this work is to use such approximations for the approximate solution of nonstationary problems. We consider Cauchy problems for the first and second order differential-operator equations in finite-dimensional Hilbert spaces. Estimates for the proximity of an approximate solution to an exact one are obtained when specifying the absolute and relative errors of the approximation of the fractional power of the operator. We construct splitting schemes based on the additive representation with a rational approximation of the operator's fractional power. The stability and accuracy of factorized two-level additive operator-difference schemes for the first order evolution equation and three-level schemes for a second order equation are established. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the idea of deferred statistical convergence of order β using fractional difference operator Δ a , b , c for the sequence of fuzzy numbers. Some results on limits of deferred statistical convergence of order β of difference fuzzy sequences have been established. Also, certain relations between this convergence and the strongly deferred-summability for the fractional difference sequences of fuzzy numbers have been discussed. As an application, we apply the idea of newly defined statistical convergence for proving fuzzy Korovkin-type approximation theorem, which allows us to approximate various fuzzy positive linear operators. Finally, for more clarifications, we provide an example involving fuzzy-type Bernstein operator in support of the applications of our work. [ABSTRACT FROM AUTHOR]

In this study, we introduce a new generalization of a Bernstein-type rational function possessing better estimates than the classical Bernstein-type rational function. We investigate its error of approximation globally and locally in terms of the first and second modulus of continuity and a class of Lipschitz-type functions. We present graphical comparisons of its approximation with illustrative examples. [ABSTRACT FROM AUTHOR]

A variant of Jessen's type inequality for a semigroup of positive linear operators, defined on a Banach lattice algebra, is obtained. The corresponding mean value theorems lead to a new family of mean-operators. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with "finite dimensional fermions," by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group Spin(2n + 1) to the fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra s o (2 n + 1). As such, they do not satisfy the canonical anticommutation relations; however, once they have been projected onto an appropriate subspace of L2(Spin(2n + 1)), these relations are satisfied. We define a free time evolution of this system of fermions in terms of a symmetric positive-definite quadratic form in the creation–annihilation operators. The realization of fermionic creation and annihilation operators brought by the (invariant) vector fields allows us to interpret this time evolution in terms of a positive self-adjoint operator that is the sum of a second order operator, which generates a stochastic diffusion process, and a first order complex operator, which strongly commutes with the second order operator. A probabilistic interpretation is given in terms of a Feynman–Kac-like formula with respect to the diffusion process associated with the second order operator. [ABSTRACT FROM AUTHOR]

This work presents a parametrized family of distances, namely the Alpha Procrustes distances, on the set of symmetric, positive definite (SPD) matrices. The Alpha Procrustes distances provide a unified formulation encompassing both the Bures-Wasserstein and Log-Euclidean distances between SPD matrices. We show that the Alpha Procrustes distances are the Riemannian distances corresponding to a family of Riemannian metrics on the manifold of SPD matrices, which encompass both the Log-Euclidean and Wasserstein Riemannian metrics. This formulation is then generalized to the set of positive definite Hilbert-Schmidt operators on a Hilbert space, unifying the infinite-dimensional Bures-Wasserstein and Log-Hilbert-Schmidt distances. In the setting of reproducing kernel Hilbert spaces (RKHS) covariance operators, we obtain closed form formulas for all the distances via the corresponding kernel Gram matrices. From a statistical viewpoint, the Alpha Procrustes distances give rise to a parametrized family of distances between Gaussian measures on Euclidean space, in the finite-dimensional case, and separable Hilbert spaces, in the infinite-dimensional case, encompassing the 2-Wasserstein distance, with closed form formulas via Gram matrices in the RKHS setting. The presented formulations are new both in the finite and infinite-dimensional settings. [ABSTRACT FROM AUTHOR]

By using a variational principle we find a necessary and sufficient condition for an operator to majorise the parallel sum of two positive definite operators. This result is then used as a vehicle to create new operator inequalities involving the parallel sum. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to study a new generalization of Lupaş-type operators whose construction depends on a real-valued function ρ by using two sequences u m $u_{m} $ and v m $v_{m}$ of functions. We prove that the new operators provide better weighted uniform approximation over [ 0 , ∞ ) $[0,\infty )$ . In terms of weighted moduli of smoothness, we obtain degrees of approximation associated with the function ρ. Also, we prove Voronovskaya-type theorem, quantitative estimates for the local approximation.

Maximum principles and uniform anti-maximum principles are a ubiquitous topic in PDE theory that is closely tied to the Krein–Rutman theorem and kernel estimates for resolvents. We take up a classical idea of Takáč – to prove (anti-)maximum principles in an abstract operator theoretic framework – and combine it with recent ideas from the theory of eventually positive operator semigroups. This enables us to derive necessary and sufficient conditions for (anti-)maximum principles in a very general setting. Consequently, we are able to either prove or disprove (anti-)maximum principles for a large variety of concrete differential operators. As a bonus, for several operators that are already known to satisfy or to not satisfy anti-maximum principles, our theory gives a very clear and concise explanation of this behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen's result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition. [ABSTRACT FROM AUTHOR]

In the present paper, we propose the modification of the Baskakov–Kantorovich operators based on μ -integral. Such operators are connected with exponential functions. We estimate moments and establish some direct results in terms of modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced a Stancu generalization of the Szasz–Mirakyan–Bernstein Operators defined on the space of continuous functions defined on a compact interval. We have given a general formula for the moments of that operators. We have used Korovkin's Theorem for uniform approximation under some restrictions. We have obtained some results for the approximation rates in terms of modulus of continuity. Finally, we gave some Voronovskaya-type theorems. [ABSTRACT FROM AUTHOR]

In this paper, under some ordered conditions, we investigate the existence of positive ω-periodic solutions for a class of second-order neutral functional differential equations with delayed derivative in nonlinearity of the form (x(t) − cx(t − δ))″ + a(t)g(x(t))x(t) = λb(t)f(t, x(t), x(t − τ1(t)), x′(t − τ2(t))). By utilizing the perturbation method of a positive operator and the fixed point index theory in cones, some sufficient conditions are established for the existence as well as the non-existence of positive ω-periodic solutions. [ABSTRACT FROM AUTHOR]

In this research article, we shall give some reverse Arithmetic-Geometric mean inequalities for unital positive linear maps on Hilbert space operators under some different conditions. Our results are sharper and more precise as compared to some recent published results. Moreover, we shall present refinements of the Lin conjecture. [ABSTRACT FROM AUTHOR]

We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric means of positive kernel operators on Banach function and sequence spaces. Some inequalities extend and refine known inequalities that have been established by several authors relatively recently. Several inequalities appear to be new even in the finite dimensional case. [ABSTRACT FROM AUTHOR]

The nonlocal boundary-value problem for a third-order partial differential equation d 3 u t dt 3 + A du t dt = f t , 0 < t < 1 , u 0 = γu λ + φ , u ′ 0 = α u ′ λ + ψ , γ < 1 , u ″ 0 = β u ″ λ + ξ , 1 + βα > α + β , 0 < λ ≤ 1 in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. As applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary-value problems for third-order partial differential equations are obtained. Numerical results for one- and two-dimensional third-order partial differential equations are provided. [ABSTRACT FROM AUTHOR]

that fix the function e − 2 x for x ≥ 0 . Then, we provide the approximation properties of these newly defined operators for different types of function spaces. In addition, we focus on the rate of convergence utilizing appropriate moduli of continuity. Then, we provide the Voronovskaya-type theorem for these new operators. Finally, in order to validate our theoretical results, we provide some numerical experiments that are produced by a MATLAB complier. [ABSTRACT FROM AUTHOR]

We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature. From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptive convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models. • We propose a density basis of the trigonometric polynomials with shape-preserving properties. • Nonparametric mixture priors are constructed using this basis for Bayesian inference. • We describe adaptive posterior convergence rates and consistency under weak assumptions. [ABSTRACT FROM AUTHOR]

Let A be a positive bounded linear operator acting on a complex Hilbert space H. Any positive operator A induces a semiinner product on H defined by ⟨x, y⟩ A:= ⟨Ax, y⟩H, ∀x, y ∈ H. For any T ∈ B (H(Ω)), its A-Berezin symbol e TA is defined on Ω by e TA:= D T bKλ, bKλ E A, λ ∈ Ω, where bKλ is the normalized reproducing kernel of H. In this paper, we introduce the notions (A, r) -adjoint of operators and A-Berezin number of operators on the reproducing kernel Hilbert space and prove some upper and lower bounds of the A-Berezin numbers of operators. In particular, we show that where |sin| A T denotes the A-sinus of angle of T. [ABSTRACT FROM AUTHOR]