ESSENTIAL NORMS OF PRODUCTS OF WEIGHTED COMPOSITION OPERATORS AND DIFFERENTIATION OPERATORS BETWEEN BANACH SPACES OF ANALYTIC FUNCTIONS

We obtain several estimates of the essential norms of the products of differen- tiation operators and weighted composition operators between weighted Banach spaces of analytic functions with general weights. As applications, we also give estimates of the es- sential norms of weighted composition operators between weighted Banach space of analytic functions and Bloch-type spaces.

PRODUCTS OF WEIGHTED COMPOSITION AND DIFFERENTIATION OPERATORS INTO WEIGHTED ZYGMUND AND BLOCH SPACES

We characterize boundedness and compactness of products of differentiation op- erators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.

Weighted Composition Followed and Proceeded by Differentiation Operators from Q_k(p,q) Space to Weighted α-Bloch Space

We study the boundedness and compactness of the weighted composition followed and proceeded by differentiation operators from Q_k（p,q）space to weighted α-Bloch space and little weighted α-Bloch space.Some necessary and sufficient conditions for the boundedness and compactness of these operators are given.

Inverse Differential Operators in Time and Space

When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.

Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

This study proposes a comprehensive,coupled thermomechanical model that replaces local spatial derivatives in classical differential thermomechanical equations with nonlocal integral forms derived from the peridynamic differential operator(PDDO),eliminating the need for calibration procedures.The model employs a multi-rate explicit time integration scheme to handle varying time scales in multi-physics systems.Through simulations conducted on granite and ceramic materials,this model demonstrates its effectiveness.It successfully simulates thermal damage behavior in granite arising from incompatible mineral expansion and accurately calculates thermal crack propagation in ceramic slabs during quenching.To account for material heterogeneity,the model utilizes the Shuffle algorithm andWeibull distribution,yielding results that align with numerical simulations and experimental observations.This coupled thermomechanical model shows great promise for analyzing intricate thermomechanical phenomena in brittle materials.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

From Control Theory to Gravitational Waves

When D:ξ→η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1:η→ξ such that Dξ = η implies D1η = 0. When D is involutive, the procedure provides successive first-order involutive operators D1,...,Dn when the ground manifold has dimension n. Conversely, when D1 is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η having the generating CC D1η = 0. If this is possible, that is when the differential module defined by D1 is “torsion-free”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D1 is parametrized by D. The parametrization is said to be “minimum” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality. We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “built in” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D1 is formally surjective, controllability just amounts to the formal injectivity of ad(D1), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, the parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G. B. Airy in 1863 for n = 2, J. C. Maxwell in 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized and the comparison needs no comment. As a byproduct, they are all based on a confusion between the so-called div operator D2 induced from the Bianchi operator and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D1 for an arbitrary n. This purely mathematical result deeply questions the origin and existence of gravitational waves, both with the mathematical foundations of general relativity. As a matter of fact, this new framework provides a totally open domain of applications for computer algebra as the quoted test can be studied by means of Pommaret bases and related recent packages.

MHD Maxwell Fluid with Heat Transfer Analysis under Ramp Velocity and Ramp Temperature Subject to Non-Integer Differentiable Operators

The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium.Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress,energy,and velocity profile.Recently,new fractional differential operators are used to define ramped temperature and ramped velocity.The obtained analytical solutions are plotted for different values of emerging parameters.Fractional time derivatives are used to analyze the impact of fractional parameters(memory effect)on the dynamics of the fluid.While making a comparison,it is observed that the fractional-order model is best to explain the memory effect as compared to classical models.Our results suggest that the velocity profile decrease by increasing the effective Prandtl number.The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.The incremental value of the M is observed for a decrease in the velocity field,which reflects to control resistive force.Further,it is noted that the Atangana-Baleanu derivative in Caputo sense(ABC)is the best to highlight the dynamics of the fluid.The influence of pertinent parameters is analyzed graphically for velocity and energy profile.Expressions for skin friction and Nusselt number are also derived for fractional differential operators.

Representations of Inverse Covariances by Differential Operators

In the cost function of three- or four-dimensional variational dataassimilation, each term is weighted by the inverse of its associated error covariance matrix and thebackground error covariance matrix is usually much larger than the other covariance matrices.Although the background error covariances are traditionally normalized and parameterized by simplesmooth homogeneous correlation functions, the covariance matrices constructed from these correlationfunctions are often too large to be inverted or even manipulated. It is thus desirable to finddirect representations of the inverses of background error correlations. This problem is studied inthis paper. In particular, it is shown that the background term can be written into ∫ dx∣Dυ(x)∣~2, that is, a squared 1/2 norm of a vector differential operator D, called theD-operator, applied to the field of analysis increment υ(x). For autoregressive correlationfunctions, the D-operators are of finite orders. For Gaussian correlation functions, the D-operatorsare of infinite order. For practical applications, the Gaussian D-operators must be truncated tofinite orders. The truncation errors are found to be small even when the Gaussian D-operators aretruncated to low orders. With a truncated D-operator, the background term can be easily constructedwith neither inversion nor direct calculation of the covariance matrix. D-operators are also derivedfor non-Gaussian correlations and transformed into non-isotropic forms.

Spectrum of a class of fourth order left-definite differential operators

The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…

SANDWICH-TYPE RESULTS FOR A CLASS OF CONVEX INTEGRAL OPERATORS

Let H（U） be the space of analytic functions in the unit disk U. For the integral operator

LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.

ON APPROXIMATION OF SMOOTH FUNCTIONS FROM NULL SPACES OF OPTIMAL LINEAR DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln（ψ） =∑k=0^n akψ（k） where the constant coefficients ak C R may be adapted to f.

Recast combination functions of coordinate and momentum operators into their ordered product forms

By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater than the existing ways.These products are very useful in obtaining some new differential relations and useful mathematical integral formulas.Further,we derive the normally ordered form of the operator(fQ+gP)^-n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate-momentum representation.In addition,general mutual transformation rules of the normal and anti-normal orderings,which have good universality,are derived and hence the anti-normal ordering of(fQ+gP)^-n is also obtained.Finally,the application of some new identities is given.

On the Spectra of General Ordinary Quasi-Differential Operators and Their L2w-Solutions

In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.

Dynamical Analysis of the Stochastic COVID-19 Model Using Piecewise Differential Equation Technique

Various data sets showing the prevalence of numerous viral diseases have demonstrated that the transmission is not truly homogeneous.Two examples are the spread of Spanish flu and COVID-19.The aimof this research is to develop a comprehensive nonlinear stochastic model having six cohorts relying on ordinary differential equations via piecewise fractional differential operators.Firstly,the strength number of the deterministic case is carried out.Then,for the stochastic model,we show that there is a critical number RS0 that can predict virus persistence and infection eradication.Because of the peculiarity of this notion,an interesting way to ensure the existence and uniqueness of the global positive solution characterized by the stochastic COVID-19 model is established by creating a sequence of appropriate Lyapunov candidates.Adetailed ergodic stationary distribution for the stochastic COVID-19 model is provided.Our findings demonstrate a piecewise numerical technique to generate simulation studies for these frameworks.The collected outcomes leave no doubt that this conception is a revolutionary doorway that will assist mankind in good perspective nature.

A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY

Using Hartogs’fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several variables.With this identity,we give new proofs of a variety of important classical formulas including Bailey’s 6ψ6 series summation formula and the Atakishiyev integral.A new transformation formula for a double q-series with several interesting special cases is given.A new transformation formula for a 3ψ3 series is proved.

Killing Operator for the Kerr Metric

When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) →F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 →F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.