Mean Value Formula Research Articles

Almansi-Type Decomposition for Slice Regular Functions of Several Quaternionic Variables

AbstractIn this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields $$2^n$$ 2 n distinct and unique decompositions for any slice function with domain in $$\mathbb {H}^n$$ H n . Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter’s Theorem in $$\mathbb {H}^n$$ H n , establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.

Mean value formulas on surfaces in Grushin spaces

We prove (sub)mean value formulas at the point \(0\in\Sigma\) for (sub)harmonic functions on a hypersurface \(\Sigma\subset\mathbb{R}^{n+1}\) where the differentiable structure and the surface measure depend on the ambient Grushin structure.

Systems involving mean value formulas on trees

Abstract In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

<p style='text-indent:20px;'>We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.</p>

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non‐smooth coefficients

AbstractWe prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C1‐boundary, the other stated for sets with finite perimeter.

A two-sided matching decision method based on interval triangular fuzzy sets

The research work on two-sided matching decision problem considering interval triangular fuzzy information is very scarce, and it needs to be further studied. Based on this, this paper proposes a two-sided matching model based on interval triangular fuzzy sets, with the background of the two-sided matching problem in the interval triangular fuzzy set environment. Firstly, the theory of two-sided matching and interval triangular fuzzy sets is given; Secondly, the comprehensive mean value formula is defined, and the interval triangular fuzzy evaluation matrix is transformed into the comprehensive mean value matrix by using the comprehensive mean value formula; Thirdly, a two-sided matching model is built with the goal of maximizing the satisfaction of each subject; Finally, the feasibility and effectiveness of the proposed method are verified by examples of investment fund institutions and financing enterprises.

On the Harnack inequality for antisymmetric s-harmonic functions

We prove the Harnack inequality for antisymmetric s-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving planes to obtain quantitative stability results for symmetry and overdetermined problems for semilinear equations driven by the fractional Laplacian.The proof is split into two parts: an interior Harnack inequality away from the plane of symmetry, and a boundary Harnack inequality close to the plane of symmetry. We prove these results by first establishing the weak Harnack inequality for super-solutions and local boundedness for sub-solutions in both the interior and boundary case.En passant, we also obtain a new mean value formula for antisymmetric s-harmonic functions.

Harnack inequality for polyharmonic equations

Some new types of mean value formulas for the polyharmonic functions were established. Based on the formulas, the Harnack inequality for the nonnegative solutions to the polyharmonic equations was proved.

Mean-Value Formula for a Hyperbolic Equation with a Factorizable Operator

A mean-value formula for a linear partial differential hyperbolic equation with an operator splitting into first-order factors is obtained. This formula can be interpreted as an extension of the Ásgeirsson principle.

Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

The fractional heat operator (∂t−Δx)s and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to (∂t−Δx)s and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.

MYMathApps Calculus: Maple Plots

I am writing an online Calculus text called MYMathApps Calculus. You can see a sample at https://mymathapps.com/mymacalc-sample/The text is highly interactive and visual. Nearly all of the graphics have been made with Maple, both 2D and 3D, static and animated. The use of plots and animated plots helps students understand concepts such as
 
 the definitions of a derivative as the limit of slopes of secant lines, an integral as limits of Riemann sums, partial derivatives as slopes of traces, curvature and torsion, tangential and normal acceleration, divergence and curl, multiple integrals, curvilinear coordinates and Jacobians.
 the proofs of the triangle inequality, the mean value theorem and formulas for applications of integrals.
 plotting functions, polar curves, and parametric curves and surfaces.
 solving applied problems involving linear approximation, related rates, max/min, area, arc length, surface area, volumes by slicing, volumes of revolution, work, mixing problems, geometric series, Taylor series, directional derivatives, Lagrange multipliers, expansion and circulation.
 how to use the right hand rule in Green’s, Stokes’ and Gauss’ theorems.

Asymptotic mean value formulas for parabolic nonlinear equations

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Ampere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.

Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fields

We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a homogeneous time-dependent magnetic field B(t)=B0(1+t/t0)−1−g, directed along the z-axis and described by means of the vector potential A(t)=B(t)[−y,x]/2. Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if g>0 (“fast” decay), while it goes asymptotically to zero if g≤0 (“slow” decay). The dependence on parameter t0 practically disappears when 1+g is close to zero value (“superslow” decay). The mean magnetic moment goes to zero for g>1, while it grows unlimitedly if g<1.

Asymptotic mean-value formulas for solutions of general second-order elliptic equations

Abstract We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k-Hessian operators.

Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids.

We consider a quantum spinless nonrelativistic charged particle moving in the plane under the action of a time-dependent magnetic field, described by means of the linear vector potential , with two fixed values of the gauge parameter : (the circular gauge) and (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for and straight lines for . We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function . They are expressed in terms of solutions to the classical equation of motion , with . Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions , when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.

Monte Carlo Algorithms for the Extracting of Electrical Capacitance

We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the capacitances are constructed on the trajectories of the Random Walk on Spheres or the Random Walk on Hemispheres. The calculation results show that the accuracy of these new algorithms does not exceed the statistical error of estimators, which is easily determined in the course of calculations. The algorithms are based on mean value formulas for harmonic functions in different domains and do not involve a transition to a difference problem. Hence, they do not need a lot of storage space.

On the spectral properties of the differential operators with involution

In this paper we deal with the problems of the eigenfunction expansions related to the differential operators with involution. The mean value formula for the eigenfunction is obtained with application of the transformation methods of the operators in the symmetric regions. The obtained formula is applied to estimate the eigenfunctions of the given differential operator in the ball. For domains with smooth boundary, the solution to these differential operator problems involves eigenfunction expansions associated with biharmonic-type operator with Navier boundary conditions.

On Mean Value formulas for solutions to second order linear PDEs

In this paper we give a general proof of Mean Value formulas for solutions to second order linear PDEs, only based on the local properties of their fundamental solution 0. Our proof requires a kind of pointwise vanishing integral condition for the intrinsic gradient of 0. Combining our Mean Value formulas with a “descent method” due to Kuptsov, we obtain formulas with improved ker- nels. As an application, we implement our general results to heat operators on stratified Lie groups and to Kolmogorov operators

A mean value formula for the iterated Dunkl-Helmoltz operator

ABSTRACT We establish a spherical mean value formula for the iterated Dunkl-Helmoltz operator, thus generalizing a result of Mejjaoli and Trimèche. We then give an application to distributions with Dunkl transform supported by the unit sphere.

A mean value formula for the variational p-Laplacian

We prove a new asymptotic mean value formula for the p-Laplace operator,Δpu=div(|∇u|p-2∇u),1<p<∞\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Delta _pu=\\text{ div }(|\\nabla u|^{p-2}\\nabla u), \\quad 1<p<\\infty \\end{aligned}$$\\end{document}valid in the viscosity sense. In the plane, and for a certain range of p, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.