Two-side Estimates Research Articles

Peeling for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the asymptotic behavior along outgoing and incoming radial geodesics, i.e. the peeling property for the tensorial Fackerell–Ipser and spin [Formula: see text] Teukolsky equations on Schwarzschild spacetime. Our method combines a conformal compactification with vector field techniques to prove the two-side estimates of the energies of tensorial fields through the future and past null infinity [Formula: see text] and the initial Cauchy hypersurface [Formula: see text] in a neighborhood of spacelike infinity [Formula: see text] far away from the horizon and future timelike infinity. Our results obtain the optimal initial data which guarantees the peeling at all orders.

Global stability of an age-structured population model on several temporally variable patches

We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number R_0 for time-independent and periodical models and establish the permanence dichotomy: if R_0le 1, extinction on all patches is imminent, and if R_0>1, permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour.

Analytical estimates of the pull-in voltage for carbon nanotubes considering tip-charge concentration and intermolecular forces

Two-side accurate analytical estimates of the pull-in parameters of a carbon nanotube switch clamped at one end under electrostatic actuation are provided by considering the proper expressions of the electrostatic force and van der Waals interactions for a carbon nanotube, as well as the contribution of the charge concentration at the free end. According to the Euler–Bernoulli beam theory, the problem is governed by a fourth-order nonlinear boundary value problem. Two-side estimates on the centreline deflection are derived. Then, very accurate lower and upper bounds to the pull-in voltage and deflection are obtained as function of the geometrical and material parameters. The analytical predictions are found to agree remarkably well with the numerical results provided by the shooting method, thus validating the proposed approach. Finally, a simple closed-form relation is proposed for the minimum feasible gap and maximum realizable length for a freestanding CNT cantilever.

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.

Finite-Difference Scheme for Initial Boundary Value Problems in Financial Mathematics

We develop unconditionally monotone nite-difference schemes of second-order of local approxi-mation on uniform grids for the initial boundary problem value for the Gamma equation. Two-sideestimates of the solution of the scheme are established. We consider the initial boundary valueproblem for the so called Gamma equation, which can be derived by transforming the nonlinearBlack-Scholes equation for option price into a quasilinear parabolic equation for the second derivativeof the option price, and for its exact solution the two-side estimates are obtained. By means of regu-larization principle, the previous results are generalized for construction of unconditionally monotonenite-difference scheme (the maximum principle is satised without constraints on relations betweenthe coeffcients and grid parameters) of second order of approximation on uniform grids for this equa-tion. With the help of difference maximum principle, the two-side estimates for difference solutionare obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in themaximum norm C is proved. It is interesting to note that the proven two-side estimates for differ-ence solution are fully consistent with differential problem, and the maximal and minimal values ofthe difference solution do not depend on the diffusion and convection coeffcients. Computationalexperiments, conrming the theoretical conclusions, are given.

On the consistent two-side estimates for the solutions of quasilinear convection–diffusion equations and their approximations on non-uniform grids

A new second-order in space linearized difference scheme on non-uniform grid that approximates the Dirichlet problem for multidimensional quasilinear convection–diffusion equation with unbounded nonlinearity is constructed. Proposed algorithm is a novel nonlinear generalization of difference schemes for linear problems developed earlier by the authors. Nontrivial two-side pointwise estimates of the solution of the scheme fully consistent with the corresponding estimates for the differential problem are established. Such estimates permit to prove the nonnegativity of the exact solution, important in physical problems, and also to find sufficient conditions on the input data when the nonlinear problem is parabolic. As a result a priori estimates of the approximate solution in the grid norm C that depend on the initial and boundary conditions and on the right-hand side only are proved.

Monotone Difference Schemes for Weakly Coupled Elliptic and Parabolic Systems

Abstract The present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is introduced and the definition of its monotonicity is given. This definition is closely associated with the property of non-negativity of the solution. Under the fulfillment of the positivity condition of the coefficients, two-side estimates of the approximate solution of these vector-difference equations are established and the important a priori estimate in the uniform norm C is given.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

Abstract In this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.

Numerical methods for a one-dimensional non-linear Biot’s model

Staggered finite difference methods for a one-dimensional Biot’s problem are considered. The permeability tensor of the porous medium is assumed to depend on the strain, thus yielding a non-linear model. Some strong two-side estimates for displacements and for pressure are provided and convergence results in the discrete L2-norm are proved. Numerical examples are given to illustrate the good performance of the proposed numerical approach.

Двобічні оцінки жорсткості і міцності центратора обсадної колони

Двобічні оцінки жорсткості і міцності центратора обсадної колони

Two-side estimates of the number of fixed points of a discrete logarithm

Lower and upper bounds are obtained for an average number of solutions to the congruence gx ≡ x (mod p) in nonnegative integer numbers x ≤ p − 1, where g is a primitive root modulo p.

Two-side estimates for sums of absolute values of Fourier coefficients of functions from H ω (T m ) with the use of partial moduli of smoothness

Two-side estimates of sums of absolute values of Fourier coefficients for functions of many variables from the classes H ω (T m ) are established in the paper with the help of partial moduli of smoothness.

On the comparison of volumes of quantum states

This paper aims to study the α-volume of , an arbitrary subset of the set of N × N density matrices. The α-volume is a generalization of the Hilbert–Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the α-volume of in terms of its Hilbert–Schmidt volume. The analogous estimates between the Bures volume and the α-volume are also established. We employ our results to obtain bounds for the α-volume of the sets of separable quantum states and of states with positive partial transpose. Hence, our asymptotic results provide answers for questions listed in Życzkowski et al (1998 Phys. Rev. A 58 883) for large N in the sense of α-volume.

Remarks on the general Funk transform and thermoacoustic tomography

We discuss properties of a generalized Minkowski-Funk transform defined for a family of hypersurfaces. We prove two-side estimates and show that the range conditions can be written in terms of the reciprocal Funk transform. Some applications to the spherical mean transform are considered.

Two-side estimates for sums of absolute values of Fourier coefficients of functions from H ω (T m )

The class of functions Hω is considered, where ω(t) is a continuity modulus monotone in the sense of Hardy and satisfying some condition C. The behavior of the value \( \mathop {\sup }\limits_{f \in H_\omega } \left\| f \right\|_{A_p } \) is obtained, where \( \left\| f \right\|_{A_p } \) is the sum of absolute values of Fourier coefficients of a function f ∈ L(Tm) in pth power.

Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

Let T ⊂ [ a , b ] be a time scale with a , b ∈ T . In this paper we study the asymptotic distribution of eigenvalues of the following linear problem − u Δ Δ = λ u σ , with mixed boundary conditions α u ( a ) + β u Δ ( a ) = 0 = γ u ( ρ ( b ) ) + δ u Δ ( ρ ( b ) ) . It is known that there exists a sequence of simple eigenvalues { λ k } k ; we consider the spectral counting function N ( λ , T ) = # { k : λ k ⩽ λ } , and we seek for its asymptotic expansion as a power of λ. Let d be the Minkowski (or box) dimension of T , which gives the order of growth of the number K ( T , ε ) of intervals of length ε needed to cover T , namely K ( T , ε ) ≈ ε d . We prove an upper bound of N ( λ ) which involves the Minkowski dimension, N ( λ , T ) ⩽ C λ d / 2 , where C is a positive constant depending only on the Minkowski content of T (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases ( d = 0 , infinite Minkowski content), and we show a family of self similar fractal sets where N ( λ , T ) admits two-side estimates.