On the Hardy-Littlewood majorant problem for random sets

We study the classical Hardy-Littlewood majorant problem for trigonometric polynomials. We show that the constant in the majorant inequality grows at most like an arbitrary small power of the degree provided the spectrum is chosen at random. We also give an example of a deterministic set where the majorant property fails, i.e., the constant grows like a fixed small power in the degree.     

Local convergence analysis of inexact Newton-like methods under majorant condition

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.     

Zygmund-type estimates for fractional integration and differentiation operators of variable order

We consider non-standard generalized Hölder spaces of functions defined on a segment of the real axis, whose local continuity modulus has a majorant varying from point to point. We establish some properties of fractional integration operators of variable order acting from variable generalized Hölder spaces to those with a “better” majorant, as well as properties of fractional differentiation operators of variable order acting from the same spaces to those with a “worse” majorant.     

The cubic semilocal convergence on two variants of Newton's method

In this paper, we discuss two variants of Newton's method without using any second derivative for solving nonlinear equations. By using the majorant function and confirming the majorant sequences, we obtain the cubic semilocal convergence and the error estimation in the Kantorovich-type theorems. The numerical examples are presented to support the usefulness and significance.     

Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.     

The asymptotic manifold of a perturbed linear differential equation as determined by the comparison principle

The asymptotic properties of the solutions of a linear homogeneous system of differential equations determine, under suitable restrictions, the asymptotic properties of a set of solutions of a nonlinear perturbation of this linear equation. The comparison principle is used here to generate an asymptotic manifold of the perturbed equation. The majorant function that is used in connection with the comparison technique is usually assumed to be nondecreasing in the dependent variable. However, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption, namely, that the majorant function is nonincreasing in the dependent variable. This type of majorant, function arises, for example, in certain gravitation problems. The main result on the structure of asymptotic manifolds which have an asymptotic uniformity is that solutions close to the manifold are either in the manifold or do not exist in the future. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 6 giugno 1970.     

Kantorovich’s majorants principle for Newton’s method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA. The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).     

Estimation of Deviations from the Exact Solution for the Reissner-Mindlin Plate Problem

In this paper, we obtain a majorant of the difference between the exact solution and any conforming approximate solution of the Reissner-Mindlin plate problem. This majorant is explicitly computable and involves constants that depend only on given data of the problem. The majorant allows us to compute guaranteed upper bounds of errors with any desired accuracy and vanishes if and only if the approximate solution coincides with the exact one. Bibliography: 12 titles. To N. N. Uraltseva with gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 145–157.     

Local convergence of Newton’s method under majorant condition

A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications.     

Harmonic and Superharmonic Majorants on the Disk

A proof is given to show that a positive function on the unitdisk admits a harmonic majorant if and only if it has a certainexplicit upper envelope that admits a superharmonic majorant.The (logarithmic) Lipschitz regularity of this superharmonicmajorant is discussed. 2000 Mathematics Subject Classification31A05.     

Local convergence analysis of inexact Gauss-Newton like methods under majorant condition

In this paper, we present a local convergence analysis of inexact Gauss-Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss-Newton like methods and some important, special cases.     

Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors.     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

Functional a posteriori error estimates for incremental models in elasto-plasticity

We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero.      

Breaking the space-time translation group in the dipole field model

The impossibility of constructing a translation-invariat physical space for the massless dipole field models in two and four dimensions follows from the Lebesgue integral theory. The method of proof may also be useful for studying space-time symmetries in other models with singular behavior in the infrared domain. To fix the Hilbert majorant structure associated with an indenfinite metric, we impose the condition that the majorant structure inherit the homogeneity properties from the metric. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115. No. 2. pp. 163 176. May. 1998.     

Global analytic regularity for non-linear second order operators on the torus

Assuming a subelliptic a-priori estimate we prove global analytic regularity for non-linear second order operators on a product of tori, using the method of majorant series.

     

On Mutually Inverse Transforms of Functions on a Half-Line

Doklady Mathematics - Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this...     

Splitting superfluid hydrodynamic equations and instability of solutions in the case of degenerate bosons

Properties of solutions to superfluid hydrodynamic equations as applied to the degenerate Bose gas are considered. The equations are split into two independent pairs of equations. One pair is written for the normal component implies the instability of solutions, which manifests itself in the majorant catastrophe with respect to the total density. The case when the thermodynamic functions depend on the difference of the normal and superfluid velocities is also considered. In that case, the system is not split; however, the instability and the majorant catastrophe occur when the initial temperature tends to absolute zero.     

Ball Convergence Theorems for Chebyshev-Halley Method in $\mathbb{B}$-Space

A local convergence analysis of Chebyshev-Halley method having third order of convergence for approximating zero of non-linear operator $f(v)=0$ by using convex majorant function and their condition in $\mathbb{B}$-space (Banach space), is presented in this article. We give the error estimate to show the efficiency of our study. Besides, we established the relation between majorant function and Kantorovich or Smale-type result as special cases of our general theory.     

The majorant method in the theory of newton-kantorovich approximations and the pták error estimates

For a certain modified Newton-Kontorovich method, sharp error estimates are obtained by menas of the majorant method. In particular, these error estimates generalize Pták's estimates for the usual Newton-Kontorovich method.