Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis

Let X be a Banach space with a Schauder basis { en }, and let Φ( I ) = Σ∞ n=1 en∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock-Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock-Kurzweil-Pettis (or Henstock, or variational Henstock) integrable function f : [0, 1] → X .     

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝR ^{n +1} be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ _q = ψ|Λ(X)|^q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|^−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov     

On a problem of H. Shapiro

Let μ be a real measure on the line such that its Poisson integral M(z) converges and satisfies|M(x+iy)|Ae^−cyα, y→+∞,for some constants A,c>0 and 0<α1. We show that for 1/2<α1 the measure μ must have many sign changes on both positive and negative rays. For 0<α1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.     

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [−1,1]. Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O(h^α|lnh|) and O(h|lnh|) in the classes of functions H^α([−1,1]) and C¹([−1,1]), respectively.     

Ratio asymptotics for orthogonal rational functions on an interval

Let {α₁,α₂,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erd s–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. We introduce rational functions _n(x) with poles {α₁,…,α_n} orthogonal on [−1,1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of _n+1(x)/_n(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions.     

On Generalized Gaussian Quadratures for Exponentials and Their Applications

We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions.We use a generalization of a representation theorem due to Carathéodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ε, selecting an eigenvalue close to ε yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm.These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval.     

On the relaxation of nonconvex superficial integral functionals

We present a new approach to the variational relaxation of functionals of the type:

where is a continuous function with growth conditions of order p≥1 but not necessarily convex. We essentially study the case when μ is the k-dimensional Hausdorff measure restricted to a suitable piece of a k-dimensional smooth submanifold of .     

General system of -maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms

In this paper, a new system of nonlinear (set-valued) variational inclusions involving (A,η)-maximal relaxed monotone and relative (A,η)-maximal monotone mappings in Hilbert spaces is introduced and its approximation solvability is examined. The notion of (A,η)-maximal relaxed monotonicity generalizes the notion of general η-maximal monotonicity, including (H,η)-maximal monotonicity (also referred to as (H,η)-monotonicity in literature). Using the general (A,η)-resolvent operator method, approximation solvability of this system based on a generalized hybrid iterative algorithm is investigated. Furthermore, for the nonlinear variational inclusion system on hand, corresponding nonlinear Yosida regularization inclusion system and nonlinear Yosida approximations are introduced, and as a result, it turns out that the solution set for the nonlinear variational inclusion system coincides with that of the corresponding Yosida regularization inclusion system. Approximation solvability of the Yosida regularization inclusion system is based on an existence theorem and related Yosida approximations. The obtained results are general in nature.     

Nonlinear equations of a timoshenko-type theory of laminated anisotropic shells

In recent years analysis of the stress—strain state of shell structures made out of composite materials has been based on refined shell theories which take into account strains in the direction normal to the reference surface. There are several approaches to the formulation of the refined theories. One can point to shell theories developed on the basis of variational principles (e.g., [1, 2]) as well as theories created with the help of iterational processes (e.g., [3–6]). A resolving system of nonlinear equations for laminated anisotropic shells has been derived in the proposed research based on the Reissner variational principle [7, 8]. A similar linear theory which takes into account the strain e₃₃ also has been developed in [1]. If the shear stiffnesses of the layers differ greatly from each other in the transverse direction, then one can treat the shell structure as a single-layer shell of nonuniform structure. In this case it is advisable to solve a problem of the type of a uniform shell with minimal stiffnesses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 501–507, May–June, 1979.     

Algorithm for solving a new class of general mixed variational inequalities in Banach spaces

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.     

An Ekeland’s variational principle for set-valued mappings with applications

In this paper, we obtain a general Ekeland’s variational principle for set-valued mappings in complete metric space, which is different from those in [G.Y. Chen, X.X. Huang, Ekeland’s ε-variational principle for set-valued mapping, Mathematical Methods of Operations Research 48 (1998) 181–186; G.Y. Chen, X.X. Huang, S.H. Hou, General Ekeland’s Variational Principle for Set-Valued Mappings, Journal of Optimization Theory and Applications 106 (2000) 151–164; S.J. Li, W.Y. Zhang, On Ekeland’s variational Principle for set-valued mappings, Acta Mathematicae Application Sinica, English Series 23 (2007) 141–148]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain, respectively. Moreover, we give some equivalent results to the variational principle.     

Creep of a thermoplastic cylindrical shell under axisymmetric loading

The solution of the problem of creep of an axisymmetrically loaded thermoplastic cylindrical shell is considered. The strains and stresses for the zero-moment zone and with allowance for the edge effect are predicted on the basis of Kachanov's variational method using a computer. An algorithm is constructed for polyethylene and PVC shells at various values of the load. The analysis of the results obtained is illustrated by the calculation data for individual variants of the program.Leningrad Mozhaiskii Military Engineering Academy. Translated from Mekhanika Polimerov, No. 3, pp. 512–518, May–June, 1969.     

The Hilbert–Carathéodory Form for Parametric Multiple Integral Problems in the Calculus of Variations

We define, and discuss some properties of, a form which bears a relation to the Carathéodory form for multiple integral problems in the calculus of variations similar to the relation that the Hilbert 1-form of Finsler geometry bears to the Poincaré–Cartan form for one-dimensional variational problems.     

Algebraic splines in locally convex spaces

In a vector space of continuous functions, a variational solution of a finite system of linear functional equations is found. The locally convex topology on the vector space and the properties of the objective functional required for obtaining the solution in the form of a decomposition in the basis dual to the family of functionals of the system are determined. The basis elements are calculated exactly and called basis algebraic splines; their linear span is called the space of algebraic splines in the corresponding locally convex space.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 339–353.Original Russian Text Copyright © 2005 by A. P. Kolesnikov.This revised version was published online in April 2005 with a corrected issue number.     

The strong Ekeland variational principle

In this paper, we consider the strong Ekeland variational principle due to Georgiev [P.G. Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988) 1–21]. We discuss it for functions defined on Banach spaces and on compact metric spaces. We also prove the τ-distance version of it.     

Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval . For a given compact interval , set

     

Galerkin solution of a singular integral equation with constant coefficients

Galerkin methods are used to approximate the singular integral equation

with solution φ having weak singularity at the endpoint −1, where a, b≠0 are constants. In this case φ is decomposed as φ(x)=(1−x)^α(1+x)^βu(x), where β=−α, 0<α<1. Jacobi polynomials are used in the discussions. Under the conditions fH^μ[−1,1] and k(t,x)H^μ,μ[−1,1]×[−1,1], 0<μ<1, the error estimate under a weighted L² norm is O(n^−μ). Under the strengthened conditions f^″H^μ[−1,1] and , 2α−<μ<1, the error estimate under maximum norm is proved to be O(n^2α−−μ+), where , >0 is a small enough constant.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Measuring and computing natural generators for homology groups

We develop a method for measuring homology classes. This involves two problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(βn³log²n) time, where n is the size of the simplicial complex and β is the Betti number of the homology group. Finally, we prove the stability of our result. The algorithm can be adapted to measure any given class.