Evaluation of certain improper integrals arising in potential problems

The use of Schwarz‐Christoffel transform of the complex variable theory is shown to be well‐suited for the evaluation of certain improper integrals arising in potential problems, in closed form.     

Geometric and transformational properties of Lipschitz domains,Semmes-Kenig-Toro domains,and other classes of finite perimeter domains

In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C¹ domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C¹. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C¹ diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C¹ diffeomorphisms. A number of other applications are also presented. Acknowledgements and Notes. The work of the authors was supported in part by NSF grants DMS-0245401, DMS-0653180, DMS-FRG0456306, and DMS-0456861.     

Some properties of certain classes of p-valent functions defined by the hadamard product

In this paper we obtain sandwich type theorems, inclusion relationships, convolution properties and coefficient estimates of certain classes of p-valent analytic functions defined by a convolution. Several other new results are also obtained.     

On extremal problems on classes of functions defined by integral moduli of continuity

We obtain lower bounds for solutions of some extremal problems on classes of functions W ^rH₁^ω with integral modulus of continuity ω(t). Some of these bounds are regarded as exact. Dneprodzerzhinsk Technical University, Dneprodzerzhinsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11. pp. 1499–1503, November, 1997.     

On best approximation in classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels

In the metrics C and L we solve the problem of best approximation by trigonometric polynomials in classes of continuous periodic functionsf(x) of the form $$f(x) = \frac{1}{\pi }\int_0^{2\pi } {K(t)} \varphi (x - t)dt,$$ where the kernel K(t) is a periodic integral of a linear combination of functions that are absolutely monotonic in the intervals (?∞, 2π) and (0, ∞), and ∥?∥≤1. A particular case of such kernels for any s>0 andαε (?∞, ∞ are kernels of the form $$K(t) = \sum\nolimits_{k = 1}^\infty {\frac{{\cos (kt - {\textstyle{{\alpha \pi } \over 2}})}}{{k^S }}} ,$$ which forα=s generate classes of periodic functions with a bounded s-th derivative in the sense of Weyl, whereas forα=s+1 they generate conjugate classes. For various values of s andα, apart from the case sε (0, 1) andα ε [0, 2]/[s, 2?s], such kernels were studied by various investigators (see [1–12]).     

Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D = {z:|z| 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0 and|f'(z)(z/f(z))~2-1|1(|z|1).Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0and|zf'(z)-f(z)|1/2(|z|1).In this article,we discuss the properties of U and Ω.     

Subordination results for classes of analytic functions related to conic domains defined by a fractional operator

Let a fractional operator (n∈N₀={0,1,2,…}, 0?α<1, λ?0) be defined by

     

The relation between invariant integrals of the linear isotropic theory of elasticity and integrals defined by the duality principle

Invariant integrals of the linear isotropic theory of elasticity, determined by a certain specified elastic field, are considered, and also invariant integrals generated by the interaction of the specified field with an arbitrary secondary field. For all types of invariant integral, generated by the interaction of the specified elastic field and an arbitrary secondary elastic field, transformations of the secondary fields are found for which the invariant integrals considered turn out to be equal to the RG-integrals, determined by the duality principle, of the specified elastic field and the transformed secondary elastic field. The invariant J-, L- and M-integrals themselves are also expressed in terms of the RG-integrals of the specified elastic field and its corresponding transformation.     

On series arising from the approximation of periodic differentiable functions by Poisson integrals

Mathematical Notes - For series of special form, we obtain an expansion in powers of the parameter. The coefficients of the expansion can be written out explicitly in terms of Bernoulli...     

Uniform approximation by spherical Fourier sums on classes of functions,defined by polyharmonic operators

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 4, pp. 543–550, April, 1989.     

Widths in L2 of classes of differentiable functions,defined by higher-order moduli of continuity

Sharp inequalities are obtained in the space L₂, connecting the best approximations of differentiable 2-periodic functions by trigonometric polynomials with integrals containing the higher-order moduli of continuity of the derivatives of these functions. The Kolmogorov widths of certain classes of functions, defined by these moduli of continuity, are computed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 125–129, January, 1991.     

Uniqueness classes of solutions of boundary problems for nondivergent second order parabolic equations in noncylindrical domains

In an unbounded noncylindrical domain G we consider classical solutions of general second order linear parabolic equations satisfying a Dirichlet boundary condition on the parabolic part of the boundary of G. On the basis of the maximum principle we single out classes of uniqueness of such solutions depending on the geometry of the domain G which are analogs of the classes of A. N. Tikhonov and S. Täcklind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 924–930, July, 1990.     

Qualitative properties of the boundary derivative of the capacity potential for special classes of annular domains

Let U(p) denote the capacity potential in an annular domain Ω (bounded by Jordan curves). We describe the qualitative behavior of ∥Δ∥ on the line segments of ?Ω which are arbitrary, radial, or ν-directional (for some vector ν ≠ 0) in the respective cases where Ω is a convex, starlike, or ≠-convex annular domain. We apply these results to some free-boundary extremal problems in which the capacity plus weighted area functional is minimized in restricted classes of annular domains Ω.     

Best approximations of classes of functions defined by means of the modulus of continuity

This paper is devoted to an exact solution of problems of best approximation in the uniform and integral metrics of classes of periodic functions representable as a convolution of a kernel not increasing the oscillation with functions having a given convex upwards majorant of the modulus of continuity. The approximating sets are taken to be the trigonometric polynomials in the case of the uniform and integral metrics, and convolutions of the kernel defining the class with polynomial splines in the case of the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 579–589, May, 1992.     

Priifer domains with class group generated by the classes of the invertible maximal ideals

In [2] and [3] we described the double coset decomposition for the modular group SL_r+s (Z) with respect to the congruence subgroup Р _r,s (m) for any positive integer m. In the present paper we obtain the double coset decomposition of SL_r+s (R) with respect to Р _r,s (I). Where R is a commutative ring with unioty 1, and I is a nonzero ideal in R such that R/I is the direct sum of local principal ideal rings; thus generalize the results obtained in [2] and [3].