Effect of the arrangement of the reinforcement on the stability of circular,cylindrically orthotropic plates with low shear strength

The Bubnov-Galerkin method is used to find the critical loads for cylindrically orthotropic circular continuous and annular plates with different edge conditions in axisymmetric buckling. The effect of the ratio of the Young moduli in the circumferential (E) and radial (E_r) directions and the ratio of E to the interlaminar shear modulus G_rz is investigated. The results are in qualitative agreement with the experimental data of [13].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 275–283, March–April, 1971.     

On the single fibre pullout and pushout problem: Effect of fibre anisotropy

Based on our recent theoretical model of the interfacial debonding and fibre pullout/pushout stresses, an improved analysis including the effect of fibre anisotropy has been performed for single fibre pullout and pushout. During fibre-matrix debonding the debonded region is considered as an interfacial crack and its extension is dependent on a fracture energy criterion being satisfied. The elastic properties of the fibre are taken to be transversely isotropic about the fibre axis and the matrix is assumed to be isotropic. The results show that the maximum debond stress for fibre pullout and the initial frictional pullout stress increase as the ratio of the Young's moduli of the fibre between the axial and radial directionsE _f /E _f ^r increases, and this is particularly so at large embedded fibre lengths. However, in the case of fibre pushout these debond and pushout stresses decrease asE _f /E _f ^r increases.     

A remark concerning the modulus of smoothness introduced by Ditzian and Totik

For each functionf(x) continuous on the segment [–1, 1], we set . We study the relationship between the ordinarykth modulus of continuity of therth derivative of the function and thekth modulus of continuity with weight _r of the rth derivativef ^(r) of the functionf introduced by Ditzian and Totik. Thus, ifr is odd andk is even, we prove that these moduli are equivalent ast0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1627–1638, December, 1995.     

Eigenfunctions and associated functions of an n-th-order linear differential operator

For n2 we consider a differential operatorL [y] z ⁿ y ⁽ⁿ⁾ +P ₁(z)z ^n–1 y ^(n–1) +P ₂ (z)z ^n–2 y ^n–2 + ...+P _n (z)y = y, p ₁ (z), ..., P _n (z) A _R : here a_r is the space of functions which are analytic in the disk ¦z¦ < R, equipped with the topology of compact convergence. We prove the existence of sequences {f_k(z)} _k =o, consisting of a finite number of associated functions of the operator L and an infinite number of its eigenfunctions; we show that the sequence forms a basis in A_r for an arbitrary r, 0 < r <- R; and we establish some additional properties of the sequence ₀ (z), ₁ (z),..., _d–1 (z), f _d (z), f _d+1 (z),... Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 869–878, December, 1976.     

A bound of the exterior arcs for a univalent mapping

In this paper we consider the intersection of the circle ¦w|=x with the image of the disc ¦z|&#x2264;r, 0f(z)=z+c₂z²+... which is univalent analytic in ¦z|<1. Earlier I. E. Bazilevich proved that for xc^/er the measure of the above intersection does not exceed the measure of the intersection produced by the functionf ^*(z)=z/(1–z)², ¦=1. In this paper I. E. Bazilevich's ideas are used to strengthen some of his results.Translated from Matematicheskie Zametki, Vol. 18, No. 3, pp. 367–378, September, 1975.     

On the diameters of certain classes of analytic functions. II

In the spaces E_q(), q1, we consider the classes W^rE_p(), p1, consisting of analytic functions f(z) E_P() the integral moduli of continuity of whose r-th derivatives are majorized by a given nonnegative nondecreasing function .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1135–1138, August, 1992.     

Characterization of Generating Ideals in Some Rings of Entire Functions

Let E be a ring of entire functions on with the operation of pointwise multiplication, and let f ₁,...,f _m be a set of nonzero elements in E. The ideal E(f ₁,...,f _m ) in E with generators f ₁,...,f _m is said to be generating if E(f ₁,...,f _m ) = E. The generating ideals in rings of entire functions on determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of d(z) max _{1 j m} d _j (z), where d _j (z) is the distance from a point to the zero set of f _j for 1 j m. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of d(z)) cannot be given.     

Power series whose coefficients form homogeneous random processes

Summary Power series f(z) = a _izⁱ are considered, where the sequence {a _i} forms a homogeneous random process. If the sequence is exchangeable and the variance of the marginal distributions exists, it is proved that r, the random radius of convergence of f(z), takes the values 0 and 1. If the sequence is a second order stationary time series then r=1 with probability 1. If {a _i} is a regular denumerable Markov chain, it can be proved that r=c=1 with probability 1, but both c=0 and c=1 can arise. A number of criteria are given for deciding the value of c in this situation.     

New extremal properties for constructing conformal mappings

Summary It is well known that for a given simply connected regionR containing zero the uniform norm attains its minimum in the class of all holomorphic functions normalized byf(0)=0 andf(0)=1 only for the conformal mappingfRD(r)={z|z|}. It is shown that this theorem is still valid if one replaces the ordinary modulus | | on by any other norm on . For instance it is possible to obtain direct mappings ofR onto parallelograms, rectangles and ellipses. For the special norms |₁ and | this leads to a simple and fast computational technique involving linear programming methods. Several numerical examples are given.     

On a Class of Operator Equations

We assume that E₁ and E₂ are Banach spaces, a:E₁ E₂ is a continuous linear surjective operator, f:E₁ E₂ is a nonlinear completely continuous operator. In this paper, we study existence problems for the equation a(x)=f(x) and estimate the topological dimension dim of the set of solutions.     

Exact constants of approximation for differentiable periodic functions

For all odd r we construct a linear operator B_r,r(f) which maps the set of 2-periodic functionsf(t) X^(r) (X^(r)=C^(r) or L₁ ^(r)) into a set of trigonometric polynomials of order not higher than n-1 such that where X is the C or L₁ metric, E_n(f)_X and (f, )_X are the best approximation by means of trigonometric polynomials of order not higher than n-1 and the modulus of continuity of the functionf in the X metric, respectively; K_r are the known Favard constants.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 21–30, July, 1973.In conclusion, the author wishes to express his deep gratitude to N. P. Korneichuk under whose guidance this paper was written.     

Problem of correctness of the best approximation in the space of continuous functions

Let W^rH _w be the subclass of those functions of C^r[a, b], for which (f ^(r),)(), where () is a given modulus of continuity, and P_n be the space of algebraic polynomials of degree at most n and _n(f) be the polynomial of best approximation for f(x) on [a, b]. Estimates for and moduli of continuity of the operators of best approximation on W^rH _w are established. For example, if ()=, then Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 351–360, March, 1978.The author thanks S. B. Stechkin for the formulation of the problem and assistance with the article.     

Multipliers on the Set of Rademacher Series in Symmetric Spaces

Let E be a symmetric space on [0,1]. Let (,E) be the space of measurable functions f such that fg E for every almost everywhere convergent series g=b _n r _n E, where (r _n) are the Rademacher functions. It was shown that, for a broad class of spaces E, the space (,E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for (,E) to be order isomorphic to L . This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which (,E) is order isomorphic to a symmetric space that differs from L . The answer is positive for the Orlicz spaces E=L _q with _q(t)=exp|t|^q-1 and 0相似文献   

The pointwise estimates on walsh overconvergence

Let f∈A_p. For any positive integer l, the quantity Δ_1,n−1(f:z) has been studied extensively. Here we give some quantitative estimates for and investigate some pointwise estimates of Δ _l,n−1 ^(r) (f;z). Supported by National Science Foundation of China     

Applications of the zero–one law to problems connected with Cauchy's functional equation

Summary Marek Kuczma's book, entitled An Introduction To The Theory Of Functional Equations And Inequalities, mentions a certain setV ₀ in several places and presents references as to where this set is discussed in the literature. The main result of this paper is a proof of the fact that the setA _M (V ₀)={xV ₀ f(x)>M} is saturated non-measurable for each additive discontinuous functionf and each real numberM. Other results aboutV ₀ are also presented. Connections between measure and category are stressed. The main tool in our proofs is a certain so-called zero–one law and its topological analogue. In addition it is shown that the zero–one law is equivalent to Smital's lemma.     

Descriptions of exceptional sets in the circles for functions from the Bergman space

Let D be a domain in $\mathbb{C}^2 $ . For w ∈ $\mathbb{C}$ , let D_w=\{z \in $\mathbb{C}$ \, \vert \, (z,w)\in D\}. If f is a holomorphic and square-integrable function in D, then the set E(D, f) of all w such that f(., w) is not square-integrable in D _w is of measure zero. We call this set the exceptional set for f. In this note we prove that for every 0 < r < 1, and every G _δ-subset E of the circle C(0,r)=\{z \in $\mathbb{C}$ \, \vert \, \vert z \vert = r \},there exists a holomorphic square-integrable function f in the unit ball B in $\mathbb{C}$ ² such that E(B, f) = E.     

Carrying capacity of rings formed by winding composites with high-modulus anisotropic fiber reinforcement

The characteristics of composites associated with the essential anisotropy of the reinforcing fibers are studied. The effect of distortion and pretensioning of the reinforcement on the axisymmetricity of the strain field, the moduli of elasticity, and the strength in the direction of the fibers under external and internal pressure are investigated with allowance for the high compliance of the material in the transverse direction. Numerical estimates are obtained for the moduli of elasticity E and E_r, expansion coefficient and _r, and initial temperature stresses _r ⁰ and ⁰ for a model material. The dependence of the bursting pressure on the thickness of the rings is described and strength criteria are formulated. In all the problems considered it is necessary to make allowance for the high compliance of the composites in the transverse direction.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 673–683, July–August, 1973.     

Asymptotic Behavior of Entire Functions with Exceptional Values in the Borel Relation

Let M _f(r) and _f (r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let l(r) be a continuously differentiable function convex with respect to ln r. We establish that, in order that ln M _f(r) ln _f (r), r +, for every entire function f such that _f (r) l(r), r +, it is necessary and sufficient that ln (rl(r)) = o(l(r)), r +.     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.