Damage behavior of laminated E-glass/epoxy beams with an initial delamination subjected to an axial impact

The damage behavior of laminated E-glass/epoxy beams, with and without an initial delamination, subjected to an axial impact by a moving bullet has been investigated experimentally and numerically. The specimens were made from a unidirectional fiber fabric stacked in the sequences [0₆] _s , [±45₃] _s , and [90₆] _s , and a delamination was created in them by locating a copper foil at a specified position. The data on the bullet speed and strain history were recorded by a laser setup, a high-speed dynamic strain indicator, and a TDS420A oscilloscope. It is shown that the delamination and the ply stacking sequence play a significant role in the dynamic response and damage behavior of laminated beams. A numerical simulation is performed by using the commercial finite-element software ABAQUS/Explicit, and the results obtained are in a good agreement with experimental observations. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 49–64, January–February, 2009.     

Circular Delaminations - the Effect of Thermal Strains on the Strain Energy Release Rate

The analysis presented focuses on the extent to which the thermal loading resulting from a difference between the curing and service temperatures and from variations in the thermal-expansion coefficients from layer to layer can affect the growth of delaminations. Square laminate plates containing circular delaminations are examined. The reinforcement combinations for the delaminated layer and the core plate considered are [0°]/[90°], [90°]/[0°], [0°]/[±45°] _s , and [±45°] _s /[0°]. The analysis is carried out using the FEM. The strain energy release rate components are calculated using the modified crack-closure integral method. The results obtained show that, for all the reinforcement combinations studied, with the exception of [±45°] _s /[0°], a temperature drop considerably reduces all the strain energy release rate components, but only minor changes in their proportion occur. In the case of the [±45°] _s /[0°] plates, all these components increase, and noticeable changes in their proportion are also observed.     

On the central value of symmetric square L-functions

Let S _k (N, χ) be the space of cusp forms of weight k, level N and character χ. For let L(s, sym² f) be the symmetric square L-function and be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym² f) and over a basis of S _k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym² f) does not vanish. The author was supported by NSERC grant 311664-05.     

Linear methods for approximation of some classes of holomorphic functions from the Bergman space

We construct a linear method {ie910-01} for the approximation (in the unit disk) of classes of holomorphic functions {ie910-02} that are the Hadamard convolutions of the unit balls of the Bergman space A _p with reproducing kernels {ie910-03}. We give conditions for ψ under which the method {ie910-04} approximates the class {ie910-05} in the metrics of the Hardy space H _s and the Bergman space A _s , 1 ≤ s ≤ p, with an error that coincides in order with the value of the best approximation by algebraic polynomials. Translated from in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 783–795, June, 2008.     

Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions

This paper deals with the Cahn–Hilliard equation subject to the boundary conditions and the initial condition ψ(0,x) = ψ₀(x) where J = (0,∞), and Ω ⊂ ℝ ⁿ is a bounded domain with smooth boundary Γ = ∂ G, n≤ 3, and Γ _s ,σ _s ,g _s > 0, h are constants. This problem has already been considered in the recent paper of R. Racke and S. Zheng (The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110, 2003), where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal L _p -regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor. Mathematics Subject Classification (2000) 82C26, 35B40, 35B65, 35Q99     

On statistical properties of finite continued fractions

Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a² + b² ≤ R², are studied. The main result is an asymptotic formula with two meaning terms for the quantity

Some dichotomy results for central Gaussian convolution semigroups on compact groups

 Consider a central Gaussian convolution semigroup (μ _t ) _{t > 0} on a connected compact semisimple group. Then either the measure μ _t is singular with respect to Haar measure for all t > 0, or there exists a time t such that μ _t is absolutely continuous with respect to Haar measure and admits a continuous density. Received: 6 November 2001 / Revised version: 13 June 2002 / Published online: 30 September 2002 Research partially supported by NSF grant DMS 0102126 Mathematics Subject Classification (2000): 28C10, 28C20, 60B15, 60G30 Keywords or phrases: Gaussian convolution semigroups – Dichotomy – Absolute continuity     

Δ 2 0 -copies of linear orderings

It is proved that, for any n ∈ ω, there exist countable linear orderings L_n whose Δ ₂ ⁰ -spectrum consists of exactly all non n-low Δ ₂ ⁰ -degrees. Properties of such orderings are examined, for n = 1 and n = 2. Supported by RFBR grant No. 02-01-00169 and by RF Ministry of Education grant No. E02-1.0-177. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 354–370, May–June, 2006.     

Direct decompositions of torsion-free Abelian groups of finite rank

It is proved that ifr ₁ ,r ₂ , ...,r _s ;l ₁ ,l ₂ , ...,l _t are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if s₀ is the number of units among the numbers r_i, while t₀ is the number of units among the numbers l_j, thenr _i ⩽n - t ₀ ,l _j ⩽n−s ₀ for all i, j. Moreover, if for some i we have r_i=n−t₀, then among the l_j's only one term is different from 1 and it is equal to n−t₀; similarly if l_j=n−s₀ for some j. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 272–285, 1987.     

Ornstein-Zernike theory for finite range Ising models above T c

 We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function 〈Σ₀Σ _x 〉_β in the general context of finite range Ising type models on ℤ ^d . The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernández, goes through in the whole of the high temperature region β<β _c . As a byproduct we obtain that for every β<β _c , the inverse correlation length ξ_β is an analytic and strictly convex function of direction. Received: 10 January 2002 / Revised version: 19 June 2002 / Published online: 14 November 2002 Partly supported by Italian G. N. A. F. A, EC grant SC1-CT91-0695 and the University of Bologna. Funds for selected research topics. Partly supported by the ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Science and Humanities. Partly supported by the Swiss National Science Foundation grant #8220-056599. Mathematics Subject Classification (2000): 60F15, 60K15, 60K35, 82B20, 37C30 Key words or phrases: Ising model – Ornstein-Zernike decay of correlations – Ruelle operator – Renormalization – Local limit theorems     

Molecular lines and continuum from W51A (I) —CO and NH3 spectra and 3 mm continuum

As for the 5′ × 4′(∼llpc × 9pc) region centered at W51 lRSl the observations of the 3.4 mm continuum, CO (J = 1-0) line and simultaneous NH₃(1,1), (2,2), (3,3), (4,4) inverse lines were made for studying the massive star formation region located in the main spiral arms of the Galaxy. In the directions of W51 IRS1, IRS2 and el/e2 in 3.4 mm continuum, analyses of the line profiles show that the absorption lines of ammonia, which arise from the gas in front of the HII region, are red-shifted with respect to the emission lines, which arise from the surrounding cloud. Furthermore, a radiation transfer and statistical equilibrium calculation of ammonia molecules show that the densities increase by 3–10 times from the eastern border to the center. These points hint that the collapse is happening in the molecular cloud core obscured in optical wavelengths. The effects of the radiation fields from radio, infrared and UCHII sources is non-negligible on the excitation of various molecules (e.g. NH₃) within the circle of radius 40″ centered at IRS1. The profiles of the COJ = 1–0 line in the circle change from double peaks ( ∼ 60, ∼ 68 km. s^-1) to triple peaks, i.e. the component ∼53 km·s⁻¹, which associates with UCHII, also appears in the spectra. There are indications that the circle of radius 40″ centered at IRSI is a region of massive star forming activity     

A note on the non-commutative neutrix product of distributions

The distributionF(x ₊, −r) Inx₊ andF(x ₋, −s) corresponding to the functionsx ₊ ^−r lnx₊ andx ₋ ^−s respectively are defined by the equations

On simultaneous diophantine approximations. Vectors of given diophantine type

For any monotone functionψ(y)=O(y ^1/s), we prove the existence of a continual family of vectors (α₁...,α_s) admitting infinitely many simultaneous ψ-approximations, but nocψ-approximations with some constantc>0. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 706–716, May, 1997. Translated by S. K. Lando     

A geometrical problem arising in a signal restoration algorithm

Let ℤ_2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ_2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ_2N→ℝ is given by φ_f(k)=arg for k ∈ ℤ_2N, where denotes the discrete Fourier transforms of f. For a fixed s∈L let K_s denote the cone of all f:ℤ_2N→ℝ which satisfy φ_f ≡ φ_s and let M_s be its linear span. The angle α_s between M_s and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:

Properly Edge-Coloured Subgraphs in Colourings of Bounded Degree

The smallest n such that every colouring of the edges of K _n must contain a monochromatic star K _1,s+1 or a properly edge-coloured K _t is denoted by f (s, t). Its existence is guaranteed by the Erdős–Rado Canonical Ramsey theorem and its value for large t was discussed by Alon, Jiang, Miller and Pritikin (Random Struct. Algorithms 23:409–433, 2003). In this note we primarily consider small values of t. We give the exact value of f (s, 3) for all s ≥ 1 and the exact value of f (2, 4), as well as reducing the known upper bounds for f (s, 4) and f (s, t) in general.     

Fundamental solution of the cauchy problem corresponding to the one-speed linear Boltzman equation for anisotropic media

We consider the fundamental solution E (t,x,s;s ₀) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t+c(s,grad _x)+γ)E(t,x,s;s ₀)=γν∫ f((s, s′))E(t,x,s′; s₀)ds′+Ωδ(t)δ(x)δ (s−s ₀) that is assumed to be valid for any (t,x)∈Rⁿ⁺¹; morevoer, for t<0 the condition E(t,x,s; s₀)=0 holds. By using the Fourier-laplace transform in space-time arguments, the problem reduces to the study of an integral equation in the variables. For 0<ν≤1, the uniqueness and existence of the solution of the original problem are proved for any fixeds in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic type (f(.)=1), the solution of the integral equation is given in explicit form. In the approximation of “small mean-free paths,” various weak limits of the solution are obtained with the help of a Tauberian-type theorem, for distributions. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 319–332. Translated by Yu. B. Yanushanets.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Interlaminar fracture of multidirectional glass/epoxy laminates under mixed-mode I + II loading

The mixed-mode I + II interlaminar fracture of multidirectional glass/epoxy laminates is investigated. Mixed-mode bending (MMB) tests were performed on specimens with delaminations in 0/θ-type interfaces, with θ varying from 0 to 90°. Preliminary three-dimensional finite-element analyses validated the beam theory model (BTM) used for analysing experimental data. The compliances measured are in a good agreement with BTM predictions. The total critical energy release rate G_c varies linearly with the mode II ratio G_II/G, although some discrepancies are observed in the high-mode II results for the 0/45 and 0/90 specimens. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 349–366, May–June, 2007.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

Kolmogorov and linear widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ∈ ℕ and let Δ ₊ ^s be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t ₀, ..., t _s ] of order s of these functions are nonnegative for all collections of s + 1 different points t ₀, ..., t _s ∈ I. For the classes Δ ₊ ^s B _p : = Δ ₊ ^s ∩ B _p , where B _p is the unit ball in L _p , we determine the orders of Kolmogorov and linear widths in the spaces _Lq for 1 ≤ q > p ≤ ∞. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.