二阶含阻尼项可化为“积分小”系数的微分方程解的振动性质

本文考虑下列二阶微分方程 (r(t)x′(t))′ q(t)x′(t) p(t)x(t)=0. (1) 和 (r(t)x′(t))′ q(t)x′(t) p(t)f(x(g(t)))=0 (2)解的振动性质。我们给出了方程(1)非振动解存在的充要条件和方程(2)存在振动解的充分判据。     

二阶非线性阻尼常微分方程的振动性定理

考虑二阶非线性阻尼微分方程(α(t)ψ(x(t))x′(t))′ p(t)x′(t) q(t)f(x(t))=0 (1)和二阶非线性微分不等式x(t){(α(t)ψ(x(t))x′(t))′ p(t)x′(t) q(t)f(x(t))}≤0,(2)其中α,p,q∈C([t_0,∞)→(-∞,∞)),ψ,f∈C(R→R),并且α(t)>0,xf(x)>0 (x≠0).此外,我们总假设方程(1)的每一个解 x(t)可以延拓于[t_0, ∞)上.在任何无穷区间[T,∞)上,x(t)不恒等于零,这样的解叫正则解.一个正则解,若它有任意大的零点,则称为振动的;否则就称为非振动的.若方程(1)的所有正则解是振动的,则称方程(1)是振动的.关于不等式(2)的振动性的定义,与方程(1)的振动性的定义完全类似,不再赘述.     

二阶非线性泛函微分方程解的振动性质

本文研究了带有扰动项的二阶非线性泛函微分方程x″(t)+p(t)k(t,x(t),x′(t))x′(t)+q(t)f(x(σ(t)))=0 (1)的解的振动性质。在一定条件下,建立了方程(1)的若干振动性定理。其结果推广和改进了已有的一些结果。     

一阶非线性偏差变元微分方程解的振动性

关于偏差变元微分方程解的振动性问题已在实际应用中提了出来.如文献[1,2].也越来越引起人们的重视,且得到了一些很好的结果,如文献[3—8],综述文献[9]在“一些问题”中提出了进一步研究方程x′(t)+p(t)f(x(g(t)))=0(1)的解的振动性的充分条件的课题.本文首先给出了较一般的滞后型方程x′(t)+p(t)F(x(g_1(t)),x(g_2(t)),…,x(g_n(t)))+h(t,x(t),x(g_1(t)),…,x(g_n(t)))=0(2)的解的振动的充分条件.把所得结果应用于方程(1),从而在很大程度上改进了文献[3]的结果.然后,又在 g_i(t)超前情形下,给出了方程(2)解振动的充分条件,把所得结果应用于某些文献[3,4]称之为超线性方程,得到了与滞后型亚线性方程解振动的类似结果.假定 x(t)在[t_x,+∞)上存在.记 g(t)=(?){g_i(t)}.     

一类含混合常数变元脉冲微分系统解的振动性

考虑下列二阶脉冲微分系统解的振动性{(r(t))(x′(t)σ)′ a(t)(x([t]))δ e(t)sgn x(t)=0,t≠n,t≥0,n∈Z ,x(n)=gn(x(n-)),x′(n)=hn(x′(n-)),t=n,n=1,2,…,其中s,d是任意给定的正奇数的商.借助脉冲微分不等式得到了保证上述系统所有有界解振动的若干充分条件,并给出例子说明定理的应用.     

变系数线性中立型微分差分方程解的振动性

本文研究了中立型微分方程 x′(t)-cx′(t-r)+sum from i=1 to n (p_i(t)x(t-r_i)=0)解的振动性。所采用的方法也适用于讨论方程 x′(t)+sum from i=1 to n (p_i(t)x(t-r_i)=0)的振动性。所得结果推广和改进了文[1—4]的主要结果。     

具有振动位势的二阶带强迫项、阻尼项的时滞微分方程的振动性

讨论了具有振动位势的二阶微分方程(k(t)x′(t))′+τ(t)x′(t)+p(t)x(τ(t))+q(t)x(σ(t))=e(t),利用其线性近似方程(k(t)x′(t))′+p(t)x(τ(t))+q(t)x(σ(t))=e(t)的振动性,给出了方程解振动的一个充分条件,所得结果推广了文献[Computer andMathematics with Applications,2006,51:1395-1404]的相关结果.     

半线性二阶阻尼微分方程的振动定理

利用积分平均技巧,得到了半线性二阶阻尼微分方程[a(t)|x′(t)|α-1x′(t)]′+p(t)k(t,x(t),x′(t))x′(t)+q(t)|x(t)|α-1x(t)=0的一些新的振动定理.这些结果改进和推广了Manojlovic J V[5]的结果.     

一类滞后量为[t]的中立型泛函微分方程的渐进性和振动性

本文讨论一类滞后量为 [t]的中立型泛涵微分方程　　　　 x′(t) - c(t) x′(t- [t]+p(t) f(x(t- [t]) ) =0　 t≥ 0的解的性质 ,得到所考虑的方程存在非振动解的充分条件和非零解的变化趋势。     

三阶非线性中立型微分方程的振动性

研究三阶中立型时滞微分方程(r(t)[x(t)+p(t)x(σ(t))]″)′+q(t)f(x(t),x[q(t)])h(x′(t))=0的振动性和渐进性.给出了方程一切解振动或者渐近趋向于零的若干充分条件.     

Zero-dimensional subschemes of ruled varieties

Here we study zero-dimensional subschemes of ruled varieties, mainly Hirzebruch surfaces and rational normal scrolls, by applying the Horace method and the Terracini method This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy)     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.     

Comments on generalized Heron polynomials and Robbins’ conjectures

Heron’s formula for a triangle gives a polynomial for the square of its area in terms of the lengths of its three sides. There is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases if A is the area of the polygon, (4A)² is a polynomial function of the square in the lengths of its edges. David Robbins in [D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (2) (1994) 223-236. MR 95g:51027; David P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (6) (1995) 523-530. MR 96k:51024] showed that for any cyclic polygon with n edges, (4A)² satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n=5 and n=6. He conjectured the degree of this polynomial for all n, and recently Igor Pak and Maksym Fedorchuk [Maksym Fedorchuk, Igor Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. 129 (2) (2005) 371-404. MR 2006f:52015] have shown that this conjecture of Robbins is true. Robbins also conjectured that his polynomial is monic, and that was shown in [V.V. Varfolomeev, Inscribed polygons and Heron polynomials (Russian. Russian summary), Mat. Sb. 194 (3) (2003) 3-24. MR 2004d:51014]. A short independent proof will be shown here.     

On the weakening of the convergence of Newton’s method using recurrent functions

We use Newton’s method to approximate a locally unique solution of an equation in a Banach space setting. We introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton’s method than before [J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1–17; I.K. Argyros, The theory and application of abstract polynomial equations, in: Mathematics Series, St. Lucie/CRC/Lewis Publ., Boca Raton, Florida, USA, 1998; I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005) 179–194; I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New York, 2008; S. Chandrasekhar, Radiative Transfer, Dover Publ., New York, 1960; F. Cianciaruso, E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003) 713–723; N.T. Demidovich, P.P. Zabrejko, Ju.V. Lysenko, Some remarks on the Newton–Kantorovich method for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993) 22–26. (in Russian); E. De Pascale, P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998) 271–280; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994) 20–24. (in Russian); B.A. Vertgeim, On conditions for the applicability of Newton’s method, (Russian), Dokl. Akad. Nauk., SSSR 110 (1956) 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957) 166–169. (in Russian); English transl.:; Amer. Math. Soc. Transl. 16 (1960) 378–382] provided that the Fréchet-derivative of the operator involved is p$p$-Hölder continuous (p∈(0,1]$p\in (0,1]$).     

Transverse compression of PPTA fibers

Results of single transverse compression testing of PPTA and PIPD fibers, using a novel test device, are presented and discussed. In the tests, short lengths of single fibers are compressed between two parallel, stiff platens. The fiber elastic deformation is analyzed as a Hertzian contact problem. The inelastic deformation is analyzed by elastic-plastic FE simulation and by laser-scanning confocal microscopy of the compressed fibers ex post facto. The results obtained are compared to those in the literature and to the theoretical predictions of PPTA fiber transverse elasticity based on PPTA crystal elasticity.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).3TEX, Inc. 109, MacKenan Drive, Cary, North Carolina 27511, USA. Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 533–544, March–April, 2000.     

Effectivity functions and simple games

The rather new notion of effectivity function is related to the notion of simple game. Every effectivity function is associated with a simple game. So theory about simple games may be applicable to effectivity functions. E.g. if the effectivity function is additive, then the associated simple game is weighted. Via a characterization of weighted simple games it is possible to characterize maximal additive effectivity functions.Finally we characterize additive effectivity functions and their associated simple games.I thank G.J. Otten for useful comments and stimulating conversations.     

Iterated function systems, Ruelle operators, and invariant projective measures

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space comes with a finite-to-one endomorphism which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets in of the same cardinality which generate complex Hadamard matrices.

Our harmonic analysis for these iterated function systems (IFS) is based on a Markov process on certain paths. The probabilities are determined by a weight function on . From we define a transition operator acting on functions on , and a corresponding class of continuous -harmonic functions. The properties of the functions in are analyzed, and they determine the spectral theory of . For affine IFSs we establish orthogonal bases in . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in .

     

Dynamic assessment of reinforced concrete beams repaired with externally bonded FRP sheets

This research deals with RC beams strengthened with FRP. An experimental research is presented which is aimed at evaluating the capability of an experimental modal analysis to assess the stiffness decrease due to damage, as well as the stiffness recovery due to strengthening. Ten beams were tested. All of them were subjected to loading cycles with increasing load levels in order to induce cracking of different severity in them. The beams were then retrofitted by externally bonded FRP sheets. Three types of composites were used. The number of layers was varied, too. Modal tests were carried out after each loading-unloading cycle. The modal frequencies and damping ratios were determined for the first four vibration modes. The results obtained indicate that an experimental modal analysis can give useful information on the severity of damage and the effectiveness of strengthening. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 1, pp. 3–20, January–February, 2006.     

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group on a -dimensional handlebody . Whenever acts freely preserving orientation on a connected -manifold , there is a -equivariant imbedding of into . There are choices of closed and Seifert-fibered for which the image of is a handlebody of a Heegaard splitting of . Provided that the genus of is at least , there are similar choices with closed and hyperbolic.

     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.