Vibrations of a rectangular membrane with an eccentric inner circular boundary: A comparison of approximate methods

Three different approaches are used in order to obtain, independently, fundamental eigenvalues of the mechanical system under study: the Ritz method, the finite element algorithm and a Fourier expansion-collocation scheme. The agreement can be considered as very reasonable, especially in view of the relatively simple formulation of the Ritz approach in which two polynomial co-ordinate functions are employed. The results may also be of interest to acousticians and microwave specialists since they are applicable to soft-walled acoustical waveguides and TM modes in electromagnetic waveguides.     

Solid surface mapping by inverse gas chromatography

Inverse gas chromatography (IGC) at infinite dilution, is a technique for characterising solid surfaces. Current practice is the injection of n-alkane homologous series to obtain the free energy of adsorption of the CH2 group, from which the London component of the solid surface free energy, gamma(d)s, is calculated. A value around 40 mJ/m2 is obtained for poly(ethylene), and 30 mJ/m2 for a clean glass fibre, while the potential surface interactivity of a glass fibre is far greater than that of poly(ethylene). A specific component of the surface, in mJ/m2, should be calculated in order to obtain significant parameters. As applied up to date, when calculating the specific component of the surface energy, the fact that W(sp)a energy values are in a totally different scale than AN or DN values is a major drawback. Consequently, Ka and Kb values obtained are in arbitrary energy units, different from those of the London component measured by injecting the n-alkane series. This paper proposes a method to obtain Ka and Kb values of the surface in the same energetic scale than the London component. The method enables us to correct the traditional London component of a solid, obtaining a new value, where the amount of WaCH2 accounting for Debye interactions with polar sites, is excluded. As a result, an approach to surface mapping is performed in several different substrate materials. We show results obtained on different solid surfaces: poly(ethylene), clean glass fibre, glass beads, chemically modified glass beads and carbon fibre.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

“Everything is Math in the Whole World”: Integrating Critical and Community Knowledge in Authentic Mathematical Investigations with Elementary Latina/o Students

This critical ethnographic study of an after-school mathematics club for elementary-aged Latina/o youth focuses on connecting critical, community, and mathematical knowledge in the context of authentic, community-based investigations. We present cases of two extended projects to highlight tensions and dilemmas that emerged, particularly tensions related to ensuring rich mathematics in the contexts of projects that were personally and socially meaningful to the students. Our analysis offers insights into critical mathematics education with elementary aged students, and has the potential to counter dominant deficit perspectives of Latina/o youth. Additionally, the findings of this study inform critical approaches to teaching mathematics in schools attended by marginalized students in order to reverse prevalent trends of our educational system failing these students.     

Try to See It My Way: The Discursive Function of Idiosyncratic Mathematical Metaphor

What are the nature, forms, and roles of metaphors in mathematics instruction? We present and closely analyze three examples of idiosyncratic metaphors produced during one-to-one tutorial clinical interviews with 11-year-old participants as they attempted to use unfamiliar artifacts and procedures to reason about realistic probability problems. Our interpretations of these episodes suggest that metaphor is both spurred by and transformative of joint engagement in situated activities: metaphor serves individuals as semiotic means of objectifying and communicating their own evolving understanding of disciplinary representations and procedures, and its multimodal instantiation immediately modifies interlocutors' attention to and interaction with the artifacts. Instructors steer this process toward normative mathematical views by initiating, modifying, or elaborating metaphorical constructions. We speculate on situation parameters affecting students' utilization of idiosyncratic resources as well as how socio-mathematical license for metaphor may contribute to effective instructional discourse.     

Asplund Operators and p-Integral Polynomials

We introduce the Banach ideals of p-integral and of p-nuclear polynomials for 1 ≤ p ≤ + ∞, extending to the polynomial setting the well known notions of p-integral and p-nuclear operators. For p = 1, we recover the Pietsch integral and nuclear polynomials, respectively. Given a Banach space E, let K be a compact Hausdorff space such that there is an embedding h : E → C(K). Let R _h be the polynomial from E into C(K) given by R _h (x) : = h(x) ^m for all ${x \epsilon E}$ . We prove that a polynomial is p-integral (1 ≤ p ≤ + ∞) if and only if it factors through a polynomial of the form R _h followed by the canonical inclusion of C(K) into L _p (K, μ) for some finite measure μ. We also prove that a polynomial P is p-integral if and only if we may write ${P = T \circ R_{h}}$ where T is a p-integral operator on a C(K) space. We show that P is ∞-integral if and only if it factors in the form ${P = T \circ R_{h}}$ where T is a weakly compact operator on a C(K) space. Analogous results are true if we replace C(K) by L _∞ (Ω, μ) for some finite measure space (Ω, Σ, μ). It is proved that a polynomial ${P \epsilon \mathcal{P}(^{m}E, F)}$ is p-integral if and only if its linearization is well defined and p-integral on ${\bigotimes ^{m}_{{\epsilon}_{s}}, s^{E}}$ . It is also shown that a p-integral polynomial may be extended to a p-integral polynomial on every larger space, and the extension has the same p-integral norm. We give a factorization theorem for p-nuclear polynomials. Finally, we prove that a polynomial P is p-nuclear if and only if it may be written in the form ${P = Q \circ A}$ where A is a compact operator and Q is a p-integral polynomial, if and only ${P = Q^{\prime} \circ H}$ with H an Asplund operator and Q′ a p-integral polynomial. This extends a result obtained by C. Cardassi in the linear case.     

Approximate reduction of non-linear discrete models with two time scales

The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system.     

Classification of bernstein algebras of type (3, n - 3)

A classification of Bernstein algebras in dimensions n ? 4 has been made by Holgate in [2], however that article contains no classification up to isomorphism, the problem is solved by Lyubich in [4] when K = R or C, and by Cortes [1] in the general case. Also Lyubich has given in [5] a classification of the regular nonexceptional Bernstein algebra of type (3,n?3) and a classification but not up to isomorphism of nonregular nonexceptional Bernstein algebras of type (3,n ? 3) when K = C. The aim of this paper it to characterize, up to isomorphism, Bernstein algebras of type(2, n ? 2) and nonexceptional of type(3, n ?3) over a infinite commutative field K whose characteristic is different from 2.