Three types of mathematical representational translations: Comparing empirical and theoretical results

Previous research has investigated the representational translation practices of high school students, high school teachers, and college preservice teachers in various mathematical contexts including linear functions. Findings from qualitative research has frequently led to new notions about participant work and understanding. Many quantitative research has investigated the degree to which some in these populations correctly perform these translations. However, it seems that only infrequently have empirical research investigated findings from qualitative studies and vice versa, and findings regarding one population are rarely compared with findings of another population. This study (a) empirically explores the frequency of success of preservice teachers (N = 80) regarding representational translations in the context of linear functions, (b) quantifies results from previous qualitative, literature‐based research regarding high school students and teachers, and (c) quantitatively compares the results. This study demonstrates that some mathematical representational translations are more difficult than others.     

Asymptotic modelling of a Signorini problem of generalized Marguerre-von Kármán shallow shells

Chacha and Bensayah [Asymptotic modeling of a Coulomb frictional Signorini problem for the von Kármán plates, C. R. Mécanique 336 (2008), pp. 846–850] have studied the asymptotic modelling of Coulomb frictional unilateral contact problem between an elastic nonlinear von Kármán plate and a rigid obstacle. The main result obtained is that the leading term of the asymptotic expansion is characterized by a two-dimensional Signorini problem but without friction. In this article, we extend this study to the case of a shallow shell under generalized Marguerre-von Kármán conditions.     

On some properties of double conjugate trigonometric Fourier series

A theorem of Ferenc Lukács [4] states that the partial sums of conjugate Fourier series of periodic Lebesgue integrable functions f diverge at logarithmic rate at the points of discontinuity of first kind of f. F. Móricz [5] proved an analogous theorem for the rectangular partial sums of bivariate functions. The present paper proves analogues of Móricz’s theorem for generalized Cesàro means and for positive linear means.     

A sufficient condition for kernel perfectness of a digraph in terms of semikernels modulo F

A kernel of a directed graph is a set of vertices which is both independent and absorbent. And a digraph is said to be kernel perfect if and only if any induced subdigraph has a kernel. Given a set of arcs F , a semikernel S modulo F is an independent set such that if some Sz-arc is not in F , then there exists a zS-arc. A sufficient condition on the digraph is given in terms of semikernel modulo F in order to guarantee that a digraph is kernel perfect. To do that we give a characterization of kernel perfectness which is a generalization of a previous result given by Neumann-Lara [Seminúcleos de una digrfica. Anales del Instituto de Matemticas 2, Universidad Nacional Autónoma de México, 1971]. And moreover, we show by means of an example that our result is independent of previous known sufficient conditions.     

A generalization of a maximum principle for the wave-operator with lower order terms

Summary It is shown that a classical maximum principles can be extended to continuous functions with piecewise continuous first and second derivatives. A simple application to the numerical solution of an initial value problem for the telegraph equation is presented. This paper was presented at the “2⁰ Seminário Brasileiro de Análise” held at the Instituto de Matematica e Estatística de Universidade de S?o Paulo in October 1975, during my stay at the program GMD-CNPQ.     

Characterizations of orthonormal scale functions: A probabilistic approach

The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L ²(R).Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in the case where the scale function is continuous. These results extend and clarify those of Cohen [2] and Hernández et al. [4]. The method also applies to more general dilation schemes that commute with translations by Z ^d .     

Extension of collineations defined on subsets of a translation plane

The T-closure of a setA in a projective translation plane is defined as the image ofA under the group generated by translations such that there exist proper points A, B with(A) = B. The sets, called quasi-anchors, make up the concepts of anchors introduced in [11]. It is proved that any collineation defined on a quasi-anchor can be extended to a collineation of the T-closure ofA. An application to the problem of uniqueness of the nomography is provided in a special case.     

On the values of the divisor function

For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : p ≤ x prime} and {τ(2ⁿ − 1) : n ≤ x}. Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia, Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia     

Decomposing Ends of Locally Finite Graphs

An important invariant of translations of infinite locally finite graphs is that of a direction as introduced by Halin . This invariant gives not much information if the translation is not a proper one. A new refined concept of directions is investigated. A double ray D of a graph X is said to be metric, if the distance metrics in D and X on V(D) are equivalent. It is called geodesic, if these metrics are equal. The translations leaving some metric double ray invariant are characterized. Using a result of Polat and Watkins , we characterize the translations leaving some geodesic double ray invariant.     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

Operators commuting with a discrete subgroup of translations

We study the structure of operators from the Schwartz space S(ℝ ⁿ ) into the tempered distributions S′(ℝ ⁿ ) that commute with a discrete subgroup of translations. The formalism leads to simple derivations of recent results about the frame operator of shift-invariant systems, Gabor, and wavelet frames.     

On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases

It is proved that if a convex density-like differential basis B is centered and invariant with respect to translations and homotheties, then the integral means of a nonnegative integrable function with respect to B can boundedly diverge only on a set of measure zero (this generalizes a theorem of Guzmán and Menarguez); it is established that both translation and homothety invariances are necessary.     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.     

The singly periodic genus-one helicoid

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77-84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: -rotation about the vertical line; rotation about the horizontal lines (the same symmetry); and their composition. Received: May 1996; revised October 1996.     

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.     

The broadest three-segment unit arc

We describe the broadest three-segment unit arc in the plane, and we conclude with some conjectures about the broadest n-segment unit arc for n > 3. Communicated by Imre Bárány     

Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.     

Projections on weak*-closed subspace of dual Banach algebras

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on M _a (S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.     

Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 A spectral set W\Omega in \RRⁿ\RR^n is a set of finite Lebesgue measure such that L² ( W)L^2 ( \Omega ) has an orthogonal basis of exponentials { e^{2 pi á\la, x ?} : \la ? \La }\{ e^{2 \pi i \langle \la, x \rangle} : \la \in \La \} restricted to W\Omega. Any such set \La\La is called a spectrum for W\Omega. It is conjectured that every spectral set W\Omega tiles \RRⁿ\RR^n by translations. A tiling set \sT\sT of translations has a \textit{ universal spectrum} \La\La if every set W\Omega that tiles \RRⁿ\RR^n by \sT\sT is a spectral set with spectrum \La\La. Recently Lagarias and Wang showed that many periodic tiling sets \sT\sT have universal spectra. Their proofs used properties of factorizations of abelian groups, and were valid for all groups for which a strong form of a conjecture of Tijdeman is valid. However, Tijdeman's original conjecture is not true in general, as follows from a construction of Szabó [17], and here we give a counterexample to Tijdeman's conjecture for the cyclic group of order 900. This article formulates a new sufficient condition for a periodic tiling set to have a universal spectrum, and applies it to show that the tiling sets in the given counterexample do possess universal spectra.     

The characterization of a class of subspace pseudoframes with arbitrary real number translations

In this article, the notion of generalized multiresolution structure is introduced. The concept of subspace pseudoframes with arbitrary real number translations is proposed. A new method for constructing a generalized multiresolution structure in Paley–Wiener subspace of L²(R) is presented. A pyramid decomposition scheme is established based on such a generalized multiresolution structure. Finally, affine frames of space L²(R) with arbitrary real number translations are obtained by virtue of the subspace pseudoframes and the pyramid decomposition scheme. Relation to some physical theories such as quarks confinement is also investigated.