The wavelet dimension function is the trace function of a shift-invariant system

In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.

     

Optimal fractional matchings and covers in infinite hypergraphs: Existence and duality

We study fractional matchings and covers in infinite hypergraphs, paying particular attention to the following questions: Do fractional matchings (resp. covers) of maximal (resp. minimal) size exist? Is there equality between the supremum of the sizes of fractional matchings and the infimum of the sizes of fractional covers? (This is called weak duality.) Are there a fractional matching and a fractional cover that satisfy the complementary slackness conditions of linear programming? (This is called strong duality.) In general, the answers to all these questions are negative, but for certain classes of infinite hypergraphs (classified according to edge cardinalities and vertex degrees) we obtain positive results. We also consider the question of the existence of optimal fractional matchings and covers that assume rational values.     

Teacher and students’ joint production of a reversible fraction conception

Within a constructivist perspective, I conducted a teaching experiment with two fourth graders to study how a teacher and students can jointly produce the reversible fraction conception. Ongoing and retrospective analysis of the data revealed the non-trivial process by which students can abstract multiplicative reasoning about fractions. The study articulates a conception in a developmental sequence of iteration-based fraction conceptions and the teacher’s role in fostering such a conception in students.     

Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λe^u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ_c, no solutions for λ > λ_c and a unique solution when λ = λ_c. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ_c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004     

Counting 1-factors in infinite graphs

F. Bry (J. Combin. Theory Ser. B 34 (1983), 48–57) proved that a locally finite infinite n-connected factorizable graph has at least (n−1)! 1-factors and showed that for n = 2 this lower bound is sharp. We prove that for n≥3 any infinite n-connected factorizable graph has at least n! 1-factors (which is a sharp lower bound).     

The COLUMBUS Standard Integral File Structure: A proposed interchange format

The integral file structure used in the COLUMBUS Program System is described. This file structure is proposed for use as an interchange format for the exchange of information between various electronic structure codes. Access to the integral and density matrix arrays stored in the file structure is simplified by a supporting subroutine library. This library is portable across various computers and is readily available to programmers from the COLUMBUS distribution files. This library provides for the efficient processing of individual records, including the use of asynchronous I/O and the vectorized processing of packed orbital labels. The individual arrays are identified in a self-defining and extensible manner, allowing for the addition of new integral types as demanded by the application. The format of the individual records is also self-defining, allowing for the use of various packing and data compression methods within each record without burdening the calling program with unnecessary complications.