Standard pairs and existence of symmetric multiscaling functions

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

Same Jeans, Same Stitch? A Comparison of Denim Production Across Three Factories in Punjab, Pakistan

In this paper, we look at denim production in three different factories in Punjab, Pakistan. We map the manufacturing process for a standard pair of denim jeans produced for an international retailer. We asked three factories of different scales and proximities to the technological frontier to stitch, finish and wash an identical pair of jeans. These firms included a large-scale exporter with established links to a major multinational brand, a medium exporter with links to regional European labels and a small producer selling primarily to the domestic market. Timing the operations ourselves, we find that the stitching time of the large-scale exporter is about one-third less than that of the medium exporter and about half the stitching time of the small firm. Of the three firms, only the large exporter pays wages based strictly on standard minute value – the time expected to complete an operation. The two smaller firms pay piece rates that reflect the market rates paid for individual operations by firms throughout the sector. Even without increases in stitching efficiency, the two smaller firms could reduce their stitching costs by 30–50 percent if they were able to switch to paying wages based on stitching times. We also calculate the labor cost savings that the two smaller firms could accrue by adopting some of the more advanced equipment used by the large exporter, along with lower piece rates. Of these, the most reasonable investment would be in better loop-making machines; the cost of equipment could be recuperated by producing 325,000–500,000 garments, which for the medium firm is four to eight months’ production at current levels. However, piece rates are entrenched and, if sticky, could reduce the incentives for firms to adopt labor-saving technologies.