partial frames provide a rich context in which to do pointfree structured and unstructured topology. a small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. these axioms are sufficiently general to include as examples bounded distributive lattices, $sigma$-frames, $ka...

this paper is a continuation of [uniformities and covering properties for partial frames (i)], in which we make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. after presenting there our axiomatization of partial frames, which we call $sels$-frames, we added structure, in th...

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

We design alternative dual frames for linearly reconstructing signals from Sigma-Delta (Σ∆) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order Sigma-Delta quantizer produces approximations where the approximation error is at most of order 1/λ , and λ > 1 is the oversampling ratio. We show that the counterpart...

In transmission, storaging and coding of digital signals we frequently perform A/D conversion using quantization. In this paper we study the maxiaml and mean square errors as a result of quantization. We focus on the sigma-delta modulation quantization scheme in the finite frame expansion setting. We show that this problem is related to the classical Traveling Salesman Problem (TSP) in the Eucl...

In this paper, we study the performance of the PCM scheme with linear quantization rule for quantizing finite unit-norm tight frame expansions for R and derive the PCM quantization error without the White Noise Hypothesis. We prove that for the class of unit norm tight frames derived from uniform frame paths the quantization error has an upper bound of O(δ) regardless of the frame redundancy. T...

The expression of sll1689, an open reading frame from the cyanobacterium Synechocystis sp. strain PCC 6803 putatively encoding a member of the sigma(70) family of sigma factors, appears to be regulated by the nitrogen control transcription factor NtcA. Disruption of sll1689 had no noticeable effect on exponential growth, identifying its product as a member of the group 2, nonessential class of ...