Iterative Construction of Fixed Points for Operators Endowed with Condition E in Metric Spaces

We consider the class of mappings endowed with the condition E in a nonlinear domain called 2-uniformly convex hyperbolic space. We provide some strong and Δ -convergence theorems for this class of mappings under the Agarwal iterative process. In order to support the main outcome, we procure an example of mappings endowed with the condition E and prove that its Agarwal iterative process is more effective than Mann and Ishikawa iterative processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT( κ ) spaces. This approach essentially provides a new setting for researchers who are working on the iterative procedures in fixed point theory and applications.

An area efficient and high throughput implementation of layered min-sum iterative construction a posteriori probability LDPC decoder

Area efficient and high speed forward error correcting codes decoder are the demand of many high speed next generation communication standards. This paper explores a low complexity decoding algorithm of low density parity check codes, called the min-sum iterative construction a posteriori probability (MS-IC-APP), for this purpose. We performed the error performance analysis of MS-IC-APP for a (648,1296) regular QC-LDPC code and proposed an area and throughput optimized hardware implementation of MS-IC-APP. We proposed to use the layered scheduling of MS-IC-APP and performed other optimizations at architecture level to reduce the area and to increase the throughput of the decoder. Synthesis results show 6.95 times less area and 4 times high throughput as compared to the standard min-sum decoder. The area and throughput are also comparable to the improved variants of hard-decision bit-flipping (BF) decoders, whereas, the simulation results show a coding gain of 2.5 over the best implementation of BF decoder in terms of error performance.

Lightweight Iterative MDS Matrices: How Small Can We Go?

As perfect building blocks for the diffusion layers of many symmetric-key primitives, the construction of MDS matrices with lightweight circuits has received much attention from the symmetric-key community. One promising way of realizing low-cost MDS matrices is based on the iterative construction: a low-cost matrix becomes MDS after rising it to a certain power. To be more specific, if At is MDS, then one can implement A instead of At to achieve the MDS property at the expense of an increased latency with t clock cycles. In this work, we identify the exact lower bound of the number of nonzero blocks for a 4 × 4 block matrix to be potentially iterative-MDS. Subsequently, we show that the theoretically lightest 4 × 4 iterative MDS block matrix (whose entries or blocks are 4 × 4 binary matrices) with minimal nonzero blocks costs at least 3 XOR gates, and a concrete example achieving the 3-XOR bound is provided. Moreover, we prove that there is no hope for previous constructions (GFS, LFS, DSI, and spares DSI) to beat this bound. Since the circuit latency is another important factor, we also consider the lower bound of the number of iterations for certain iterative MDS matrices. Guided by these bounds and based on the ideas employed to identify them, we explore the design space of lightweight iterative MDS matrices with other dimensions and report on improved results. Whenever we are unable to find better results, we try to determine the bound of the optimal solution. As a result, the optimality of some previous results is proved.

Synthesis of α,α’-linked penta- and septaheterocycles by tandem Suzuki coupling

The controlled synthesis of oligoaromatics can provide materials of wide utility. Here, we describe the preparation of higher order oligoheterocycles via a tandem Suzuki cross-coupling protocol. This method has allowed for the iterative construction of fluorescent [Formula: see text],[Formula: see text]-linked penta- and septaheterocyclic systems with modification of the terminal moiety allowing for fine-tuning of the emission features.