Consider a random source which evolves on a finite set. It follows from existing literature, see for example [1] and [2] (Pages 491-500, in particular, see Definition (9.8.3) and Theorem 9.8.3 for achievability), that the limit of the normalized rate-distortion functions of block-independent approximations of a a stationary, ergodic source is equal to the rate-distortion function of the source....

The zero-temperature XX chain is studied with emphasis on the properties of a block of L spins inside the chain. We investigate the quantum fluctuations resulting from the entanglement of the block with the rest of the chain using analytical as well as numerical (density matrix renormalization group) methods. It is found that the rest of the chain acts as a thermal environment and an effective ...

The zero-temperature XX chain is studied with emphasis on the properties of a block of L spins inside the chain. We investigate the quantum fluctuations resulting from the entanglement of the block with the rest of the chain using analytical as well as numerical (density matrix renormalization group) methods. It is found that the rest of the chain acts as a thermal environment and an effective ...

The DNA-templated polymerization of synthetic building blocks provides a potential route to the laboratory evolution of sequence-defined polymers with structures and properties not necessarily limited to those of natural biopolymers. We previously reported the efficient and sequence-specific DNA-templated polymerization of peptide nucleic acid (PNA) aldehydes. Here, we report the enzyme-free, D...

Mining is the process of adding blocks of transaction to the Bitcoin’s block chain. The block chain serves to confirm transactions in the network. Nodes use the block chain to verify the legality of transactions. Each block contains a proof of work to be considered valid. This proof of work is verified by other nodes in the network. Mining is designed to be difficult so that the frequency of bl...