There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of D-finite generating functions, a class which contains algebraic. We consider several different t...

In this paper we consider relations between the operation of word insertion and primitivity. A necessary and sufficient condition under which the insertion u ← u of the word u into itself has maximum cardinality is obtained. The notion of insertion sequence is introduced and sufficient conditions under which an insertion sequence is a special type of language (regular, context-free, biprefix co...

The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρk,n : i→ ki (mod kn + 1), i ∈ {1,2, . . . ,kn}. We uncover the cycle structu...

Shuffle operation on trajectories is useful in modeling parallel composition of words and languages. In this work, a new class of P systems with shuffle operation and catalytic-like rules is presented. Such a system has a membrane structure, where language-objects and shuffleoperation rules are placed in its regions. It can be used as a language generator. In this study, we propose a variant P ...

We show that shuffle, the problem of determining whether a string w can be composed from an order preserving shuffle of strings x and y, is not in AC, but it is in AC. The fact that shuffle is not in AC is shown by a reduction of parity to shuffle and invoking the seminal result of Furst et al., while the fact that it is in AC is implicit in the results of Mansfield. Together, the two results p...

In TCC 2007, Adida and Wikström proposed a novel approach to shuffle, called a public shuffle, in which a shuffler can perform shuffle publicly without needing information kept secret. Their scheme uses an encrypted permutation matrix to shuffle ciphertexts publicly. This approach significantly reduces the cost of constructing a mix-net to verifiable joint decryption. Though their method is suc...

Human Interferon β (INF-β) is a member of cytokines family which different studies have shown its immunomodulatory and antiviral activities. In this study an expression vector was designed and constructed for expression of human INF-β-1b either in shake flasks and bench top bioreactor. The designed vector was constructed based upon pET-25b(+) with T7 promoter. Recombinant human beta interferon ...

In [6] it was shown that shuffle languages are contained in one-way-NSPACE(log n) and in P. In this paper we show that nondeterministic one-way logarithmic space is in some sense the lower bound for accepting shuffle languages. Namely, we show that there exists a shuffle language which is not accepted by any deterministic one-way Turing machine with space bounded by a sublinear function, and th...

1. Review of the previous lecture This section highlights the problem of routing a hard problem on the butterfly network. 2. Valiant’s Algorithm This section describes the idea behind Valiant’s algorithm for solving a hard problem on the butterfly network. 3. Ranade’s Algorithm This section briefly introduces Ranade’s algorithm. 4. Shuffle-exchange graph This section explains the properties of ...

Efficient implementation of bit-level matrix transpose operations using a new family of perfect shuffle instructions in a word-based micro-processor architecture is investigated. Using a tensor representation of the perfect shuffle index mapping, we show how bit matrix of arbitrary size may be transposed using a minimal sequence of shuffle and pack instructions, and prove the correctness of the...