We first find simple characterizations of $rac{1}{N} mathbb{Z}$-invariance of arbitrary principal shift-invariant space $V(phi)$. Then we find several equivalent conditions for $V(phi)$ to admit periodic oversampling for a class of continuous frame generators $phi$. In particular, when $phi$ is band-limited and $hat{phi}$ is piecewise continuous, we find very simple and general sufficient conditions for $V(phi)$ to admit periodic oversampling, which involve the extra invariance of $V(phi)$, together with an illustrating example.

This paper proposes homogeneous neural networks (HNNs), in which each neuron has identical weights. HNNs can realize shift-invariant associative memory, that is, HNNs can associate not only a memorized pattern but also its shifted ones. The transition property of HNNs is analyzed by the statistical method. We show the probability that each neuron outputs correctly and the error-correcting ability. Further, we show that HNNs cannot memorize over the number,, of patterns, where m is the number of neurons and k is the order of connections.

For fixed initial and destination states (i.e., markings), M0 and Md, there exist generally infinite firing count vectors in a Petri net. In this letter, it is shown that all fundamental particular solutions as well as all minimal T-invariants w.r.t. firing count vectors are needed to express an arbitrary firing count vector for the fixed M0 and Md. An algorithm for finding a special firing count vector which is expressed by using the only one specified fundamental particular solution is also given.

This paper proposes a public-key cryptography by applying RSA and Petri nets. We introduce RSA and a Petri net based private-key cryptography and then taking the advantages of these two cryptography, we propose a new public-key cryptography, PNPKC. To compare with RSA on security as well as computation order, we do simulation experiments. As the results, the security of PNPKC is as strong as RSA cryptography, and the encryption and decryption of PNPKC are in average 239 times as fast as RSA cryptography from our experiments. Besides, to see if our current PNPKC program can be practically used, we do comparative experiment with PGP, which shows PNPKC takes computation time in average as much as 36 times of PGP cryptography. That means our PNPKC program still needs to be technically improved.

This paper describes a modeling of the cryptography based on a concept of Petri nets. Movement of tokens in the net model shows a dynamic behavior of systems. On the other hand, the cryptography is considered as a bit operation, so that we can point out a common property between the net structure and the cryptography, which provides our idea that movement of tokens of the net model corresponds to a bitoperation of the cryptography. Some effective keys in the net model are considered by means of the net elements, which are based on T-invariant and net structures. It is shown that the keys of the net structured cryptography provide reasonable strength comparing with the data encryption standard (DES).