Additive Hermitian idempotent preservers between operator algebras

Let L be an additive map between (real or complex) matrix algebras sending $n\times n$ Hermitian idempotent matrices to $m\times m$ Hermitian idempotent matrices. We show that there are nonnegative integers $p,q$ with $n(p+q)=r\le m$ and an $m\times m$ unitary matrix U such that$L(A)=U[(I_p?A)\oplus (I_q?A^t)\oplus 0_{m?r}]U^{?},\text{for any}n\times n\text{Hermitian}A\text{with rational trace}.$ We also extend this result to the (complex) von Neumann algebra setting, and provide a supplement to the Dye-Bunce-Wright Theorem asserting that every additive map of Hermitian idempotents extends to a Jordan ?-homomorphism.     

On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

In this work, we prove the existence of positive solution for the following class of problems where λ>0 and is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C¹ functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V^?1({0})).     

Some extensions of the operator entropy type inequalities

In this paper, we define the generalised relative operator entropy and investigate some of its properties such as subadditivity and homogeneity. As application of our result, we obtain the information inequality. In continuation, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond–Pe?ari? method.     

Explosive synchronization in a mobile network in the presence of a positive feedback mechanism

Synchronization is a process that describes the coherent dynamics of a large ensemble of interacting units.The study of explosive synchronization transition attracts considerable attention.Here,I report the explosive transition within the framework of a mobile network,while each oscillator is controlled by global-order parameters of the system.Using numerical simulation,I find that the explosive synchronization(ES)transition behavior can be controlled by simply adjusting the fraction of controlled oscillators.The influences of some parameters on explosive synchronization are studied.Moreover,due to the presence of the positive feedback mechanism,I prevent the occurrence of the synchronization of continuous-phase transition and make phase transition of the system a first-order phase transition accompanied by a hysteresis loop.     

Equivalences among Z2s-linear Hadamard codes

The ${\mathbb{Z}}_{2^s}$-additive codes are subgroups of ${\mathbb{Z}}_{2^s}^n$, and can be seen as a generalization of linear codes over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$. A ${\mathbb{Z}}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ${\mathbb{Z}}_{2^s}$-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some ${\mathbb{Z}}_{2^s}$-linear Hadamard codes of length $2^t$ are equivalent, once $t$ is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to $t=11$, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on $s\in \{2,3\}$, the full classification of the ${\mathbb{Z}}_{2^s}$-linear Hadamard codes of length $2^t$ is established by giving the exact number of such codes.     

A novel chaotic image encryption scheme using DNA sequence operations

In this paper, we propose a novel image encryption scheme based on DNA (Deoxyribonucleic acid) sequence operations and chaotic system. Firstly, we perform bitwise exclusive OR operation on the pixels of the plain image using the pseudorandom sequences produced by the spatiotemporal chaos system, i.e., CML (coupled map lattice). Secondly, a DNA matrix is obtained by encoding the confused image using a kind of DNA encoding rule. Then we generate the new initial conditions of the CML according to this DNA matrix and the previous initial conditions, which can make the encryption result closely depend on every pixel of the plain image. Thirdly, the rows and columns of the DNA matrix are permuted. Then, the permuted DNA matrix is confused once again. At last, after decoding the confused DNA matrix using a kind of DNA decoding rule, we obtain the ciphered image. Experimental results and theoretical analysis show that the scheme is able to resist various attacks, so it has extraordinarily high security.