In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.

Past few years have seen an extensive use of RAM chips with wide I/O data (e.g. 16, 32, 64 bits) in computer memory systems. These chips are highly vulnerable to a special type of byte error, called an m-spotty byte error, which can be effectively detected or corrected using byte errorcontrol codes. The MacWilliams identity provides the relationship between the weight distribution of a code and...

we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

In this paper, we give a matrix construction method for designing DNA codes that come from group rings. We show with our one can obtain reversible $$G^k$$ -codes of length kn, where $$k, n \in \mathbb {N},$$ over the finite commutative Frobenius ring R. employ to many $$\mathbb {F}_4$$ satisfy Hamming distance, reverse, reverse-complement and fixed GC-content constraints. Moreover, improve lowe...

The AKS conjecture states that (ζ − 1) ≡ ζ − 1 (mod n) for ζ an r-th root of unity with r prime, only if n is a prime and r n − 1. If n is a prime, n-th power map would be a Frobenius automorphism of Fn(ζ), and thus the congruence will hold true. In general, n-th power map is not a field automorphism, but under suitable conditions, it can be a group automorphism of the multiplicative group of c...

<p style='text-indent:20px;'>In this work, we define a modification of bordered construction for self-dual codes which utilises <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-circulant matrices. We provide the necessary conditions to produce over finite commutative Frobenius rings characteristic 2. Using modi...

The classes of clean and nil-clean rings are closed with respect standard constructions as direct products and (triangular) matrix rings, cf. [12] resp. [4], while the classes of weakly (nil-)clean rings are not closed under these constructions. Moreover, while all matrix rings over fields are clean, [12] when we consider nil-clean rings there are strongly restrictions: if a matrix ring over a ...

We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.