Waldemar Hołubowski

There are several graphs naturally associated with rings. The unitary Cayley graph of a ring R is the graph with vertex set R, where two elements x,y∈R are adjacent if and only if x−y is a unit of R. We show that the unitary Cayley graph CTn(F) of the ring Tn(F) of all upper-triangular matrices over a finite field F is isomorphic to a semistrong product of a complete graph and the antipodal graph of a Hamming graph. In particular, when |F|=2, the graph CTn(F) has a highly symmetric structure: it is the union of 2^{n−1} complete bipartite graphs. Moreover, we prove that the clique number and the chromatic number of C_Tn(F) are both equal to |F|, and we establish tight upper and lower bounds for the domination number of C_Tn(F).

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

Data transmission is the most critical operation for mobile sensors networks in term of energy waste. Particularly in pervasive healthcare sensors network it is paramount to preserve the quality of service also by means of energy saving policies. Communication and data transmission are among the most critical operation for such devises in term of energy waste. In this paper we present a novel approach to increase battery life-span by means of shorter transmission due to data compression. On the other hand, since this latter operation has a non-neglectable energy cost, we developed a compression efficiency estimator based on the evaluation of the absolute and relative entropy. Such algorithm provides us with a fast mean for the evaluation of data compressibility. Since mobile wireless sensor networks are prone to battery discharge-related problems, such an evaluation can be used to improve the electrical efficiency of data communication. In facts the developed technique, due to its independence from the string or file length, is extremely robust both for small and big data files, as well as to evaluate whether or not to compress data before transmission. Since the proposed solution provides a quantitative analysis of the source’s entropy and the related statistics, it has been implemented as a preprocessing step before transmission. A dynamic threshold defines whether or not to invoke a compression subroutine. Such a subroutine should be expected to greatly reduce the transmission length. On the other hand a data compression algorithm should be used only when the energy gain of the reduced transmission time is presumably greater than the energy used to run the compression software. In this paper we developed an automatic evaluation system in order to optimize the data transmission in mobile sensor networks, by compressing data only when this action is presumed to be energetically efficient. We tested the proposed algorithm by using the Canterbury Corpus as well as standard pictorial data as benchmark test. The implemented system has been proven to be time-inexpensive with respect to a compression algorithm. Finally the computational complexity of the proposed approach is virtually neglectable with respect to the compression and transmission routines themselves.

We solve an old Sus̆kevic̆’s problem on right zero divisors in the ring T(∞,K) of infinite ℕ×ℕ upper triangular matrices over a field K. In the solution we use a notion of strong linear independence.

Let