Some remarks on progress with finite elements

Information is the most valuable but least valued tool that professionals have. The amount of data in science and technology grows so rapidly that broad-coverage compilations cannot be maintained but concentrate on the coverage of specialized topics. The volume of finite element literature in the form of books, conference proceedings and journal papers, as well as a number of developed finite element codes, has been growing at a prodigious rate. It is almost impossible to be up to date with all the relevant information. A bibliometric study is presented; the author takes the number of published papers on finite elements as a measure of the research activity in the field of finite element techniques and investigates some engineering fields/topics where these techniques have been/are used.     

Some subsets of automata over a finite ring

Sets of Mealy and Moore automata over an arbitrary finite commutative-associative ring are investigated in which transition and output functions are linear combinations of functions of automaton states and functions of inputs. Subsets of strongly connected, reduced, and reversible automata and automata with permutation transition functions are characterized.     

Some remarks on the optimal choice of finite element grids

The results of Prager's Note [1]of similar title are discussed and expanded. By minimizing the potential energy associated with an assumed approximate deformation field, general criteria are developed for the optimal grid spacing of bars under varying axial forces and of beams under lateral loads, both with arbitrary variation of the cross section. It is shown that Prager's tentative energy criterion is indeed confined to the special case treated in his Note, as originally conjectured by himself.For constant cross section all optimality conditions assume a simple and easily realizable form. It is shown that in this case the computed nodal displacements are exact, irrespective of the grid spacing or of the value of the potential energy. Nevertheless, optimal spacing is still useful in reducing the average error in the computed stress values.     

Some algebraic properties of measure-once two-way quantum finite automata

Quantum finite automata (QFA) can be divided into four kinds depend upon the head-directions and the measure times. They are measure-once one way QFA (MO-1QFA) introduced by Moore and Crutchfield (Theor Comput Sci 237: 275–306, 2000); measure-many one way QFA (MM-1QFA) and measure-many two-way QFA (MM-2QFA) introduced by Kondacs and Watrous (Proceedings of the 38th IEEE annual symposium on 433 foundations of computer science, 66–75, 1997); and measure-once two-way QFA (MO-2QFA) which were not given until now. The purpose of this work is mainly to discuss one kind of QFA, which is called MO-2QFA for brief. First of all, the definition of MO-2QFA is given and the conditions for preserving unitary properties are shown. Then, we analysis the basic algebraic properties of the class of languages which can be recognized by MO-2QFA, such as the union, intersection, complement and reversal operations. As well, we consider the catenation operation on the class of quantum languages recognized by MO-2QFA is closed in the generalized conditions.      

Controlled finite automata

This paper discusses finite automata regulated by control languages over their states and transition rules. It proves that under both regulations, regular-controlled finite automata and context-free-controlled finite automata characterize the family of regular languages and the family of context-free languages, respectively. It also establishes conditions under which any state-controlled finite automaton can be turned into an equivalent transition-controlled finite automaton and vice versa. The paper also demonstrates a close relation between these automata and programmed grammars. Indeed, it proves that finite automata controlled by languages generated by propagating programmed grammars with appearance checking are computationally complete. In fact, it demonstrates that this computational completeness holds even in terms of these automata with a reduced number of states.     

Some remarks on the discrete maximum-principle for finite elements of higher order

The discrete maximum principle for finite element approximations of standard elliptic problems in the plane is discussed. Even in the case Δu=0 a slightly stronger version of the principle does not hold with piecewise quadratic elements for all but some very special triangularisation geometries.     

Independent finite automata on Cayley graphs

In the setting of symbolic dynamics on discrete finitely generated infinite groups, we define a model of finite automata with multiple independent heads that walk on Cayley graphs, called group-walking automata, and use it to define subshifts. We characterize the torsion groups (also known as periodic groups) as those on which the group-walking automata are strictly weaker than Turing machines, and those on which the head hierarchy is infinite.     

A note on three-dimensional finite automata

This paper investigates the relationships between the accepting powers of three-dimensional six-way finite automata (3-FA's) and three-dimensional five-way Turing machines (5WTM's), where the input tapes of these automata are restricted to cubic ones. A 3-FA (5WTM) can be considered as a natural extension of the two-dimensional four-way finite automaton (two-dimensional three-way Turing machine) to three dimensions. The main results are: (1) n²logn (n³) space is necessary and sufficient for deterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's; (2) n² (n²) space is necessary and sufficient for nondeterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's.     

Complexity of control on finite automata

We consider control questions for finite automata viewed as input/output systems. In particular, we find estimates of the minimal number of states of an automaton able to control a given automaton. We prove that, on average, feedback closed-loop control automata do not have fewer states than open-loop control automata when the control objective is to steer the controlled automaton to a target state. We compare our approach to other ways of formalizing of formalizing analogous control objectives.     

Alternating finite automata on ω-words

Alternating finite automata on ω-words are introduced as an extension of nondeterministic finite automata which process infinite sequences of symbols. The classes of ω-languages defined by alternating finite automata are investigated and characterized under four types of acceptance conditions. It is shown that for one type of acceptance condition alternation increases the power in comparison with nondeterminism and for other three acceptance conditions nondeterministic finite automata on ω-words have the same power as alternating ones.     

Path-bounded three-dimensional finite automata

The comparative study of the computational powers of deterministic and nondeterministic computations is one of the central tasks in complexity theory. This paper investigates the computational power of nondeterministic computing devices with restricted nondeterminism. There are only few results measuring the computational power of restricted nondeterminism. In general, there are three possibilities to measure the amount of nondeterminism in computation. In this paper, we consider the possibility to count the number of different nondeterministic computation paths on any input. In particular, we deal with five-way three-dimensional finite automata with multiple input heads operating on three-dimensional input tapes. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008