On the theory of fugoid motions

In 1891 Zhukovslii in his paper “On soaring of birds” [1] solved the problem of the motion of a body of high lift — drag ratio in an atmosphere of constant density. In [2] this problem was considered in greater detail, but the basic assumption of a constant density was made here as well. There have recently appeared numerous papers concerning the analytical solution of the problem of entry into the atmosphere with orbital and escape velocities [3 to 5]. But these studies were concerned primarily with the problems of ballistic entry and entry with low lift — drag ratio. In considering oscillatory states, the authors limited their treatment to small angles between the trajectory and local horizon. In the present paper we consider the problem without imposing any limitations on the slope of the trajectory or initial velocity. The case examined will be that of a hypothetical glider spacecraft of sufficiently high lift — drag ratio. It is interesting to note that the solution of this problem reduces to the solution of Zhukovskii's problem, but for an atmosphere of variable density. The associated trajectories are termed “fugoid”. All of our assumptions about the parameters of such a glider are of a particular hypothetical character.     

Advances in the study of elementary cellular automata regular language complexity

Cellular automata (CA) are discrete dynamical systems that, out of the fully local action of its state transition rule, are capable of generating a multitude of global patterns, from the trivial to the arbitrarily complex ones. The set of global configurations that can be obtained by iterating a one‐dimensional cellular automaton for a finite number of times can always be described by a regular language. The size of the minimum finite automaton corresponding to such a language at a given time step provides a complexity measure of the underlying rule. Here, we study the time evolution of elementary CA, in terms of such a regular language complexity. We review and expand the original results on the topic, describe an alternative method for generating the subsequent finite automata in time, and provide a method to analyze and detect patterns in the complexity growth of the rules. © 2015 Wiley Periodicals, Inc. Complexity 21: 267–279, 2016     

Exact enumeration of acyclic deterministic automata

A linear recurrence relation is derived for the number of unlabelled initially connected acyclic automata. The coefficients of this relation are determined by another, alternating, recurrence relation. The latter determines, in particular, the number of acyclic automata with labelled states. Certain simple enumerative techniques developed by the author for counting initially connected automata and acyclic digraphs are combined and applied. Calculations show that the results obtained in this paper improve recent upper bounds for the number of minimal deterministic automata (with accepting states) recognizing finite languages. Various related questions are also discussed.     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Cellular automata study of high burn-up structures

We propose a new approach, based on the cellular automata, for processing and modeling the structure dynamics of UO₂ at different cross-section averaged burn-ups. Micrographs of the material surface, subjected to both “as-polished” and “as-etched” treatments, with a magnification of 1250× have been used in our study. It has been shown that this approach provides efficient tools for investigation of the surface structure dynamics both at local and global levels.     

Using the wigner function for quantum transport in device simulation

The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the cost of dealing with generalized functions in place of proper functions.     

Impact of memory size on graph exploration capability

A mobile agent (robot), modeled as a finite automaton, has to visit all nodes of a regular graph. How does the memory size of the agent (the number of states of the automaton) influence its exploration capability? In particular, does every increase of the memory size enable an agent to explore more graphs? We give a partial answer to this problem by showing that a strict gain of the exploration power can be obtained by a polynomial increase of the number of states. We also show that, for automata with few states, the increase of memory by even one state results in the capability of exploring more graphs.     

无穷自动机的几个等价命题

本文证明了具有无穷栈符合的实时确定下推自动机与无穷自动机的等价性，并且将有限状态自动机的Ｍｙｈｉｌｌ－Ｎｅｒｏｄｅ定理推广到了无穷自动机和具有无穷栈符号的实时确定下推自动机。     

Nonlinear observer for autonomous switching systems with jumps

This work deals with nonlinear observer synthesis for a particular class of Hybrid Dynamic Systems (HDS): autonomous switching systems with jumps. The jumps can result from the system’s dynamics or from the diffeomorphism which makes it possible to lead the system to an observability canonical form. The contribution of this work relates to the design of a second order sliding mode based observer (“Super Twisting Algorithm”). It allows estimating both continuous and discrete states related to the system active dynamic. On the other hand these observers ensure a finite time convergence of the estimation error.     

The Lamplighter Group ℤ3≀ℤ Generated by a Bireversible Automaton

We construct a bireversible self-dual automaton with three states over an alphabet with three letters which generates the lamplighter group ?₃??. In particular, this fact shows that not all groups defined by birevirsible automata are finitely presented.     

Decidability of the word problem in Yamamura’s HNN extensions of finite inverse semigroups

We prove that the word problem is decidable in Yamamura’s HNN extensions of finite inverse semigroups, by providing an iterative construction of approximate automata of the Schützenberger automata of words relative to the standard presentation of Yamamura’s HNN-extensions. This work was done in the framework of the group GNSAGA of INDAM, the PRIN “Automi e Linguaggi Formali: aspetti matematici e applicativi” and the ESF project Automatha.     

Spherical rarefaction flow in the neighbourhood of a reflection point of a “boundary” characteristic

The structure of the one-dimensional steady spherically symmetric rarefaction flow of an ideal (inviscid and non-heat-conducting) gas in the neighbourhood of a reflection point of a “boundary” C⁻-characteristic is investigated in principal order. The “boundary” C⁻-characteristic separates the gas at rest from the flow due to the outward motion of a piston which confines the gas. In the rt plane, where r is the distance from the centre of symmetry and t is the time, the reflection point, which coincides with the point of arrival on the t axis of the boundary characteristic, coincides with the origin of coordinates. The initial velocity of the piston may be zero (for positive acceleration) or finite. When two symmetrical plane pistons advance, the “derived” derivatives of all the flow parameters on the C⁻-characteristic at the origin of coordinates, which in this case lies on the plane of symmetry, are finite. When a cylindrical and spherical piston advance, the derived derivative of the pressure (velocity) of the gas on the C⁻-characteristic at the origin of coordinates becomes minus (plus)-infinity although without intersecting characteristics of the same family [1–4].     

Interval orders based on weak orders

One definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “ ”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “ ”, a six-element crown or a six-element fence.     

The Steiner ratio of several discrete metric spaces

Steiner's Problem is the “Problem of shortest connectivity”, that means, given a finite set of points in a metric space (X,ρ), search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST). Steiner's Problem is very hard as well in combinatorial as in computational sense, but, on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees:

which is called the Steiner ratio (of (X,ρ)). We look for estimates and exact values for the Steiner ratio in several discrete metric spaces. Particularly, we determine the Steiner ratio for spaces of words, and we estimate the Steiner ratio for specific graphs.     

Random majority percolation

We shall consider the discrete time synchronous random majority‐vote cellular automata on the n by n torus, in which every vertex is in one of two states and, at each time step t, every vertex goes into the state the majority of its neighbors had at time t ‐ 1 with a small chance p of error independently of all other events. We shall show that, if n is fixed and p is sufficiently small, then the process spends almost half of its time in each of two configurations. Further more, we show that the expected time for it to reach one of these configurations from the other is Θ(1/pⁿ⁺¹) despite the actual time spent in transit being O(1/p³). Unusually, it appears difficult to obtain any results for this regime by simulation. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Array approach to fuzzy logic

Promising results from applying an array-based approach to two-valued logic suggests its application to fuzzy logic. The idea is to limit the domain of truth-values to a discrete, finite domain, such that a logical relationship can be evaluated by an exhaustive test of all possible combinations of truth-values. The paper presents a study of the topic from an engineer's viewpoint. As an example 31 logical sentences valid in two-valued logic were tested in three-valued logic using the nested interactive array language, Nial. Out of these, 24 turned out to be valid in a three-valued extension based on the well-known S^* implication operator, also called “Gödel's implication operator”. Applications to automated approximate reasoning and fuzzy control are also illustrated.     

Automata and Square Complexes

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedońska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.     

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the “best” choice for an initial stepsize, as well as developing effective strategies for stepsize control—the same, of course, must be carried out in the stochastic case.

In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge–Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.