a b s t r a c tThe palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).
We prove that, in general, a given two-dimensional inhomogeneous potential V(z,y) does not allow for the creation of homogeneous families of orbits. Yet, depending on the case at hand, if the given potential satisfies certain conditions, this potential is compatible either with one (or two) monoparametric homogeneous families of orbits or at most with five such families. The orbits are then found on the grounds of the given potential. Key words: inhomogeneous potentials -homogeneous orbitsA A A subject classification: 042 I n t r o d u c t i o nThe two-dimensional inverse problem of dynamics seeks all the potentials V(x,y) which can give rise to a preassigned monoparametric family of curves f(z, y) = c, traced by a unit mass material point. If the total energy dependence E = E ( f ( z , y)) is not given in advance, the connection between orbits and potentials is established by a partial differential equation of the second order (Bozis 1984). The equation is linear in V(z, y), of the hyperbolic type with coefficients depending merely on the given orbits. The above equation, if rearranged adequately, can also serve to face the direct problem, i.e. given a potential to seek all monoparametric families which can be created by this potential, for adequate initial conditions, of course.Indeed, it turns out that for a function y(x, y) = fy/jz, related to the slope of the given orbits, the second order partial equation is now nonlinear in the unknown function y(z, y). The direct problem then requires the solution of a harder to solve differential equation.In the framework of the direct problem, one expects that additional information regarding the orbits will generally facilitate its solution. Such information is e.g. the homogeneity of the family of orbits, i.e., the property of the family to include geometrically similar orbits. The case of having homogeneous families produced by homogeneous potentials has been studied by Bozis and Stefiades (1993) and Bozis and Grigoriadou (1993) and led to an ordinary differential equation.In the present paper we study the following version of the direct problem: in the system of Cartesian coordinates O x y , a purely inhomogeneous potential V is given. Are there any homogeneous families of orbits satisfying the system of differential equations ii=-v,, y=-vy, i.e., created by this potential?For an affirmative answer we find that: (i) In general, certain conditions have to be satisfied by the given potential. In this case there exist no more than two homogeneous monoparametric families of orbits consistent with the given potential. They correspond to the common roots of a quadratic and a quintic algebraic equation.
The particular version of the inverse problem of dynamics considered here is: given the 'slope function' γ = f y /f x , representing uniquely a family of planar curves f (x, y) = c, find, if possible, potentials of the form V (x, y) = v(γ (x, y)) which give rise to this family. Such potentials V will then have as equipotential curves the isoclinic curves γ = const of the family f (x, y) = c. We show that, for the problem of admitting a solution, a necessary and sufficient condition must be satisfied by the given γ (x, y). Inferring by reasoning from particular to more general forms, we find analytically a very rich set of slope functions γ (x, y) satisfying this condition. In contrast to the (not always solvable) general case V = V (x, y), in all these cases we can find the potential v = v(γ ) analytically by quadratures. Several examples of pairs (γ , v(γ )) are presented.
The version of the inverse problem of dynamics considered here is: given a family of planar curves f (x, y) = c, find the potentials V (x, y) which give rise to this family. Its solution is based on two linear partial differential equations satisfied by V : one of first order, containing the total energy function E(f ), given by Szebehely in 1974, and the other one of second order, derived by Bozis in 1984 by eliminating the energy from Szebehely's equation. In this paper, Bozis' partial differential equation is obtained directly by eliminating the time derivatives of x(t) and y(t) up to the third order between seven differential relations based on the equations of motion and on the given family. Szebehely's equation is then derived as a consequence. This shows the importance of Bozis' equation, which is traditionally considered as following from Szebehely's one. The connection with the nonconservative case is emphasized.
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