The local boundedness of the perturbed motions of a gyroscope in gimbals with dissipative and accelerating forces

The motion of an unbalanced gyroscope in gimbals in a central Newtonian field of forces is considered, taking the masses of the suspension rings into account. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings, and there is an accelerating (electromagnetic) moment applied to the axis of rotation axis of the other ring. The equations of motion have a partial solution such that the mean velocity of the outer ring is perpendicular to the direction from the centre of gravitation S to the stationary point O, the middle plane of the inner ring contains this direction, and the gyroscope rotates about SO with an arbitrary constant angular velocity.     

Multifrequency oscillations of singularly perturbed systems

We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i.e., all of its eigenvalues have nonzero real parts.     

Sperner''s theorem with constraints

A set X of subsets of an n-element set S is called an anti-chain if no two elements of X are related by set-wise inclusion. Sperner showed [8] that max |X|=(ⁿ_[n/2]), where |X| denotes the number of elements in X and the maximum is taken over all anti-chains of subsets of S.

Let non-negative integers i_o<n and m_io≠0, m_io+1,…m_n be given. In this paper we give an algorithm for calculating max |X| where the maximum is taken only over anti-chains containing exactly m_i i-element subsets of S for i_o i n.     

Instabilities of robot motion

Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let X denote the space of all admissible configurations of a mechanical system. A motion planner is given by a splitting X×X=F₁F₂F_k (where F₁,…,F_k are pairwise disjoint ENRs, see below) and by continuous maps s_j :F_j→PX, such that Es_j=1_Fj. Here PX denotes the space of all continuous paths in X (admissible motions of the system) and E :PX→X×X denotes the map which assigns to a path the pair of its initial–end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets F_j in any motion planner in X. We also introduce a new notion of order of instability of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space X. We study a number of specific problems: motion of a rigid body in R³, a robot arm, motion in R³ in the presence of obstacles, and others.     

Explicit formulas for the Dumont-Foata polynomial

Dumont and Foata have defined a polynomial F_n(x, y, z) recursively. They proved that F_n(x, y, z) is symmetric in x, y, z and that F_n(1, 1, 1) = G_2n+2 the Genocchi number. Moreover, they gave an elegant combinatorial interpretation for the coefficients of F_n(x, y, z). In the present paper explicit formulas and generating functions for F_n(x, y, z) are obtained.     

On the convergence of parallel chaotic nonlinear multisplitting Newton-type methods

A class of parallel chaotic nonlinear multisplitting Newton-type methods for solving the nonlinear system of equations F(x) = 0(F : D Rⁿ → Rⁿ) is established and its local convergence theory is presented.     

The Hess case in rigid-body dynamics

Generalizations of the Hess integral are presented for different forms of the equations of motions of rigid body. The general conditions for the existence of this integral, which is due to the presence of additional explicit symmetries of the equations of motion, are pointed out. Problems of reducing the order, of the explicit integration and the qualitative analysis of the motion of a rigid body subject to these conditions are considered. Analogues of Hess cases for a gyroscope in gimbals and the Chaplygin equations describing the fall of a rigid body in a fluid are indicated for the first time.     

Quadratic expansions of spectral functions

A function, F, on the space of n×n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A)=F(UAU^T) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on : those that are invariant under arbitrary swapping of their arguments. In this paper we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessian' of the spectral function. In the case of convex functions we show that a positive definite ‘Hessian' of f implies positive definiteness of the ‘Hessian' of F.     

Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U₁, …, U_m) of vector subspaces of W be uniformly distributed; and consider the statistics X_m,1 dim_F(∑_i=1^m U_i) and X_m,2 dim_F (∩_i=1^m U_i). If F is finite of cardinality q, we determine lim E(X_m,1^k), and lim E(X_m,2^k), and hence, lim var(X_m,1) and lim var(X_m,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.     

Polyhedral graphs with extreme numbers of types of faces

A face F of a polyhedral graph G(V,E,F) is an a₁,a₂,…,a_l-face if is an l-gon and the degrees d(x_i) of the vertices x_iV incident with in the cyclic order are a_i,i=1,2,…,l. The lexicographic minimum b₁,b₂,…,b_l such that is a b₁,b₂,…,b_l-face is the type of . All polyhedral graphs having only one type of faces are listed. It is proved that the set of triangulations having only faces of different types is non-empty and finite.     

Exponential estimates for singularly perturbed linear fuctional differential equations

Exponential estimates on the fundamental matrix, uniform on the perturbation parameter, are obtained for singularly perturbed systems of linear retarded functional differential equations, under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts. The exponential rates in the estimates are computable from upper bounds on the real parts of the characteristic values of the system or of associated simpler equations. Differences between differential-difference equations and equations with distributed delays are emphasized.     

Estimating a mixing distribution in a multiple observation setting

Given unobservable i.i.d. r.v.'s M_i with c.d.f. F, and observable r.v.'s Y_ij with E(Y_ij|M_i) = M_i; we study a nonparametric estimate of F based on the Y_ij's by conditioning and kernel techniques. This investigation extends previous works of Blischke (1965) and James (1978).     

Distributional chaos for triangular maps

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2^∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smítal [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φ_xy and Φ_xy^* generated by F such that Φ_xy^*≡1 and Φ_xy(0₊)<1, and no distribution functions Φ_uv, and Φ_uv^* such that Φ_uv^*≡1 and Φ_uv(t)=0 whenever 0<t<ε, for some ε>0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.     

Estimating the spectral measure of an extreme value distribution

Let (X₁, Y₁), (X₂, Y₂),…, (X_n, Y_n) be a random sample from a bivariate distribution function F which is in the domain of attraction of a bivariate extreme value distribution function G. This G is characterized by the extreme value indices and its spectral measure or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure. In this paper, we construct an empirical measure, based on the sample, which is a consistent estimator of the spectral measure. We also show for positive extreme value indices the asymptotic normality of the estimator under a suitable 2nd order strengthening of the bivariate domain of attraction condition.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

A characterization on n-critical economical generalized tic-tac-toe games

There is a so called generalized tic-tac-toe game playing on a finite set X with winning sets A₁, A₂,…, A_m. Two players, F and S, take in turn a previous untaken vertex of X, with F going first. The one who takes all the vertices of some winning set first wins the game. Erd s and Selfridge proved that if |A₁|=|A₂|==|A_m|=n and m<2ⁿ⁻¹, then the game is a draw. This result is best possible in the sense that once m=2ⁿ⁻¹, then there is a family A₁, A₂,…, A_m so that F can win. In this paper we characterize all those sets A₁,…, A_2n−1 so that F can win in exactly n moves. We also get similar result in the biased games.     

Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations

In this paper, stability and local bifurcation behaviors for a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plane excitations in the uniform thermal environment are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of Maple and normal form theory, the explicit expressions of transition curves are obtained, which may lead to static bifurcation, Hopf bifurcation and 2-D torus bifurcation. Finally, the numerical solutions obtained by using fourth-order Runge–Kutta method agree with the analytic predictions.     

Concerning two conjectures on the set of fixed points of a complete rotation of a Cayley digraph

In 1996, J.C. Bermond, T. Kodate, S. Perennes and N. Marlin conjectured that the set F_σ of fixed points of some complete rotation σ of the toroidal mesh TM(p)^k is not separating (that is F_σ does not disconnect TM(p)^k). They also conjectured that the set F_ω of fixed points of any complete rotation ω of any Cayley digraph is not separating. In this paper, we prove the first conjecture and disprove the second one.     

Existence results for doubly near resolvable (υ, 3, 2)-BIBDs

Let V be a set of υ elements. A (1, 2; 3, υ, 1)-frame F is a square array of side v which satisfies the following properties. We index the rows and columns of F with the elements of V, V={x₁,x₂,…,x_υ}. (1) Each cell is either empty or contains a 3-subset of V. (2) Cell (x_i, x_i) is empty for i=1, 2,…, υ. (3) Row x_i of F contains each element of V−{x_i} once and column x_i of F contains each element of V−{x_i} once. (4) The collection of blocks obtained from the nonempty cells of F is a (υ, 3, 2)-BIBD. A (1, 2; 3, υ, 1)-frame is a doubly near resolvable (υ, 3, 2)-BIBD. In this paper, we first present a survey of existence results on doubly near resolvable (υ, 3, 2)-BIBDs and (1, 2; 3, υ, 1)-frames. We then use frame constructions to provide a new infinite class of doubly near resolvable (υ, 3, 2)-BIBDs by constructing (1, 2; 3, υ, 1)-frames.     

Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application

Let Pⁿ be a simple n-polytope with a Z²-characteristic function λ. And h is a Morse function over Pⁿ. Then the small cover Mⁿ(λ) corresponding to the pair (Pⁿ, λ) has a cell structure given by h. From this cell structure we can derive a cellular chain complex of Mⁿ(λ) with integer coefficients. In this paper, firstly, we discuss the highest dimensional boundary morphism ∂_n of this cellular chain complex and get that ∂_n=0 or 2 by a natural way. And then, from the well-known result that the submanifold corresponding to (F, λ_F) is naturally a small cover with dimension k, where F is any k-face of Pⁿ and λ_F is the restriction of λ on F, we get that ∂_k=0 or ±2 for 0 ≤ k < n. Finally, by using the definition of inherited characteristic function which is the restriction of λ on the faces of Pⁿ, we get a way to calculate the homology groups of Mⁿ(λ). Applying our result to a 3-small cover we have that the homology groups of any 3-small cover is torsion-free or has only 2-torsion.