K-dimensional Torus Research Articles

Estimation of Lyapunov exponents for quasi-stable attractors of dynamical systems with time delay

Lyapunov exponents play an important role among the invariant characteristics of dynamical systems. Analysis of the spectrum of Lyapunov exponents is widely used to study the complex dynamics in systems of ordinary differential equations and models that can be reduced to maps. In this paper we consider the question of numerical evaluation of Lyapunov exponents for delay differential equations. We used a new algorithm with FFT, tested it on Hutchinson equation with known Lyapunov exponents spectrum and compared with results of the old algorithm. The so-called quasi-stable behavior is typical for a number of gene networks models and neuronal associations that have been studied recently. The phenomenon of cycle (k-dimensional torus) quasistability in dynamical system is characterized by the fact that some of its multipliers are asymptotically close to the unit circle, and the remaining multipliers are modulo less than one (with the exception of a simple root 1 (k such roots)). In some cases, it is possible to prove the existence and give an asymptotic estimate of the multipliers of considered system by using the large parameter methods. However, if these methods are not applicable, it is necessary to obtain a tool for the numerical estimation of multipliers. Such a tool is provided by algorithms for Lyapunov exponents estimation. Given that the equations with delay are often applied in the models of neural and gene networks, the algorithm for Lyapunov exponents estimation for such systems will be in demand.

Optimal Control, Everywhere Dense Torus Winding, and Wolstenholme Primes

An optimal control problem is constructed so that its control runs over an everywhere dense winding of a k-dimensional torus for arbitrary natural k ≤ 249 998 919 given in advance. The construction is based on Galois theory and the Wolstenholme primes distribution.

Non-closed isometry-invariant geodesics

Let c be a non-closed and bounded geodesic in a complete Riemannian manifold M. Assume that c is invariant under an isometry A of M and that c is not contained in the set of fixed points of A. We prove that, for some $${k\ge 2}$$ , the geodesic flow line ċ corresponding to c is dense in a k-dimensional torus N embedded in TM. In particular, every geodesic with initial vector in N is A-invariant.

Algebras of the equivariant cohomologies of an $\mathfrak{F}$ -classifying T k -spaces

We consider equivariant cohomologies generated by the Borel functor \(E_\mathfrak{F} \) for the family of orbit types \(\mathfrak{F}\) ⊂ ConjG, which translates equivariant homotopy category EQUIV-HOMOT in \(\mathfrak{F}\)-isovariant homotopy category \(ISOV_\mathfrak{F} \)-HOMOT. Due to the effect of concentration of isovariant absolute extensors \(ISOV_\mathfrak{F} \)-AE we calculate in explicit form the algebra of equivariant cohomologies of an \(\mathfrak{F}\)-classifying G-spaces for finite families of orbit types \(\mathfrak{F}\) ⊂ ConjG in the case of actions of k-dimensional torus G = Tk.

Smooth almost Δ-bundles over polyhedrons

In this paper, we construct and study a category of principal fiber bundles with the following properties: 1) The base of a bundle is a polyhedron and the structure group is a k-dimensional torus. 2) The transition functions of the bundle atlas are smooth on simplexes of the base. 3) On the base, a simplicial action of a finite group Δ is given which has a multivalued lifting to the total space of the bundle. We study invariant connections and construct integer-valued realizable characteristic classes.

Graph embeddings and simplicial maps

An undirected graph is viewed as a simplicial complex. The notion of a graph embedding of a guest graph in a host graph is generalized to the realm of simplicial maps. Dilation is redefined in this more general setting. Lower bounds on dilation for various guest and host graphs are considered. Of particular interest are graphs that have been proposed as communication networks for parallel architectures. Bhatt et al. provide a lower bound on dilation for embedding a planar guest graph in a butterfly host graph. Here, this lower bound is extended in two directions. First, a lower bound that applies to arbitrary guest graphs is derived, using tools from algebraic topology. Second, this lower bound is shown to apply to arbitrary host graphs through a new graph-theoretic measure, called bidecomposability. Bounds on the bidecomposability of the butterfly graph and of the k-dimensional torus are determined. As corollaries to the main lower bound theorem, lower bounds are derived for embedding arbitrary planar graphs, genus g graphs, and k-dimensional meshes in a butterfly host graph.

Irregularities of distributions with respect to polytopes

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.

TWO-LOOP APPROACH TO THE EFFECTIVE ACTION IN QUANTUM GRAVITY

For the first time we present the general formalism and results of calculation of the two-loop effective action in Einstein quantum gravity on the background MN × Tk, where MN is Minkowski space and Tk is a k-dimensional torus. We discuss the case of a zero cosmological constant as well as of a nonzero one. The method of calculating variations of the action on a metric tensor and the technique of calculating momentum integrals in dimensional regularization are presented. Some applications to spontaneous compactification are discussed, as well as some prospects.

Two-loop effective action in quantum gravity

The two-loop effective action in Einstein quantum gravity on the background M N × T k (where M N is Minkowski space, T k is a k-dimensional torus) is calculated. The spontaneous compactification with account of quantum-gravitational corrections is discussed.

Rigidity and other topological aspects of compact nonpositively curved manifolds

Rigidity and other topological aspects of compact nonpositively curved manifolds

C-uniform distribution on compact metric spaces

holds for all continuous, real-valued functions f‘on X. In the case of the k-dimensional torus ([W/Z)k this concept was introduced by H. Weyl [9 J. (See also [5,6].) Later E. Hlawka [3] introduced a quantitative measure for C-uniform distribution on ([W/Z)k = [0, l)k with respect to the k-dimensional Lebesgue measure ,I, : the function cr): rW: -+ X is C, A,-ud. if and only if the discrepancy D>(w) := sup I 5 $ ,: xco. .x,(dO) A - ~k(CO, xl) (1.2) Y t [O, I 1” tends to 0 if T tends to infinity, ~t~,.~) denotes the characteristic function of the k-dimensional interval 10, x) = [O, x,) x ...

Polynomially bounded multisequences and analytic continuation

Given a polynomially bounded multisequence {fm}, where m = (m1, …, mk) ∈ ℤk, we will consider 2k power series in exp(iz1), …, exp(izk), each representing a holomorphic function within its domain of convergence. We will consider this same multisequence as a linear functional on a class of functions defined on the k-dimensional torus by a Fourier series, , with the proper convergence criteria. We shall discuss the relationships that exist between the linear functional properties of the multisequence and the analytic continuation of the holomorphic functions. With this approach we show that a necessary and sufficient condition that the multisequence be given by a polynomial is that each of the power series represents, up to a unit factor, the same function that is entire in the variables

Bordismengruppen unit�rer Torusmannigfaltigkeiten

Let G be a k-dimensional torus. Let U * G , denote the homotopical unitary bordism theory. We show that restriction to the fixed point set determines an element of U * G . This implies that the bordism class of a unitary G-manifold is determined by its characteristic numbers in equivariant K-theory.