Normal numbers and selection rules

Given anormal number x=0,x ₁ x ₂ ··· to base 2 and aselection rule S ?{0, 1}*=∪ _n=0 ^/t8 {0, 1} ⁿ , we define a subsequencex,=0, \(\chi _{t_1 } \chi _{t_2 } \) ·· where {t ₁<t ₂<···}={i; x ₁ x ₂···x _i?1 εS}.x _s is called aproper subsequence ofx if lim_i/∞ ti/S is said topreserve normality if for any normal numberx such thatx _s is a proper subsequence ofx, x _s is also a normal number. We prove that ifS/～ _s is a finite set, where ～ _s is an equivalence relation on {0, 1}* such that ξ ～ _s η if and only if {ζ; ξζ εS}={ζ; ηζ εS}, thenS preserves normality. This is a generalization of the known result in finite automata case, where {0, 1}*/～ _s is a finite set (Agafonov [1]).     

Contractive properties for the heat equation in Sobolev spaces

Consider the heat equation ?_ru ? Δ_xu = 0 in a cylinder Ω × [0,T] ? $\text{R}$ⁿ⁺¹ smooth lateral boundary under zero Neumann or Dirichlet conditions. Geometric conditions for Ω are given that guarantee that for a given P, 6▽_xu(·, t)6_L^p will be non-increasing for any solution. Decay rates are also given. For arbitrary Ω and p, it is shown how to construct an equivalent L^p-norm, such that ▽_x(·, t) is non-increasing in this norm.     

Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i < 0 for all i, then there is a constant α such that every solution of (1) is O(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that 0 < Re b₁ ? Re b₂ ? ··· ? Re b_n and λ times Re b_n < Re b₁, then every solution of (1) is O(e^bn^t) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that Re b₁ ? Re b₂ ? ··· ? Re b_n < 0 and λ Re b_n < Re b₁, then no solution x(t) of (1), except the identically zero solution, is 0(e^b1^t) as t → ∞.     

Normed Topological Pseudovector Groups

A normed topological pseudovector group (NTPVG for short) is a valued topological group (V,?+?,||·||) (not necessarily Abelian) endowed with a continuous scalar multiplication \({\mathbb R}_+ \times V \ni (t,x) \mapsto t \cdot x \in V\) such that 0 ·x?=?e (e denotes the neutral element of V), 1 ·x?=?x, (st) ·x?=?s ·(t ·x), t ·(x?+?y)?=?(t ·x)?+?(t ·y) and ||t ·x||?=?t ||x|| for each t, \(s \in {\mathbb R}_+\) and x, y?∈?V. It is shown that every valued topological group can be isometrically and group-homomorphically embedded in a NTPVG as a closed subset by means of a functor. Locally compact NTPV groups are fully classified. It is shown that the (unbounded) Urysohn universal metric space can be endowed with a structure of a NTPV group of exponent 2.     

Backward stochastic differential equations and applications to optimal control

We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?_t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?_t-adapted process, andX is ?_t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.     

Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

We are concerned with Friedrichs's scheme for an initial value problem u_t(t, x) = A(t, x)u_x(t, x), u(0, x) = u₀(x), where u₀(x) belongs to L_∞, not to L₂. We show that Friedrichs's scheme is stable in the maximum norm ·_L∞, provided that the system is regularly hyperbolic and that the eigenvalues d_i(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λd_i(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio.     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

Kinematical conservation laws in a space of arbitrary dimensions

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ω_t in d-dimensional x-space, where x = (x ₁, x ₂,..., x _d) ∈ R^d. The KCL are derived in a specially defined ray coordinates (ξ = (ξ₁, ξ₂,..., ξ_d?1), t), where ξ₁, ξ₂,..., ξ_d?1 are surface coordinates on Ω_t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω_t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω_t when Ω_t is a curve in R² and is a curve on Ω_t when it is a surface in R³. Across a kink the normal n to Ω_t and normal velocity m on Ω_t are discontinuous.     

A new family of Padé-type approximants in

This paper gives the definition and some properties of a new family of Padé-type approximants (PTA) for k-variate formal power series (FPS). These PTA have the form P(t)/Q(t) where Q(t) = Π^r_{i = 0}(1 ? x(i)·t), {x(i), 0 ? i ? r} being a given set of points in

, and x·t is the scalar product of x and t in

. Some results about the approximation order, the unicity and some invariance properties of these PTA are proved together with a convergence result when the FPS is defined by a Stieltjes integral.     

Optimal discounted linear control of the wiener process

The following stochastic control problem is considered: to minimize the discounted expected total cost $$J(x;u) = E\int_0^\infty {\exp ( - at)[\phi } (x_l ) + |u_l (x)|]dt,$$ subject todx _t =u _t (x)dt+dw _t ,x ₀=x, |u _t |≤1, (w _t ) a Wiener process, α>0. All bounded by unity, measurable, and nonanticipative functionalsu _t (x) of the state processx _t are admissible as controls. It is proved that the optimal law is of the form $$\begin{gathered} u_t^* (x) = - 1,x_t > b, \hfill \\ u_t^* (x) = 0,|x_t | \leqslant b, \hfill \\ u_t^* (x) = 1,x_t< - b, \hfill \\ \end{gathered}$$ for some switching pointb > 0, characterized in terms of the function ø(·) through a transcendental equation.     

A new method of proof of Filippov’s theorem based on the viability theorem

Filippov??s theorem implies that, given an absolutely continuous function y: [t ₀; T] ?? ? ^d and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x ₀, there exists a solution x(·) to the differential inclusion x??(t) ?? F(t, x(t)) starting from x ₀ at the time t ₀ and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ where the function ??(·) is the estimation of dist(y??(t), F(t, y(t))) ?? ??(t). Setting P(t) = {x ?? ? ⁿ : |x ?y(t)| ?? r(t)}, we may formulate the conclusion in Filippov??s theorem as x(t) ?? P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ?? DP(t, x)(1) ?? ?. It allows to obtain Filippov??s theorem from a viability result for tubes.     

Averaging of systems of differential inclusions with slow and fast variables

We consider the system of differential inclusions

$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$

, where F,G: D → Kυ (\(R^{m_1 } \)), Kυ (\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces \(R^{m_1 } \) and \(R^{m_2 } \), respectively, D = R ₊ × \(R^{m_1 } \) × \(R^{m_2 } \) × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem \(\dot u\) ∈ µF ₀(u), u(0) = x ₀, F ₀(u) ∈ Kυ (\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ₀] and any solution x _µ(·), y _µ(·) of the original problem, there exists a solution u _µ(·) of the averaged problem such that ∥x _µ(t) ? y _µ(t) ∥ ≤ ? for t ∈ [0, 1/µ]. Furthermore, for each solution u _µ(·)of the averaged problem, there exists a solution x _µ(·), y _µ(·) of the original problem with the same estimate.

     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

On Asymptotically Optimal Methods of Prediction and Adaptive Coding for Markov Sources

The problem of predicting a sequence x₁, x₂, … generated by a discrete source with unknown statistics is considered. Each letter x_t+1 is predicted using information on the word x₁x₂…x_t only. In fact, this problem is a classical problem which has received much attention. Its history can be traced back to Laplace. To estimate the efficiency of a method of prediction, three quantities are considered: the precision as given by the Kullback–Leibler divergence, the memory size of the program needed to implement the method on a computer, and the time required, measured by the number of binary operations needed at each time instant. A method is presented for which the memory size and the average time are close to the minimum. The results can readily be translated to results about adaptive coding.     

Subordination,rank, and determinism of multivariate stationary sequences

Necessary and sufficient conditions for an arbitrary q-variate stationary sequence x_t, t ∈ $\text{Z}$, to be deterministic are presented. A characterization of the rank r(x) of x_t, t ∈ $\text{Z}$, and a method to construct the Wold-Cramér decomposition for x_t, t ∈ $\text{Z}$, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.     

Effective computation of singularities of parametric affine curves

Let k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x₁=f₁(t),…,x_n=f_n(t) over an arbitrary ground field k, where f₁,…,f_n∈k[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.     

Output stabilization for a class of uncertain systems

The system

$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$

where v = C*x is an output, u = S*y is a control, A(·) ∈ R ^{n × n} , B(·) ∈ R ^{n × (n–p)}, C ∈ R ^{n × (n–p)}, and D ∈ R ^{n × (n–p)}, is considered. The elements α_ij(·) and β_ij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions

$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$

It is assumed that A(·) ∈ Z ₁ ∪ Z ₃ and A*(·) ∈ Z ₁ ∪ Z ₃, where Z ₁ is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z ₃ differs from Z _t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.

     

Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient. The case of control by displacement at one endpoint with the other endpoint being fixed

We consider the problem of boundary control by displacement at one boundary point x = 0 for a process described by the Klein-Gordon-Fock equation with a variable coefficient on a finite interval 0 ≤ x ≤ l with the Dirichlet condition u(l, t) = 0 at the other boundary point. For the critical time interval T = 2l, we show that there exists a unique boundary function u(0, t) = µ(t) bringing the system from an arbitrary initial state into an arbitrary terminal state.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

Permutation Polynomials Modulo 2w

We give an exact characterization of permutation polynomials modulo n=2^w, w≥2: a polynomial P(x)=a₀+a₁x +···+a_dx^d with integral coefficients is a permutation polynomial modulo n if and only if a₁ is odd, (a₂+a₄+a₆+···) is even, and (a₃+a₅+a₇+···) is even. We also characterize polynomials defining latin squares modulo n=2^w, but prove that polynomial multipermutations (that is, a pair of polynomials defining a pair of orthogonal latin squares) modulo n=2^wdo not exist.