Minimum asymptotic error of algorithms for solving ODE

We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) × ^s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n^−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n^−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f.     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

Sur l'approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifies

We study here a new kind of modified Bernstein polynomial operators on L¹(0, 1) introduced by J. L. Durrmeyer in [4]. We define for f integrable on [0, 1] the modified Bernstein polynomial M_n f: M_nf(x) = (n + 1) ∑ⁿ_{k = o}P_nk(x)∝¹₀ P_nk(t) f(t) dt. If the derivative d^r f/dx^r with r 0 is continuous on [0, 1], d^r/dx^rM_n f converge uniformly on [0,1] and sup_xε[0,1] ¦M_n f(x) − f(x)¦ 2ω_f(1/trn) if ω_f is the modulus of continuity of f. If f is in Sobolev space W_l,p(0, 1) with l 0, p 1, M_n f converge to f in w^l,p(0, 1).     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

A sufficient condition for pairing problem of generators in symmetrizable Kac–Moody algebras

In a symmetrizable Kac–Moody algebra g(A), let α=∑_i=1ⁿk_iα_i be an imaginary root satisfying k_i>0 and α,α_i<0 for i=1,2,…,n. In this paper, it is proved that for any x_αg_α{0}, satisfying [x_α,f_n]≠0 and [x_α,f_i]=0 for i=1,2,…,n−1, there exists a vector y such that the subalgebra generated by x_α and y contains g′(A), the derived subalgebra of g(A).     

A self-normalized Erd s—Rényi type strong law of large numbers

The original Erd s—Rényi theorem states that max_0kn∑^k+[clogn]_i=k+1X_i/[clogn]→α(c),c>0, almost surely for i.i.d. random variables {X_n, n1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erd s—Rényi theorem, and the function α(c) uniquely determines the distribution function of X₁. We prove that if the normalizing constant [c log n] is replaced by the random variable ∑^k+[clogn]_i=k+1(X²_i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric.     

Hermite Interpolation and Sobolev Orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g) _S = V(f) A V(g) ^T + <u, f ^(N) g ^(N)V(f) =(f(c ₀), f "(c ₀), ..., f ^{(n – 1)} ₀(c ₀), ..., f(c _p ), f "(c _p ), ..., f ^{(n – 1)} _p(c _p )) u is a regular linear functional on the linear space P of real polynomials, c ₀, c ₁, ..., c _p are distinct real numbers, n ₀, n ₁, ..., n _p are positive integer numbers, N=n ₀+n ₁+...+n _p , and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

Multiset Permutations and Loopless Generation of Ordered Trees with Specified Degree Sequence

An ordered tree with specified degree sequence and n internal nodes has a_i nodes of degree i, where a₀ = 1 + ∑_i = 1(i − 1)a_i and n = ∑_i = 0a_i. This paper presents the first loopless algorithm for generating all ordered trees with specified degree sequence. It uses a new version of the algorithm for generating multiset permutations. When a_k = N, a₀ = (k − 1)N + 1, and all other a_i's are 0, all N node k-ary trees are generated.     

On Approximate ℓ 1 Systems in Banach Spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i=1 in X which satisfy ∑_iA±x_i|A|−f(|A|), for all finite A and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.     

On the Uniform Approximation of a Class of Analytic Functions by Bruwier Series

For a class of analytic functions f(z) defined by Laplace–Stieltjes integrals the uniform convergence on compact subsets of the complex plane of the Bruwier series (B-series) ∑^∞_n=0 λ_n(f) , λ_n(f)=f⁽ⁿ⁾(nc)+cf⁽ⁿ⁺¹⁾(nc), generated by f(z) and the uniform approximation of the generating function f(z) by its B-series in cones |arg z|< is shown.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

Counting cycles and finite dimensional L norms

We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both ∑_i=1ⁿx_i^k and ∑_i=1ⁿ|x_i|^k, subject to the constraints that ∑_i=1ⁿx_i²=C and ∑_i=1ⁿx_i=0.     

Approximation of the unit step function by a linear combination of exponential functions

We approximate the unit step function, which equals 1 if t ε [0, T] and equals 0 if t > T, by functions of the form ∑_{n = 1}^N Ax_n^N e^−λ_n^t/T, where each λ_n is a given positive constant. We find the coefficients A_n^(N) by minimizing the integrated square of the difference between the unit step function and the approximating function. We first solve the specialized case where each λ_n = n. The resulting sum can be shown to converge in the mean to the unit step function as N → ∞. The general case is then solved and some interesting properties of the numbers A_n^(N) are noted.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

New Coins from Old, Smoothly

Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ _p (N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ _p (N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C ^α [0,1] if and only if a simulation scheme as above exists with ℙ _p (N>n)≤C(Δ _n (p)) ^α , where \varDelta _n(x):=max{?{x(1-x)/n},1/n}\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}. The key to the proof is a new result in approximation theory: Let B⁺_n\mathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C ^α [0,1] if and only if f has a series representation ?_n=1^￥F_n\sum_{n=1}^{\infty}F_{n} with F_n ? B⁺_nF_{n}\in \mathcal{B}^{+}_{n} and ∑ _k>n F _k (x)≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some j_n ? B⁺_n\varphi_{n}\in\mathcal{B}^{+}_{n} satisfy |f(x)−φ _n (x)|≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1, then f∈C ^α [0,1].     

Expansion of derivatives in one-dimensional dynamics

We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ _f attracts a finite setCrit of non-flat critical points. Assuming that for eachcεCrit, either |D f ⁿ(f(c))|→∞ (iff is real) orb _n·|Df ⁿ(f(c))|→∞ for some summable sequence {b _n} (iff is complex; this is equivalent to summability of |D f ⁿ(f(c))|⁻¹), we show that for everyxεJ _f\U _i f ⁻ⁱ(Crit), there existℓ(x)≤max _c ℓ(c) andK′(x)>0

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).