Convergence of Modified Lagrange Interpolation toLp-Functions Based on the Zeros of Orthonormal Polynomials with Freud Weights

We consider the “Freud weight”W²_Q(x)=exp(−Q(x)). let 1<p<∞, and letL*_n(f) be a modified Lagrange interpolation polynomial to a measurable functionf∈{f; ess sup_x∈ |f(x)| W_Q(x)(1+|x|)^α<∞},α>0. Then we have lim_n→∞ ∫^∞_−∞ [|f(x)−L*_n(f; x)| W_Q(x)(1+|x|)^−Δ]^p dx=0, whereΔis a constant depending onpandα.     

Real Solving of Elementary-Algebraic Systems

In this paper we present a new method for enclosing all the real roots in a bounded box of a system of n elementary-algebraic equations depending on n variables. This system is denoted by h(x)=P(x,f ₁(x ₁),...,f _k (x ₁),f ₁(x ₂),...,f _k (x _n )), where x=(x ₁,x ₂,...,x _n ) R ⁿ , P is a system of polynomials depending on n+kn variables and the univariate functions f _i are simple. This method arises from the exclusion method of Dedieu and Yakoubsohn. We provide both a theoretical complexity bound and some numerical experiments.     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

Error estimates for a class of quadrature formulae

The problem considered is that of estimating the error of a class of quadrature formulae for _–1 ¹ w _r (x)f(x)dx, (w _r (x) being a positive weight-function), where only values off(x) in (–1,1) and off(x) and its derivatives at the end-points of the interval are considered.     

On the neutrix composition of the distributions x −s ln m |x| and x r

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F _n (f)}, where F _n (x) = F(x) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^?s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^?rs ln ^m |x| for r, s, m = 2, 3….     

Quasi-periodic solutions for the general semilinear Duffing equations with asymmetric nonlinearity and oscillating potential

In this paper,we prove the existence of quasi-periodic solutions and the boundedness of all the solutions of the general semilinear quasi-periodic differential equation x′′+ax~+-bx~-=G_x(x,t)+f (t),where x~+=max{x,0},x~-=max{-x,0},a and b are two different positive constants,f(t) is C~(39) smooth in t,G(x,t)is C~(35) smooth in x and t,f (t) and G(x,t) are quasi-periodic in t with the Diophantine frequency ω=(ω_1,ω_2),and D_x~iD_t~jG(x,t) is bounded for 0≤i+j≤35.     

Moser's second one-dimensional inequality for a class of integral operators

Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? L^p(-￥ ,￥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò₀^xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò_-￥^￥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant C_pC_p, such that ò_-￥^￥ exp[a |F(x)|^q-|x|]  dx ￡ C_p\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||_p ￡ 1||f||_p\le 1 and every a ￡ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

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.     

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Complete rank theorem of advanced calculus and singularities of bounded linear operators

Let E and F be Banach spaces, f: U ⊂ E → F be a map of C ^r (r ⩾ 1), x ₀ ∈ U, and ft (x ₀) denote the FréLechet differential of f at x ₀. Suppose that f′(x ₀) is double split, Rank(f′(x ₀)) = ∞, dimN(f′(_{x 0})) > 0 and codimR(f′(x ₀)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x ₀) near x ₀. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x ₀ is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.      

Error bounds for multistep methods revisited

For linear multistep methods with constant stepsize we consider error bounds in terms of weightedL ₂-norms ofh ^px^(p) rather than ofh ^px^(p+1). The bounds apply to stiff systemsx'=Ax+f(t,x) where the spectrum ofA lies in a sector andf is of moderate size.     

An algorithm of MCMC method for solving F(X) =0

An algorithm of continuous stage-space MCMC method for solving algebra equation f(x)=0 is given. It is available for the case that the sign of f(x) changes frequently or the derivative f′(x) does not exist in the neighborhood of the root, while the Newton method is hard to work. Let n be the number of random variables created by computer in our algorithm. Then after m=O(n) transactions from the initial value x ₀,x* can be got such that |f(x*)|<e ^−cm |f(x ₀)| by choosing suitable positive constant c. An illustration is also given with the discussion of convergence by adjusting the parameters in the algorithm. Supported by the National Natural Science Foundation of China (70171008).     

Eigenvalue Intervals for 2mth Order Sturm—Liouville Boundary Value Problems

Values of λ are determined for which there exist positive solutions of the 2mth order differential equation on a measure chain, (-1)^m x ^?2m (t)=λa(t)f(u(σ(t))), y? [0,1], satisfying α _i+1 u ^?21(0)+0, γ _i+1 u ^?21(σ(1))=0, 0≤i≤m?1 with αi,βi,γi,δi≥0, where a and f are positive valued, and both lim _x-0+ (f(x)/x) and lim _x-0+ (f(x)/x) exist.     

Characterising the Accuracy of Multivariable Refinable Functions via the Symbol Function

Suppose that f(x) = (f ₁(x),...,f _r (x)) ^T , x∈R ^d is a vector-valued function satisfying the refinement equation f(x) = ∑_Λ c _κ f(2x−κ) with finite set Λ of Z ^d and some r×r matricex c _κ. The requirements for f to have accuracy p are given in terms of the symbol function m(ξ). Supported by NSFC     

A counterexample in copositive approximation

The present paper gives a converse result by showing that there exists a functionf ∈C _[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE _n ⁽⁰⁾ (f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E _n(f) is the ordinary best polynomial approximation off of degreen.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

On a multiplicative functional transformation arising in nonlinear filtering theory

Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(x_t)¦y _s, st] where {x _t} is a Markov process with infinitesimal generator A and {y _t} is an observation process given by dy _t=h(x_t)dt +dw_twhere {w _t} is a Brownian motion. If h(x_t) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T _s,t ^y obtained by a certain y-dependent multiplicative functional transformation of the signal process {x _t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A _t ^y f=e^y(t)h(A– 1/2h²)(e^–y(t)h f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology