Quadrature rules for Prandtl's integral equation

In this paper we construct an interpolatory quadrature formula of the type $$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 \frac{{f'(x)}}{{y - x}}dx \approx \sum\limits_{i = 1}^n {w_{ni} (y)f(x_{ni} )} ,$$ wheref(x)=(1?x)^α(1+x)^β f _o(x), α, β>0, and {x _ni} are then zeros of then-th degree Chebyshev polynomial of the first kind,T _n (x). We also give a convergence result and examine the behavior of the quantity \( \sum\limits_{i = 1}^n {|w_{ni} (y)|} \) asn→∞.     

Alcuni problemi relativi alle formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the form $$\int\limits_{ - 1}^1 {(1 - x)^a (1 + x)^\beta f(x)dx = \sum\limits_{i = 0}^{r - 1} {[A_i f^{(i)} ( - 1) + B_i f^{(i)} (1)] + K_n (\alpha ,\beta ;r)\sum\limits_{i = 1}^n {f(x_{n,i} ),} } } $$ (α>?1, β>?1), with realA _i ,B _i ,K _n and real nodesx _n,i in (?1,1), valid for prolynomials of degree ≤2n+2r?1. In the first part we prove that there is validity for polynomials exactly of degree2n+2r?1 if and only if α=β=?1/2 andr=0 orr=1. In the second part we consider the problem of the existence of the formula $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(x)dx = A_n f( - 1) + B_n f(1) + C\sum\limits_{i = 1}^n {f(x_{n,i} )} }$$ for polynomials of degree ≤n+2. Some numerical results are given when λ=1/2.     

Modifikationen des Pollard-Algorithmus

The factorization algorithm of Pollard generates a sequence in ? _n by $$x_0 : = 2;x_{i + 1} : = x_i^2 - 1(\bmod n),i = 1,2,3,...$$ wheren denotes the integer to be factored. The algorithm finds an factorp ofn within \(0\left( {\sqrt p } \right)\) macrosteps (=multiplications/divisions in ? _n ) on average. An empirical analysis of the Pollard algorithm using modified sequences $$x_{i + 1} = b \cdot x_i^\alpha + c(\bmod n),i = 1,2,...$$ withx ₀,b,c,α∈? and α≥2 shows, that a factorp ofn under the assumption gcd (α,p-1)≠1 now is found within $$0\left( {\sqrt {\frac{p}{{ged(\alpha ,p - 1}}} } \right)$$ macrosteps on average.     

Sulle formule di quadratura di Tschebyscheff

For the Tschebyscheff quadrature formula: $$\int\limits_{ - 1}^1 {\left( {1 - x^2 } \right)^{\lambda - 1/2} f(x) dx} = K_n \sum\limits_{k = 1}^n {f(x_{n,k} )} + R_n (f), \lambda > 0$$ it is shown that the degre,N, of exactness is bounded by: $$N \leqslant C(\lambda )n^{1/(2\lambda + 1)} $$ whereC(λ) is a convenient function of λ. For λ=1 the complete solution of Tschebyscheff's problem is given.     

Sul grado di precisione di formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the types (1) $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = C_n^{ (\lambda )} \sum\limits_{i = 1}^n f (x_{n,i} ) + R_n \left[ f \right]} ,$$ (2) $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = A_n^{ (\lambda )} \left[ {f\left( { - 1} \right) + f\left( 1 \right)} \right] + K_n^{ (\lambda )} \sum\limits_{i = 1}^n f (\bar x_{n,i} ) + \bar R_n \left[ f \right]} ,$$ with 0<λ<1, and we obtain inequalities for the degreeN of their polynomial exactness. By using such inequalities, the non-existence of (1), with λ=1/2,N=n+1 ifn is even andN=n ifn is odd, is directly proved forn=8 andn≥10. For the same value λ=1/2 andN=n+3 ifn is evenN=n+2 ifn is odd, the formula (2) does not exist forn≥12. Some intermediary results regarding the first zero and the corresponding Christoffel number of ultraspherical polynomialP _n ^(λ) (x) are also obtained.     

Formule di quadratura «chiuse» di tipo gaussiano: Tabulzione dei nodi e dei coefficienti

The coefficientsA _hi and the nodesx _mi for «closed” Gaussian-type quadrature formulae $$\int\limits_{ - 1}^1 {f(x)dx = \sum\limits_{h = 0}^{2_8 } {\sum\limits_{i = 0}^{m + 1} {A_{hi} f^{(h)} (x_{mi} ) + R\left[ {f(x)} \right]} } } $$ withx _m0 =?1,x _{m, m+1} =1 andR[f(x)]=0 iff(x) is a polinomial of degree at most2m(s+1)+2(2s+1)?1, have been tabulated for the cases: $$\left\{ \begin{gathered} s = 1,2 \hfill \\ m = 2,3,4,5 \hfill \\ \end{gathered} \right.$$ .     

On positive function series

Function series of the form $$f(x) = \sum\limits_{n = 0}^N {c_n f_n (x)} $$ are considered under the constraintf(x)≥0 in a given intervala≤x≤b. The cone in teh spaceR ^N+1 of the coefficientsc _n which is determined by the positivity constraint is approximated numerically by a polyhedral cone. A numerical estimate for the error involved is given and it is shown how it may be reduced. A special series of Jacobi polynomials is discussed and new estimates for the range of parameters for which this series is non-negative are obtained.     

The addition of bounded quantification and partial functions to a computational logic and its theorem prover

We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects in the logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as ‘mapping functions’. The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as ‘capturing’, ‘bottom’, or ‘continuity’. Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are:

? classic quantifier manipulation theorems, such as $$\sum\limits_{{\text{l}} = 0}^{\text{n}} {{\text{g}}({\text{l}}) + {\text{h(l) = }}} \sum\limits_{{\text{l = }}0}^{\text{n}} {{\text{g}}({\text{l}})} + \sum\limits_{{\text{l = }}0}^{\text{n}} {{\text{h(l)}};} $$

? elementary theorems involving quantifiers, such as the Binomial Theorem: $$(a + b)^{\text{n}} = \sum\limits_{{\text{l = }}0}^{\text{n}} {\left( {_{\text{i}}^{\text{n}} } \right)} \user2{ }{\text{a}}^{\text{l}} {\text{b}}^{{\text{n - l}}} ;$$

? elementary theorems about ‘mapping functions’ such as: $$(FOLDR\user2{ }'PLUS\user2{ O L) = }\sum\limits_{{\text{i}} \in {\text{L}}}^{} {{\text{i}};} $$

? termination properties of many partial recursive functions such as the fact that an application of the partial function described by $$\begin{gathered} (LEN X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F ({\rm E}QUAL X NIL) \hfill \\ {\rm O} \hfill \\ (ADD1 (LEN (CDR X)))) \hfill \\ \end{gathered} $$ terminates if and only if the argument ends in NIL;

? theorems about functions satisfying unusual recurrence equations such as the 91-function and the following list reverse function: $$\begin{gathered} (RV X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F (AND (LISTP X) (LISTP (CDR X))) \hfill \\ (CONS (CAR (RV (CDR X))) \hfill \\ (RV (CONS (CAR X) \hfill \\ (RV (CDR (RV (CDR X))))))) \hfill \\ X). \hfill \\ \end{gathered} $$

     

On the convergence rate of SOR: A worst case estimate

LetA be any real symmetric positive definiten×n matrix, and κ(A) its spectral condition number. It is shown that the optimal convergence rate $$\rho _{SOR}^* = \mathop {\min }\limits_{0< \omega< 2} \rho (M_{SOR,\omega } )$$ of the successive overrelaxation (SOR) method satisfies $$\rho _{SOR}^* \leqslant 1 - \frac{1}{{\alpha _n \kappa (A)}}, \alpha _n \approx \log n.$$ This worst case estimate is asymptotically sharp asn→∞. The corresponding examples are given by certain Toeplitz matrices.     

Convergence of quadratures for Cauchy principal value integrals

Quadrature formulas based on the “practical” abscissasx _k=cos(k π/n),k=0(1)n, are obtained for the numerical evaluation of the weighted Cauchy principal value integrals $$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 (1 - x)^\alpha (1 + x)^\beta (f(x))/(x - a)){\rm E}dx,$$ where α,β>?1 andaε(?1, 1). An interesting problem concerning these quadrature formulas is their convergence for a suitable class of functions. We establish convergence of these quadrature formulas for the class of functions which are Hölder-continuous on [?1, 1].     

Valutazione dell'errore per una formula di quadratura alla Tchebycheff di tipo chiuso

In this paper we study the remainder term of a quadrature formula of the form $$\int\limits_{ - 1}^1 {f(x)dx = A_n \left[ {f( - 1) + f(1)} \right] + C_n \sum\limits_{i = 1}^n {f(x_{n,i} ) + R_n \left[ f \right],} } $$ , withx _x,i ∈^-1,1, andR _n [f]=0 whenf(x) is a polynomial of degree ≤n+3 ifn is even, or ≤n+2 ifn is odd. Such a formula exists only forn=1(1)11. It is shown that, iff(x)∈ ^C(h+1) [-1,1], (h=n+3 orn+2), thenR _n [f]=f ^h+1 (τ)·± _n . The values α _n are given.     

Global wire routing in two-dimensional arrays

We examine the problem of routing wires of a VLSI chip, where the pins to be connected are arranged in a regular rectangular array. We obtain tight bounds for the worst-case “channel-width” needed to route ann×n array, and develop provably good heuristics for the general case. Single-turn routings are proved to be near-optimal in the worst-case. A central result of our paper is a “rounding algorithm” for obtaining integral approximations to solutions of linear equations. Given a matrix A and a real vector x, then we can find an integral x such that for alli, ¦x _i -x _i ¦ <1 and (Ax) _i -(Ax) _i <Δ. Our error bound Δ is defined in terms of sign-segregated column sums of A: $$\Delta = \mathop {\max }\limits_j \left( {\max \left\{ {\sum\limits_{i:a_{ij} > 0} {a_{ij} ,} \sum\limits_{i:a_{ij}< 0} { - a_{ij} } } \right\}} \right).$$      

A transformation of series

The transformation (*) $$\sum\limits_{\nu = 0}^\infty {t_\nu z^\nu \to } \sum\limits_{\nu = 0}^\infty {\left\{ {\sum\limits_{\tau = 0}^{h - 1} {z^\tau } \Delta ^\nu t_{h\nu + \tau } + \frac{{z^h }}{{1 - z}}\Delta ^\nu t_{h(\nu + 1)} } \right\}} \left( {\frac{{z^{h + 1} }}{{1 - z}}} \right)^\nu$$ whereh≥0 is an integer and Δ operates upon the coefficients {t _v } of the series being transformed, is derived. Whenh=0, the above transformation is the generalised Euler transformation, of which (*) is itself a generalisation. Based upon the assumption that \(t_\nu = \int\limits_0^1 {\varrho ^\nu d\sigma (\varrho ) } (\nu = 0, 1,...)\) , where σ(?) is bounded and non-decreasing for 0≤?≤1 and subject to further restrictions, a convergence theory of (*) is given. Furthermore, the question as to when (*) functions as a convergence acceleration transformation is investigated. Also the optimal valne ofh to be taken is derived. A simple algorithm for constructing the partial sums of (*) is devised. Numerical illustrations relating to the case in whicht _v =(v+1) ^?1 (v=0,1,...) are given.     

Symbolic solution of nonhomogeneous linear ordinary differential equations in terms of power series

For a nonhomogeneous linear ordinary differential equation Ly(x) = f(x) with polynomial coefficients and a holonomic right-hand side, a set of points x = a is found where a power series solution $y(x) = \sum\nolimits_{n = 0}^\infty {c_n (x - a)} ^n $ with hypergeometric coefficients exists (starting from some number, the ratio c _{n + 1}/c _n is a rational function of n).     

Quadrature formulas for cauchy principal value integrals

Quadrature formulas of the Clenshaw-Curtis type, based on the “practical” abscissasx _k=cos(kπ/n),k=0(1)n, are obtained for the numerical evaluation of Cauchy principal value integrals \(\int\limits_{ - 1}^1 {(x - a)^{ - 1} } f(x) dx, - 1< a< 1\) .     

A note on balanced immunity

For a finite alphabet ∑ we define a binary relation on \(2^{\Sigma *} \times 2^{2^{\Sigma ^* } } \) , called balanced immunity. A setB ? ∑* is said to be balancedC-immune (with respect to a classC ? 2^Σ* of sets) iff, for all infiniteL εC, $$\mathop {\lim }\limits_{n \to \infty } \left| {L^{ \leqslant n} \cap B} \right|/\left| {L^{ \leqslant n} } \right| = \tfrac{1}{2}$$ Balanced immunity implies bi-immunity and in natural cases randomness. We give a general method to find a balanced immune set'B for any countable classC and prove that, fors(n) =o(t(n)) andt(n) >n, there is aB εSPACE(t(n)), which is balanced immune forSPACE(s(n)), both in the deterministic and nondeterministic case.     

On some weight functions admitting (0, 2) quadrature formulae with a high algebraic degree of precision

The purpose of this paper is to find a class of weight functions μ for which there exist quadrature formulae of the form (1) $$\int_{ - 1}^1 {\mu (x) f(x) dx \approx \sum\limits_{k = 1}^n {(a_k f(x_k ) + b_k f''(x_k ))} }$$ , which are precise for every polynomial of degree 2n.     

Esistenza,unicità e calcolo della soluzione di un sistema non lineare

LetX be a point of the realn-dimensional Euclidean space ? ⁿ ,G(X) a given vector withn real components defined in ? ^u ,U an unknown vector withs real components,K a known vector withs real components andA a given reals×n matrix of ranks. Assuming that, for every pair of pointsX ₁ , X₂of ? ⁿ ,G(X) satisfies the conditions $$(G(X_1 ) - G(X_2 ), X_1 - X_2 ) \geqslant o (X_1 - X_2 , X_1 - X_2 )$$ and $$\left\| {(G(X_1 ) - G(X_2 )\left\| { \leqslant M} \right\|X_1 - X_2 )} \right\|$$ wherec andM are positive constants, we prove that a unique solution of the system $$\left\{ \begin{gathered} G(X) + A ^T U = 0 \hfill \\ AX = K \hfill \\ \end{gathered} \right.$$ exists and we show a method for finding such a solution     

Stability Criteria for Finite Difference Approximations to Parabolic Systems—An Update

In a recent series of papers, Goldberg [G1, G2] and Sun and Yuan [SY] studied the L ₂-stability of a well-known family of finite difference approximations for the initial-value problem associated with the multispace-dimensional parabolic system $$\frac{{\partial {\text{u(x,}}\;t{\text{)}}}}{{\partial t}} = \sum\limits_{1 \leqslant {\kern 1pt} p{\kern 1pt} \leqslant {\kern 1pt} q{\kern 1pt} \leqslant {\kern 1pt} {\kern 1pt} s} {A_{pq} \frac{{\partial ^2 {\text{u(x,}}\;t{\text{)}}}}{{\partial x_p \partial x_q }}} + \sum\limits_{1 \leqslant {\kern 1pt} p{\kern 1pt} \leqslant {\kern 1pt} s} {B_p \frac{{\partial {\text{u(x,}}\;t{\text{)}}}}{{\partial x_p }} + C{\text{u(x,}}\;t{\text{)}}} $$ where A _pq ,B _p and C are constant matrices, A _pq being Hermitian. In the present paper we discuss these earlier results and complete the underlying theory by answering four open questions.     

A technical note on AC-Unification. The number of minimal unifiers of the equation αx1 + ⋯ + αxp ≐AC βy1 + ⋯ + βyq

We show in this note that the equation αx₁ + #x22EF; +αx_p?_ACβy₁ + α +βy_q where + is an AC operator and αx stands for x+...+x (α times), has exactly $$\left( { - 1} \right)^{p + q} \sum\limits_{i = 0}^p {\sum\limits_{j = 0}^q {\left( { - 1} \right)^{1 + 1} \left( {\begin{array}{*{20}c} p \\ i \\ \end{array} } \right)\left( {\begin{array}{*{20}c} q \\ j \\ \end{array} } \right)} 2^{\left( {\alpha + \begin{array}{*{20}c} {j - 1} \\ \alpha \\ \end{array} } \right)\left( {\beta + \begin{array}{*{20}c} {i - 1} \\ \beta \\ \end{array} } \right)} } $$ minimal unifiers if gcd(α, β)=1.