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.     

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.     

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.     

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: $$\left\{ \begin{gathered} s = 1,2 \hfill \\ m = 2,3,4,5 \hfill \\ \end{gathered} \right.$$ .     

Numerical evaluation of a solution of a special mixed-type differential-difference equation

The function * $$f(t) = \frac{{e^{ - \alpha \gamma } }}{\pi }\int\limits_0^\infty {\cos t \xi e^{\alpha Ci(\xi )} \frac{{d\xi }}{{\xi ^\alpha }},t \in R,\alpha > 0} $$ [Ci(x)=cosine integral, γ=Euler's constant] is studied and numerically evaluated;f is a solution to the following mixed type differential-difference equation arising in applied probability: ** $$tf'(t) = (\alpha - 1)f(t) - \frac{\alpha }{2}[f(t - 1) + f(t + 1)]$$ satisfying the conditions: i) $$f(t) \geqslant 0,t \in R$$ , ii) $$f(t) = f( - t),t \in R$$ , iii) $$\int\limits_{ - \infty }^{ + \infty } {f(\xi )d\xi = 1} $$ . Besides the direct numerical evaluation of (*) and the derivation of the asymptotic behaviour off(t) fort→0 andt→∞, two different iterative procedures for the solution of (**) under the conditions (i) to (iii) are considered and their results are compared with the corresponding values in (*). Finally a Monte Carlo method to evaluatef(t) is considered.     

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.     

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.     

Formule di quadratura di tipo gaussiano con valori delle derivate dell'integrando

The coefficientsA _hi (m,s) and the nodesx _i (m,s) for Gaussian-type quadrature formulae

$$\int\limits_{ - 1}^1 {f(x)dx = \mathop \sum \limits_{h = 0}^{2s} \mathop \sum \limits_{i = 1}^m } A_{hi} \cdot f^{(h)} (x_i )$$     

Caratteristiche numeriche della forma integrale delle equazioni differenziali di ordineN

A numerical performance of integral form for the linear ordinary differential equations $$y^{(n)} = \sum\limits_{i = 0}^{n - 2} { a_{i + 2} (x) y^{(n - 2 - i)} (x)}$$ is proved. Three numerical experiments are also given.     

Quantitative theorems on approximation by Bernstein-Stancu operators

In 1972 D. D. Stancu introduced a generalization \(L_{mp} ^{< \alpha \beta \gamma > }\) of the classical Bernstein operators given by the formula $$L_{mp}< \alpha \beta \gamma > (f,x) = \sum\limits_{k = 0}^{m + p} {\left( {\begin{array}{*{20}c} {m + p} \\ k \\ \end{array} } \right)} \frac{{x^{(k, - \alpha )} \cdot (1 - x)^{(m + p - k, - \alpha )} }}{{1^{(m + p, - \alpha )} }}f\left( {\frac{{k + \beta }}{{m + \gamma }}} \right)$$ . Special cases of these operators had been investigated before by quite a number of authors and have been under investigation since then. The aim of the present paper is to prove general results for all positiveL _mp ^<αβγ> 's as far as direct theorems involving different kinds of moduli of continuity are concerned. When applied to special cases considered previously, all our corollaries of the general theorems will be as good as or yield improvements of the known results. All estimates involving the second order modulus of continuity are new.     

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→∞.     

On the variance of gaussian quadrature formulae in the ultraspherical case

For ultraspherical weight functions ω_λ(x)=(1–x²)^λ–1/2, we prove asymptotic bounds and inequalities for the variance Var(Q _n ^G ) of the respective Gaussian quadrature formulae Q _n ^G . A consequence for a large class of more general weight functions ω and the respective Gaussian formulae is the following asymptotic result, $$\mathop {lim}\limits_{n \to \infty } n \cdot Var\left( {Q_n^G } \right) = \pi \int_{ - 1}^1 {\omega ^2 \left( x \right)\sqrt {1 - x^2 } dx.} $$      

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.     

Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor

The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:

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     

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: $$\begin{gathered} (RV X) \hfill \\ \Leftarrow \hfill \\ ({\rm I}F (AND (LISTP X) (LISTP (CDR X))) \hfill \\ (CONS (CAR (RV (CDR X))) \hfill \\ (RV (CONS (CAR X) \hfill \\ (RV (CDR (RV (CDR X))))))) \hfill \\ X). \hfill \\ \end{gathered} $$

     

Generalizzazione di un metodo alle differenze finite per una classe di problemi al contorno di tipo ellittico

In a previous paper the numerical solution of a particular boundary-value problem for the «weakly linear» equation $$\Delta \left[ {u(P)} \right] = f(P,u)$$ was obtained and the convergence of a suitable finite-difference scheme was proved. This paper is concerned with the more general equation $$L\left[ {u(P)} \right] = f(P,u)$$ where $$L \equiv - \left[ {a\frac{\partial }{{\partial x^2 }} + c\frac{\partial }{{\partial y^2 }} + d\frac{\partial }{{\partial x}} + e\frac{\partial }{{\partial y}}} \right]$$ ; the solution is obtained using the same finite-difference scheme as in the previous paper, and sufficient condition for its convergence are given for this new case     

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.     

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.     

Predicting the future of functions on flows

Let Ω be a topological space,S _t ∈ R (R the reals) a homeomorphism group on Ω andμ a Borel measure invariant with respect toS _t , (μ(Ω)=1); forP ∈Ω putS _t (P)=P _t . Assumef ∈L ₂(Ω,μ); according to E. Hopf there is for almost everyP ∈ Ω a well-determined spectral function σ(P,λ),λ ∈ R with lim \(T^{ - 1} \int_0^T {f(P_{t + s} )\overline {f(P_t )} dt = \int_{ - \infty }^{ + \infty } {e^{i\lambda s} d\sigma (P\lambda )} }\) . The question to be considered is:^*) if for a fixedP ∈ Ω we know the “past”f(P _t ), t ≦ 0, is it then possible to compute (or “predict”) the future valuesf(P _t ), t > 0? By using ideas from linear prediction theory we show that if \(\int_{ - \infty }^{ + \infty } {(1 + \lambda ^2 )\log \frac{d}{{d\lambda }}\sigma (P,\lambda )} d\lambda = - \infty\) then the prediction required by^*) is possible. An algorithm is described which accomplishes the prediction.