Numerical analysis of singular weighted integrals

In this article we investigate the numerical aspects of integrals of the form (1) $$\int_a^b {f(x)\psi (x)dx} $$ wheref is an unobjectionable function and ψ is singular, i.e. ψ is oscillating with high frequency, is discontinuous or unbounded. Suitable integration algorithms are presented.     

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.     

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→∞.     

Convergence,stability and truncation error estimation of a method for the numerical integration of the initial value problemY″=F(X,Y)

In this paper a detailed study of the convergence and stability of a numerical method for the differential problem $$\left\{ \begin{gathered} y'' = f(x,y) \hfill \\ y(x_0 ) = y_0 \hfill \\ y'(x_0 ) = y_0 ^\prime \hfill \\ \end{gathered} \right.$$ has carried out and its truncation error estimated. Some numerical experiments are described.     

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.     

On evolution of open dynamical systems: Some algebraic methods

Open dynamical systems which are governed by a finite number of ordinary differential equations with controls (time-dependent control parameters) constitute a large and important class of models for practical purposes. In the last few years, there has been considerable interest and progress in algebraic methods for solving the equations of the form (*) $$\dot x\left( t \right) = L_0 x\left( t \right) + \sum\limits_{j = 1}^r {u\left( t \right)L_i x\left( t \right)} ,$$ i.e. bilinear models. In this paper, intended as an expository introduction to the main results of system-theoretic approach to the modelling of open systems, a new “polynomial” representation of solutions to (*) is discussed.     

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

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals \(I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx\) and \(I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx\) . These integrals are not defined in the regular sense; \(I^{(1)}[g]\) is defined in the sense of Cauchy Principal Value while \(I^{(2)}[g]\) is defined in the sense of Hadamard Finite Part. With \(h=(b-a)/n, \,n=1,2,\ldots \) , and \(t=a+kh\) for some \(k\in \{1,\ldots ,n-1\}, \,t\) being fixed, the numerical quadrature formulas \({Q}^{(1)}_n[g]\) for \(I^{(1)}[g]\) and \(Q^{(2)}_n[g]\) for \(I^{(2)}[g]\) are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ We provided a complete analysis of the errors in these formulas under the assumption that \(g\in C^\infty [a,b]\) . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ the constants \(c^{(k)}_i\) being independent of \(h\) . In this work, we apply the Richardson extrapolation to \({Q}^{(k)}_n[g]\) to obtain approximations of very high accuracy to \(I^{(k)}[g]\) . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function \(g(x)\) , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.     

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.     

On the computation of Tricomi's ψ function

Two methods for calculating Tricomi's confluent hypergeometric function are discussed. Both methods are based on recurrence relations. The first method converges like $$\exp ( - \alpha |\lambda |^{1/3} n^{2/3} )for some \alpha > 0$$ and the second like $$\exp ( - \beta |\lambda |^{1/2} n^{1/2} )for some \beta > 0.$$ Several examples are presented.     

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.     

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.     

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.     

Quantum random walk polynomial and quantum random walk measure

In the paper, we introduce a quantum random walk polynomial (QRWP) that can be defined as a polynomial $\{P_{n}(x)\}$ , which is orthogonal with respect to a quantum random walk measure (QRWM) on $[-1, 1]$ , such that the parameters $\alpha _{n},\omega _{n}$ are in the recurrence relations $$\begin{aligned} P_{n+1}(x)= (x - \alpha _{n})P_{n}(x) - \omega _{n}P_{n-1}(x) \end{aligned}$$ and satisfy $\alpha _{n}\in \mathfrak {R},\omega _{n}> 0$ . We firstly obtain some results of QRWP and QRWM, in which case the correspondence between measures and orthogonal polynomial sequences is one-to-one. It shows that any measure with respect to which a quantum random walk polynomial sequence is orthogonal is a quantum random walk measure. We next collect some properties of QRWM; moreover, we extend Karlin and McGregor’s representation formula for the transition probabilities of a quantum random walk (QRW) in the interacting Fock space, which is a parallel result with the CGMV method. Using these findings, we finally obtain some applications for QRWM, which are of interest in the study of quantum random walk, highlighting the role played by QRWP and QRWM.     

On the Exact Complexity of Evaluating Quantified k -CNF

We relate the exponential complexities 2 ^s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a 3-cnf in $\vec{x}$ variables and $\vec{y}$ variables) and show that s(∞) (the limit of s(k) as k→∞) is at most $s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))$ . Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ running in time 2 ^cn with c<1. On the other hand, a nontrivial exponential-time algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ would provide a $\textsc {$k$-sat}$ solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ have nontrivial algorithms, and provide strong evidence that the hardest cases of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n?o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable $\textsc {$k$-cnf}$ s and the application of the Sparsification lemma.     

Small-amplitude limit cycles of Liénard systems

This paper is concerned with the study of second order differential equations of Liénard type: (A) $$\ddot x + f(x)\dot x + g(x) = 0$$ where f and g are polynomials. The equation (A) can also be written as a system of the form (B) $$\dot x = y - F(x),\dot y = - g(x),$$ , where \(F(x) = \mathop \smallint \limits_0^x f(\xi )d\xi \) . The results described here are mainly concerned with small amplitude limit cycles; that is, limit cycles which may be bifurcated from the origin on perturbation of the coefficients of F and g. The problem is to estimate the maximum number of limit cycles which various classes of systems of the form (B) can have; this is a special case of the second part of Hilbert’s sixteenth problem. Most of the calculations have been carried out on a computer using the REDUCE symbolic manipulation package.     

On the Gap Between Trivial and Nontrivial Initial Segment Prefix-Free Complexity

An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if the prefix-free Kolmogorov complexity of each initial segment of X is the same as the complexity of the sequence of 0s of the same length, up to a constant. We study the gap between the minimum complexity K(0 ⁿ ) and the initial segment complexity of a nontrivial sequence, and in particular the nondecreasing unbounded functions f such that ? for a nontrivial sequence X, where K denotes the prefix-free complexity. Our first result is that there exists a $\varDelta^{0}_{3}$ unbounded nondecreasing function f which does not have this property. It is known that such functions cannot be $\varDelta^{0}_{2}$ hence this is an optimal bound on their arithmetical complexity. Moreover it improves the bound $\varDelta^{0}_{4}$ that was known from Csima and Montalbán (Proc. Amer. Math. Soc. 134(5):1499?C1502, 2006). Our second result is that if f is $\varDelta^{0}_{2}$ then there exists a non-empty $\varPi^{0}_{1}$ class of reals X with nontrivial prefix-free complexity which satisfy (?). This implies that in this case there uncountably many nontrivial reals X satisfying (?) in various well known classes from computability theory and algorithmic randomness; for example low for ??, non-low for ?? and computably dominated reals. A special case of this result was independently obtained by Bienvenu, Merkle and Nies (STACS, pp. 452?C463, 2011).     

Deterministic function computation with chemical reaction networks

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising language for the design of artificial molecular control circuitry. Nonetheless, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. CRNs have been shown to be efficiently Turing-universal (i.e., able to simulate arbitrary algorithms) when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, have been shown to decide only the semilinear predicates (a multi-dimensional generalization of “eventually periodic” sets). We introduce the notion of function, rather than predicate, computation by representing the output of a function \({f:{\mathbb{N}}^k\to{\mathbb{N}}^l}\) by a count of some molecular species, i.e., if the CRN starts with \(x_1,\ldots,x_k\) molecules of some “input” species \(X_1,\ldots,X_k, \) the CRN is guaranteed to converge to having \(f(x_1,\ldots,x_k)\) molecules of the “output” species \(Y_1,\ldots,Y_l\) . We show that a function \({f:{\mathbb{N}}^k \to {\mathbb{N}}^l}\) is deterministically computed by a CRN if and only if its graph \({\{({\bf x, y}) \in {\mathbb{N}}^k \times {\mathbb{N}}^l | f({\bf x}) = {\bf y}\}}\) is a semilinear set. Finally, we show that each semilinear function f (a function whose graph is a semilinear set) can be computed by a CRN on input x in expected time \(O(\hbox{polylog} \|{\bf x}\|_1)\) .     

Metodi pseudo Runge-Kutta ottimali

For the numerical integration of the problem with initial value $$y' = f(x,y),y(x_0 ) = y_0 ,\begin{array}{*{20}c} {\begin{array}{*{20}c} x \\ {x_0 } \\ \end{array} \in [a,b],} \\ \end{array} $$ the pseudo R. K. methods of second kind are taken again and approximations are drawn, that in particular casef(x, y)≡f(x) are reduced to quadrature formulae of Radau and Lobatto. The limits of the trancation's error and the stability's intervals of the pseudo R. K. methods of the first and second species with the approximations of the same order of R. K. are determined and compared. At the end of that, a numerical example is taken.