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.     

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.     

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.     

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

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.     

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

     

Some properties of Rényi entropy over countably infinite alphabets

We study certain properties of Rényi entropy functionals $H_\alpha \left( \mathcal{P} \right)$ on the space of probability distributions over ?₊. Primarily, continuity and convergence issues are addressed. Some properties are shown to be parallel to those known in the finite alphabet case, while others illustrate a quite different behavior of the Rényi entropy in the infinite case. In particular, it is shown that for any distribution $\mathcal{P}$ and any r ∈ [0,∞] there exists a sequence of distributions $\mathcal{P}_n$ converging to $\mathcal{P}$ with respect to the total variation distance and such that $\mathop {\lim }\limits_{n \to \infty } \mathop {\lim }\limits_{\alpha \to 1 + } H_\alpha \left( {\mathcal{P}_n } \right) = \mathop {\lim }\limits_{\alpha \to 1 + } \mathop {\lim }\limits_{n \to \infty } H_\alpha \left( {\mathcal{P}_n } \right) + r$ .     

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.     

On a direct algorithm for the generation of log-normal pseudo-random numbers

The random variable \(\left( {\prod {_{i = 1}^n {{X_i } \mathord{\left/ {\vphantom {{X_i } {X_{i + n} }}} \right. \kern-0em} {X_{i + n} }}} } \right)^{{1 \mathord{\left/ {\vphantom {1 {\sqrt {2n} }}} \right. \kern-0em} {\sqrt {2n} }}}\) is used to generate standard log-normal variables Λ(0, 1), where theX _i are independent uniform variables on [0, 1].     

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→∞.     

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.     

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.     

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.     

On the k-th root in circular arithmetic

The representation of thek-th root of a complex circular intervalZ={c;r} is considered in this paper. Thek-th root is defined by the circular intervals which include the exact regionZ ^1/k={z:z ^k ∈Z}. Two representations are given: (i) the centered inclusive disks \( \cup \{ c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} ; \mathop {\max }\limits_{z \in Z} |z^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} - c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} |\} \) and (ii) the diametrical inclusive disks with the diameter which is equal to the diameter of the regionZ ^1/k.     

Sull'analisi numeerica delle equazioni integrali di volterra

In this note we examine a fourth-order Runge-Kutta procedure for the nn-meric resolution of Volterra integral equation:

$$y(t) = f(t) - \int\limits_{t_0 }^t {\varphi (t, \tau , y (\tau )) } d\tau$$     

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.     

Converging multistep methods for initial value problems involving multivalued maps

Every consistent and strongly stable multistep method of stepnumberk yields a solution, of the setvalued initial value problem \(\dot y \in F(t,y),y(t_0 ) = y_0 \) . The setF(t, z) is assumed to be nonvoid, convex and closed. Upper semicontinuity of F with respect to both variables is not required everywhere. If the initial value problem is uniquely solvable, the solutions of the multistep method will converge to the solution of the continuous problem. These results carry over to functional differential equations \(\dot y \in F(t,M_t y)\) of Volterra type and to discontinuous problems \(\dot y(t) = f(t,M_t y)\) in the sense of A.F. Filippov. A difference method is applied to the discontinuous delay equation \(\ddot x(t) + 2D\dot x(t) + \omega ^2 x(t) = = - \operatorname{sgn} (x(t - \tau ) + \dot x(t - \tau ))\) . In the limit τ→0 we obtain results for the problem \(\ddot x + 2D\dot x + \omega ^2 x = = - \operatorname{sgn} (x + \dot x)\) which cannot be solved classically everywhere.     

Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.     

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.