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.     

A note on a result of R. Kemp on R-typly rooted planted plane trees

R. Kemp has shown that the average height of r-tuply rooted planted plane trees is $$\sqrt {\pi n} - \frac{1}{2}(r - 2) + O(\log (n)n^{1/2 - \varepsilon } ), \varepsilon > 0, n \to \infty ,$$ assuming that all such trees withn nodes are equally likely. We give a quite short proof of this result (with an error term ofO (1)).     

The correct exponent for the Gotsman–Linial Conjecture

We prove new bounds on the average sensitivity of polynomial threshold functions. In particular, we show that for f, a degree-d polynomial threshold function in n variables that $$\mathbb{AS}(f) \leq \sqrt{n}(\log(n))^{O(d log(d))}2^{O(d^2 log(d))}.$$ This bound amounts to a significant improvement over previous bounds, and in particular, for fixed d gives the same asymptotic exponent of n as the one predicted by the Gotsman–Linial Conjecture.     

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.     

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 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, $$\mathop {lim}\limits_{n \to \infty } n \cdot Var\left( {Q_n^G } \right) = \pi \int_{ - 1}^1 {\omega ^2 \left( x \right)\sqrt {1 - x^2 } dx.} $$      

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send $\tilde{\varOmega}(m)$ messages and incur $\tilde{\varOmega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\tilde{\varOmega}(m)$ messages and $\tilde{\varOmega}(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using ${\tilde{O}}(\sqrt {n} + D)$ time) requires O(m+n ^3/2) messages, and the best known message-optimal algorithm (using ${\tilde{O}}(m)$ messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.     

The average height of r-typly rooted planted plane trees

In this paper we generalize a result of de Bruijn, Knuth und Rice concerning the average height of planted plane trees withn nodes. First we compute the number of allr-typly rooted planted plane trees (r-trees) withn nodes and height less than or equal tok. Assuming that all planted plane trees withn nodes are equally likely, we show, that in the average a planted plane tree is a 3-tree for largen; for this distribution we compute also the cumulative distribution function and the variance. Finally, we shall derive an exact expression and its asymptotic equivalent for the average height \(\bar h_r \) (n) of anr-tree withn nodes. We obtain for all ε>0 $$\bar h_r (n) = \sqrt {\pi n} - \frac{1}{2}(r - 2) + O(1n(n)/n^{1/2 - \varepsilon } ).$$      

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

     

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.     

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.     

Linear-Space Data Structures for Range Mode Query in Arrays

A mode of a multiset S is an element a∈S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i,j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (Proceedings of the International Symposium on Algorithms and Computation (ISAAC), pp. 517–526, 2003), requires \(\varTheta (\sqrt{n}\log\log n)\) query time in the worst case. We improve their result and present an O(n)-space data structure that supports range mode queries in \(O(\sqrt{n/\log n})\) worst-case time. In the external memory model, we give a linear-space data structure that requires \(O(\sqrt{n/B})\) I/Os per query, where B denotes the block size. Furthermore, we present strong evidence that a query time significantly below \(\sqrt{n}\) cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two \(\sqrt{n} \times \sqrt{n}\) matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for the dynamic problem (queries and updates in near O(n ^3/4) time), for orthogonal range mode in d dimensions (queries in near O(n ^1?1/2d ) time) and for half-space range mode in d dimensions (queries in \(O(n^{1-1/d^{2}})\) time). Finally, we complement our dynamic data structure with a reduction from the multiphase problem, again supporting that we cannot hope for much more efficient data structures.     

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     

A parametric algorithm for semigroup computation on mesh with buses

In this paper, we present a new parametric parallel algorithm for semigroup computation on mesh with reconfigurable buses (MRB). Givenn operands, our parallel algorithm can be performed in $O(2^{(2c^2 + 3c)/(4c + 1)} n^{1/(8c + 2)} )$ , time on a $2^{(c^2 - c)/(8c + 2)} n^{(5c + 1)/(8c + 2)} \times 2^{(c - c^2 )/(8c + 2)} n^{(3c + 1)/(8c + 2)} $ MRB ofn processors, where $0 \leqslant c \leqslant O(\sqrt {\log _2 n} )$ . Specifically, whenc=0, it takes $O(\sqrt n )$ time on the $\sqrt n \times \sqrt n $ MRB and is equal to the result on the mesh-connected computers; whenc=1, it takesO(n ^1/10) time on then ^3/5×n ^2/5 MRB and is equal to the previous result on the mesh-connected computers with segmented multiple buses; whenc=2, it takesO(n ^1/18) time on the 2^1/9 n ^11/18×2^(?1/9) n ^7/18 MRB; when $O(\sqrt {\log _2 n} )$ , it takesO(log₂ n) time and is equal to the previous result on the MRB. Consequently, our results can be viewed as a unification of some best known results on different parallel computational models.     

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.     

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.     

An algorithm for division of powerseries

An algorithm is given to compute a solution (b ₀, ...,b _n) of $$\sum\limits_0^n {a_i t^i } \sum\limits_0^n {b_i t^i } \equiv \sum\limits_0^n {c_i t^i } (t^{n + 1} )$$ froma ₀, ..., a_n, c₀, ..., c_n. It needs less than 7n multiplications, where multiplications with a skalar from an infinite subfield are not counted.     

The Rate of Convergence to the Limit of the Probability of Encountering an Accidental Similarity in the Presence of Counter Examples

This paper refines the main result of [1], where the limit \( - {e^{ - a}} - a{e^{ - a}}\left[ {1 - {e^{ - c\sqrt a }}} \right]\) was proved for the probability of encountering an accidental similarity between two parent examples without \(m = c\sqrt n \) counter examples if each parent example and counter example is described by a series of \(\sqrt n \) independent Bernoulli trials with success probability \(p = \sqrt {a/n} \). In this paper, we prove that the rate of convergence to the limit is proportional to \({n^{\frac{1}{2}}}\).     

The exact region of stability for MacCormack scheme

Let the two dimensional scalar advection equation be given by $$\frac{{\partial u}}{{\partial t}} = \hat a\frac{{\partial u}}{{\partial x}} + \hat b\frac{{\partial u}}{{\partial y}}.$$ We prove that the stability region of the MacCormack scheme for this equation isexactly given by $$\left( {\hat a\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} + \left( {\hat b\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} \leqslant 1$$ where Δ _t , Δ _x and Δ _y are the grid distances. It is interesting to note that the stability region is identical to the one for Lax-Wendroff scheme proved by Turkel.     

Quantum search algorithm for set operation

The operations of data set, such as intersection, union and complement, are the fundamental calculation in mathematics. It’s very significant that designing fast algorithm for set operation. In this paper, the quantum algorithm for calculating intersection set ${\text{C}=\text{A}\cap \text{B}}$ is presented. Its runtime is ${O\left( {\sqrt{\left| A \right|\times \left| B \right|\times \left|C \right|}}\right)}$ for case ${\left| C \right|\neq \phi}$ and ${O\left( {\sqrt{\left| A \right|\times \left| B \right|}}\right)}$ for case ${\left| C \right|=\phi}$ (i.e. C is empty set), while classical computation needs O (|A| × |B|) steps of computation in general, where |.| denotes the size of set. The presented algorithm is the combination of Grover’s algorithm, classical memory and classical iterative computation, and the combination method decrease the complexity of designing quantum algorithm. The method can be used to design other set operations as well.